Post on 31-Jan-2022
Extremum Seeking for Spark Advance Calibration under Tailpipe Emissions Constraints
Miguel Antonio Ramos Herrera
ORCID 0000-0002-2578-4345
Submitted in total fulfilment of the requirements of the degree of Master of Philosophy
Department of Mechanical Engineering The University of Melbourne
October 2016
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Abstract
Engine control parameters are calibrated on a test rig laboratory by a series of experimental methods. The resulting parameters are obtained using fuels with a fixed composition and tested for compliance with the emission standard on legislated drive cycle. As fuel composition and driving behaviour may vary in the real world, there is motivation in considering methods of continually calibrating online for optimal performance (in some sense) subject to emissions legislation.
In this regard, extremum seeking (ES) is a potential non-model-based adaptive control strategy to achieve the online calibration of automotive engines. The technique has been used in tuning the engineβs spark timing to minimize fuel consumption. Spark timing also plays a role in emission formation, which has not been considered previously.
This research proposes an approach to extend extremum seeking control for online optimisation of dynamic systems by explicitly considering output constraints. The proposed controller formulation required a slight relaxation to provide average constraint satisfaction in the limiting case. The stability of the proposed approach is investigated under a range of circumstances.
The novel formulation is then applied to the problem of fuel consumption optimisation subject to emissions constraints in a high-fidelity engine model with a three-way catalytic aftertreatment system. The manipulated input was the spark timing, brake specific fuel consumption was chosen to be the metric function, and the distance-based NOx tailpipe emission was treated as the constrained output. Results showed that it is possible to obtain the optimal spark timing whilst satisfying on average the Euro-3 emission limit for NOx under different operating points.
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Declaration
This is to certify that:
(i) the thesis comprises only my original work towards the master except where indicated,
(ii) due acknowledgement has been made in the text to all other material used,
(iii) the thesis is less than 50,000 words in length, exclusive of tables, maps, bibliographies and appendices.
Signed,
________________________
Miguel Antonio Ramos Herrera
19th October 2016
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Contents
1 Introduction ........................................................................................1
1.1 Background and motivation ............................................................... 1
1.1 Thesis outline ..................................................................................... 3
2 Literature review .................................................................................5
2.1 Application perspective ...................................................................... 6
2.2 Extremum-seeking controller .............................................................. 7
2.3 Continuous-time algorithms for optimisation .................................... 12
2.4 Extremum seeking for engine calibration .......................................... 13
2.5 Integrated engine and aftertreatment models ................................... 15
2.6 Conclusions ...................................................................................... 16
3 Extremum seeking for systems with output constraints ...................... 19
3.1 Problem formulation ........................................................................ 20
3.2 An algorithm for off-line optimisation of static maps ....................... 21
3.3 Extremum seeking for dynamic plants with output constraints ........ 31
3.4 Conclusions ...................................................................................... 40
4 Extremum seeking of spark timing under tailpipe emissions constraints ......................................................................................... 43
4.1 Simulation environment: Plant description ....................................... 44
4.2 Problem formulation ........................................................................ 48
4.3 Simulation set-up and engine mapping ............................................. 50
4.4 Simulation results ............................................................................. 54
4.5 Conclusions ...................................................................................... 69
5 Contribution and future work ............................................................ 71
5.1 Contribution ..................................................................................... 71
5.2 Future work ..................................................................................... 72
A. Spark ignition engine model ............................................................... 75
A.1 Throttle and intake manifold............................................................ 77
A.2 The engine........................................................................................ 80
A.3 Exhaust manifold ............................................................................. 86
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A.4 Connecting pipe ............................................................................... 87
B. Three-way catalytic converter model .................................................. 91
B.1 Reaction kinetics of the three-way catalyst ...................................... 93
B.2 Enthalpy of reaction ......................................................................... 96
B.3 Heat and mass transfer coefficients ................................................... 97
B.4 Solution scheme ................................................................................ 99
Bibliography ........................................................................................ 103
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List of Figures
Figure 2.5: (a) BSFC maps for two fuel compositions at 1500 rpm and 6% throttle position, [5]. (b) Maps of spark advance and brake torque for CNG A (solid line) and Gas B (dashed line) at 800 rpm and 23 Nm, [6]. .................................................................................................... 6
Figure 2.7: ES scheme used in [13] ....................................................................... 8
Figure 2.8: (a) Extremum seeking paradigm for black-box and grey-box problems [23]. (b) Generalised black-box extremum seeking framework [24]. ....................................................................................... 10
Figure 3.1: (a) Discontinuous gradient system, equation (3.11). (b) Smooth gradient at the boundary of the constraint set πποΏ½ . ................................ 24
Figure 3.2: Lyapunov functions with critical points π’π’οΏ½β on the optimal
solution π’π’οΏ½β. (a) π’π’οΏ½
β β πποΏ½. (b) π’π’οΏ½β β πππποΏ½. ................................................... 25
Figure 3.3: Cost and constraint function for example 1. ..................................... 28
Figure 3.4: Smooth gradient system for the example 1 with different values of πΌπΌ. ....................................................................................................... 28
Figure 3.5: Convergence results in example 1 with different values of πΌπΌ. .......... 29
Figure 3.6: Cost function for the equivalent unconstrained optimisation for example 1. .............................................................................................. 29
Figure 3.7: Convergence result for the example 2 with different values of πΌπΌ. ...... 31
Figure 3.8: Extremum seeking scheme with output constraints .......................... 32
Figure 3.9: Extremum seeking with output constraint: No integral action. ........ 39
Figure 3.10: Extremum seeking with output constraint and integral action ....... 40
Figure 4.1: Engine-aftertreatment model in Matlab/Simulinkβ’. ........................ 45
Figure 4.2: Discretization of the catalytic converter. ππ = 1,2,3, β¦ ,60 .................. 46
Figure 4.3: Engine brake torque and speed from a chassis dynamometer test with NEDC condition [58]. The (*) corresponds to the operating point used in this thesis. ......................................................................... 51
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Figure 4.4: Calculated steady-state conversion efficiency of the TWC at 1395 rpm and 53.2 Nm with respect to ππ ................................................ 52
Figure 4.5: Steady-state maps, ππ = 1395 rpm, πποΏ½,οΏ½οΏ½οΏ½ = 53.2 Nm,πποΏ½οΏ½οΏ½οΏ½ =30, ππ = 1.01............................................................................................. 53
Figure 4.6: π΅π΅πππ΅π΅π΅π΅ and ππππ maps, ππ = 1395 rpm, πποΏ½,οΏ½οΏ½οΏ½ = 53.2 Nm, πποΏ½οΏ½οΏ½οΏ½ = 30, ππ = 1.01 ................................................................................. 53
Figure 4.7: Unconstrained extremum seeking scheme to minimise fuel consumption ........................................................................................... 55
Figure 4.8: UES simulation result.(1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap) ........................................................................... 56
Figure 4.9: TWC states: node 2 to 20. (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap) .............................................................. 57
Figure 4.10: TWC states: node 22 to 40. (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap) .............................................................. 58
Figure 4.11: TWC states: node 42 to 60. (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap) .............................................................. 59
Figure 4.12: Tailpipe ππππ emission and conversion efficiency with UES. ............ 60
Figure 4.13: Constrained extremum seeking scheme to minimise fuel consumption subject to ππππ emission constraint. .................................... 61
Figure 4.14 CES simulation result, (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smooth parameter) .......................................... 62
Figure 4.15: Tailpipe ππππ emission and conversion efficiency with CES ............. 63
Figure 4.16: CES result for different smoothing parameter πΌπΌ ............................. 63
Figure 4.17: Gradient convergence with CES ..................................................... 64
Figure 4.18: Constrained Extremum seeking with integral action, (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter) .............................................................................................. 66
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Figure 4.19: Sigmoid function with smoothing parameter πΌπΌ = 5 ......................... 68
Figure 4.20: Constrained Extremum seeking with integral action, (1299 rpm engine speed, 31.2 Nm, 49.9 vehicle speed, 1.01 AFR, 80 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter, 0.25 integral gain) ................................................................. 69
Figure A.1 Integrated engine model: throttle, intake and exhaust manifold, torque controller, connecting pipe, and the TWC ................................... 76
Figure B.1: Discretisation along the catalytic converterβs length, ππ = 1,2,3, β¦ ,60 ....................................................................................... 100
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List of tables
Table 4.1: EURO-3 limits for emission in gasoline passenger cars ..................... 49
Table 4.2: Average ππππ emission and spark timing for different πΌπΌ values at 200 s. ...................................................................................................... 64
Table 4.3: Average ππππ emission and spark timing for different integral gains at 250 s of simulation time. CES with integral action, engine at 1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter. ............................................................................. 67
Table A.1: Ford Falcon engine specification ....................................................... 75
Table A.2: Molar mass of gases. ......................................................................... 77
Table A.3: Parameters defining the throttle and intake manifold model ............ 77
Table A.4: Inputs, outputs and states for the throttle and intake manifold model ..................................................................................................... 78
Table A.5: Volumetric efficiency parameters ...................................................... 79
Table A.6: Parameters for the engineβs model .................................................... 81
Table A.7: Inputs, outputs and states for the engineβs model ............................. 81
Table A.8: The net indicated efficiency coefficients ............................................ 82
Table A.9: Parameters for the exhaust manifold model ...................................... 86
Table A.10: Inputs, outputs and states and for the exhaust manifold subsystem ............................................................................................... 86
Table A.11: Parameters for the exhaust manifold model .................................... 88
Table A.12: Inputs, states and outputs for the connecting pipe ......................... 88
Table B.1: Parameters for the catalyst model .................................................... 91
Table B.2: Inputs and output of the TWC model .............................................. 92
Table B.3: Pre-exponential factors and activation energy for the reaction model of the three-way catalyst.* Parameters obtained from experiments [58]. ** Data taken from [60]. ............................................. 92
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Table B.4: Reaction mechanism in the three-way catalytic converter [60]. ......... 94
Table B.5: Ideal-gas specific heat coefficients of various gases [61] ..................... 97
Table B.6: Standard molar enthalpy of formation of the given species [61] ........ 97
Table B.7: Diffusion volumes ............................................................................. 98
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Chapter 1
1 Introduction
1.1 Background and motivation
In production engines, the control of parameters such as spark timing, exhaust gas recirculation (EGR), intake and exhaust valves is usually carried out by an open loop engine controller. The calibration procedure to obtain such parameters is normally done in dynamometer and emission test rigs where the compliance of vehicles to emissions standards is tested over drive cycles such as the New European Drive Cycle (NEDC) [1], [2].
One of the major factors that can potentially jeopardize the aforementioned calibration process and lead to a suboptimal engine operation with performance impact is the fuel composition variation. It is well known that the environmental impact of fossil fuels has sparked interest in the use of alternative fuels to gasoline and diesel. Compressed natural gas (CNG), liquefied petroleum gas (LPG) and several blending levels of ethanol with gasoline are popular alternatives to conventional fuels. Nevertheless, these alternative fuels introduce a variety of new challenging problems related to their variable compositions. For instance, LPG can vary from propane-butane ratios of 25:75 to 100:1. Similarly natural gas is in fact a mixture of hydrocarbons, mainly of methane varying from 85% to 96 % [3], [4]. As a consequence, fuels with variable compositions might show different engine operating characteristic curves [5], [6], [7], [8].
Another important consideration is the difference between the drive cycle and the use of the actual engine whilst driving. On-road emissions vary depending on the route type, operation mode, and ambient conditions [9]. This fact has been experimentally evidenced by several scientific commissions who used portable emission measurement systems (PEMS) over a significant sample of commercial vehicles, [10], [11]. This actual driving condition can make the calibrated open
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loop controller partially redundant and compromise the emission performance. It would be desirable to ensure that real world emissions replicate the level of the legislative test cycle.
The commonality of the effect of variable fuel composition and real-world driving means that some form of online calibration is desirable. In principle online calibration would involve a constrained optimisation described by means of the following high level requirements:
1. To tune the engine inputs (spark timing, fuel injection duration, air/fuel ratio, valve timing) such that the engine operates in the neighbourhood of the optimum brake specific fuel consumption (BFSC).
2. To achieve point 1 while satisfying the emission standard.
One of the possible approaches to accomplish the online calibration of automotive engine is by extremum seeking (ES). This is an optimal control architecture that is suitable for dynamic plants where only limited knowledge of the system is available, but the plant inputs and outputs are measured.
Despite the on-going research work on extremum seeking control, it is essentially used for unconstrained optimisation of dynamic plants [12], [13]. Successful results in both theoretical and experimental contexts are well documented. Different issues are addressed such as increasing convergence speed and exploiting partial information of the plant to improve the overall controller performance. But an inability to handle plant output constraints in the control loop still remains as a limitation of this approach. The incorporation of output constraints is a difficult problem since it requires designing an extremum seeking controller that guarantees the closed-loop stability whilst the output does not violate the constraint. And more importantly, it is required to do so without an exact knowledge of the system dynamics, cost and constraints functions. Thus, if the extremum seeking can be appropriately extended to incorporate these constraints, it will be possible to achieve an online constrained calibration of the engine that simultaneously meets the requirement 1 and 2.
With that in mind, the high level aim of this research is to extend the extremum seeking controller for output-constrained optimisation of dynamic plants. With this new approach the online engine calibration subject to emission regulation can be achieved. In addition, this extension of extremum seeking would also be of value to other applications where output constraints need to be considered.
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1.1 Thesis outline
Chapter 2 introduces the extremum seeking controller for non-model based dynamic plants optimisation. This is then followed by a discussion of the recent research work on constrained extremum seeking. In continuation, a brief summary of the constrained optimisation algorithms from the perspective of continuous-time methods are also discussed. Similarly, the application of extremum seeking control for engine optimisation is reviewed. Some limitations of the current approaches that motivate this research are highlighted. Conclusions and research objectives are stated at the end of the chapter.
In Chapter 3 the development of a constrained extremum seeking scheme is presented. A continuous-time optimisation algorithm is tailored to handle the plant output constraint. The stability analysis of the proposed controller is provided.
In Chapter 4, the proposed extremum seeking controller is demonstrated on a validated engine model with a three-way catalyst aftertreatment system. The controller is used to tune the spark timing for optimal fuel economy subject to legislated limits for tailpipe emissions.
Chapter 5 contains the conclusions of this research and some potential future works are highlighted.
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Chapter 2
2 Literature review
This literature review discusses extremum-seeking control for real-time optimisation of dynamic systems. The recent research work that has improved it is also covered. In addition, limitations of extremum seeking controllers for plants with output constraints are also explored.
In order to shed light on these limitations, the above is followed by a brief summary of constrained optimisation methods, which are well developed for static maps. This aims to find algorithms which can be used within one of the existing extremum seeking framework. Consequently, special attention is given to the family of continuous-time optimisation methods which are expressed by means of ordinary differential equations. Some difficulties of these approaches are stated.
From an application-centric perspective, this chapter reviews recent research works on extremum seeking controller for online engine optimisation. Although most of the research has been conducted in laboratory conditions and promising results are reported, the consideration of tailpipe emissions is a common omission in these works.
As discussed in Chapter 1, these problems justify the goals of this research and the proposed solution will be demonstrated in a high fidelity engine model. The last section, a literature review is undertaken in order to select a suitable engine and aftertreatment model.
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2.1 Application perspective
In current production engines, the control of parameters such as spark timing and cam timing is carried out by an open-loop engine controller. The calibration procedure to obtain such parameters is normally conducted using a dynamometer test cell. The engine calibration process commonly requires the use of optimisation strategies. A standard procedure that involves optimisation techniques for engines calibration is documented in [1]. The engine operating envelope in the speed-torque plane is covered by a grid. A selected node in the grid represents a particular engine operating point. At each node, an optimisation method finds the optimal value of the engine parameters (e.g. spark timing) that minimises a metric function. The resulting optimised parameters are stored in the engine control unit in the form of calibration tables, which map the speed and torque to the optimal engine parameters.
The engine calibration parameters also play a role in emission formation. experimental results produced to evaluate the effect of spark timing on the fuel consumption and emissions generation of an automotive engine were obtained in [14]. This article clearly showed that advancing the spark timing towards the minimum BSFC increases the NOx and HC emissions with little effect on CO at the indicated engine operating point.
The utilisation of calibrated maps may result in suboptimal engine performance as the fuel composition changes. In recent years, the stringent environmental regulation has led to use alternative fuels to gasoline and diesel for internal combustion engines. Among these alternative fuels, Compressed Natural Gas (CNG) is widely used in current automotive engines due to its worldwide availability, attractive price and clean combustion [4].
Figure 2.1: (a) BSFC maps for two fuel compositions at 1500 rpm and 6% throttle position, [5]. (b) Maps of spark advance and brake torque for CNG A (solid line) and Gas B (dashed line) at 800 rpm and 23 Nm, [6]
(a) (b)
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However, the composition of CNG can vary as a mixture of hydrocarbons varying from 85% to 96 % of methane with the remainder as varying proportions of ethane (π΅π΅2π»π»6), propane (π΅π΅3π»π»8) and inert gases such as carbon dioxide (π΅π΅ππ2) and molecular nitrogen (ππ2), [4].
Figure 2.1 shows experimental results reported in [5] and [6] in which two fuels comprised of CNG were tested in a Ford Falcon engine with aftermarket CNG conversion kit under fixed speed and load. This demonstrates how fuels with variable composition may affect the optimal values of fuel injection duration and spark timing to minimise fuel economy and brake torque respectively. The aforementioned articles indicate a risk of suboptimal engine operation if the engine control unit stores calibrated maps obtained for fuels with fixed compositions.
These issues have generated interest in the development of online calibration for automotive engine in order to achieve optimal fuel efficiency while ensuring tailpipe emissions remain within legislated limits. One approach to online calibration is by using extremum seeking control. The next section discusses this control strategy and its application for engine optimisation.
2.2 Extremum-seeking controller
Extremum seeking (ES) is an adaptive control technique for online optimisation of dynamic plants where only limited knowledge of the plant is available, but its inputs and outputs are measurable. To illustrate the key concept of extremum seeking consider the following plant:
π₯π₯Μ = ππ(π₯π₯, π’π’), (2.1)
π¦π¦ = β(π₯π₯). (2.2)
Suppose there exists some state π₯π₯β, such that, π¦π¦β = β(π₯π₯β) is an extremum of the mapping β(β ). The main objective in extremum-seeking control is to force the solutions of the system (2.1) to eventually converge to a neighbourhood of π₯π₯β, π¦π¦β. And more important, to do so without the knowledge of the functional relationship of the map β(β ).
Figure 2.2 shows a basic scheme of extremum seeking control, [15]. In order to provide sufficient persistent excitation, the plant is probed using a sinusoidal signal. The observed output is then processed to obtain a gradient estimate. This is then used to drive the plantβs input, and subsequently the plantβs output close to their optimum operating values.
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Figure 2.2: ES scheme used in [15]
A rigorous stability proof of the control scheme shown in Figure 2.2 can also be found in [15]. The assumptions utilised in this article are summarised as follows.
Consider a family of control laws of the following form:
π’π’ = πΌπΌ(π₯π₯, ππ). (2.3)
The closed-loop system (2.1)-(2.2) and (2.3) is then
π₯π₯Μ = πποΏ½π₯π₯, πΌπΌ(π₯π₯, ππ)οΏ½, (2.4)
Assumption 1 [15]. There is a smooth function ππ: β β βοΏ½ such that
ππ(π₯π₯, πΌπΌ(π₯π₯, ππ)) = 0, if and only if π₯π₯ = ππ(ππ), (2.5)
Assumption 2 [15]. For each ππ β β, the equilibrium π₯π₯ = ππ(ππ) of the system (2.4) is locally exponentially stable with decay and overshoot constants uniform in ππ.
Assumption 3 [15]. There exists ππβ β β such that
(β β ππ)β²(ππβ) = 0, (2.6)
(β β ππ)β²β²(ππβ) < 0. (2.7)
This means that the steady-state input-output map π¦π¦ = β(ππ(ππ)) has a maximum at ππ = ππβ. Tuning parameters of the ES are ππ, ππβ, πποΏ½, ππ and ππ. Under these assumptions, local exponential stability was demonstrated. That is, if the initial condition is chosen sufficiently close to the extremum and βsmallβ values for the controller parameters are chosen, then ππ converges to a small neighbourhood of ππβ.
Unlike the local stability result presented in [15], non-local stability properties of ES control were demonstrated in [16] and [17]. Moreover, in these, a simplified extremum seeking scheme was used in which both the low-pass and high-pass filter were removed. In this simplified scheme, the controllerβs tuning parameters
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are only ππ, ππ, and ππ. The stability analysis was based on stronger assumptions compared to those in [15]:
Assumption 2 [16]. For each constant ππ β β, the corresponding equilibrium of the system (2.4) is globally asymptotically stable, uniformly in ππ.
Assumption 3 [16]. The steady-state input-output map has a global optimum (maximum or minimum).
Under these stronger assumptions, semi-global practical asymptotical stability (SPA) was demonstrated. That is, for each pair of strictly positive numbers Ξ, ππ, it is possible to adjust the controller parameter ππ, ππ, ππ such that all solutions starting in π΅π΅β={ππ β β||ππ β ππβ| β€ Ξ} converge to the ball π΅π΅οΏ½ = {ππ β β||ππ β ππβ| β€ Ξ½}.
In a similar vein, other extremum seeking schemes have also been proposed under different assumptions, plants and optimisation algorithms [12]. Furthermore, ES control has been used in wide range of applications such as ABS control in automobiles [18], axial gas compressor [19], direct-heated solar thermal power plant [20] and bioreactors [21], [22], which sometimes lead to new ES schemes with variations. This prompted the development of a unified and systematic design of ES controllers.
Such unified framework for the analysis and design of extremum seeking controllers was proposed in [13]. This framework is concentrated on black-box and gray-box problems in continuous time setting as illustrated in Figure 2.3 (a). In the former case, the plants model is unknown whereas in the latter case the plant may be parameterised with a series of unknown parameters.
The scheme in Figure 2.3 (b) is a special case of the abovementioned paradigm. This corresponds to the black-box extremum seeking framework for dynamic plants. The framework assumes that the plant, ππ(π₯π₯, π’π’), possesses an asymptotically stable equilibrium surface π₯π₯ = ππ(π₯π₯, π’π’), uniformly in π’π’. In addition, the steady-state input-output map ππ(π’π’) = β(ππ(π’π’)) is a ππ times continuously differentiable function and has a global extremum (maximum or minimum). The derivatives are denoted by
π·π·οΏ½(ππ) = οΏ½πππππππ’π’
, β― , πποΏ½ππ
πππ’π’οΏ½ οΏ½οΏ½
. (2.8)
The framework decomposes the closed loop into a gradient estimator and an optimiser. This offers a convenient flexibility since the practitioner may select a suitable off-the-shelf optimisation algorithm in combination with a large class of gradient estimators. Optimisation algorithms include continuous-time gradient
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ascent, continuous Newton methods, and Levenberg-Marquardt methods. The derivative estimation (π·π·οΏ½(β )οΏ½ ) may be realised by means of first order filters [23].
Figure 2.3: (a) Extremum seeking paradigm for black-box and grey-box problems [13]. (b) Generalised black-box extremum seeking framework [23]
The discussed extremum-seeking schemes do not consider constraints. But there are situations where the optimisation of a plant should be carried out while satisfying some constraint set. The next section reviews some of the works in constrained extremum seeking and highlights limitations of the current techniques.
2.2.1 Constrained extremum seeking
In general, a plant may have inputs, states, and outputs constraints. These, appear natural in the context of guaranteeing performance, and/or meeting regulated standards. Since the extremum seeking control is essentially a control strategy to optimise dynamic plants, there in merit in investigating the current approaches which extend the capacity of ES to handle plant constraints. For reasons that will become apparent in the next section, output constraints are only considered in this work. However, for the sake of completeness, input as well as state constraints in the context of extremum-seeking control are reviewed below.
An extreme seeking scheme to manage input constraints was proposed in [24]. This article states that in some situations the extremum seeking controllerβs optimiser produces update for the inputs of plant that wander outside their physical operational limits (input constraints). In order to address this, a saturation function is used to limit the input. Moreover, the original problem is reformulated as a classic constrained optimisation where a penalty function is carefully chosen. By doing so, the original cost function is augmented with the penalty function, thus generating an auxiliary problem with no constraints. The above approach implied that the saturated function and in turn the saturation limits are known in advance.
(a) (b)
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Another approach in [23] states that input constraints in extremum-seeking frameworks can be addressed by using a projection operator. This is a mechanism to force the inputs to reside in the constrained set by directly mapping the output of the optimiserβs integrator onto the feasible set. With the same spirit, input constraints are also studied in [25] in which a constrained extremum seeking controller is developed for a single-input-single-output static map. The cost function was assumed to be unknown, but with a general quadratic form. The controller incorporates an orthogonal projection operator. It was shown that the extremum seeking algorithm converges to the optimum by driving the input within the feasible region. That is, it prohibits the input from leaving the feasible set.
An extremum seeking controller to achieve optimum points while maintaining feasibility of the state constraints was discussed in [26]. In this article the plant state equations were known in advance, but they were expressed in terms of some unknown parameters (grey-box plant). Thereby, this work uses system identification techniques to estimate the model parameters to implement classical constrained optimisation methods by mean of augmenting the cost function with both penalties and barrier functions.
An extremum-seeking control approach to handling output constraints was proposed in [27]. In order to preserve feasibility of the systemsβ trajectories, a barrier function formulation is proposed to transform the constrained optimisation problem into an unconstrained problem. The closed-loop system was shown to converge exponentially to a neighbourhood of the unknown local minimiser of the constrained optimisation problem. The size of the neighbourhood was shown to be dependent not only on the tuning parameters of the extremum-seeking controller, but also on some additional parameters of the barrier function which need to be chosen. When the optimal solution resides on the boundary of the constraint set, there is still a possibility that the closed-loop system converges outside the feasible region.
There are optimisation problems in dynamical systems where transient constraint violations are allowed to some degree, but the feasibility is required after some time has elapsed. In addition, the constraint may be met on average rather than every time instant. This raises the question of whether it is possible to develop an extremum seeking controller to satisfy the plantβs output constraint on average. In the previous section, the flexibility provided by the unified framework for black-box extremum seeking suggests that it is possible to choose a continuous-time off-line optimisation algorithm to handle output constraints. Moreover, the algorithm may not require the augmentation of the cost function with constraints, thus avoiding the potential ill effect of the barrier function near
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the constraint boundary. The next section briefly discusses some off-line optimisation algorithms.
2.3 Continuous-time algorithms for optimisation
Having considered the extremum seeking limitation in terms of handling plant output constraints, there is merit in investigating the approaches in the optimisation field that can potentially be used in the extremum seeking architecture. The framework provided in [13] shows that a number of continuous-time algorithms for off-line unconstrained optimisation may be used. The simplest method is the continuous-time gradient algorithm in which the solutions of a dynamical system are driven by the gradient of the cost function. These solutions asymptotically converge to a neighbourhood of the optimum. Then it is worth investigating continuous-time algorithms for solving constrained optimisation problems.
In the past, several continuous-time off-line optimisation algorithms based on ordinary differential equations (ODE) were proposed to solve nonlinear constrained optimisation problems. Tanabe [28] proposed a continuous version of the gradient projection method discussed in [29] and [30]. These algorithms project the gradient field of the cost function onto a tangent space of the feasible set. These algorithms are suited to optimisation problems with equality constraints. To incorporate inequality constraints a space transformation is required, however, this introduces slack variables to convert the inequality into equalities constraints. Consequently, the dimension of the original problem increases as well as the computational effort required to solve it.
The algorithm presented in [28] only converges to the optimal solution as long as the initial condition starts within the feasible set. Although the strategy stated in that article contained a self-correction when the trajectories violate the constraint as they approach the optimal point, the initial state must still be within the feasible set for the theorem of convergence to be valid. In order to extend this result, [31] improved the algorithm by a more elaborate analysis of the global behaviour of the solutions.
Another ODE-based algorithm which did not require feasibility of the initial state was proposed in [32]. This method uses the classic space transformation to generate a new system of differential equations and incorporates certain matrices which act as βbarrier-projectionsβ. This approach prevents the trajectories from making excursions outside the feasible set. In this way, the algorithm may be initialized in the infeasible region, but trajectories are guaranteed to eventually enter the feasible set as they move towards the optimal point.
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On the other hand, there are algorithms specifically tailored to handle only equality constraints including [33], [34], [35] and [36]. Their approaches are based on different sets of differential equations whose solutions convergence to the optimum. However, these articles do not provide any stability analysis to prove the claimed convergence properties of the algorithm.
Nevertheless, the algorithms discussed above are not suitable for extremum-seeking control since they require the explicit knowledge of both the cost and constraint function, which by definition are not available for problems where the black-box ES framework is employed. In the framework proposed in Section 2.2, the practitioner only has access to measured inputs and outputs. But the gradient and higher order derivatives with respect to the plantβs input may be obtained from the outputs if the appropriate derivate estimator is implemented.
An algorithm that may exploit these gradients to find a constrained optimum is known as the βhemstitchingβ or βboundary-following methodβ in [37]. This algorithm is an iterative method that originated from nonlinear programming and only requires the gradient of the cost function and the constrained output to approach the optimal solution. This simple but effective algorithm may form the basis of an appropriate optimiser to be used in the black-box extremum-seeking control framework and extend its capacity for handling output constraints. Extending this algorithm in this way requires further research.
2.4 Extremum seeking for engine calibration
In the presence of stringent regulation, car manufacturers have added additional devices in order meet legislated emission limits whilst improving fuel economy. The exhaust gas recirculation and variable cam timing are some of the additional methodologies employed. This increases not only the degrees of freedom for the engine control unit to manage, but also the time to calibrate the engine parameters and determine the maps of the engine variables.
In order to reduce the experimental burden for engine calibration, extremum seeking was demonstrated to locate optimal parameters on a dual-independent variable cam timing engine, [38]. This approach minimises BSFC by finding the optimal values of spark and cam timings. Several optimisation algorithms were implemented and tested on a dynamometer at different operation conditions (speed/load). Experimental results show the extremum seeking controller converges to a local minimum in 15 minutes.
Section 2.1 illustrated that fuel composition variation is a factor that may result in suboptimal calibrated maps. This motivated research into online engine calibration approach. With this regard, an extremum seeking controller was
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utilised to operate the engine close the maximum fuel efficiency in the presence of fuel composition variation, [39]. This research implemented an in-cylinder pressure sensor to estimate the net fuel specific consumption and experiments were conducted on a spark ignition flex fuel engine at different engine operating conditions such as load, speed and gasoline blends, specifically E70 and E85. Spark timing was manipulated to minimise fuel consumption. According to the experimental results, the controller achieved convergence to the optimum spark advance in approximately 20 seconds under fixed load condition.
Section 2.1 also shows that compressed natural gas is an alternative fuel to gasoline and diesel. Unlike these fuels, the natural gas composition varies significantly, which directly impact the engine performance [3]. With that in mind, extremum seeking for the optimisation of spark timing of a natural gas fuelled engine was presented in [6]. From the theoretical viewpoint, this research was based on a general framework for grey-box ES, which was previously introduced in [13]. The grey-box ES approach exploits the available partial information about the plant in order to speed-up the closed-loop convergence. Spark timing was shown to converge to its optimum value in approximately 20 seconds.
Another example of extremum seeking for natural gas engines optimisation is reported in [5]. The manipulated input was the fuel injection duration and BSFC was used as the cost function. Initial experiments were conducted to approximate the engine dynamic with a Hammerstein structure. By utilising the knowledge of the plant structure, it was possible to avoid a time scale separation in the closed-loop system and consequently speed up the convergence rate.
The previous literature review on extremum-control for online engine calibration did not find any research which considered tailpipe emissions constraints. The limits of these emissions are currently legislated to improve the air quality. Since engine parameters (e.g. spark timing) play a role in the formation of NOx, HC, and CO emissions, it is desirable to achieve online engine calibration for optimal engine performance subject to the legislated limits for these pollutants.
The combination of the engine and the aftertreatment systems may be considered as a black-box dynamic plant in which tailpipe emissions are viewed as outputs. Since the compliance with regulated emissions is assessed on an average basis, this configures a real-world scenario for applying an extremum-seeking control strategy for dynamic plants subject to output constraints. If such a controller can be developed as mentioned in Section 2.2.1, then it may be
15
possible to extend the online engine calibration with ES while satisfying tailpipe emission limits on average.
2.5 Integrated engine and aftertreatment models
In order to carry out engine optimisation studies, the model of a spark ignition engine and the three-way catalytic converter (TWC) are required. This simulation environment represents the platform to assess the performance of the control strategies that optimise the engine fuel efficiency subject to regulated tailpipe emissions. Like other models, the level of detail encapsulated in the model depends on the problem to be studied and its final use. From the control objective pursued in this research, it is desirable to have a model with low computational complexity. Otherwise, the problems to be studied could be intractable or the solution could not be obtained in a reasonable time.
A common approach in automotive engine modelling is the use of mean value engine models (MVEM). These models describe the average behaviour of the engine over several of engine cycles. Examples of mean value engine models can be found in Hendricks and Sorensen, [40]; Powell et al [41]; Muller et al., [42]; Eriksson et al., [43]. These are low-order physics-based models in which the dynamic equations are commonly obtained from thermodynamics and heat transfer principles. These models are distinguished by their modular structure, which implies that they are self-consistent and compact. However, they lack the aftertreatment system. As a result, an exploration of the literature of an integrated MVEM which include the three-way catalytic converter was undertaken.
Combined engine and aftertreatment models can be found in [44]. The MVEM consisted of two submodels; an air path model and fuel path model. However, the effect of spark timing on the engineβs performance was not considered. In addition, the TWC was a single-state phenomenological catalyst model but did not provide tailpipe emissions concentrations. An integrated powertrain model that calculates tailpipe emissions was presented in [45]. Nevertheless, emissions after catalyst were calculated by using empirical look-up tables (static maps) and the engineβs exhaust temperature. The effect of engine inputs such as spark timing and air-fuel ratio on tailpipe emission was not included. Conversely, a model that allows studying the effect of the aforementioned control inputs on emissions was reported in [46]. However, the TWC model did not include dynamics, but a series of static maps for calculating efficiency conversion, and HC was the only pollutant considered.
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A low order physics-based model of a spark ignition engine and the aftertreatment system was proposed in [47]. This high-fidelity integrated mean value engine model includes the transient model for the intake manifold, combustion chamber, exhaust manifold, connecting pipelines, and the three-way-catalytic converter. Unlike the literature discussed above, the aftertreatment system is a transient PDE model that describes the mass and energy transfer due to chemical reactions that take place inside the TWC, thus providing transient responses of HC, NOx, and CO tailpipe emissions. This integrated model allows optimisation studies with a variety of engine controls inputs such as spark timing, air-fuel ratio, intake valve closing position, valve overlap, engine speed and load. Consequently, the effect of different control strategies on fuel consumption and tailpipe emissions can be investigated. Based on these features, the described integrated model above is suitable for the interests pursued in this research work.
2.6 Conclusions
Extremum-seeking control strategies were studied as possible candidates to accomplish online calibration of engines. It was found that the reported extremum seeking control algorithms developed for online engine optimisation did not consider engine tailpipe emissions. This gap in the research motivates the development of new approaches that enable the extremum seeking controller to handle dynamic plant with output constraints.
A possible source of solution to handling output constraints is optimisation algorithms that may be used in the extremum-seeking control architecture. Based on reported literature, an extremum seeking framework that relies on continuous-time optimisation algorithms was identified. In this framework a large class of optimisers may be taken off-the-shelf from the rich literature of static map optimisation. Although the reviewed continuous-time optimisers for constrained optimisation are not suitable for the extremum seeking framework, it is possible to modify an existing optimisation algorithm to handle constraints such as the boundary-following method and then use it in the aforementioned framework.
As a consequence, to solve the overall aim of this research project it is necessary to extend the existing research in two ways. Firstly, a conventional extremum seeking algorithm needs to be extended to incorporate the consideration of output constraints. Because the work is motivated by satisfying emissions regulations, only average constraint satisfaction needs to be considered, which conveniently avoids many infeasibility issues. The second aspect is the deployment of the proposed scheme to a high fidelity engine and aftertreatment model. These aims are more explicitly detailed in the following subsection.
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2.6.1 Research Objectives
1. To develop an extremum seeking controller for dynamic plants subject to output constraints.
There are situations where the optimisation of a plant should be carried out by satisfying some constraints. In general, a system may have constraints in the inputs, states and/or outputs. They are frequently related to safety conditions or operational characteristics of the system. Therefore, they are essential in many practical applications and control systems must take them into account to guarantee a desirable performance.
In contrast to other control approaches, where the objective function is known, extremum-seeking control (ESC) is an adaptive control scheme that locates extremum of the plant input-output map without the explicit knowledge of the mapβs functional relationship. This feature raises some theoretical challenges when it comes to a possible extension to handle output constraints. For instance, ensuring feasibility of the obtained solution, guaranteeing the closed-loop controller stability without a precise knowledge of the system dynamics, cost functions and incorporating constraints are still topics of current investigation.
The aim of this first research objective is to develop an approach to bridge the gap in the constrained extremum seeking field, especially for handling dynamic plants with output constraints. In this research, the constrained extremum seeking controller is developed for dynamic plants with a single input and two outputs where one input-output map is considered as the cost function and the other is function representing the constraint. In addition, the problem formulation is relaxed such that the controller enforces the constraint satisfaction on average.
2. To demonstrate in a high fidelity simulation environment the proposed constrained extremum seeking controller. The metric considered is the specific fuel consumption of a spark-ignition engine and it will be minimised subject to tailpipe emission constraints.
The constrained extremum seeking controller is used to carry out the online optimal calibration of a passenger car automotive engine while meeting regulated emission. To this end, the controller is implemented on a simulation environment by using a validated integrated physics-based spark ignition engine model. This is composed of the engine transient model for the intake manifold, the engine block, combustion chamber, the exhaust manifold, pipelines and the three-ways-catalytic converter with chemical reactions modelled using second order partial differential equations.
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From the perspective of the proposed extremum seeking scheme, the plant input is the spark timing. The BSFC is one of the calculated plant outputs and it is considered as the process cost function. The second output (representing the constraint function) is the tailpipe emission. The goal of the proposed constrained extremum-seeking controller is to tune the spark timing to minimise the engine fuel consumption subject to legislated limits for tailpipe emissions.
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Chapter 3
3 Extremum seeking for systems with output constraints
Extremum seeking is a non-model based adaptive control technique for on-line optimisation of the steady-state input-output characteristic of a plant. This control strategy has been used for problems where only limited knowledge of the system is available, but the plant input and output signals are measured. According to the literature review in Section 2.2, extremum seeking has been extended for constrained optimisation problems. The vast majority of the available schemes are focused on input constraints and require the system to be initialized within a feasible region. However, handling the optimisation of dynamic plants with output constraints remains as a limitation of this approach and a topic of interest for further research.
With that in mind, this chapter proposes an approach to extend the extremum seeking control for on-line optimisation of the steady-state characteristics of dynamic systems with output constraints. This chapter begins with the problem description. This is followed by the development of a continuous-time algorithm for off-line constrained optimisation and its stability analysis. Numerical examples are provided to demonstrate the efficacy of the developed algorithm and the effect of its parameters on its convergence. The last section introduces the extremum seeking scheme in which the previous proposed algorithm is combined with a derivative estimator for optimising dynamic plants subject to output constraints. In addition, the benefits of augmenting the proposed extremum seeking control with integral action are discussed and demonstrated with simulations.
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3.1 Problem formulation
Consider the dynamic plant
π₯π₯Μ = ππ(π₯π₯, π’π’) (3.1)
with the following two outputs
π¦π¦οΏ½ = βοΏ½(π₯π₯), π¦π¦1 = β1(π₯π₯),
(3.2)
where ππ: βοΏ½ Γ β β βοΏ½, βοΏ½: βοΏ½ β β and β1: βοΏ½ β β. Let π·π·οΏ½1 (β ) and π·π·οΏ½
2(β ) be the first and second derivative operators with respect to π’π’.
Assumption 1: ππ(β ), βοΏ½(β ), and β1(β ) are smooth.
Assumption 2: There exists a differentiable function ππ: β β βοΏ½, such that ππ(π₯π₯, π’π’) = 0 if and only if π₯π₯ = ππ(π’π’).
Assumption 3: For each constant π’π’ β β, the equilibrium π₯π₯ = ππ(π’π’) is asymptotically stable, uniformly in π’π’.
These assumptions imply two reference-to-output maps at the equilibrium,
π¦π¦οΏ½,οΏ½οΏ½ = βοΏ½(ππ(π’π’)) = βοΏ½ β ππ(π’π’) = πποΏ½(π’π’), π¦π¦1,οΏ½οΏ½ = β1(ππ(π’π’)) = β1 β ππ(π’π’) = ππ1(π’π’),
(3.3)
Assumption 4: πποΏ½(β ) and ππ1(β ) are twice continuously differentiable. Moreover, π·π·οΏ½
2πποΏ½(π’π’) > 0, π·π·οΏ½2ππ1(π’π’) > 0 β π’π’ β β. Thus, πποΏ½(β ) and ππ1(β ) are strictly convex
functions.
Assumption 5: πποΏ½(β ) has a strict local minimum, π’π’β β β, such that π·π·οΏ½1πποΏ½(π’π’β) = 0.
Let ππ be the constraint set defined by
ππ = {π’π’ β β|ππ1(π’π’) β€ 0}. (3.4)
The set πποΏ½οΏ½οΏ½ = {π’π’ β β|ππ1(π’π’) < 0} is the interior of ππ and its boundary is denoted by ππππ = {π’π’ β β|ππ1(π’π’) = 0}.
Assumption 6: The set ππ is not empty.
The objective is to steer the system (3.1) to the equilibrium π₯π₯οΏ½β and π’π’οΏ½
β which solves the inequality constrained optimisation problem given by
minοΏ½βοΏ½
πποΏ½(π’π’), (3.5)
and to do so without any explicit knowledge about π₯π₯οΏ½β, π’π’οΏ½
β, ππ, βοΏ½, β1. In addition, the initial condition for π’π’ is not required to be close to π’π’οΏ½
β. However, on the basis of state dynamics can make instantaneous constraint satisfaction untenable, the
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problem (3.5) is relaxed by satisfying (3.4) on average, and only after some time period has elapsed. Thus, the relaxed problem is formulated as follows:
minοΏ½βοΏ½οΏ½
πποΏ½(π’π’), (3.6)
where
πποΏ½ = οΏ½π’π’ β β| limοΏ½ββ
1ππ
οΏ½ ππ1(π’π’)πππποΏ½
οΏ½βοΏ½β€ 0οΏ½. (3.7)
Here ππ depends on a design parameter of the proposed control strategy to be yet discussed.
In order to develop a solution for this problem systematically, the next section discusses the developing of an optimisation algorithm for plants with no dynamics and the relaxed average constraint is not used. To this end, the constrained optimisation problem for static maps is formulated with the appropriate assumptions. Then, the aforementioned algorithm will be used to propose a solution to the original problem (3.6)-(3.7).
3.2 An algorithm for off-line optimisation of static maps
Consider a plant with no dynamics with one input and two outputs. The static maps are given by
π¦π¦οΏ½ = πποΏ½(π’π’), π¦π¦1 = ππ1(π’π’),
(3.8)
where πποΏ½: β β β, ππ1: β β β.
The goal is to solve the following constrained optimisation problem:
minοΏ½βοΏ½οΏ½
πποΏ½(π’π’), (3.9)
πποΏ½ = {π’π’ β β|ππ1(π’π’) β€ 0}. (3.10)
Suppose πποΏ½(β ), ππ1(β ), and πποΏ½ satisfy Assumptions 4-6. Moreover, πποΏ½(β ), ππ1(β ), and their respective derivatives π·π·οΏ½
1πποΏ½(β ) and π·π·οΏ½1ππ1(β ) are known. Since the
minimization of a strictly convex function over a convex set has a unique solution [48], then the solution to the problem (3.9)-(3.10), namely π’π’οΏ½
β, is unique.
The objective of this section is to propose a continuous-time optimisation algorithm whose solutions asymptotically converge to π’π’οΏ½
β, regardless of whether it resides in the interior of the constraint set πποΏ½,οΏ½οΏ½οΏ½ or on its boundary πππποΏ½. In addition to this, the initial condition for π’π’ could be specified in any of these sets.
The following update law is proposed for the input to the static maps.
οΏ½ΜοΏ½π’ = οΏ½βπ·π·οΏ½1πποΏ½(π’π’) ππππ ππ1(π’π’) < 0
βπ·π·οΏ½1ππ1(π’π’) ππππ ππ1(π’π’) β₯ 0
(3.11)
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This dynamical system can be viewed as a continuous-time version of a class of iterative algorithms called boundary-following methods [49]. Equation (3.11) can be restated as
οΏ½ΜοΏ½π’ = β[1 β π π οΏ½ππ1(π’π’)οΏ½] β π·π·οΏ½1πποΏ½(π’π’) β π π οΏ½ππ1(π’π’)οΏ½ β π·π·οΏ½
1ππ1(π’π’), (3.12)
where π π : β β β , and ππ β β, such that
π π (ππ) = οΏ½0 ππππ ππ < 01 ππππ ππ β₯ 0 (3.13)
However, due to the piece-wise definition of the above dynamical system, the right-hand side of (3.12) is discontinuous with respect to π’π’. There exists a significant research body that addresses the stability analysis of such systems that are not locally Lipchitz on π’π’, [50], [51], [52]. A common approach adopted in these references is the utilisation of differential inclusions, generalised gradients, and nonsmooth stability analysis. However, to simplify the analysis, a smooth version for (3.12) is proposed here.
Notice that the abrupt transition of π·π·οΏ½1πποΏ½(π’π’) to π·π·οΏ½
1ππ1(π’π’) as π’π’ approaches πππποΏ½ can be eliminated by approximating the step function in (3.13) with a smooth function. There are many possibilities to achieve this. Let πΌπΌ β β>0, ππ β β and ππ: β Γ β>0 Γ β β β. Some smooth approximations for (3.13) are listed below.
1) The sigmoid function:
ππ(ππ, πΌπΌ, ππ) = 1
1 + ππβ(οΏ½βοΏ½)οΏ½
(3.14)
2) The twice differentiable spline function:
ππ(ππ, πΌπΌ, ππ) =
β©βͺβ¨
βͺβ§ 0 ππππ ππ β ππ β€ βπΌπΌ316
οΏ½ππ β πππΌπΌ
οΏ½5β 5
8οΏ½ππ β ππ
πΌπΌ οΏ½
3+ 15
16οΏ½ππ β ππ
πΌπΌοΏ½ + 1
2 ππππ β πΌπΌ β€ ππ β ππ β€ πΌπΌ
1 ππππ πΌπΌ β€ ππ β ππ
(3.15)
3) The hyperbolic tangent function:
ππ(ππ, πΌπΌ, ππ) = 0.5 + 0.5 tanhοΏ½ππ β πππΌπΌ
οΏ½ (3.16)
Note that for ππ = 0 all these approximations are centred with respect to ππ = 0, where the discontinuity occurs in (3.13). In order to facilitate the analysis presented in this chapter, the sigmoid approximation will be used here.
Equation (3.12) can be now approximated with a continuous locally Lipchitz right-hand side as:
οΏ½ΜοΏ½π’ = β[1 β ππ(ππ1(π’π’), πΌπΌ, 0)]π·π·οΏ½1πποΏ½(π’π’) β ππ(ππ1(π’π’), πΌπΌ, 0)π·π·οΏ½
1ππ1(π’π’). (3.17)
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For simplicity and where the context allows, the functionβs argument of ππ will be dropped, so that
οΏ½ΜοΏ½π’ = β(1 β ππ)π·π·οΏ½1πποΏ½(π’π’) β πππ·π·οΏ½
1ππ1(π’π’). (3.18)
3.2.1 Equilibrium points
As mentioned earlier, the required property of the optimisation algorithm (3.18) is that all solutions converge to π’π’οΏ½
β. Before establishing this, it is necessary to show that the system possesses an equilibrium π’π’οΏ½ = π’π’οΏ½
β. Note that depending on the particular problem, π’π’οΏ½
β can be an element of πππποΏ½ or πποΏ½,οΏ½οΏ½οΏ½.
First, consider the case for π’π’οΏ½β β πποΏ½,οΏ½οΏ½οΏ½. In the interior of the constraint set πποΏ½,
the smooth function ππ β 0, thereby, the equation (3.18) reduces to following dynamic system:
οΏ½ΜοΏ½π’ = βπ·π·οΏ½1πποΏ½(π’π’). (3.19)
According to Assumption 5, πποΏ½(β ) has a strict local minimum π’π’β and π·π·οΏ½
1πποΏ½(π’π’β) = 0. Furthermore, by Assumption 4 πποΏ½(β ) is strictly convex, then π’π’β = π’π’οΏ½
β and the system (3.19) has an equilibrium π’π’οΏ½ = π’π’οΏ½β.
Now consider the case for π’π’οΏ½β β πππποΏ½. The system in (3.18) is designed to
smooth the gradients as π’π’ approaches πππποΏ½ where the constraint is active. Figure 3.1 graphically illustrates this feature in a neighbourhood of the boundary of the constraint. Let, for two real π’π’οΏ½ and π’π’οΏ½, π’π’οΏ½ < π’π’οΏ½, such that οΏ½ΜοΏ½π’(π’π’οΏ½) and οΏ½ΜοΏ½π’(π’π’οΏ½) are of opposite signs. Then there exists a π’π’οΏ½ β [π’π’οΏ½, π’π’οΏ½] with οΏ½ΜοΏ½π’(π’π’οΏ½) = 0 (Bolzanoβs Theorem). Thus when π’π’οΏ½
β β πππποΏ½ and π·π·οΏ½1ππ1(π’π’) β π·π·οΏ½
1πποΏ½(π’π’) < 0 in a region of π’π’οΏ½β, the
system (3.19) has an equilibrium π’π’οΏ½.
In order to show that π’π’οΏ½ = π’π’οΏ½β, some conditions must be satisfied. Consider the
equation (3.18). Since ππ1(π’π’) = 0 for π’π’ β πππποΏ½, then ππ = 0.5 and (3.18) reduces to
οΏ½ΜοΏ½π’ = β0.5π·π·οΏ½1πποΏ½(π’π’)β0.5π·π·οΏ½
1ππ1(π’π’). (3.20)
It follows that for π’π’ β πππποΏ½ the gradients must satisfy π·π·οΏ½1πποΏ½(π’π’) = βπ·π·οΏ½
1ππ1(π’π’) for the system to have an equilibrium π’π’οΏ½ = π’π’οΏ½
β. This is a strong condition and in general difficult to meet since the derivatives of ππ1(β ) and πποΏ½(β ) depend on the problem itself. However, for the purpose of illustrating the main ideas of this section, the following stability analysis assumes the abovementioned condition holds. This will be followed by an analysis of the general case in which π·π·οΏ½
1πποΏ½(π’π’) β βπ·π·οΏ½1ππ1(π’π’) on πππποΏ½.
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Figure 3.1: (a) Discontinuous gradient system, equation (3.11). (b) Smooth gradient at the boundary of the constraint set πποΏ½
3.2.2 Stability analysis of the proposed algorithm
Although the existence of an equilibrium point for the system (3.18) was discussed in the previous section, this is not sufficient for investigating its stability property. In this section, the Lyapunov direct method is applied to investigate the stability of the equilibrium point π’π’οΏ½.
The system (3.18) can be expressed in the following form
οΏ½ΜοΏ½π’ = βπ΅π΅(π’π’), (3.21)
where
π΅π΅(π’π’) = (1 β ππ)π·π·οΏ½1πποΏ½(π’π’) + πππ·π·οΏ½
1ππ1(π’π’). (3.22)
Let π·π· β β be a domain containing the equilibrium point of (3.21). Let ππ:π·π· β β be an antiderivative of π΅π΅(π’π’), so that
π·π·οΏ½1ππ(π’π’) = π΅π΅(π’π’). (3.23)
Then (3.21) can be written as follows:
οΏ½ΜοΏ½π’ = βπ·π·οΏ½1ππ(π’π’). (3.24)
In other words, the system (3.21) can be expressed as the smooth gradient system (3.24). These Gradient Dynamical Systems (GDS) have special properties that simplify their analysis; for instance, the real-valued function ππ(β ) is a natural Lyapunov function for (3.21) [53].
The time derivative of ππ(π’π’) along the trajectories of (3.24) is given by:
ππ(Μπ’π’) = π·π·οΏ½1ππ(π’π’)π’π’,Μ (3.25)
ππ(Μπ’π’) = β[π·π·οΏ½1ππ(π’π’)]2. (3.26)
0
u$c
_u
0
u$c
_u
βπ·π·οΏ½1ππ1(π’π’)
βπ·π·οΏ½1πποΏ½(π’π’)
ππ1(π’π’) < 0
ππ1(π’π’) > 0
(ππ) (ππ)
25
It is evident from (3.26) that ππ(Μπ’π’) is zero at π’π’οΏ½ and negative definite in π·π· β {π’π’οΏ½}. Then, π’π’οΏ½ is an asymtotically stable equilibrium point of (3.21). Furthermore, according to Section 3.2.1, if π’π’οΏ½
β β πποΏ½,οΏ½οΏ½οΏ½, or π’π’οΏ½β β πππποΏ½ and
π·π·οΏ½1πποΏ½(π’π’οΏ½
β) = βπ·π·οΏ½1ππ1(π’π’οΏ½
β), then π’π’οΏ½ = π’π’οΏ½β. Consequently, π’π’οΏ½
β is an asymtotically stable equilibrium point of (3.21) under these conditions.
Figure 3.2: Lyapunov functions with critical points π’π’οΏ½
β on the optimal solution π’π’οΏ½β. (a)
π’π’οΏ½β β πποΏ½. (b) π’π’οΏ½
β β πππποΏ½
Note that the critical point of ππ(π’π’), namely π’π’οΏ½β, is the equilibrium point of
(3.24). Therefore, π’π’οΏ½ = π’π’οΏ½β. It follows that, under the aforementioned conditions,
π’π’οΏ½β = π’π’οΏ½
β. Figure 3.2 is a sketch to graphically illustrate this property of the proposed optimisation algorithm. To summarise, solving the constrained optimisation problem (3.9) with the proposed algorithm (3.18) is equivalent to solving the following unconstrained optimisation problem:
minοΏ½
ππ(π’π’), (3.27)
which in turn, it is the minimisation of the Lyapunov function of the system (3.21).
0_u 0_u
V V
π’π’οΏ½β
π’π’οΏ½β π’π’οΏ½
β
π’π’οΏ½β
π π ππππ πποΏ½ πποΏ½
(ππ) (ππ)
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Remark. As mentioned earlier, the solution of the convex problem (3.9) is unique and according to the previous section, the proposed algorithm is designed to have an equilibrium precisely at π’π’οΏ½
β such that all solutions of (3.21) asymptotically converge to π’π’οΏ½
β if π’π’οΏ½β β πποΏ½,οΏ½οΏ½οΏ½. A similar stability result was
concluded when π’π’οΏ½β β πππποΏ½, however π·π·οΏ½
1πποΏ½(π’π’οΏ½β) = βπ·π·οΏ½
1ππ1(π’π’οΏ½β) must hold for the
solution to converge to π’π’οΏ½β.
Now, consider the case when π·π·οΏ½1πποΏ½(π’π’) β βπ·π·οΏ½
1ππ1(π’π’) on πππποΏ½. If π’π’οΏ½β β πποΏ½,οΏ½οΏ½οΏ½, then
the smooth function ππ β 0, therefore, the equation (3.18) reduces to (3.19) and follows the same analysis as above to conclude that π’π’οΏ½ = π’π’οΏ½
β. On the other hand, if π’π’οΏ½
β β πππποΏ½, then ππ1(π’π’οΏ½β) = 0 and, ππ = 0.5, whereby (3.18) reduces to
οΏ½ΜοΏ½π’ = β0.5π·π·οΏ½1πποΏ½(π’π’)β0.5π·π·οΏ½
1ππ1(π’π’). (3.28)
However, since π·π·οΏ½1πποΏ½(π’π’) β βπ·π·οΏ½
1ππ1(π’π’) and π·π·οΏ½1ππ1(π’π’)π·π·οΏ½
1πποΏ½(π’π’) < 0 on πππποΏ½, (3.28) does not possess and equilibrium point on πππποΏ½. That is, π’π’οΏ½ β π’π’οΏ½
β.
Despite this, it is possible to have an equilibrium point in a neighbourhood of π’π’οΏ½
β by choosing a suitable value for the parameter πΌπΌ in the smooth function (3.14). To demonstrate this, consider the equilibria of (3.18):
0 = β(1 β πποΏ½)π·π·οΏ½1πποΏ½(π’π’οΏ½) β πποΏ½π·π·οΏ½
1ππ1(π’π’οΏ½), (3.29)
solving for πποΏ½ yields
πποΏ½ = π·π·οΏ½1πποΏ½(π’π’οΏ½)
(π·π·οΏ½1πποΏ½(π’π’οΏ½) β π·π·οΏ½
1ππ1(π’π’οΏ½)), (3.30)
substituting (3.14) in (3.30) gives:
1
1 + ππβοΏ½1(οΏ½οΏ½)οΏ½
= π·π·οΏ½1πποΏ½(π’π’οΏ½)
(π·π·οΏ½1πποΏ½(π’π’οΏ½) β π·π·οΏ½
1ππ1(π’π’οΏ½)). (3.31)
Now solving for ππ1(π’π’οΏ½):
ππ1(π’π’οΏ½) = βπΌπΌπππΌπΌοΏ½βπ·π·οΏ½1ππ1(π’π’οΏ½)
π·π·οΏ½1πποΏ½(π’π’οΏ½)
οΏ½. (3.32)
Consider the following two cases for the equation (3.32). If |π·π·οΏ½1πποΏ½(π’π’οΏ½)| >
|π·π·οΏ½1ππ1(π’π’οΏ½)|, then ππ1(π’π’οΏ½)>0 and the equilibrium point of (3.18) lies outside the
constrained set. That is π’π’οΏ½ β πποΏ½. Now, if |π·π·οΏ½1πποΏ½(π’π’οΏ½)| < |π·π·οΏ½
1ππ1(π’π’οΏ½)| then ππ1(π’π’οΏ½) < 0 and π’π’οΏ½ β πποΏ½,οΏ½οΏ½οΏ½. As a consequence, if π’π’οΏ½
β β πππποΏ½ and π·π·οΏ½1πποΏ½(π’π’οΏ½
β) β βπ·π·οΏ½
1ππ1(π’π’οΏ½β), then the proposed algorithm might converge to an equilibrium point
that either violates the constraint or resides in πποΏ½οΏ½οΏ½. However, by reducing the parameter πΌπΌ in (3.32), it is possible to construct a ππ(πΌπΌ)-sized neighbourhood centred at ππ1(π’π’οΏ½
β) = 0, such that
|ππ1(π’π’οΏ½) β ππ1(π’π’οΏ½β)| β€ πππΌπΌ, (3.33)
27
for some ππ > 0. Then, by continuity:
|π’π’οΏ½ β π’π’οΏ½β| β€ ππ. (3.34)
It is concluded that the practitioner may choose a sufficiently small πΌπΌ such that π’π’οΏ½ is arbitrarily close to π’π’οΏ½
β, and consequently, ππ1(π’π’οΏ½) resides in a small region of ππ1(π’π’οΏ½
β).
3.2.3 Numerical examples
Example 1. Consider the following problem:
minοΏ½
(π’π’ + 1)2
s. t. 2(π’π’ β 2)2 β 2 β€ 0,
(3.35)
Let πποΏ½(π’π’) = (π’π’ + 1)2 and ππ1(π’π’) = 2(π’π’ β 2)2 β 2. The constraint set is πποΏ½ = [1,3]. To solve this problem, the following input update law for π’π’ is used.
οΏ½ΜοΏ½π’ = β[(1 β ππ)π·π·οΏ½1πποΏ½(π’π’) + πππ·π·οΏ½
1ππ1(π’π’)], (3.36)
where
ππ = 1
1 + ππβοΏ½1(οΏ½)οΏ½
. (3.37)
Assumptions 4 to 6 are satisfied. From Figure 3.3 it is apparent that the unique constrained optimum occurs at π’π’οΏ½
β = 1 and resides on the boundary of πποΏ½, that is, ππ1(π’π’οΏ½
β ) = 0. Figure 3.4 shows the proposed dynamical system as a function of π’π’ for three different values of the parameter πΌπΌ.
Note that the upper diagonal line corresponds to βπ·π·οΏ½1ππ1(π’π’), while βπ·π·οΏ½
1πποΏ½(π’π’) is represented by the lower diagonal. If π’π’ was only driven by βπ·π·οΏ½
1ππ1(π’π’), then for all π’π’(0) β β, the solutions would asymptotically converge to π’π’ = 2. Conversely, if π’π’ was driven by βπ·π·οΏ½
1πποΏ½(π’π’), then the solutions would asymptotically converge to π’π’ = β1. None of these is the constrained optimum. However, since π·π·οΏ½
1ππ1(1) =βπ·π·οΏ½
1ππo(1), then for any πΌπΌ β β>0 the proposed dynamical system (3.36) possesses an equilibrium point at π’π’οΏ½ = 1, which is the optimal solution of (3.35).
The convergence to the optimal solution for π’π’(0) = 4 and different πΌπΌ values is shown in Figure 3.5. Note that this condition implies an initialisation outside πποΏ½ and violates the constraint. With this initialisation, it is possible to demonstrate the efficacy of the proposed optimisation algorithm to drive π’π’ towards the feasible set [1,3]. To do so, π’π’ is initially driven by the gradient of the constraint, βπ·π·οΏ½
1ππ1(π’π’). Once the trajectory approaches π’π’ = 3 the gradient transition to βπ·π·οΏ½
1πποΏ½(π’π’) and π’π’ enters into the feasible set [1,3]. Note that π’π’ = 3 is indeed a feasible solution of (3.35), but it is not the optimal solution. Solutions of (3.36) do not converge to this point, since π’π’ = 3 is not an equilibrium point.
28
Figure 3.3: Cost and constraint function for example 1
Figure 3.4: Smooth gradient system for the example 1 with different values of πΌπΌ
-2 -1 0 1 2 3 4 5 6-10
0
10
20
30
40
50
u
f o(u
);f 1
(u)
fo(u)
f1(u)
-2 -1 0 1 2 3 4 5 6-20
-15
-10
-5
0
5
10
15
20
_u=!
F(u
)
u
, = 0:1, = 1, = 2
29
Figure 3.6: Cost function for the equivalent unconstrained optimisation for example 1
Figure 3.5: Convergence results in example 1 with different values of πΌπΌ
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
4
u(t
)
, = 0:1, = 1, = 2
0 0.5 1 1.5 2-8
-7
-6
-5
-4
-3
-2
-1
0
1
_u(t
)
t (s)
0 0.5 1 1.5 20
5
10
15
20
25
f o(t
)
0 0.5 1 1.5 2-2
-1
0
1
2
3
4
5
6
f 1(t
)
t (s)
-2 -1 0 1 2 3 4 5 6-10
-5
0
5
10
15
20
25
30
35
40
u
V
, = 0:1, = 1, = 2
30
The effect of πΌπΌ in the speed of convergence to the constrained optimum is also shown in Figure 3.5. Decreasing πΌπΌ increases the rate of convergence of the input to π’π’ = 1, and consequently the rate of convergence to πποΏ½(1) = 4 and ππ1(1) = 0. To explain this, consider again Figure 3.4. With πΌπΌ = 0.1, the gradient transition occurs when π’π’ is close to either π’π’ = 1 or π’π’ = 3. In the interval (1,3), ππ β 0 and the dynamic system (3.36) rapidly approximates the gradient descent βπ·π·οΏ½
1πποΏ½(π’π’), which is represented by the lower diagonal in the same figure. Conversely, for πΌπΌ = {1,2}, the algorithm provides a smoother gradient transition, and this occurs before π’π’ approaches the boundaries of the constraint set. However, in the interior of [1,3], ππ β 0, then, a lower |οΏ½ΜοΏ½π’| is obtained compared to the case with πΌπΌ = 0.1, thus causing a slow convergence speed to the constrained optimum.
A Lyapunov function of (3.36) is shown in Figure 3.6 for different values of πΌπΌ. This function is obtained by solving the following antiderivative:
ππ(π’π’) = οΏ½π΅π΅(π’π’) πππ’π’, (3.38)
ππ(π’π’) = οΏ½{[1 β ππ]π·π·οΏ½1πποΏ½(π’π’) + πππ·π·οΏ½
1ππ1(π’π’)} πππ’π’, (3.39)
ππ(π’π’) = οΏ½οΏ½οΏ½1 β 1
1 + ππβοΏ½1(οΏ½)οΏ½
οΏ½π·π·οΏ½1πποΏ½(π’π’) + 1
1 + ππβοΏ½1(οΏ½)οΏ½
π·π·οΏ½1ππ1(π’π’)οΏ½πππ’π’. (3.40)
As mentioned in Section 3.3.2, the system (3.36) can be viewed as a dynamical system which solves the equivalent unconstrained optimisation problem of minimising a real valued function ππ. As expected, the optimal solution of minimising πποΏ½(π’π’) over ππ is the critical point of ππ(β ), that is, π’π’οΏ½
β = π’π’οΏ½β.
Example 2, There are situations where π·π·οΏ½1πποΏ½(π’π’οΏ½
β) β βπ·π·οΏ½1ππ1(π’π’οΏ½
β). In such cases π’π’οΏ½ β π’π’οΏ½
β and the solution of the dynamical system (3.36) will not converge to the constraint optimum, but arbitrarily close to it if an appropriate πΌπΌ is chosen. To illustrate this, consider a modification of the cost function used in example 1. The optimisation problem is stated as:
minοΏ½
2(π’π’ + 1)2
s. t. 2(π’π’ β 2)2 β 2 β€ 0,
(3.41)
Let πποΏ½(π’π’) = 2(π’π’ + 1)2, ππ1(π’π’) = 2(π’π’ β 2)2 β 2. The constraint set is πποΏ½ = [1,3]. Note that the optimal solution is still π’π’οΏ½
β = 1 but now π·π·οΏ½1πποΏ½(π’π’οΏ½
β) > βπ·π·οΏ½1ππ1(π’π’οΏ½
β).
The convergence of π’π’, πποΏ½(π’π’), and ππ1(π’π’) for π’π’(0) = 4 is shown in Figure 3.7 Clearly, all solutions converge to an equilibrium point such that π’π’οΏ½ < π’π’οΏ½
β, and the constraint is not satisfied, that is, ππ1(π’π’οΏ½) > 0. However, by decreasing πΌπΌ, π’π’οΏ½ might reside in a small neighbourhood |π’π’οΏ½ β π’π’οΏ½
β| β€ ππ(πΌπΌ), thus π’π’οΏ½ will approach π’π’οΏ½β from
below.
31
Although the ππ(πΌπΌ)-neighbourhood of π’π’οΏ½
β could be arbitrarily small, decreasing πΌπΌ also results in a fast gradient transition. This is observed in the response of οΏ½ΜοΏ½π’ in Figure 3.7. The sharp transition might not be of any concern for this βtoyβ example, however, it may introduce issues in real systems. For example, if π’π’ represents an actuator, this would require a large controller effort to cope with the high rate of change of π’π’ as it approaches the boundary of πποΏ½.
3.3 Extremum seeking for dynamic plants with output constraints
In this section, the proposed extremum seeking scheme in Figure 3.8 is used to solve the problem (3.6). In this scheme, the plantβs input and outputs are only available. Thus, it is treated as a black box system. A sinusoid signal is added to the input to probe the plant and the observed outputs are used to estimate the gradient of the steady state input-output maps. This is achieved with the use of low-pass filters within the dashed box in Figure 3.8. The derivate estimates are then used by the optimiser to drive the input to a neighbourhood of its optimum.
Figure 3.7: Convergence result for the example 2 with different values of πΌπΌ
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
4
u(t
)
, = 1, = 3, = 5
0 0.5 1 1.5 2
-10
-8
-6
-4
-2
0
_u(t
)
t (s)
0 0.5 1 1.5 25
10
15
20
25
30
35
40
45
50
f o(t
)
0 0.5 1 1.5 2-2
-1
0
1
2
3
4
5
6
f 1(t
)
t (s)
32
Figure 3.8: Extremum seeking scheme with output constraints
For convenience, the proposed optimisation algorithm is repeated below with the smooth gradient system written as a member of a large family of optimisation algorithms:
οΏ½ΜοΏ½π’ = π΅π΅οΏ½οΏ½ οΏ½π’π’,π·π·οΏ½1ππ(π’π’)οΏ½. (3.42)
Without loss of generality, consider the case for minimization, and the optimiser:
π΅π΅οΏ½οΏ½ οΏ½π’π’,π·π·οΏ½1ππ(π’π’)οΏ½ = β π·π·οΏ½
1ππ(π’π’), (3.43)
where
π·π·οΏ½1ππ(π’π’) = (1 β ππ) β π·π·οΏ½
1πποΏ½(π’π’) + ππ β π·π·οΏ½1ππ1(π’π’). (3.44)
3.3.1 Singular perturbation and averaging analysis
The controller parameters in Figure 3.8 are ππ,π€π€οΏ½,πΌπΌ, ππ, Throughout this chapter ππ = πΏπΏπ€π€π€π€οΏ½ for some πΏπΏ > 0. Let the smooth function be:
ππ = 1
1 + ππββ1(οΏ½)οΏ½
, (3.45)
and ππ(βοΏ½(π₯π₯), β1(π₯π₯), ππ) = [βοΏ½(π₯π₯)π π πππΌπΌ (ππ), β1(π₯π₯)π π πππΌπΌ (ππ)]β€, ππ = [πποΏ½, ππ1]β€. Let ππ(β ) be given by:
ππ(ππ, ππ, ππ) = β(1 β ππ) β πποΏ½(ππ, ππ) β ππ β ππ1(ππ, ππ), (3.46)
where πποΏ½(ππ, ππ) = 2ππβ1πποΏ½, and ππ1(ππ, ππ) = 2ππβ1ππ1, which on steady-state aproximate π·π·οΏ½
1πποΏ½(π’π’) and π·π·οΏ½1ππ1(π’π’) [54]. Consequently, ππ(ππ, ππ, ππ) approximates βπ·π·οΏ½
1ππ(π’π’).
The closed-loop equations of the system in Figure 3.8 with the gradient estimator ππ(β ) are given by
π₯π₯Μ = ππ(π₯π₯, π’π’) π¦π¦οΏ½ = βοΏ½(π₯π₯) π¦π¦1 = β1(π₯π₯)
π€π€οΏ½π€π€π π + π€π€οΏ½π€π€
π’π’οΏ½οΏ½οΏ½Μ = πππ΅π΅οΏ½οΏ½(π’π’οΏ½οΏ½οΏ½, ππ(ππ, ππ, ππ))
π π πππΌπΌ (π€π€ππ)
π’π’οΏ½ +
π’π’ = π’π’οΏ½οΏ½οΏ½ + πππ π πππΌπΌ (π€π€ππ)
πππ π πππΌπΌ (π€π€ππ)
π¦π¦1
πποΏ½
ππ1
π¦π¦οΏ½
π€π€οΏ½π€π€π π + π€π€οΏ½π€π€
33
π₯π₯Μ = ππ(π₯π₯, π’π’οΏ½οΏ½οΏ½ + πππ π πππΌπΌ (π€π€ππ)), (3.47)
π’π’οΏ½οΏ½οΏ½Μ = πΏπΏπ€π€π€π€οΏ½π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½οΏ½οΏ½, ππ(ππ, ππ, ππ)οΏ½, (3.48)
ππΜ = βπππποΏ½[ππ β ππ(βοΏ½(π₯π₯), β1(π₯π₯),π€π€ππ)]. (3.49)
Let οΏ½ΜοΏ½π’ = π’π’οΏ½οΏ½οΏ½ β π’π’οΏ½β and ππβ² = ππππ. The system (3.47)-(3.49) is brought into the
standard singular perturbation form
π€π€ πππ₯π₯ππππβ²
= ππ(π₯π₯, οΏ½ΜοΏ½π’ + π’π’οΏ½β + πππ π πππΌπΌ (ππβ²)), (3.50)
πποΏ½ΜοΏ½π’ππππβ²
= πΏπΏπ€π€οΏ½π΅π΅οΏ½οΏ½οΏ½οΏ½ΜοΏ½π’ + π’π’οΏ½β, ππ(ππ, ππ, ππ)οΏ½, (3.51)
ππππππππβ²
= βπποΏ½[ππ β ππ(βοΏ½(π₯π₯), β1(π₯π₯), ππβ²)]. (3.52)
Setting π€π€ = 0 βfreezesβ π₯π₯ at its equilibrium π₯π₯= ππ(οΏ½ΜοΏ½π’ + π’π’οΏ½β + πππ π ππ πΌπΌ(ππβ²)) to obtain
the following reduced system:
πππ’π’οΏ½ππππβ²
= πΏπΏπ€π€οΏ½π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½ + π’π’οΏ½β, ππ(ππ, ππ, πποΏ½)οΏ½, (3.53)
πππποΏ½ππππβ²
= βπποΏ½[πποΏ½ β ππ(πποΏ½(οΏ½ΜοΏ½π’ + π’π’οΏ½β + πππ π πππΌπΌ (ππβ²)), ππ1(οΏ½ΜοΏ½π’ + π’π’οΏ½
β + πππ π πππΌπΌ (ππβ²), ππβ²)], (3.54)
and its average system is
πππ’π’οΏ½οΏ½ππππβ²
= πΏπΏπ€π€οΏ½π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ(ππ, ππ, πποΏ½οΏ½οΏ½)οΏ½, (3.55)
πππποΏ½οΏ½ππππβ²
= βπποΏ½[πποΏ½οΏ½οΏ½ β ππ(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ))], (3.56)
where
ππ(π’π’οΏ½οΏ½ + π’π’οΏ½β , ππ) = [πποΏ½(π’π’οΏ½οΏ½ + π’π’οΏ½
β, ππ), ππ1(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ)]β€, (3.57)
and
πποΏ½(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ) = 1
2πποΏ½ βοΏ½(π’π’οΏ½οΏ½ + π’π’οΏ½
β + πππ π πππΌπΌ(ππβ²)), ππβ²)2οΏ½
0π π πππΌπΌ (ππβ²)ππππβ² (3.58)
ππ1(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ) = 1
2πποΏ½ β1(π’π’οΏ½οΏ½ + π’π’οΏ½
β + πππ π πππΌπΌ(ππβ²)), ππβ²)2οΏ½
0π π πππΌπΌ (ππβ²)ππππβ². (3.59)
By introducing the new time scale π π = πΏπΏπποΏ½ππ, the system (3.55)-(3.56) is transformed into standard singular perturbation form
πππ’π’οΏ½οΏ½πππ π
= π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ(ππ, ππ, πποΏ½οΏ½οΏ½)οΏ½, (3.60)
πΏπΏ πππποΏ½οΏ½οΏ½πππ π
= β[πποΏ½οΏ½οΏ½ β ππ(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ))]. (3.61)
Setting πΏπΏ = 0 βfreezesβ πποΏ½οΏ½ at its equilibrium: πποΏ½οΏ½= ππ(π’π’οΏ½οΏ½ + π’π’οΏ½β, ππ), and the
following reduced system is obtained:
34
πππ’π’οΏ½πππ π
= π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½ + π’π’οΏ½β, ππ(ππ, ππ, ππ(π’π’οΏ½ + π’π’οΏ½
β , ππ))οΏ½, (3.62)
which can be written in the new coordinates π’π’οΏ½οΏ½οΏ½οΏ½ = π’π’οΏ½ + π’π’οΏ½β as follow.
πππ’π’οΏ½οΏ½οΏ½οΏ½πππ π
= π΅π΅οΏ½οΏ½οΏ½π’π’οΏ½οΏ½οΏ½οΏ½, ππ(ππ, ππ, ππ(π’π’οΏ½οΏ½οΏ½οΏ½, ππ))οΏ½. (3.63)
Note that
ππ(ππ, ππ, ππ(π’π’οΏ½οΏ½οΏ½οΏ½, ππ)) = β(1 β ππ) β πποΏ½οΏ½ππ, πποΏ½(π’π’οΏ½οΏ½οΏ½οΏ½, ππ)οΏ½ β ππ β ππ1οΏ½ππ,ππ1(π’π’οΏ½οΏ½οΏ½οΏ½, ππ)οΏ½, (3.64)
where πποΏ½(ππ,πποΏ½(π’π’οΏ½οΏ½οΏ½οΏ½, ππ))=π·π·οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 πποΏ½(π’π’οΏ½οΏ½οΏ½οΏ½) , and ππ1(ππ, ππ1(π’π’οΏ½οΏ½οΏ½οΏ½, ππ))=π·π·οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
1 ππ1(π’π’οΏ½οΏ½οΏ½οΏ½). The difference
βββπ·π·οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½1 πποΏ½(π’π’οΏ½οΏ½οΏ½οΏ½) β π·π·οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
1 πποΏ½(π’π’οΏ½οΏ½οΏ½οΏ½)βββ can be made arbitrarily small by reducing the parameter ππ, [54]. As a result
ππ(ππ, ππ, ππ(π’π’οΏ½οΏ½οΏ½οΏ½, ππ)) = βπ·π·οΏ½οΏ½1ππ(π’π’οΏ½οΏ½οΏ½οΏ½). (3.65)
Therefore equation (3.63) can be written as
πππ’π’οΏ½οΏ½οΏ½οΏ½πππ π
= π΅π΅οΏ½οΏ½ οΏ½π’π’οΏ½οΏ½οΏ½οΏ½,π·π·οΏ½οΏ½1ππ(π’π’οΏ½οΏ½οΏ½οΏ½)οΏ½. (3.66)
In other words, the average reduced system (3.66) of the proposed extremum seeking scheme approximates the gradient descent law given by (3.42). Moreover, according to the previous section, the algorithm (3.66) behaves as a dynamic system for unconstrained optimisation. Hence, the average reduced system (3.66) also approximates the unconstrained optimisation problem to minimise the steady-state scalar function ππ(π’π’).
3.3.2 Stability
Since the proposed extremum seeking scheme solves the equivalent unconstrained optimisation problem of minimising ππ(π’π’), it suggests that it is possible to use the stability results of the unified framework for analysis and design of extremum seeking controllers, [13]. In order to do so, it is important to verify the assumptions of this framework.
Assumption 1 [13]. Consider the static map given by π¦π¦ = ππ(π’π’), in general π¦π¦ β β, π’π’ β βοΏ½. Suppose ππ(β ) has a strict local minimum, π’π’β β βοΏ½, such that π·π·οΏ½
1ππ(π’π’β) = 0, π·π·οΏ½2ππ(π’π’β) > 0. Moreover, π·π·οΏ½
1ππ(β ) and π·π·οΏ½2ππ(β ) are continuous.
Suppose the map ππ is ππ times continuously differentiable.
Assumption 2 [13]. A large class of off-line optimisation methods can be formulated in the following form
ππ Μ = π΅π΅οΏ½ππ, π·π·οΏ½ππ(ππ)οΏ½, (3.67)
35
where ππ β βοΏ½ and π·π·οΏ½(β ) is the vector of the iterated derivatives of ππ with respect to its argument. The system (3.67) possesses an equilibrium ππβ = π’π’β and π’π’β is the strict local minimum of ππ defined in Assumption 1.
The continuous-time optimisation algorithm typically must satisfy:
Property 2 [13]. There exists π½π½οΏ½ β πΎπΎβ, and Ξ > 0, such that the following holds: for any ππ > 0, there exists ππβ > 0 such that for any ππ β (0, ππβ) and |π€π€|β <ππ, the solutions of the following system:
ππ Μ = π΅π΅(ππ, π·π·οΏ½ππ(ππ) + π€π€), (3.68)
satisfy
|ππ(ππ) β ππβ| β€ π½π½οΏ½(|ππ(0) β ππβ|, ππ) + ππ, (3.69)
for all |ππ(0) β ππβ| β€ Ξ.
With regards to Assumption 1-2 and Property 2, Section 3.2.2 showed that for the equivalent unconstrained optimisation problem of (3.9), the scalar function ππ(π’π’) has a strict local minimum π’π’οΏ½
β, such that π·π·οΏ½1ππ(π’π’οΏ½
β) = 0, and π·π·οΏ½2ππ(π’π’οΏ½
β) > 0, thus Assumption 1 holds.
The proposed update law for π’π’ given by (3.42) has the form of (3.67) for ππ = 1. Moreover, the Section 3.2.2 showed that this system has an asymptotically stable equilibrium at π’π’οΏ½
β. Then, Assumption 2 holds. In addition, the asymptotic stability implies local input to state stability [55], whereby Property 2 holds as well. Assumption 4 and 5 in the framework shown in [13] are equal to Assumption 2 and 3 stated in Section 3.1. Then, the Theorem 2 in [13] can be invoked and used to follow its prescribed procedure for choosing πΏπΏ, πποΏ½ and ππ, such that π’π’ will converge to a small neighbourhood of π’π’οΏ½
β. Let π§π§1 = ππ β ππ(π’π’οΏ½οΏ½οΏ½, ππ), and π§π§2 = π₯π₯ β ππ(π’π’οΏ½οΏ½οΏ½ + πππ π πππΌπΌ (ππβ²)), Theorem 2 in [13] is stated as follows:
Theorem 2 [13]: Suppose that Property 2 and Assumption 4, 5 hold. Then there exist and π½π½οΏ½, π½π½οΏ½ and π½π½οΏ½ β πΎπΎβ and Ξ > 0 such that the following holds: for any positive ππ, there exist ππβ > 0 and πποΏ½
β > 0, such that for any ππ β (0, ππβ) and πποΏ½ β (0, πποΏ½
β ), there exists πΏπΏβ(ππ) > 0 such that for any πΏπΏ β (0, πΏπΏβ(ππ)), there exists π€π€β(ππ,π€π€οΏ½, πΏπΏ) > 0 such that for any π€π€ β (0,π€π€β(ππ,π€π€οΏ½, πΏπΏ)), the solutions of the system (3.47)-(3.49) satisfy
|οΏ½ΜοΏ½π’(ππ)| β€ π½π½οΏ½οΏ½|οΏ½ΜοΏ½π’(ππ0)|, πΏπΏπποΏ½ππ(ππ β ππ0)οΏ½ + ππ, (3.70)
|π§π§1(ππ)| β€ π½π½οΏ½οΏ½|π§π§1(ππ0)|, πποΏ½ππ(ππ β ππ0)οΏ½ + ππ, (3.71)
|π§π§2(ππ)| β€ π½π½οΏ½οΏ½|π§π§2(ππ0)|, (ππ β ππ0)οΏ½ + ππ, (3.72)
for all |(οΏ½ΜοΏ½π’(ππ0), π§π§1(ππ0), π§π§2(ππ0))| < Ξ , and ππ β₯ ππ0 β₯ 0.
36
Thus, for any ππ > 0, it is possible to tune πΏπΏ, πποΏ½, ππ such that π’π’οΏ½οΏ½οΏ½ will converge to a ππ-sized ball centred on π’π’οΏ½
β. Since in general π’π’οΏ½β will be in a small region of
optimal solution π’π’οΏ½β, it is necessary to discuss the convergence of π’π’οΏ½οΏ½οΏ½ with respect
to π’π’οΏ½β.
With this in mind, consider the case for π’π’οΏ½β β πποΏ½οΏ½οΏ½. As mentioned in Section
3.2.1, the proposed optimisation algorithm possesses the property that π’π’οΏ½β = π’π’οΏ½
β. According to (3.70), it follows that π’π’οΏ½οΏ½οΏ½ converges to the ball β¬οΏ½οΏ½οΏ½οΏ½οΏ½ = {π’π’οΏ½οΏ½οΏ½ β β|π’π’οΏ½οΏ½οΏ½ β π’π’οΏ½
β| β€ππ}, and β¬οΏ½οΏ½οΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½, provided a sufficiently small ππ.
The average of the constraint output is given by
ππ1,οΏ½οΏ½(π’π’) = 1ππ
οΏ½ ππ1(π’π’οΏ½οΏ½οΏ½ + πππ π πππΌπΌ (π€π€ππ))ππππ,οΏ½
οΏ½βοΏ½ (3.73)
where ππ = 2ππ/π€π€. Using a Taylor series expansion, the integrand in (3.73) can be written in the following form:
ππ1(π’π’οΏ½οΏ½οΏ½ + πππ π πππΌπΌ (π€π€ππ)) = ππ1(π’π’οΏ½οΏ½οΏ½) + πππ·π·οΏ½1ππ1(π’π’οΏ½οΏ½οΏ½)π π πππΌπΌ (π€π€ππ) + οΏ½2
2 π·π·οΏ½2ππ1(π’π’οΏ½οΏ½οΏ½) π π πππΌπΌ2(π€π€ππ) + Ξ(ππ3), (3.74)
thereby, for a sufficiently small ππ, ππ1,οΏ½οΏ½(π’π’) in (3.73) can be approximated by
ππ1,οΏ½οΏ½(π’π’) β ππ1(π’π’οΏ½οΏ½οΏ½), (3.75)
Since π’π’ converges to the ball β¬οΏ½οΏ½οΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½, then, the constrained is satisfied on average.
On the other hand, in the case of π’π’οΏ½β β ππππ, Section 3.2.1, showed that π’π’οΏ½
β is in the ball β¬οΏ½οΏ½
β = {π’π’οΏ½β β β|π’π’οΏ½
β β π’π’οΏ½β| β€ ππ(πΌπΌ)}. According to (3.70), π’π’οΏ½οΏ½οΏ½ converges to the
set β¬οΏ½οΏ½οΏ½οΏ½οΏ½ = {π’π’οΏ½οΏ½οΏ½ β β|π’π’οΏ½οΏ½οΏ½ β π’π’οΏ½β | β€ ππ}. Thus, π’π’οΏ½οΏ½οΏ½ converges to an ππ(πΌπΌ + ππ)-sized
neighbourhood of π’π’οΏ½β and ππ1,οΏ½οΏ½(π’π’) will be in an ππ(πΌπΌ + ππ)-sized neighourhood of
ππ1(π’π’οΏ½β) = 0. Hence, if π’π’οΏ½
β β ππππ, π’π’οΏ½οΏ½οΏ½ may converge to an equilibrium point that resides outside the constraint set ππ and consequently ππ1,οΏ½οΏ½(π’π’) > 0. Although this constraint violation could still be βsmallβ if the parameters ππ, πΏπΏ, πποΏ½, ππ, πΌπΌ are appropriately tuned, an approach to address this issue is presented in the next subsection.
3.3.3 Incorporation of integral action
As discussed, the proposed extremum seeking drives π’π’οΏ½οΏ½οΏ½ close to the optimal solution of the problem. However, when π’π’οΏ½
β β ππππ, π’π’οΏ½οΏ½οΏ½ might converge to an equilibrium point that on average violates the constraint. To overcome this, integral action is incorporated into the controller to modify the properties of the system by changing the equilibrium point of (3.42) such that, if π’π’οΏ½
β β ππππ, then π’π’οΏ½
β = π’π’οΏ½β. This eliminates the need of having the ball β¬οΏ½οΏ½
β in the first place.
37
To incorporate integral action in the controller, consider the proposed update law for π’π’ with a modification in the smooth function (3.45) given by
οΏ½ΜοΏ½π’ = βπ·π·οΏ½1ππ(π’π’), (3.76)
where
π·π·οΏ½1ππ(π’π’) = (1 β ππ) β π·π·οΏ½
1πποΏ½(π’π’) + ππ β π·π·οΏ½1ππ1(π’π’), (3.77)
and
ππ = 1
1 + ππβ[οΏ½1(οΏ½)βοΏ½]οΏ½
. (3.78)
Here ππ β β. The benefits of this parameter can be appreciated by considering the equilibrium point of (3.76),
0 = (1 β ππ) β π·π·οΏ½1πποΏ½(π’π’) + ππ β π·π·οΏ½
1ππ1(π’π’, ), (3.79)
solving for ππ:
ππ = π·π·οΏ½1πποΏ½(π’π’)
(π·π·οΏ½1πποΏ½(π’π’) β π·π·οΏ½
1ππ1(π’π’)). (3.80)
Since π’π’οΏ½β β ππππ , then, the goal is to have an equilibrium point on ππππ. Note that
ππ1(π’π’οΏ½β) = 0. Suppose in general that π·π·οΏ½
1ππ1(π’π’οΏ½β)π·π·οΏ½
1πποΏ½(π’π’οΏ½β) < 0, Then by subtituting
(3.78) in (3.80):
11 + πποΏ½
οΏ½= π·π·οΏ½
1πποΏ½(π’π’οΏ½β)
(π·π·οΏ½1πποΏ½(π’π’οΏ½
β) β π·π·οΏ½1ππ1(π’π’οΏ½
β)), (3.81)
solving for ππ:
ππ = πΌπΌπππΌπΌ οΏ½βπ·π·οΏ½1ππ1(π’π’οΏ½
β)π·π·οΏ½
1πποΏ½(π’π’οΏ½β)
οΏ½. (3.82)
Then, for any πΌπΌ > 0, there is an offset ππ such that (3.76) possesses an equilibrium point π’π’οΏ½ = π’π’οΏ½
β. Moreover, as discussed in Section 3.2.2, one of the properties of the gradient system (3.76) is that ππ(β ) has a strict local minimum π’π’οΏ½
β = π’π’οΏ½, and therefore, π’π’οΏ½β = π’π’οΏ½
β. Note that when the optimiser (3.76) is used within the proposed extremum seeking scheme, the gradients in (3.82) are not available, so are estimated with an approximation error. Despite this, it is still possible to tune ππ such that (3.79) has an equilibrium point on the boundary of the constraint.
The potential benefit of the offset ππ motivates the use of an online update law for this parameter. And so, the following proposition is made:
Proposition 1: The sigmoid function (3.45) is modified and the dynamical system in (3.47)-(3.49) is augmented with the integral action by
ππ = 1
1 + ππβ[οΏ½1(οΏ½)βοΏ½]οΏ½
(3.83)
38
ππ Μ = ππ β π¦π¦1(π₯π₯) (3.84)
ππ = πποΏ½ππ. (3.85)
The reference is set to ππ = 0 since the integral action aims to regulate the system output at π¦π¦1 = 0. For the sake of simplicity, the feedback control law is chosen to be ππ = πποΏ½ππ, where πποΏ½ is the integral gain.
3.3.4 Numerical example
To demonstrate the proposed extremum seeking controller for dynamic plants with output constraints, consider the following very simple example:
π₯π₯Μ = 100(π’π’ β π₯π₯), (3.86)
π¦π¦0 = (π₯π₯ + 1)2, (3.87)
π¦π¦1 = 2(π₯π₯ β 4)2 β 8. (3.88)
The steady state input-output maps are
πποΏ½(π’π’) = (π’π’ + 1)2, (3.89)
ππ1(π’π’) = 2(π’π’ β 4)2 β 8. (3.90)
The objective is to solve the following problem:
minοΏ½
πποΏ½(π’π’(ππ))
s. t. limοΏ½ββ
1ππ
οΏ½ ππ1(π’π’(ππ))πππποΏ½
οΏ½βοΏ½β€ 0.
(3.91)
The solution of this constrained optimisation is π’π’οΏ½β = 2, πποΏ½(2) = 9, ππ1(2) = 0.
This problem is firstly solved by using the proposed dynamical system (3.47)-(3.49) with no integral action. The controllerβs parameter are chosen to be ππ = 0.1, ππ = 3πππ π β1, πποΏ½ = 0.005, ππ = 0.1, and πΌπΌ = 2. The initial conditions are π’π’οΏ½οΏ½οΏ½(0) = 7, π₯π₯(0) = 7, πποΏ½(0) = 0, ππ1(0) = 0, and the period for averaging is ππ = 2ππ/ππ.
Figure 3.9 shows the simulation results. Notice that π’π’οΏ½οΏ½οΏ½ converges within a small region around π’π’οΏ½
β. In this particular case, the average of the constraint is -0.48. It is inside the feasible set, albeit slightly suboptimal. By augmenting the system with the integral action, it is possible for π’π’οΏ½οΏ½οΏ½ to converge closer to π’π’οΏ½
β. This is accomplished by updating the parameter ππ. The Figure 3.10 illustrates the effect of the integral action. π’π’οΏ½οΏ½οΏ½ is still cycling but converges on average to π’π’οΏ½
β. Moreover, the constraint is met with an average value of -0.0013, and ππ = 0.562.
39
ππ(π π ) Figure 3.9: Extremum seeking with output constraint: No integral action
0 20 40 60 80 100 120 140 1600
5
10
7 u(t
)
0 20 40 60 80 100 120 140 1600
20
40
60
80
fo(t
)
0 20 40 60 80 100 120 140 160-10
0
10
20
30
f1(t
);f1;a
ve(t)
f1(t)
f1;ave(t)
150 152 154
-1
0
1
f 1
150 152 1541.8
2
2.2
7u
40
Figure 3.10: Extremum seeking with output constraint and integral action
3.4 Conclusions
In this chapter an extremum seeking controller was developed for the constrained optimisation of dynamic plants with a single input and two outputs. The problem formulation was relaxed such that the controller enforces the output constraint satisfaction on average. The controller combined online gradient estimators with a proposed smooth continuous-time optimisation algorithm that takes into account the plantβs output constraint.
0 20 40 60 80 100 120 140 1600
5
10
7u(t
)
0 20 40 60 80 100 120 140 1600
20
40
60
80
fo(t
)
0 20 40 60 80 100 120 140 160-10
0
10
20
30
f 1(t
);f 1
;ave(t
)
f1(t)
f1;ave(t)
0 20 40 60 80 100 120 140 160-10
-5
0
5
"(t)
t (s)
150 152 154-1
0
1
f 1
150 152 1541.9
2
2.1
7u
41
Singular perturbation and averaging techniques were used to show that the reduced average system behaves as a smooth gradient system to solve an equivalent unconstrained optimisation problem. The cost function in this equivalent problem is a continuously differentiable Lyapunov function whose local minimum point is the solution to the original constrained optimisation problem, provided certain conditions hold. Thus, an existing framework for analysis and design of extremum seeking was used to assess the stability property of the closed-loop.
The convergence property of the closed-loop was analysed for cases in which the constrained optimum resides in the interior of the constrained set or on its boundary. In the latter case, the solution converges to a larger neighborhood compared to the former case. This was found to be primarily caused by the smoothing parameter. To overcome this, the proposed extremum-seeking control scheme was augmented with integral action whose potential benefit was demonstrated by means of simulations.
43
Chapter 4
4 Extremum seeking of spark timing under tailpipe emissions constraints
It is current Industry practice for tuning parameters of vehicleβs engines to be conventionally calibrated in a test rig laboratory. This is typically a lengthy process that utilises a series of experimental methods. These engine parameters are determined for a set of predefined engine operating points. The resulting optimal parameters are recorded in the engine control unit in the form of look-up tables (also referred to as 'engine maps'). A recognised shortcoming of this method is that these engine maps are only valid for the range of operating conditions that were used in their determination. Specifically, these maps are accurate at the fuel composition used in the experimental methods.
Chapter 1 identified two potential factors that may result in suboptimal calibrated maps. One of these factors is the fuel composition variation. Historically, automotive engines have utilised gasoline or diesel fuel. In order to reduce the tailpipe emission of automotive engines, many are now powered with a variety of alternative fuels such as natural gas, liquefied petroleum gas and gasoline blended with ethanol. These alternative fuels exhibit composition variation that may lead to suboptimal engine performance [3], [6]. The other important factor is the difference between the engine performance for emission compliance under controlled conditions in a test rig and the vehicle tailpipe emission in real-world driving situations. As discussed in Chapter 1, on-road emissions depend on the route type, operation mode, and ambient conditions, which all potentially compromise the emission performance [11].
44
These issues have generated interest in the use of some form of online engine calibration. In the literature review, extremum seeking was found to be a potential non-model-based adaptive control strategy to achieve the on-line calibration of automotive engines [5], [39]. This technique has been proven to optimise the engineβs spark timing, which is a parameter that affects the efficiency of the combustion, fuel consumption, tailpipe emissions, and the engine knock [56], [57]. The metric function considered in the reviewed literature of extremum seeking is essentially the brake specific fuel consumption (BSFC). But the tailpipe emission is not accounted for the online calibration of the spark timing to improve the fuel consumption.
Having developed an extremum seeking controller for dynamic plant optimisation with output constraints, this strategy can now be applied for on-line optimisation of engines. The novel goal of the proposed constrained extremum seeking controller is to tune the spark timing in order to minimise the fuel consumption subject to a legislated tailpipe emission. To demonstrate it, this chapter is organized as follows: first, the simulation environment of the engineβs transient model and its subcomponents are described. It is then followed by the problem formulation. Some open loop tests to explore the input-output maps are presented in conjunction with the proposed extremum scheme to solve to formulated problem. The chapter concludes with simulation results and conclusions.
4.1 Simulation environment: Plant description
The plant is a high-fidelity Mean Value Engine Model (MVEM) of the Ford Falcon engine, which has been experimentally calibrated at the Advanced Centre for Automotive Research and Testing (ACART) at the University of Melbourne. Additional sub-models augmenting the MVEM include the transient model for the intake manifold, combustion chamber, exhaust manifold, connecting pipelines, and three-way-catalytic converter. Every model is governed by mathematical equations and semi-empirical maps that capture the fundamental principles involved. These include physics, thermodynamics, heat transfer, fluid mechanics, and the aftertreatment dynamics. The structure of this integrated engine model is shown in Figure 4.1 and the modelβs equations can be found in the Appendix A.
The engine model is simulated using the software Matlab/Simulinkβ’. Inside this environment, the engineβs inputs are the normalized air-fuel ratio ππ, spark timing ππ, intake valve closing πποΏ½οΏ½οΏ½, valve overlap πποΏ½οΏ½οΏ½οΏ½, engine speed ππ, desired load πποΏ½,οΏ½οΏ½οΏ½, and vehicle speed πποΏ½οΏ½οΏ½.
45
Figure 4.1: Engine-aftertreatment model in Matlab/Simulinkβ’
Connecting pipe
Ξ»
V int (CAD-ABDC)
Vovlap (CAD)
Vspd (km/h)
NO (mg/km)
Tb,ref (Nm)
ΞΈ (CAD-BTDC)
N(RPM)
BSFC (g/kWh)
-C-
[Vspd]
Goto
u
Ξ±t
pim
mfuel
mcyl
Throttle & manifold Model
mcyl
mfuel
pim
Vspd
u Tcyl
Teng
mcyl
NOxd
Eoe
BSFC
Tbrake
Ford Falcon Mean Value Model
[Vspd]
From1
Tem,i
mcyl
Tem
Tem,o
mcyl
Exhaust manifold
mcyl
Tcp,i
Tcp
Tcp,o
mcyl
[Tb_r]
Goto3
[Vspd]
From
Eoe
Tg,in
mcyl
Vspd
mNO
mCO
mHC
Ξ·%
TWC
Tb,ref
Tb
Ξ± t
PI Torque controller
[Tb_r]
From2
T1
T2
rpm
r/s
r/s2
conversion
[Tem]
Goto2
[Tcp]
Goto4
.
..
...
-C-
-C-
-C-
-C-
-C-
-C-
BSFC
NO
46
Figure 4.2: Discretization of the catalytic converter. ππ = 1,2,3, β¦ ,60
A PI controller is used to adjust the engine throttle position (πΌπΌοΏ½) to track πποΏ½,οΏ½οΏ½οΏ½
for a specific engine speed. Once these inputs are specified, it is possible to calculate other variables in the model including the intake manifold pressure πποΏ½οΏ½, exhaust gas temperature πποΏ½οΏ½οΏ½, and engine-out emission πΈπΈοΏ½οΏ½. The engine out emission is a vector that contains the concentration of oxygen ππ2, nitric oxide ππππ, carbon monoxide π΅π΅ππ, hydrogen π»π»2, and hydrocarbon π»π»π΅π΅.
ππππ (ππ) β’ 1in β’ 2 β’ 3 β’ 4
ππππ (ππ) β’ 5 β’ 6 β’ 7 β’ 8
π΅π΅ππ,ππ2
(ππ) β’ 9 in β’ 10 β’ 11 β’ 12
π΅π΅ππ,π΅π΅ππ (ππ) β’ 13 in β’ 14 β’ 15 β’ 16 out
π΅π΅ππ,π»π»2
(ππ) β’ 17 in β’ 18 β’ 19 β’ 20
π΅π΅ππ,ππππ (ππ) β’ 21 in β’ 22 β’ 23 β’ 24 out
π»π»π΅π΅ππ,πππππ π ππ (ππ) β’ 25 in β’ 26 β’ 27 β’ 28 out
π»π»π΅π΅ππ,π π πππ π π€π€ (ππ) β’ 29 in β’ 30 β’ 31 β’ 32 out
π΅π΅ππ,ππ2
(ππ) β’ 33 β’ 34 β’ 35 β’ 36
π΅π΅ππ,π΅π΅ππ (ππ) β’ 37 β’ 38 β’ 39 β’ 40
π΅π΅ππ,π»π»2
(ππ) β’ 41 β’ 42 β’ 43 β’ 44
π΅π΅ ππ,ππππ(ππ) β’ 45 β’ 46 β’ 47 β’ 48
π»π»π΅π΅ππ,πππππ π ππ (ππ) β’ 49 β’ 50 β’ 51 β’ 52
π»π»π΅π΅ππ,π π πππ π π€π€ (ππ) β’ 53 β’ 54 β’ 55 β’ 56
ππ (ππ) β’ 57 β’ 58 β’ 59 β’ 60
Gas and substratetemperature
Species concentrationsin the gas
Species in the Substrate
Oxygen storage level
Nodes
πΏπΏ=143.5mm
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
47
The aftertreatment system for this engine is a three-way catalytic converter (TWC). Unlike the MVEM and connected sub-models described above, the aftertreatment system is a transient 1-D PDE model that describes the mass and energy transfer due to the chemical reactions that occur inside the TWC. In order to solve the equations of the TWC, the PDE model is discretised along the spatial domain with a uniform increment and a set of ODEs is obtained after applying the method of lines (MOL) [58]. The number of increments used in the discretisation effects the computational burden in solving the resulting ODE system. In [47], the TWCβs length πΏπΏ was divided into three increments as shown in Figure 4.2 as this does not significantly compromise the accuracy. More details about the TWC and the solution scheme can be found in Appendix B.
In Figure 4.2, the variables of interest are classified into four groups. The group of the temperatures contains the temperatures of the exhaust gas and catalytic substrate, πποΏ½ and πποΏ½ respectively. This group of temperatures encompasses the nodes 1-8. Similarly, the concentrations of the species in the gas π΅π΅οΏ½,οΏ½, and in the TWCβs washcoat (substrate) π΅π΅οΏ½,οΏ½ are located along nodes 9-32, and 33-56, respectively. Note that the hydrocarbons (π»π»π΅π΅) are modelled as slow-oxidising π»π»π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½, π»π»π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½ and fast-oxidising fuel, π»π»π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½, π»π»π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½ in addition, the engine-out hydrocarbon emission is assumed to comprise of 85% fast-oxidising and 15% of slow oxidising hydrocarbons. The chemical properties of the fast and slow oxidising fuels are approximated by those of π΅π΅3π»π»6 and π΅π΅π»π»4 respectively [59]. Finally, the oxygen storage level ππ is located along nodes 57-60. This oxygen storage level is used to model the capacity of the TWC to chemically store excess oxygen in its washcoat under a lean-burn engine condition and releases it when rich combustion occurs, thus providing the oxygen required to react with unburnt gasses.
Figure 4.2 also shows the inputs and outputs of interest. At the inlet, the TWC receives the exhaust mass flow rate οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ at the temperature πποΏ½, which is considered as the temperature at the front, thus driving the node 1. In a similar vein, the exhaust mass flow contains the engine-out emissions and specifies the concentration at the nodes 9, 13, 17, 21, 25, 29. These inputs are time-varying and drive the remaining 53 nodes by mean of 53 coupled non-linear differential equations (one per node). Consequently, TWC model consists of 53 states. Among these states, the nodes 16, 24, 28, 32 are the outputs that specify the concentrations of the legislated tailpipe emissions.
48
4.2 Problem formulation
The metric function to be optimised is the BSFC in g/kWh, which reflects the fuel efficiency of the engine. Figure 4.1 shows this as one of the outputs of the engine model and is calculated by
BSFC = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½οΏ½ππ, (4.1)
where οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ denotes the fuel mass flow rate in kg/s, ππ is the engine speed in rad/s and πποΏ½οΏ½οΏ½οΏ½οΏ½ is the produced engine torque in Nm. The high level goal is to adjust the engineβs parameters to minimise (4.1) whilst taking into account the legislated tailpipe emissions. These pollutants are the outputs of the TWC: ππππ,π΅π΅ππ, π»π»π΅π΅. In order to mathematically formulate this goal, it is necessary to define the outputs to be used in the optimisation process. To that end, the mass flow rates in kg/s of the legislated tailpipe emission are calculated according to:
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½πποΏ½οΏ½πποΏ½
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½), (4.2)
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½πποΏ½οΏ½πποΏ½
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½), (4.3)
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ οΏ½πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πποΏ½π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) +
πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πποΏ½π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½)οΏ½, (4.4)
where the exhaust gas mass flow rate οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ is given in kg/s, πποΏ½ is its molar mass, and πποΏ½οΏ½,πποΏ½οΏ½,πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½,πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½ are the molar masses of π΅π΅ππ, ππππ, π΅π΅3π»π»6 and π΅π΅π»π»4 respectively. The emissions at the end of the TWC are calculated from the concentration at the nodes 16, 24, 28, 32:
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½ (16) , (4.5)
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½ (24) , (4.6)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
(28) , (4.7)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
(32) . (4.8)
The emission standard used in this thesis is based on the European legislation. This legal framework consists in a series of directives that progressively tighten the limits on tailpipe emission over time. The TWC model used in this research is taken from [56], which utilized experimental result from a Euro-3 aftertreatment system to calibrate the model. Consequently, Euro-3 levels are used throughout this chapter. These tailpipe emissions limits for gasoline-passenger cars are reported in the following table.
49
Emission Tier: Euro-3
Limits (ππππ/ππππ)
ππππ 150
π΅π΅ππ 2300 π»π»π΅π΅ 200
Table 4.1: EURO-3 limits for emission in gasoline passenger cars
Note that tailpipe concentrations in (4.2)-(4.4) are expressed in kg/s but the regulations are distance-based limits in mg/km. οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½, οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ and οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ can be converted into mg/km by taking into account the vehicle speed, πποΏ½οΏ½οΏ½,
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½, (4.9)
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½, (4.10)
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½, (4.11)
where πποΏ½οΏ½οΏ½ β 0 is the vehicle speed in km/h.
These three outputs can be used for constrained optimisation. However, since the proposed extremum seeking strategy in Chapter 3 was developed to handle only one output constraint, it is necessary to choose one of the three outputs. Nowadays, NO and PM sensors to measure pollutants on a vehicle are already used in the aftertreatment system of diesel engine and so could be considered currently available. Consequently, the distance-based nitric oxide is the emission considered in this research.
Having defined BSFC as the cost function to minimise in (4.1) and πποΏ½οΏ½ in (4.10) as the output for constrained optimisation under the Euro-3 standard, it is now possible to state the mathematical formulation of the engine optimisation problem. According to Figure 4.1, the engine has several inputs candidates that might be used for optimisation studies. However, for the purpose of demonstrating the proposed extremum seeking controller to handling output constraints, the spark timing ππ is chosen to be the only manipulated input used. This choice is made on the basis of the strong influence of the spark timing to the engine performance and emission [39], [47], [57]. The optimisation problem is thus to find the optimal spark timing that minimises the fuel consumption subject to the tailpipe nitric oxide emission constraint. This is expressed mathematically as:
50
ππβ = arg min οΏ½βοΏ½οΏ½οΏ½οΏ½
BSFC (ππ), (4.12)
where
π΅π΅οΏ½οΏ½οΏ½ = οΏ½ππ β Ξ β β|πποΏ½οΏ½ β€ πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½. (4.13)
Here Ξ is the physical range for ππ and πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½ is based on the Euro-3 legislated emission limit. But, this formulation requires meeting the emission constraint at every time instant. Equation (4.13) might be relaxed for two main reasons. (1) the current vehicle tests for emission compliance are based on average tailpipe emission, and (2) the proposed extremum seeking controller uses a periodic dither signal to probe the plant and induce sufficient excitation for the gradient estimation of the plantβs outputs. As a consequence, a periodic output can be observed in this type of control strategy. For these reasons (4.13) is relaxed to allow the controller to enforce the constraint compliance on average. Thus, the set used in (4.12) is replaced by
π΅π΅οΏ½οΏ½οΏ½,οΏ½ = οΏ½ππ β Ξ β β| limοΏ½ββ
1ππ
οΏ½ πποΏ½οΏ½ πππποΏ½
οΏ½βοΏ½β€ πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½, (4.14)
where ππ is a suitable designed period calculated from the dither frequency, which is discussed in the following section.
4.3 Simulation set-up and engine mapping
Before running simulated tests of extremum seeking, it is necessary to specify the engine operating point. In addition, since this chapter aims to optimise the spark timing, it is required to set appropriate values for the remaining inputs. This is critical since as it will become apparent in the following section, incorporating tailpipe emission constraints in the optimisation might require not only the optimal tuning of spark timing ππ, but also other engine inputs such as the intake valve closing πποΏ½οΏ½οΏ½. Consequently, several open loop tests were conducted in order to determine suitable simulation conditions. Results are presented in terms of steady-state input-output maps, which are then used to find a region where there is a spark timing value such that the constraint (4.13) in not violated on average.
4.3.1 Engine operating point
The engine speed and load are prescribed inputs of the engine model. Moreover, as the emissions are expressed in mg/km, it is also necessary to specify the vehicle speed. The engine model used in this research does not include the model of the gearbox. Therefore, the abovementioned inputs are chosen from a
51
chassis dynamometer test in which a Ford Falcon vehicle was subject to the New European Drive Cycle (NEDC). The results of this test are shown in Figure 4.3. The operating point given by ππ = 1395 rpm, brake torque πποΏ½,οΏ½οΏ½οΏ½ = 53.2 Nm and vehicle speed πποΏ½οΏ½οΏ½ = 70.8 km/h is chosen at 853 s in this drive cycle. This operating point is inside the extra-urban driving cycle with high speed driving modes at a constant speed cruises. An alternative constant operating point at 1299 rpm, 31.2 Nm, 49.9 km/h was chosen for comparison of the proposed controller.
Figure 4.3: Engine brake torque and speed from a chassis dynamometer test with NEDC condition [56]. The (*) corresponds to the operating point used in this thesis
Since the air-fuel ratio (ππ) plays an important role in the pollutants formation
and therefore in tailpipe emissions, preliminary simulated tests in open loop were conducted in the engine model for a fixed speed of 1395 rpm, and desired load 53.2 Nm. Figure 4.4 shows a sweep of ππ with the corresponding effect in the conversion efficiency of the engine aftertreatment system. As discussed in the previous section, the ππππ concentration is the emission considered due to the current availability of sensors to measure this pollutant.
0 100 200 300 400 500 600 700 800 900 1000 1100 12000
25
50
75
100
125
150
Veh
icle
spee
d(k
m/h)
0 100 200 300 400 500 600 700 800 900 1000 1100 1200500
750
1000
1250
1500
1750
2000
Engin
esp
eed
(rev
/m
in)
0 100 200 300 400 500 600 700 800 900 1000 1100 1200-100-50
050
100150200250300
Bra
keto
rque
(Nm
)
t (s)
52
The ππ set point was chosen to be ππ = 1.01. This is leaner than would normally be considered in driving vehicle operations, and thus represents a harder test for NO-conversion whilst better BSFC improvement is possible. Furthermore, this operating point enables high conversion efficiency of HC and CO, which is beneficial, given that neither is considered as a constraint in the existing problem formulation.
Figure 4.4: Calculated steady-state conversion efficiency of the TWC at 1395 rpm and 53.2 Nm with respect to ππ
A second test was conducted in order to select the intake valve closing
position (πποΏ½οΏ½οΏ½). In this case, values for spark timing ππ and πποΏ½οΏ½οΏ½ are spanned by an 81-point grid. At each point of this grid, the simulation is run for sufficient time until all the engine and three-way catalytic converter states reach their steady-state condition. The resulting input-output maps of this test are illustrated in Figure 4.5. These maps depend on the engine operating point, input parameters, and the fuel composition. In addition, these steady-state maps are in general non-convex, whereby the optimisation problem becomes harder. Despite this, the contour levels of these maps include a shaded region, which is a non-empty feasible set such that
πποΏ½οΏ½οΏ½ = οΏ½(ππ, πποΏ½οΏ½οΏ½)|πποΏ½οΏ½ β€ πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½, (4.15)
where πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½ is 150 mg/km for ππππ emission. As a result, by fixing the intake valve timing πποΏ½οΏ½οΏ½ at any value in the interval [93.7, 98], the set π΅π΅οΏ½οΏ½οΏ½ in (4.13) is non-empty and it is possible to find a solution to the problem (4.12). This can be observed in the third round of tests conducted to obtain the steady-state maps
0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.0410
20
30
40
50
60
70
80
90
100Con
vers
ion
e/ci
ency
(%)
6 (!)
NOCOHC
53
between the spark timing versus BSFC and the distance-based ππππ emission (πποΏ½οΏ½), Figure 4.6.
Figure 4.5: Steady-state maps, ππ = 1395 rpm, πποΏ½,οΏ½οΏ½οΏ½ = 53.2 Nm, πποΏ½οΏ½οΏ½οΏ½ = 30, ππ = 1.01
4659
7285
98
212
2232
420
500
1000
1500
Vint
(CADABDC)
3
(CADBTDC)
mN
O(m
g=k
m)
4659
7285
98
212
2232
42300
400
500
600
700
Vint
(CADABDC)
3
(CADBTDC)
BSF
C(g
=kW
h)
410411
413413
415415
418
418
421
421
431
431
441
441
451
451
461
461
Vint (CADABDC)
3(C
AD
BTD
C)
46 52.5 59 65.5 72 78.5 85 91.5 9817
22
27
32
37
42
200
300
400
400
500
500
600
600
700
800
900
150
Vint (CADABDC)
3(C
AD
BTD
C)
46 52.5 59 65.5 72 78.5 85 91.5 9817
22
27
32
37
42
Figure 4.6: π΅π΅πππ΅π΅π΅π΅ and ππππ maps, ππ = 1395 rpm, πποΏ½,οΏ½οΏ½οΏ½ = 53.2 Nm, πποΏ½οΏ½οΏ½οΏ½ = 30, ππ = 1.01
10 15 20 25 30 35 40 45 50400
420
440
460
480
500
520
540
BS
FC
(g=kW
h)
3 (CADBTDC)
Vint = 94Vint = 96
Vint = 98
10 15 20 25 30 35 40 45 50100
150
200
250
300
350
400
450
mN
O(m
g=km
)
3 (CADBTDC)
Vint = 94Vint = 96
Vint = 98Euro-3 Constraint
54
The BSFC clearly exhibits a local extremum at the spark timing value of 42.5 CAD-BTDC. Moreover, notice these maps were obtained from a specific set of engine parameters and conditions. As a result, different maps can be expected if some of these engineβs parameters vary. For instance, the three different intake valve closing positions {94, 96, 98} displace the πποΏ½οΏ½ maps and the corresponding optimal spark timing that minimise the fuel consumption, whilst complying the Euro-3 regulation. For this reason, the extremum seeking is going to be demonstrated to adaptively locate the constrained optimal value of spark timing without the knowledge of these maps.
Remarks. It is worth mentioning that extremum seeking could be applied to find simultaneously the constrained optimal values of ππ and πποΏ½οΏ½οΏ½ in (4.15) by using the extremum seeking for multiple inputs. This would allow the controller to directly find the two optimal parameters in the shaded region depicted in Figure 4.5. However, this high-dimensional problem will most likely require the incorporation of additional constraints into the problem. For instance, a set to accommodate the physical (hardware) limits of the actuators πποΏ½ = {(ππ, πποΏ½οΏ½οΏ½)|ππ β[2,42], πποΏ½οΏ½οΏ½ β [46,96]}, is needed to close and bound πποΏ½οΏ½οΏ½ in (4.15), and to obtain a well-posed optimisation problem. Handling multiple inputs and output constraints is not considered in this research project, and it is a topic for future work.
4.4 Simulation results
This section presents simulation results to demonstrate the proposed extremum seeking scheme to solve the problem (4.12)-(4.14). For comparative purposes, this section begins with the result of the conventional extremum seeking scheme, which is essentially used for unconstrained optimisation, and henceforth will be referred to as UES. Then, the proposed approach to handling output constraints (CES) is introduced. The section ends by demonstrating the CES for a different engine operating point.
4.4.1 Unconstrained extremum seeking (UES)
Figure 4.7 shows the diagram of the conventional extremum seeking architecture to minimise BSFC alone. This scheme is based on the framework for analysis and design of extremum seeking control proposed in [13]. The transient model of the engine, in combination with the aftertreatment system, is enclosed in a block, thus, the plant is treated as a black-box system. A PI controller is
55
used to manipulate the throttle angle position πΌπΌοΏ½ to effectively emulate a driver pushing the pedal such that the engineβs brake torque is regulated at the desired load.
This UES strategy consists in a continuous-time optimisation algorithm, a
derivative estimator and a dither signal to introduce plant excitation. In this section, the gradient descent algorithm is selected, the dither signal is a sinusoid, and the gradient estimator is realised by mean of a low pass filter. The equations of the UES for the dynamic system are given by,
π₯π₯Μ = πποΏ½π₯π₯, ππ Μ + πππ π πππΌπΌ(π€π€ππ) , πΌπΌοΏ½οΏ½, (4.16)
ππ Μ Μ = πππ΅π΅οΏ½οΏ½ οΏ½ππ,Μ π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½, (4.17)
πποΏ½Μ = βπππποΏ½[πποΏ½ β π΅π΅πππ΅π΅π΅π΅ β π π πππΌπΌ(ππππ)], (4.18)
where ππ = πΏπΏπππποΏ½ and,
π΅π΅οΏ½οΏ½ οΏ½ππ,Μ π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½ = βπ·π·οΏ½ Μ
1οΏ½ (π΅π΅πππ΅π΅π΅π΅), (4.19)
π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅) = 2ππβ1πποΏ½. (4.20)
The extremum seeking parameters ππ, πΏπΏ, ππ, πποΏ½ need to be adjusted to obtain a desirable convergence of the closed loop. These parameters are chosen at sufficiently small values such that time scale separation among the plant (fastest
Figure 4.7: Unconstrained extremum seeking scheme to minimise fuel consumption
ππ Μ Μ = πππ΅π΅οΏ½οΏ½ οΏ½ππ,Μ π·π·οΏ½Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½
πππ π πππΌπΌ(π€π€ππ)
ππ = ππ Μ + πππ π πππΌπΌ(π€π€ππ)
πποΏ½,οΏ½οΏ½οΏ½
ππππππππππππ
PI TWC Plant
Engine/
ππ Μ +
π΅π΅πππ΅π΅π΅π΅
πΌπΌοΏ½
π€π€οΏ½π€π€π π + π€π€οΏ½π€π€
2ππ
π π πππΌπΌ(π€π€ππ)
Derivative Estimator
Optimiser
π·π·οΏ½Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅)
πποΏ½
56
time scale), the derivate estimator (medium time scale) and the optimiser (slowest time scale) is obtained. To do so, the extremum seeking controllerβs parameters are set to ππ = 1.5, ππ = 2πππ π β1, πποΏ½ = 0.01, ππ = 0.09. The simulation is then run at 1395 rpm engine speed, 53.2 Nm load, and 70.8 km/h vehicle speed. The rest of the inputs are the intake valve closing πποΏ½οΏ½οΏ½ = 94 CAD-ABDC, valve overlap πποΏ½οΏ½οΏ½οΏ½ = 30 CAD, and air-fuel ratio with ππ = 1.01. Figure 4.8 shows the results at this operating point.
Figure 4.8: UES simulation result.(1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap)
As can be seen, the spark timing converges to a neighborhood centred at the
optimal spark timing 42 CAD-BTDC where the minimum BSFC is attained. However, the average tailpipe ππππ emission is well above the Euro-3 limit.
To explain this tailpipe violation, it is necessary to further investigate the convergence of the plantβs internal states with the simulation environment. Although in a practical context the access to those states is not available in general, here some of the internal states of the TWC are shown to establish a background for further discussion in this section.
0 20 40 60 80 100 120 140 160 180 20010
20
30
40
50
3(C
AD
BT
DC
)
0 50 100 150 200 250 300 350 400400
425
450
475
500
525
550
BS
FC
(g=kW
h)
0 50 100 150 200 250 300 350 400100
150
200
250
300
350
mN
O(m
g=km
)
t (s)
Euro-3 limit
57
Figure 4.9 to Figure 4.11 show that the TWC states converge to equilibrium points during the optimisation of spark timing using the scheme shown in Figure 4.7. The numbering of nodes corresponds to that of the TWC discretization grid in Figure 4.2. Almost all the species concentration in either the gas phase or in the TWCβs washcoat settles down in 150 s. On the other hand, the exothermal reaction in the TWC produces increments in the temperature of the gas and substrate that exhibit a slower dynamic response compared to that of the species concentration.
Figure 4.9: TWC states: node 2 to 20. (1395 rpm engine speed,53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap)
0 50 100 150 200 250 300 350 400400
600
800
Tg(K
)
node 2node 3node 4
0 50 100 150 200 250 300 350 400400
600
800
Ts(K
)
node 6node 7node 8
0 50 100 150 200 250 300 350 4000.15
0.2
0.25
Cg;
O2(%
)
node 10node 11node 12
0 50 100 150 200 250 300 350 4000
0.02
0.04
Cg;
CO
(%)
node 14node 15node 16
0 50 100 150 200 250 300 350 4000
2
4
x 10-3
Cg;
H2( %
)
t (s)
node 18node 19node 20
58
Figure 4.10: TWC states: node 22 to 40. (1395 rpm engine speed ,53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap)
The tailpipe emission concentration at the nodes 16, 24, 28, 32 show that the concentration of π»π»π΅π΅ and π΅π΅ππ are reduced significantly but the tailpipe ππππ concentration is increased. This does not come as any surprise since the engine was set at 1.01 air-fuel ratio, thus excess oxygen is expected under the lean combustion. TWC is saturated with oxygen, which is observed by looking at the oxygen storage level ππ reaching 100% along the nodes 58-60. In this circumstance,
it is likely to expect a violation of the Euro-3 limit.
0 50 100 150 200 250 300 350 4000
200
400
Cg;
NO
(ppm
)
node 22node 23node 24
0 50 100 150 200 250 300 350 4000
100
200
300
HC
g;f
ast
(ppm
)
node 26node 27node 28
0 50 100 150 200 250 300 350 4000
100
200
300
HC
g;s
low
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59
Note that Figure 4.12 shows that optimal spark timing obtained with USC produces a pre-catalyst concentration of 600 ppm. But the conversion efficiency is only around 35% and the TWC cannot sufficiently reduce the emission. Therefore, when this emission is expressed in distance-based units the average tailpipe ππππ emission is above the Euro-3 limit. Next section proposes an approach to handle the ππππ output constraint into the optimisation of fuel consumption.
Figure 4.11: TWC states: node 42 to 60. (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap)
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Figure 4.12: Tailpipe ππππ emission and conversion efficiency with UES
4.4.2 Extremum seeking to handle the tailpipe nitric oxide constraint
To solve the problem (4.12)-(4.14), the developed extremum seeking architecture in Chapter 3 is going to be used. Figure 4.13 shows the proposed extremum seeking scheme. It shares many features with the UES controller introduced in the previous section. One of the main differences is the additional plantβs output of ππππ emission (πποΏ½οΏ½) in this architecture, which in turn, requires an additional low pass filter to estimate its gradient. For convenience, the output is shifted to the origin by changing the variables as ΞπποΏ½οΏ½ = πποΏ½οΏ½ β πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½.
The equations of the CES for the dynamic system are given by
π₯π₯Μ = πποΏ½π₯π₯, ππ Μ + πππ π πππΌπΌ(π€π€ππ) , πΌπΌοΏ½οΏ½, (4.21)
ππ Μ Μ = πππ΅π΅οΏ½οΏ½ οΏ½ππ,Μ ππ(πΌπΌ,βπποΏ½οΏ½),π·π·οΏ½ Μ 1οΏ½ (βπποΏ½οΏ½),π·π·οΏ½ Μ
1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½, (4.22)
πποΏ½Μ = βπππποΏ½[πποΏ½ β π΅π΅πππ΅π΅π΅π΅ β π π πππΌπΌ(ππππ)], (4.23)
ππ1Μ = βπππποΏ½[ππ1 β ΞπποΏ½οΏ½ β π π πππΌπΌ(ππππ)], (4.24)
where ππ = πΏπΏπππποΏ½ and
π΅π΅οΏ½οΏ½ = β[1 β ππ(ΞπποΏ½οΏ½,πΌπΌ)] β π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅) β ππ(ΞπποΏ½οΏ½,πΌπΌ) β π·π·οΏ½ Μ
1οΏ½ (βπποΏ½οΏ½), (4.25)
ππ(ΞπποΏ½οΏ½,πΌπΌ) = 1
1 + ππβΞοΏ½οΏ½οΏ½οΏ½, (4.26)
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post-catalyst withUES
0 50 100 150 200 25020
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E/
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61
Figure 4.13: Constrained extremum seeking scheme to minimise fuel consumption subject to ππππ emission constraint
where
π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅) = 2ππβ1πποΏ½, (4.27)
π·π·οΏ½ Μ 1οΏ½ (βπποΏ½οΏ½) = 2ππβ1ππ1. (4.28)
The extremum seeking parameters ππ, πΏπΏ, ππ, πποΏ½ and the smoothing parameter πΌπΌ > 0 need to be adjusted to obtain a desirable convergence on the closed loop. The procedure to tune the parameters ππ, πΏπΏ, ππ, πποΏ½ is the same as that in UES to provide the scale separation. The parameter πΌπΌ is used to control the transition between the gradient estimates given in (4.27) and (4.28) as the spark timing ππ evolves and approaches the constraint boundary, that is, ΞπποΏ½οΏ½ = 0. In addition, πΌπΌ is also used to reduce the steady-state error between the post-catalyst average ππππ emission and the emission limit.
Figure 4.14 demonstrates the proposed extremum architecture for optimising the spark timing under the tailpipe nitric oxide emission constraint. Note the spark timing still started outside the feasible region and violates the constraint. Eventually the spark timing approaches 19 CAD, where the constraint is met on average (πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = 150 mg/km). This can be observed in the πποΏ½οΏ½ plot, which is on average equal to 150 mg/km before the 20 seconds. However, 19 CAD spark timing is a feasible solution but not the optimal, thus the trajectory of ππ Μ evolves until it asymptotically converges to 28.55 CAD. At this point πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ =
Engine/ TWC Plant
π€π€οΏ½π€π€π π + π€π€οΏ½π€π€
ππ Μ Μ = πππ΅π΅οΏ½οΏ½ οΏ½ππ,Μ ππ(πΌπΌ, βπποΏ½οΏ½),π·π·οΏ½Μ 1οΏ½ (βπποΏ½οΏ½), π·π·οΏ½Μ
1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½
π΅π΅πππ΅π΅π΅π΅
π€π€οΏ½π€π€π π + π€π€οΏ½π€π€
πποΏ½οΏ½
πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½
+
β
π·π·οΏ½Μ 1οΏ½ (βπποΏ½οΏ½)
π·π·οΏ½Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅)
π π πππΌπΌ (π€π€ππ)
πποΏ½ +
ππ = ππ Μ + πππ π πππΌπΌ (π€π€ππ)
πππ π πππΌπΌ (π€π€ππ)
PI πποΏ½,οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½οΏ½
πΌπΌοΏ½ βπποΏ½οΏ½
2ππ 2
ππ
ππ0
ππ1
62
149.43 mg/km, which is slightly below Euro-3 limit. Although the obtained spark timing is suboptimal, it represents a significant reduction of tailpipe emission compared to the case with UES.
Figure 4.14 CES simulation result, (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smooth parameter)
Figure 4.15 provides important insights to understand how the emission limit
is now met. Comparatively, the constrained optimisation with the proposed CES controller induces a pre-catalyst ππππ concentration lower than that obtained with the UES. Although the engine-out ππππ conversion efficiency is around 35% for both of the controllers, the CES retards the spark timing such that TWC converts ππππ from a lower inlet concentration and the resulting post-catalyst emission is within the Euro-3 limit.
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Figure 4.16: CES result for different smoothing parameter πΌπΌ
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Figure 4.15: Tailpipe ππππ emission and conversion efficiency with CES
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64
Smooth parameter
πΌπΌ
Average emission ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ (mg/km)
Average emission πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ (mg/km)
Spark timing ππ Μ CAD-BTDC)
0.1 -0.34 149.65 28.69
1 -0.36 149.63 28.68
5 -0.57 149.43 28.55 10 -0.88 149.12 28.42
Table 4.2: Average ππππ emission and spark timing for different πΌπΌ values at 200 s
To study the effect of the parameter πΌπΌ in the convergence properties of the proposed CES, consider the Figure 4.16 . For a small πΌπΌ a faster respond with a greater overshoot in the output response is observed. Table 4.2 summarizes the average ππππ emission (ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½) and optimal spark timings for different πΌπΌ. The average ππππ emission approaches to 150 mg/km from below if a small πΌπΌ is chosen. Based on the theoretical analysis of the proposed CES in Chapter 3, this behaviour implies that in a neighbourhood of the constrained optimal spark timing the gradient of the constraint π·π·οΏ½ Μ
1(ΞπποΏ½οΏ½) is, in absolute value, greater than the gradient of the cost function π·π·οΏ½ Μ
1(π΅π΅πππ΅π΅π΅π΅). This can be appreciated by observing the steady-state values of these gradients in Figure 4.17. As a consequence, for any πΌπΌ > 0 the spark timing converges to values that are on average inside the constrained set and they approach the optimum by choosing a sufficiently small πΌπΌ.
Figure 4.17: Gradient convergence with CES
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65
Note that a choice of a small πΌπΌ may potentially introduce undesirable effects. Consider the update law π΅π΅οΏ½οΏ½ in (4.25). The Figure 4.17 shows the average of this gradient-based system for an πΌπΌ value of 0.1 and 5. With the former, π΅π΅οΏ½οΏ½ clearly exhibits several sudden rates of change during the first 50 seconds of the transient response. With this πΌπΌ, the algorithm will introduce a sharp transition between the gradients at the vicinity of the boundary of the constraint where βπποΏ½οΏ½ = 0. Depending on the application at hand, this might require considerable actuator effort. To alleviate this, the CES should be tuned to a higher πΌπΌ value. For instance, consider the test with πΌπΌ = 5 in Figure 4.17. This parameter effectively provides a smoother transient response at the price of obtaining suboptimal solutions according to Table 4.2.
4.4.3 Incorporation of integral action
In the previous section the smooth parameter πΌπΌ was shown to play a role in the constrained solution of CES. Although the resulting suboptimal solutions in Table 4.2 might be considered sufficiently close to the optimal spark timing that meet the Euro-3 emission standard, this section introduces integral action to demonstrate it as a potential alternative to reduce the average steady-state error in βπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½. To do so, the smoothing function is slightly modified and the closed- loop equations (4.25)-(4.28) are augmented with an integrator state ππ. The equations of the CES equipped with integral action are given by
π₯π₯Μ = πποΏ½π₯π₯, ππ Μ + πππ π πππΌπΌ(π€π€ππ) , πΌπΌοΏ½οΏ½, (4.29)
ππ Μ Μ = πππ΅π΅οΏ½οΏ½ οΏ½ππ,Μ ππ(πΌπΌ,βπποΏ½οΏ½, ππ),π·π·οΏ½ Μ 1οΏ½ (βπποΏ½οΏ½),π·π·οΏ½ Μ
1οΏ½ (π΅π΅πππ΅π΅π΅π΅)οΏ½, (4.30)
ππ Μ = βππΞπποΏ½οΏ½, (4.31)
πποΏ½Μ = βπππποΏ½[πποΏ½ β π΅π΅πππ΅π΅π΅π΅ β π π πππΌπΌ(ππππ)], (4.32)
ππ1Μ = βπππποΏ½[ππ1 β ΞπποΏ½οΏ½ β π π πππΌπΌ(ππππ)], (4.33)
where ππ = πΏπΏπππποΏ½ and
π΅π΅οΏ½οΏ½ = β[1 β ππ(πΌπΌ, βπποΏ½οΏ½, ππ)] β π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅) β ππ(πΌπΌ, βπποΏ½οΏ½, ππ) β π·π·οΏ½ Μ
1οΏ½ (βπποΏ½οΏ½), (4.34)
ππ(ΞπποΏ½οΏ½,πΌπΌ, ππ) = 1
1 + ππβΞοΏ½οΏ½οΏ½βοΏ½οΏ½
, (4.35)
where
π·π·οΏ½ Μ 1οΏ½ (π΅π΅πππ΅π΅π΅π΅) = 2ππβ1πποΏ½, (4.36)
π·π·οΏ½ Μ 1οΏ½ (βπποΏ½οΏ½) = 2ππβ1ππ1, (4.37)
66
ππ = πποΏ½ππ. (4.38)
Once again, πΏπΏ, ππ, πποΏ½, πΌπΌ and πποΏ½ are design parameters. By properly choosing the integral gain, the feedback control law (4.38) updates the offset parameter ππ. This parameter is fed into the sigmoid function in (4.35), which is used to modify the properties of the continuous-time optimisation algorithm in (4.34) by changing its equilibrium point.
Figure 4.18: Constrained Extremum seeking with integral action, (1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter, 0.25 integral gain)
In the following simulation result, the CES augmented with integral action was tested with an integral gain given by πποΏ½ = 0.25. The controller parameters
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πΏπΏ, ππ, πποΏ½ and the engine operating variables are the same as those used in the previous section. An πΌπΌ = 5 is selected as this ensured a smooth behavior of π΅π΅οΏ½οΏ½. This may represent a practical situation where reducing πΌπΌ is not possible and the average output is required to be on the boundary of the constraint, that is, ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = 0. Figure 4.18 shows the dynamic respond of spark timing ππ,Μ average emission πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½, and offset parameter ππ with the CES. The average ππππ emission πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ converges to 149.97 mg/km in 250 s. To further investigate the effect of the integral gains, additional simulation tests were conducted and the result are summarised in Table 4.3.
Integral Gain πποΏ½
Average emission ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ (mg/km)
Average emission πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ (mg/km)
Offset parameter
ππ
Spark timing ππ(Μ CAD-BTDC)
0.1 -0.18 149.81 0.56 28.688
0.35 -0.01 149.98 0.81 28.742
0.55 -0.002 149.99 0.842 28.7469
0.85 0.00 150.00 0.868 28.7479 Table 4.3: Average ππππ emission and spark timing for different integral gains at 250 s of simulation time. CES with integral action, engine at 1395 rpm engine speed, 53.2 Nm, 70.8 km/h vehicle speed, 1.01 AFR, 94 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter
As expected, among the integral gains, the highest value in the test (πποΏ½ = 0.85)
produces a faster convergence to the optimal spark timing where ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = 0 within 250 s of simulation. This result agrees with the analysis provided in Chapter 3 that justified the incorporation of the integral action to modify the properties of the proposed optimisation algorithm. The integral action is changing the equilibrium point of the system such that ππ Μ converges to the constrained optimum. To illustrate it, consider the modified sigmoid function defined in (4.35) and note that ππ converges to a positive value (0.868) with πποΏ½ = 0.85. With no integral action the offset is ππ = 0 and the constrained spark timing converges into a feasible point such that ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = β0.57 mg/km at steady-state (Table 4.2 for πΌπΌ = 5), as a result, ππ(β0.57,5,0) = 0.471. The location of this point along the sigmoid curve can be easily visualized in Figure 4.19 (solid line). With integral action, on the other hand, the steady-state value of ππ = 0.868 translates the sigmoid curve to the right where ππ = 0.456, such that the dynamic system (4.34) has an equilibrium at the constrained optimal spark timing and consequently ΞπποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = 0.
68
Figure 4.19: Sigmoid function with smoothing parameter πΌπΌ = 5
4.4.4 Simulation results for a different engine operating point
On the other hand, the proposed constrained extremum seeking equipped with integral action was shown for engine optimisation subject to ππππ tailpipe emission constraint at one engine operating point. To demonstrate the efficacy of the proposed extremum seeking control strategy for a different engine condition, the engine operating point is now set to 1299 rpm, 31.2 Nm, vehicle speed 49.9 km/h. The rest of the inputs are 1.01 air-fuel ratio, 80 CAD-ABDC intake valve closing, and 30 CAD valve overlap. The constrained extremum seeking controllerβs tuning parameters are chosen to be ππ = 1.5, ππ = 2πππ π β1, πποΏ½ = 0.01, ππ = 0.09, πΌπΌ = 5, and πποΏ½ = 0.35. Figure 4.20 shows the simulation results under these new conditions. The controller effectively drives the spark timing at 35.167 CAD-BTDC, under which the average tailpipe emission is πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½=150.00 mg/km, and π΅π΅π΅π΅πππ΅π΅ =575.73 g/kWh.
-15 -10 -5 0 5 10 150
0.1
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Sm
ooth
funct
ion<
"mNO
Sigmoid for " = 0
Sigmoid for " = 0:868
(!0:57; 0:471)
(0; 0:456)
69
Figure 4.20: Constrained Extremum seeking with integral action, (1299 rpm engine speed, 31.2 Nm, 49.9 vehicle speed, 1.01 AFR, 80 CAD-ABDC intake valve closing, 30 CAD valve overlap, 5 smoothing parameter, 0.35 integral gain)
4.5 Conclusions
The proposed extremum seeking controller for the optimisation of dynamics plants with output constraints was demonstrated for enginesβ park timing optimisation under tailpipe emission constraints. The demonstration was conducted in a high-fidelity engine/TWC model for two engine operating points. Results showed that it is possible to obtain the optimal spark timing that minimises fuel consumption while satisfying on average the Euro-3 emission constraint. In addition to that, a reasonable approximation of the optimal spark
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timing was achieved with the simplest version of the proposed scheme, that is, without updating the offset parameter ππ. However, the controller equipped with integral action demonstrated benefits in terms of providing spark timings closer to the constrained optimum.
71
Chapter 5
5 Contribution and future work
5.1 Contribution
In this thesis an approach was developed to optimise the steady-state input-put map of dynamical plants subject to average output constraints. This approach was based on an existing extremum seeking framework, which was originally developed for unconstrained optimisation purposes. The major contributions of this thesis are summarised as follows:
1. Developing an extremum seeking controller for the optimisation of dynamic plants subject to output constraints.
The development of the extremum seeking controller was achieved by constructing a suitable continuous-time optimisation algorithm. This approach relied upon smoothing techniques to incorporate plantβs output constraint into the controller. These smoothing techniques also facilitate the stability analysis of the proposed controller.
2. The capability of this extremum seeking controller has been demonstrated in a high-fidelity simulation environment for the spark timing of an automotive engine subject to regulated tailpipe emissions.
The proposed extremum seeking controller was used to minimise the specific fuel consumption of a gasoline fueled engine while conforming to legislated limits of ππππ emissions. In the simulation, the engine was tested at a fixed engine speed and load while the spark timing was varied by the controller. This controller was able to find the optimal spark timing whilst satisfying, on average, the Euro-3 emission limit.
72
5.2 Future work
The following discussion outlines both theoretical and applied research opportunities to further develop the concepts presented in this thesis.
1. Working with a larger class of non-convex input-output maps
The proposed controller was developed and analysed under strong assumptions such as the convexity of the steady-state input-output maps. Therefore, the convergence to the constrained optimum using the proposed approach may not be guaranteed in a context where the strict convexity is difficult if not impossible to meet. Thus, there are opportunities to investigate optimisation problems under non-convex and potentially non-differentiable maps.
2. Incorporation of other optimisation algorithms.
In this research, a continuous-time optimisation algorithm was constructed to handle output constraints. This algorithm is based on dynamical systems whose solutions converge to the constrained optimum under certain conditions. However, there is still room to propose new algorithms for improving the convergence speed or explore the possibility of combining existing fast extremum seeking algorithms with the ideas presented in this thesis. In addition, algorithms to prevent the trajectory of the manipulated variable from making excursions outside of a given set deserve further research.
3. Working with multiple inputs and multiple output constraints.
The proposed extremum seeking controller was developed for a dynamic plant with only one input and two outputs. One the outputs was the performance metric to be optimised whereas the other represented a constraint. This approach can be extended to those cases with multiple inputs and multiple outputs constraints by generalising some of the ideas introduced in this thesis.
On the other hand, potential future work from the application viewpoint includes:
1. Experimental validation of the proposed approach on an engine.
There is a potential opportunity to test the proposed extremum seeking by conducting experiments in an engine with its aftertreatment system. An in-cylinder pressure sensor can be installed in an engine to compute the net indicated work per cycle from the cylinder pressure and known volume. Provided the injected fuel measurement is available, then the net specific fuel consumption
73
can be estimated. This directly reflects the fuel consumption of the engine. Additionally, a commercially available ππππ sensor can be used to monitor this tailpipe output and enable the enforcement of the regulated emission constraints. With this experiment set-up, spark timings for optimal performance while meeting regulated emissions can be investigated.
2. Application of multi-variable extremum seeking for engine optimisation subject to regulated tailpipe emissions.
In order to achieve benefits in terms of fuel consumption and ππππ emission it may be necessary to optimally tune more than one engine parameter. This could be potentially accomplished by extending the extremum-seeking control for a multi-variable scenario. In addition, apart from emission constraints, inputs constraints set by actuator saturation must also be adhered, although this is relatively straightforward using projection operators.
75
Appendix A
A. Spark ignition engine model
This appendix presents the mean value model of a production 4L Ford Falcon engine. The model assumes ambient air temperature and adiabatic flow inside the intake manifold, and the engine is modeled as lumped thermal system. This model was experimentally validated at the advanced centre for automotive research and testing (ACART) at the University of Melbourne. The equations and parameters that are introduced in this appendix are reproduced from [47], [56].
The structure of the engine-aftertreatment model is shown in Figure A.1. This appendix is devoted to the engine model and its subsystems before the TWC. The aftertreatmet model is included in appendix B. The engineβs basic specifications are given in Table A.1.
Parameter Description Manufacturer Ford Australia
Cylinders In line 6 Capacity 3984 ππππ3
Bore 92.25 ππππ Stroke 99.31 ππππ
Conrod length 153.85 ππππ
Valve train Dual independent variable cam timing (di-VCT)
Aftertreatment system
Three-way catalytic converter
Table A.1: Ford Falcon engine specification
76
Figure A.1 Integrated engine model: throttle, intake and exhaust manifold, torque controller, connecting pipe, and the TWC
Connecting pipe
Ξ»
V int (CAD-ABDC)
Vovlap (CAD)
Vspd (km/h)
NO (mg/km)
Tb,ref (Nm)
( )
N(RPM)
BSFC (g/kWh)
-C-
[Vspd]
Goto
u
Ξ±t
pim
mfuel
mcyl
Throttle & manifold Model
mcyl
mfuel
pim
Vspd
u Tcyl
Teng
mcyl
NOxd
Eoe
BSFC
Tbrake
Ford Falcon Mean Value Model
[Vspd]
From1
Tem,i
mcyl
Tem
Tem,o
mcyl
Exhaust manifold
mcyl
Tcp,i
Tcp
Tcp,o
mcyl
[Tb_r]
Goto3
[Vspd]
From
Eoe
Tg,in
mcyl
Vspd
mNO
mCO
mHC
Ξ·%
TWC
Tb,ref
Tb
Ξ± t
PI Torque controller
[Tb_r]
From2
T1
T2
rpm
r/s
r/s2
conversion
[Tem]
Goto2
[Tcp]
Goto4
.
..
...
-C-
-C-
-C-
-C-
-C-
-C-
BSFC
NO
SK (CAD-BTDC)
77
The Table A.2 contains the molar mass of some gases that are used in calculations within different sub-sections of the engine-TWC model.
Gas πποΏ½ [ππππ/πππ π ππ] π΄π΄ππππ 28.96 Γ 10β3 ππππ 30.01 Γ 10β3 ππ2 32 Γ 10β3 π΅π΅ππ 28.01 Γ 10β3 π»π»2 2.016 Γ 10β3 ππ2 28.0134 Γ 10β3
π΅π΅3π»π»6 42.0797 Γ 10β3 π΅π΅π»π»4 16.043 Γ 10β3 πποΏ½οΏ½οΏ½οΏ½ 13.82 Γ 10β3
Table A.2: Molar mass of gases.
A.1 Throttle and intake manifold
The parameters for the models of the throttle and the intake manifold are listed in Table A.3, whereas the inputs, outputs and systemβs states are given in Table A.4.
Name Value Description πΌπΌοΏ½ 6 Number of cylinders [β] πποΏ½ 6.637 Γ 10β4 Cylinder swept volume [ππ3] πποΏ½οΏ½ 0.004 Intake manifold volume [ππ3] πποΏ½ 1.3381 Γ 10β3 Throttle parameter [β] πποΏ½ 4.6367 Γ 10β4 Throttle parameter [β] πποΏ½ 9.5868 Γ 10β6 Throttle parameter [β]
πποΏ½οΏ½οΏ½(β ) Variable Volumetric efficiency [β] π π οΏ½ 8.31 Universal gas constant [π½π½/πππ π πππΎπΎ] πΎπΎ 1.44 Ratio of specific heats πποΏ½/πποΏ½
πποΏ½οΏ½οΏ½ 101000 Ambient pressure [ππππ] πποΏ½οΏ½οΏ½ 298 Ambient temperature [πΎπΎ] π΄π΄π΅π΅π π οΏ½ 14.50 Stoichiometric air fuel ratio for gasoline
Table A.3: Parameters defining the throttle and intake manifold model
78
Name Type Description
ππ Input
Normalized air/fuel ratio [β] ππ Input
Engine speed [ππππππ/π π ]
πποΏ½οΏ½οΏ½ Input Intake valve closing [π΅π΅π΄π΄π·π· π΄π΄π΅π΅πππ·π·π΅π΅ ] πποΏ½οΏ½οΏ½οΏ½ Input Valve overlap [π΅π΅π΄π΄π·π·] πΌπΌοΏ½ Input Throttle position [ππππππ] πποΏ½οΏ½ State Intake manifold pressure [ππππ]
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ Output Fuel mass flow rate [ππππ/π π ] οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ Output Total mas flow rate of the mixture [ππππ/π π ]
Table A.4: Inputs, outputs and states for the throttle and intake manifold model
The air flow through the throttle plate can be approximated by the flow across a nozzle,
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ = π΅π΅οΏ½π΄π΄οΏ½(πΌπΌοΏ½)
οΏ½π π πποΏ½οΏ½οΏ½οΏ½ πποΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½
1οΏ½οΏ½ 2πΎπΎ
πΎπΎ β 1οΏ½1 β οΏ½ πποΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½
(οΏ½β1)/οΏ½
οΏ½οΏ½
12
, (A.1)
for the pressure ratios πποΏ½οΏ½/πποΏ½οΏ½οΏ½ greater than the critical value, which is defined as
πποΏ½οΏ½ = οΏ½ 2πΎπΎ + 1
οΏ½οΏ½
οΏ½β1, (A.2)
and
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ = π΅π΅οΏ½π΄π΄οΏ½(πΌπΌοΏ½)
οΏ½π π πποΏ½οΏ½οΏ½οΏ½πΎπΎοΏ½ 2
πΎπΎ + 1οΏ½
οΏ½+12(οΏ½β1)
, (A.3)
when the pressure ratio is less than πποΏ½οΏ½. The product of the throttle open area and the discharge coefficient is determined by
π΅π΅οΏ½π΄π΄οΏ½(πΌπΌοΏ½) = 10β3 Γ (πποΏ½πΌπΌοΏ½2 + πποΏ½πΌπΌοΏ½ + πποΏ½), (A.4)
whose coefficients {πποΏ½, πποΏ½, πποΏ½} were experimentally obtained and are reported in Table A.3. The mass flow rate of air into the cylinder is calculated by the following equation:
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ = πποΏ½οΏ½πΌπΌοΏ½πποΏ½ππ4πππ π πποΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½,ππ, πποΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,πποΏ½, (A.5)
where the volumetric efficiency for the Ford Falcon engine was approximated by the polynomial:
79
where
πποΏ½οΏ½ = 10β3πποΏ½οΏ½, (A.7)
πποΏ½οΏ½οΏ½οΏ½ = πποΏ½οΏ½οΏ½ β 58.5, (A.8)
πποΏ½οΏ½οΏ½ = ππ 602ππ
. (A.9)
Here, the manifold pressure is expressed in (kPa), intake valve closing position πποΏ½οΏ½οΏ½ relative to 58.5 CAD-ABDC, and engine speed ππ in (rpm). Based on this change of variables, the coefficients πποΏ½οΏ½οΏ½,οΏ½,οΏ½,οΏ½,οΏ½,οΏ½ were experimentally obtained and are given in Table A.5.
πποΏ½οΏ½οΏ½ parameters ππππ00000 0.35689 ππππ01001 β1.67540 Γ 10β5 ππππ00001 1.8431 Γ 10β4 ππππ01010 β9.5532 Γ 10β4 ππππ00002 β3.6965 Γ 10β8 ππππ01100 1.12060 Γ 10β3 ππππ00010 1.6316 Γ 10β3 ππππ02000 1.12920 Γ 10β1 ππππ00011 1.78080 Γ 10β6 ππππ10000 8.8101 Γ 10β3 ππππ00020 β9.6684 Γ 10β5 ππππ10001 β8.2443 Γ 10β7 ππππ00100 β2.7419 Γ 10β4 ππππ10010 2.9296 Γ 10β5 ππππ00101 3.48180 Γ 10β7 ππππ10100 1.4431 Γ 10β5 ππππ00110 β1.3453 Γ 10β4 ππππ11000 1.5496 Γ 10β4 ππππ00200 β1.1835 Γ 10β4 ππππ20000 β5.2594 Γ 10β5 ππππ01000 β0.20586
Table A.5: Volumetric efficiency parameters
(A.6) can be expanded as:
πποΏ½οΏ½οΏ½ = (
ππππ02000ππ2 + ππππ01100πππποΏ½οΏ½οΏ½οΏ½ + ππππ01001πππποΏ½οΏ½οΏ½ + ππππ11000πππποΏ½οΏ½ + ππππ01010 πππποΏ½οΏ½οΏ½οΏ½ + ππππ01000 ππ + ππππ00200πποΏ½οΏ½οΏ½οΏ½
2 + ππππ00101πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ + ππππ10100πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½ + ππππ00110πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ + ππππ00100πποΏ½οΏ½οΏ½οΏ½ + ππππ00002πποΏ½οΏ½οΏ½
2 + ππππ10001 πποΏ½οΏ½οΏ½πποΏ½οΏ½ + ππππ00011πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ + ππππ00001πποΏ½οΏ½οΏ½ + ππππ20000 πποΏ½οΏ½
2 + ππππ10010 πποΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ + ππππ10000πποΏ½οΏ½ + ππππ00020πποΏ½οΏ½οΏ½οΏ½
2 + ππππ00010πποΏ½οΏ½οΏ½οΏ½ + ππππ00000),
(A.10)
The fuel mass flow rate is given by
πποΏ½οΏ½οΏ½ = οΏ½οΏ½οΏ½ οΏ½ οΏ½ πποΏ½οΏ½οΏ½,οΏ½,οΏ½,οΏ½,οΏ½,οΏ½
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0πποΏ½οΏ½
οΏ½ πποΏ½πποΏ½οΏ½οΏ½οΏ½ οΏ½ πποΏ½οΏ½οΏ½οΏ½
οΏ½ πποΏ½οΏ½οΏ½ οΏ½ , (A.6)
80
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ =οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½
ππ(π΄π΄π΅π΅π π οΏ½), (A.11)
and the total mass flow rate of the mixture by
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ + οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½. (A.12)
Finally the differential equation that governs the intake manifold pressure is given by
πππποΏ½οΏ½ππππ
= πΎπΎπ π οΏ½πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½πποΏ½οΏ½
(οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ β οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½) (A.13)
A.2 The engine
This section is split into three main subsections: (1) the engine exhaust gas temperature, (2) torque production and friction, and (3) the engine-out emission. The parameters of the model, inputs, outputs and states and are given in Table A.6 and Table A.7.
The steady flow energy balance applied to the fluid inside the control volume delimited by the combustion chamber is,
οΏ½ΜοΏ½ποΏ½οΏ½ β οΏ½ΜοΏ½ποΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½π₯π₯βοΏ½, (A.14)
the heat due to the fuel combustion and heat exchanged with the walls is given by,
οΏ½ΜοΏ½ποΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ + π΄π΄οΏ½οΏ½οΏ½βοΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½οΏ½, (A.15)
and the work output by
οΏ½ΜοΏ½ποΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½, πππΎπΎ, ππ, πποΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,πποΏ½. (A.16)
Then (A.14) can be written as
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½1 β πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½, πππΎπΎ, ππ, πποΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,πποΏ½οΏ½ +π΄π΄οΏ½οΏ½οΏ½βοΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ π₯π₯βοΏ½.
(A.17)
In (A.17), three variables need to be specified: ΞβοΏ½, βοΏ½οΏ½ and πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½(β ). The term π₯π₯βοΏ½ is determined by approximating the gasesβ properties in the combustion chamber as those of air modelled as ideal gas,
π₯π₯βοΏ½ = 28.1(πποΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½) + 1.9670 Γ 10β3οΏ½πποΏ½οΏ½οΏ½
2 β πποΏ½οΏ½οΏ½ 2 οΏ½
2+
4.8020 Γ 10β6οΏ½πποΏ½οΏ½οΏ½ 3 β πποΏ½οΏ½οΏ½
3 οΏ½3
β1.9 Γ 10β9οΏ½πποΏ½οΏ½οΏ½
4 β πποΏ½οΏ½οΏ½ 4 οΏ½
4,
(A.18)
81
Name Value Description π½π½οΏ½οΏ½οΏ½οΏ½οΏ½ 0.15 Crankshaft moment of inertia [ππππ ππ2] π΄π΄οΏ½οΏ½οΏ½ 0.04215 Total combustion chamber area [ππ2] π΅π΅ 0.9225 Cylinder bore diameter [ππ]
πποΏ½οΏ½οΏ½ 44 Γ 106 Lower heating value of the fuel [π½π½/ππππ] πποΏ½ 69 Cylinder wall heat transfer parameter πποΏ½ 0.2 Cylinder wall heat transfer parameter
πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ 115000 Thermal mass of the engine [π½π½/πΎπΎ] πΎπΎοΏ½οΏ½ 3.5 Equilibirum constant of water-gas shift reaction πποΏ½οΏ½ 1.8 Fuel π»π» to π΅π΅ ratio πποΏ½2 0.2095 Proportion of ππ2 in air [πππ π ππ/πππ π ππ]
πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½(β ) Variable Net indiacated efficiency [β] πποΏ½οΏ½(β ) Variable π΅π΅ππ emissions [πππ π ππ/(ππππ πππ’π’ππππ)] πποΏ½οΏ½(β ) Variable ππππ emissions [πππ π ππ/(ππππ πππ’π’ππππ)] πποΏ½οΏ½(β ) Variable π»π»π΅π΅ emissions [πππ π ππ/(ππππ πππ’π’ππππ)]
Table A.6: Parameters for the engineβs model
Name Type Description
ππ Input
Normalized air/fuel ratio [β] ππ Input
Engine speed [ππππππ/π π ]
οΏ½ΜοΏ½π Input
Engine acceleration [ππππππ/π π 2] πποΏ½οΏ½οΏ½ Input Intake valve closing [π΅π΅π΄π΄π·π· β π΄π΄π΅π΅πππ·π·π΅π΅ ] πποΏ½οΏ½οΏ½οΏ½ Input Valve overlap [π΅π΅π΄π΄π·π·] πππΎπΎ Input Spark timing [π΅π΅π΄π΄π·π· β π΅π΅πππ·π·π΅π΅] πποΏ½οΏ½ Input Intake manifold pressure [ππππ]
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½ Input Fuel mass flow rate [ππππ/π π ] οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ Input Total mas flow rate of the mixture [ππππ/π π ] πποΏ½οΏ½οΏ½ State Lumped engine temperature [πΎπΎ] πποΏ½οΏ½οΏ½ Output Exhaust temperature [πΎπΎ]
πποΏ½οΏ½οΏ½οΏ½οΏ½ Output Engine brake torque [ππππ] π΅π΅πππ΅π΅π΅π΅ Output Brake specific fuel consumption [ππ/ππππβ] πΈπΈοΏ½οΏ½ Output Engine-out emissions β β5 [πππ π ππ/πππ π ππ]
Table A.7: Inputs, outputs and states for the engineβs model
82
The heat transfer coefficient, βοΏ½οΏ½ is obtained by using,
βοΏ½οΏ½ = πποΏ½πποΏ½οΏ½οΏ½
π΅π΅ οΏ½
4οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
πΌπΌοΏ½ πππ΅π΅πποΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½
, (A.19)
where the thermal conductivity is given by
πποΏ½οΏ½οΏ½ = 3.4288 Γ 10β11πποΏ½οΏ½οΏ½ 3 β 9.1803 Γ 10β8πποΏ½οΏ½οΏ½
2 +
1.294 Γ 10β4πποΏ½οΏ½οΏ½ β 5.2076 Γ 10β3, (A.20)
and the dynamic viscosity is by
πποΏ½οΏ½οΏ½ = 1.066 Γ 10β14πποΏ½οΏ½οΏ½ 3 β 3.6432 Γ 10β11πποΏ½οΏ½οΏ½
2 +
6.6706 Γ 10β8πποΏ½οΏ½οΏ½ + 1.433 Γ 10β6. (A.21)
The indicated thermal efficiency for the Ford Falcon engine was approximated by the polynomial,
πποΏ½οΏ½οΏ½οΏ½ =
οΏ½οΏ½οΏ½ οΏ½ οΏ½ οΏ½ πποΏ½,οΏ½,οΏ½,οΏ½,οΏ½,οΏ½πποΏ½οΏ½ οΏ½ πποΏ½πππΎπΎ οΏ½πποΏ½οΏ½οΏ½οΏ½
οΏ½ πποΏ½οΏ½οΏ½οΏ½ οΏ½ πποΏ½οΏ½οΏ½
οΏ½2
οΏ½=0
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0
2
οΏ½=0, (A.22)
where the variables πποΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½ and πποΏ½οΏ½οΏ½, were given in (A.7)-(A.9). The coefficients πποΏ½,οΏ½,οΏ½,οΏ½,οΏ½,οΏ½ were experimentally obtained and are reported in Table A.8.
πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½ parameters
ππ000000 β8.4001 Γ 10β1 ππ002000 β1.2642 Γ 10β4 ππ000001 3.4125 Γ 10β5 ππ010000 1.4883 ππ000002 β6.2929 Γ 10β9 ππ010001 β4.1641 Γ 10β5 ππ000010 1.536 Γ 10β3 ππ010010 β2.7417 Γ 10β3 ππ000011 6.8762 Γ 10β7 ππ010100 β3.6511 Γ 10β3 ππ000020 β5.1696 Γ 10β5 ππ011000 8.8654 Γ 10β3 ππ000100 1.211 Γ 10β3 ππ020000 β9.4752 Γ 10β1 ππ000101 5.2989 Γ 10β7 ππ100000 8.2314 Γ 10β3 ππ000110 β8.6462 Γ 10β5 ππ100001 β4.9836 Γ 10β7 ππ000200 β6.1361 Γ 10β5 ππ100010 4.4968 Γ 10β5 ππ001000 8.6231 Γ 10β4 ππ100100 6.3069 Γ 10β5 ππ001001 1.2589 Γ 10β6 ππ101000 β8.5707 Γ 10β5 ππ001010 4.4078 Γ 10β5 ππ110000 8.5054 Γ 10β3 ππ001100 5.3959 Γ 10β5 ππ200000 β1.1767 Γ 10β4
Table A.8: The net indicated efficiency coefficients
The net indicated efficiency is then expanded as
83
πποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½ = ( ππ000200 πποΏ½οΏ½οΏ½οΏ½
2 + ππ000101πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ + ππ010100πποΏ½οΏ½οΏ½οΏ½ +ππ100100πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½ + ππ001100πποΏ½οΏ½οΏ½οΏ½πππΎπΎ + ππ000110πποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ +ππ000100πποΏ½οΏ½οΏ½οΏ½ + ππ000002πποΏ½οΏ½οΏ½
2 + ππ010001πποΏ½οΏ½οΏ½ππ + ππ100001πποΏ½οΏ½οΏ½πποΏ½οΏ½ +ππ001001πποΏ½οΏ½οΏ½πππΎπΎ + ππ000011πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ + ππ000001πποΏ½οΏ½οΏ½ + ππ020000ππ2 + ππ110000πππποΏ½οΏ½ + ππ011000ππ(πππΎπΎ) + ππ010010πππποΏ½οΏ½οΏ½οΏ½ + ππ010000ππ
+ππ200000πποΏ½οΏ½ 2 + ππ101000 πποΏ½οΏ½πππΎπΎ + ππ100010πποΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½ + ππ100000 πποΏ½οΏ½
+ππ002000 πππΎπΎ 2 + ππ001010 πππΎπΎοΏ½πποΏ½οΏ½οΏ½οΏ½οΏ½ + ππ001000πππΎπΎ + ππ000020 πποΏ½οΏ½οΏ½οΏ½ 2
+ ππ000010πποΏ½οΏ½οΏ½οΏ½ + ππ000000),
(A.23)
A.2.1 Torque production and friction
Newtonβs second law applied to the engine crankshaft yields,
πποΏ½οΏ½οΏ½οΏ½οΏ½ = πποΏ½οΏ½οΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½οΏ½ β π½π½οΏ½οΏ½οΏ½οΏ½οΏ½ππππππππ
, (A.24)
where
πποΏ½οΏ½οΏ½οΏ½οΏ½ =οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½
πππποΏ½οΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½πποΏ½οΏ½, πππΎπΎ,ππ, πποΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,πποΏ½, (A.25)
and
πποΏ½οΏ½οΏ½οΏ½ =πΌπΌοΏ½οΏ½οΏ½πποΏ½
4πππποΏ½οΏ½οΏ½οΏ½ππ, πποΏ½οΏ½οΏ½οΏ½. (A.26)
The frictional effective pressure, πποΏ½οΏ½οΏ½, is approximated by
πποΏ½οΏ½οΏ½ = πποΏ½1πποΏ½οΏ½οΏ½οΏ½ + πποΏ½2πποΏ½οΏ½οΏ½οΏ½ 2 + πποΏ½3πποΏ½οΏ½οΏ½ + πποΏ½4, (A.27)
where πποΏ½οΏ½οΏ½οΏ½ is the lumped engine temperature πποΏ½οΏ½οΏ½ in β and,
πποΏ½1 = β7881 (A.28)
πποΏ½2 = 50.596 (A.29)
πποΏ½3 = 39.107 (A.30)
πποΏ½4 = 324910.6 (A.31)
The brake specific fuel consumption in [ππ/ππππβ] is given by
π΅π΅πππ΅π΅π΅π΅ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½
πποΏ½οΏ½οΏ½οΏ½οΏ½ππ, (A.32)
and the energy balance applied to the engine yields,
πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ πππποΏ½οΏ½οΏ½
ππππ= π΄π΄οΏ½οΏ½οΏ½βοΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½οΏ½ + πππποΏ½οΏ½οΏ½οΏ½. (A.33)
84
A.2.2 Engine-out emissions
Engine-out emissions such as π΅π΅ππ, ππππ and π»π»π΅π΅ were obtained as static maps, which depend on several engine parameters,
πποΏ½οΏ½(ππ) (A.34)
πποΏ½οΏ½(πποΏ½οΏ½, πππΎπΎ, ππ,ππ) (A.35)
πποΏ½οΏ½(πποΏ½οΏ½, πππΎπΎ,ππ, πποΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,ππ) (A.36)
where πποΏ½ is the normalized emission of the compound ππ in [πππ π πποΏ½/ππππ πππ’π’ππππ]. By introducing the change of variables (A.7)-(A.9), the normalized emission are given by the following calibrated polynomials,
(A.37)
πποΏ½οΏ½οΏ½πποΏ½οΏ½, πππΎπΎ, ππ,πποΏ½οΏ½οΏ½οΏ½ =
(A.38)
πποΏ½οΏ½(πποΏ½οΏ½, πππΎπΎ,ππ, πποΏ½οΏ½οΏ½οΏ½, πποΏ½οΏ½οΏ½οΏ½,πποΏ½οΏ½οΏ½) =
πππ΅π΅ππ(ππ ) = (15430.2129ππ5 β 82708.4092ππ4 + 175681.3381ππ3 β 184575.6623ππ2 + 95730.3456ππ β 19554.9757)
(0.00026607 (ππππππ )2 + 0.043891 ππππππ ππ β 0.00071808 ππππππ πππΎπΎ
+ 2.3068 Γ 10β5ππππππ ππππππππ β 0.094552 ππππππ + 7.2269 ππ2
+ 0.071725 πππππΎπΎ β 0.0018909 ππππππππππ β 17.9685 ππβ 0.00044268 (πππΎπΎ)2 β 3.1327 Γ 10β6πππΎπΎ ππππππππ + 0.0067427πππΎπΎ
+ 8.0627 Γ 10β7οΏ½ππππππππ οΏ½2 β 0.0023642 ππππππππ + 15.5957)
οΏ½β8.6207 Γ 10β6(ππππππ )3 + 0.0015661 (ππππππ )2ππ β 2.5081 Γ 10β5(ππππππ )2πππΎπΎ
+ 1.0709 Γ 10β5(ππππππ )2πππππΌπΌππππ + 2.0362 Γ 10β5(ππππππ )2πππ π ππππππ β 1.3387 Γ 10β7(ππππππ )2ππππππππ β 0.00019104 (ππππππ )2 β 0.40518 ππππππ ππ2
+ 0.006661 ππππππ πππππΎπΎ β 0.0027099 ππππππ πππππππΌπΌππππ β 0.0021044 ππππππ πππππ π ππππππ
β 2.1798 Γ 10β5ππππππ ππππππππππ + 0.72644 ππππππ ππ β 5.9171 Γ 10β6ππππππ (πππΎπΎ)2
+ 1.0742 Γ 10β5ππππππ πππΎπΎ πππππΌπΌππππ + 2.299 Γ 10β5ππππππ πππΎπΎ πππ π ππππππ + 3.1168 Γ 10β8ππππππ πππΎπΎ ππππππππ β 0.0034456 ππππππ πππΎπΎ β 1.6157 Γ 10β5ππππππ (πππππΌπΌππ ππ )2
β 3.9845 Γ 10β5ππππππ πππππΌπΌππππ πππ π ππππππ + 4.1223 Γ 10β7ππππππ πππππΌπΌππππ ππππππππ
+ 0.0017333 ππππππ πππππΌπΌππππ β 1.495 Γ 10β5ππππππ οΏ½πππ π ππππππ οΏ½2 + 2.5882 Γ 10β7ππππππ πππ π ππππππ ππππππππ + 0.00020505 ππππππ πππ π ππππππ + 1.6978
Γ 10β9ππππππ οΏ½ππππππππ οΏ½2 + 1.3202 Γ 10β5ππππππ ππππππππ β 0.35554 ππππππ
β 36.5564 ππ3 β 0.24804 ππ2πππΎπΎ + 0.24286 ππ2πππππΌπΌππππ+ 0.2056 ππ2πππ π ππππππ β 0.0038141 ππ2ππππππππ + 121.9179 ππ2
+ 0.0019239 ππ(πππΎπΎ)2 β 0.0035712 πππππΎπΎ πππππΌπΌππππ β 0.0026637 πππππΎπΎ πππ π ππππππ
+ 5.3709 Γ 10β5πππππΎπΎ ππππππππ + 0.20893 πππππΎπΎ β 0.0019114 ππ(πππππΌπΌππππ )2
85
(A.39)
The overall reaction of the combustion of gasoline is given by
πΌπΌοΏ½οΏ½οΏ½οΏ½π΅π΅π»π»οΏ½οΏ½οΏ½ + πΌπΌοΏ½οΏ½οΏ½οΏ½πποΏ½2ππ2 + οΏ½1 β πποΏ½2
οΏ½ππ2οΏ½
β πΌπΌοΏ½οΏ½π΅π΅π»π»οΏ½οΏ½οΏ½+ πΌπΌοΏ½οΏ½π΅π΅ππ + πΌπΌοΏ½οΏ½ππππ + πΌπΌοΏ½οΏ½2
π΅π΅ππ2
+ πΌπΌοΏ½2ππ2 + πΌπΌοΏ½2
ππ2 + πΌπΌοΏ½2π»π»2 + πΌπΌοΏ½2οΏ½π»π»2ππ,
(A.40)
and the water-gas shift reaction by
π΅π΅ππ2 + π»π»2 β π΅π΅ππ + π»π»2ππ. (A.41)
Thus, it is possible to estimate the other products of the reaction (A.40), [56].
πποΏ½οΏ½2= 1
πποΏ½οΏ½οΏ½οΏ½β πποΏ½οΏ½ β πποΏ½οΏ½, (A.42)
πποΏ½2οΏ½ = 0.5πποΏ½οΏ½
1πποΏ½οΏ½οΏ½οΏ½
β πποΏ½οΏ½
1 + πποΏ½οΏ½πΎπΎοΏ½οΏ½πποΏ½οΏ½2
, (A.43)
πποΏ½2=
πποΏ½2
πποΏ½οΏ½οΏ½π΄π΄π΅π΅π π οΏ½ππ β 0.5πποΏ½οΏ½ β 0.5πποΏ½οΏ½ β πποΏ½οΏ½2
β 0.5πποΏ½2οΏ½ , (A.44)
πποΏ½2=
πποΏ½οΏ½πποΏ½2οΏ½
πΎπΎοΏ½οΏ½πποΏ½οΏ½2
, (A.45)
where
β 0.0024392 πππππππΌπΌππππ πππ π ππππππ + 2.8594 Γ 10β5πππππππΌπΌππππ ππππππππ β 0.25166 πππππππΌπΌππππ
β 0.0021685 πποΏ½πππ π ππππππ οΏ½2 + 7.3965 Γ 10β5πππππ π ππππππ ππππππππ β 0.25842 πππππ π ππππππ
β 4.1835 Γ 10β7πποΏ½ππππππππ οΏ½2 + 0.0061968 ππππππππππ β 131.0003 ππ β 9.5166 Γ 10β6(πππΎπΎ)3 + 7.8065 Γ 10β6(πππΎπΎ)2πππππΌπΌππππ + 1.5802 Γ 10β5(πππΎπΎ)2πππ π ππππππ
+ 2.0864 Γ 10β7(πππΎπΎ)2ππππππππ β 0.0013541 (πππΎπΎ)2 β 2.8959 Γ 10β5πππΎπΎ (πππππΌπΌππππ )2 β 5.2006 Γ 10β5πππΎπΎ πππππΌπΌππππ πππ π ππππππ + 1.0303 Γ 10β7πππΎπΎ πππππΌπΌππππ ππππππππ + 0.0039259 πππΎπΎ πππππΌπΌππππ β 2.5525
Γ 10β5πππΎπΎ οΏ½πππ π ππππππ οΏ½2 + 1.8598 Γ 10β7πππΎπΎ πππ π ππππππ ππππππππ + 0.002035 πππΎπΎ πππ π ππππππ
β 4.1617 Γ 10β9πππΎπΎ οΏ½ππππππππ οΏ½2 β 5.2757 Γ 10β5πππΎπΎ ππππππππ
β 0.039895 πππΎπΎ + 2.0968 Γ 10β5(πππππΌπΌππππ )3 + 4.3798 Γ 10β5(πππππΌπΌππππ )2πππ π ππππππ β 1.5891 Γ 10β7(πππππΌπΌ ππππ )2ππππππππ
+ 0.0011484 (πππππΌπΌππππ )2 + 4.6537 Γ 10β5πππππΌπΌππππ οΏ½πππ π ππππππ οΏ½2 β 4.1651 Γ 10β7πππππΌπΌππππ πππ π ππππππ ππππππππ + 0.0021636 πππππΌπΌππππ πππ π ππππππ β 1.0669
Γ 10β8πππππΌπΌππππ οΏ½ππππππππ οΏ½2 + 1.6185 Γ 10β6πππππΌπΌππππ ππππππππ + 0.021872 πππππΌπΌππππ
+ 1.4185 Γ 10β5οΏ½πππ π ππππ ππ οΏ½3 β 8.0137 Γ 10β8οΏ½πππ π ππππππ οΏ½2ππππππππ
+ 0.0013281 οΏ½πππ π ππππππ οΏ½2 β 2.1102 Γ 10β8πππ π ππππππ οΏ½ππππππππ οΏ½2 β 7.0696
Γ 10β6πππ π ππππππ ππππππππ + 0.078225 πππ π ππππππ + 1.5873 Γ 10β10 οΏ½ππππππππ οΏ½3 + 2.0081
Γ 10β7οΏ½ππππππππ οΏ½2 β 0.0026787 ππππππππ + 46.1231οΏ½
86
π΄π΄π΅π΅π π οΏ½ = 1 + 0.25πποΏ½οΏ½πποΏ½2
οΏ½ πποΏ½οΏ½οΏ½πποΏ½οΏ½οΏ½οΏ½
οΏ½. (A.46)
The engine-out gases needed for the three-way catalytic converter model are ππ2,π΅π΅ππ,π»π»2,ππππ and π»π»π΅π΅. Their concentration are given by
π΅π΅οΏ½,οΏ½ = πποΏ½πποΏ½οΏ½οΏ½
1 + πππ΄π΄π΅π΅π π οΏ½, (A.47)
where π΅π΅οΏ½,οΏ½ is the exhaust concentration of the species X in [πππ π ππ/πππ π ππ]. Then the engine-out emissions of interest for the TWC can be represented by
πΈπΈοΏ½οΏ½ = οΏ½π΅π΅οΏ½,οΏ½2,π΅π΅οΏ½,οΏ½οΏ½,π΅π΅οΏ½,οΏ½2
,π΅π΅οΏ½,οΏ½οΏ½,π΅π΅οΏ½,οΏ½οΏ½οΏ½β€, (A.48)
A.3 Exhaust manifold
The experimentally obtained parameters for the exhaust manifold are given in Table A.9. Similarly, the input and outputs of this subsystem are reported in Table A.10.
Name Value Description πποΏ½οΏ½1 0.0179 Nusselt paramter inner convection πποΏ½οΏ½1 0.95 Nusselt paramter inner convection πποΏ½οΏ½2 0 Nusselt paramter outer convection π·π·οΏ½οΏ½οΏ½ 0.060 Inner diameter at outlet [ππ] π΄π΄οΏ½οΏ½1 0.181 Inner surface area [ππ2] π΄π΄οΏ½οΏ½2 0.199 Outer surface area [ππ2]
πποΏ½οΏ½πποΏ½οΏ½ 4140 Exhasut manifold thermal mass [π½π½/πΎπΎ]
Table A.9: Parameters for the exhaust manifold model
Name Type Description
πποΏ½οΏ½,οΏ½ Input Exhaust manifold inlet temperature [πΎπΎ] οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ Input Total mass flow rate of the mixture [ππππ/π π ] πποΏ½οΏ½ State Lumped exhaust manifold temperature [πΎπΎ] πποΏ½οΏ½,οΏ½ Output Exhaust manifold outlet temperature [πΎπΎ]
Table A.10: Inputs, outputs and states and for the exhaust manifold subsystem
By using a steady-state approximation of the gas temperature inside the exhaust manifold, the energy balance for the gas phase in the exhaust manifold equation reduces to
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½π₯π₯βοΏ½οΏ½,οΏ½ = βοΏ½οΏ½,οΏ½οΏ½π΄π΄οΏ½οΏ½1οΏ½πποΏ½οΏ½ β πποΏ½οΏ½,οΏ½οΏ½, (A.49)
where
87
πποΏ½οΏ½,οΏ½ βπποΏ½οΏ½,οΏ½ + πποΏ½οΏ½,οΏ½
2, (A.50)
and
π₯π₯βοΏ½οΏ½,οΏ½ = 28.1(πποΏ½οΏ½,οΏ½ β πποΏ½οΏ½,οΏ½) + 1.9670 Γ 10β3οΏ½πποΏ½οΏ½,οΏ½
2 β πποΏ½οΏ½,οΏ½ 2 οΏ½
2+
4.8020 Γ 10β6οΏ½πποΏ½οΏ½,οΏ½
3 β πποΏ½οΏ½,οΏ½ 3 οΏ½
3β
1.9 Γ 10β9οΏ½πποΏ½οΏ½,οΏ½ 4 β πποΏ½οΏ½,οΏ½
4 οΏ½4
. (A.51)
In order to compute βοΏ½οΏ½,οΏ½οΏ½, the Nusselt correlation is firstly approximated by
πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½= πποΏ½οΏ½1οΏ½π π πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½1, (A.52)
where
π π πποΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½=
4οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
πππ·π·οΏ½οΏ½οΏ½πποΏ½οΏ½,οΏ½πΌπΌοΏ½. (A.53)
Finally
βοΏ½οΏ½,οΏ½οΏ½ =πποΏ½οΏ½,οΏ½
π·π·οΏ½οΏ½οΏ½1πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½οΏ½
. (A.54)
The conductivity and dynamic viscosity are calculated by,
πποΏ½οΏ½,οΏ½ = 3.4288 Γ 10β11οΏ½πποΏ½οΏ½,οΏ½οΏ½3 β 9.1803 Γ 10β8οΏ½πποΏ½οΏ½,οΏ½οΏ½
2 +1.29410β4οΏ½πποΏ½οΏ½,οΏ½οΏ½ β 5.2076 Γ 10β3, (A.55)
πποΏ½οΏ½,οΏ½ = 1.066 Γ 10β14οΏ½πποΏ½οΏ½,οΏ½οΏ½3 β 3.6432 Γ 10β11οΏ½πποΏ½οΏ½,οΏ½οΏ½
2
+ 6.6706 Γ 10β8οΏ½πποΏ½οΏ½,οΏ½οΏ½ + 1.433 Γ 10β6, (A.56)
The energy balance in the solid phase of the exhaust manifold body reduces to
πποΏ½οΏ½πποΏ½οΏ½πππποΏ½οΏ½ππππ
= βοΏ½οΏ½,οΏ½οΏ½π΄π΄οΏ½οΏ½1οΏ½πποΏ½οΏ½,οΏ½ β πποΏ½οΏ½οΏ½. (A.57)
A.4 Connecting pipe
The connecting pipe model parameters, inputs and outputs are presented in Table A.11 and Table A.12 respectively.
The energy balance in steady state for the gas phase inside the connecting pipe equation reduces to
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½οΏ½π₯π₯βοΏ½οΏ½,οΏ½ = βοΏ½οΏ½,οΏ½οΏ½π΄π΄οΏ½οΏ½1οΏ½πποΏ½οΏ½ β πποΏ½οΏ½,οΏ½οΏ½, (A.58)
Where
πποΏ½οΏ½,οΏ½ βπποΏ½οΏ½,οΏ½ + πποΏ½οΏ½,οΏ½
2 (A.59)
88
Name Value Description πποΏ½οΏ½1 0.0641 Nusselt paramter inner convection πποΏ½οΏ½1 0.789 Nusselt paramter inner convection πποΏ½οΏ½2 529 Nusselt paramter outer convection π·π·οΏ½οΏ½1 0.060 Inner diameter [ππ] π·π·οΏ½οΏ½2 0.064 Outer diameter [ππ] π΄π΄οΏ½οΏ½1 0.0826 Inner surface area [ππ2] π΄π΄οΏ½οΏ½2 0.0882 Outer surface area [ππ2]
Table A.11: Parameters for the exhaust manifold model
Name Type Description
πποΏ½οΏ½,οΏ½ Input Connecting pipe inlet temperature [πΎπΎ] οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ Input Total mass flow rate of the mixture [ππππ/π π ] πποΏ½οΏ½ State Lumped pipe temperature [πΎπΎ] πποΏ½οΏ½,οΏ½ Output Connecting pipe outlet temperature [πΎπΎ]
Table A.12: Inputs, states and outputs for the connecting pipe
and
π₯π₯βοΏ½οΏ½,οΏ½ = 28.1(πποΏ½οΏ½,οΏ½ β πποΏ½οΏ½,οΏ½) +1.9670 Γ 10β3οΏ½πποΏ½οΏ½,οΏ½
2 β πποΏ½οΏ½,οΏ½ 2 οΏ½
2+
4.8020 Γ 10β6οΏ½πποΏ½οΏ½,οΏ½
3 β ππcp,οΏ½ 3 οΏ½
3β
1.9 Γ 10β9οΏ½πποΏ½οΏ½,οΏ½ 4 β πποΏ½οΏ½,οΏ½
4 οΏ½4
. (A.60)
The convective heat transfer coefficient between the gas and the internal wall of the pipe is calculated by
βοΏ½οΏ½,οΏ½οΏ½ =πποΏ½οΏ½,οΏ½1
π·π·οΏ½οΏ½1πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½1
, (A.61)
where, the Nusseltβs number is approximated by
πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½1= πποΏ½οΏ½1 οΏ½π π πποΏ½οΏ½,οΏ½οΏ½οΏ½1
οΏ½οΏ½οΏ½οΏ½1
, (A.62)
and the Reynoldβs number by
π π πποΏ½οΏ½,οΏ½οΏ½οΏ½1=
4οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
πππ·π·οΏ½οΏ½1πποΏ½οΏ½,οΏ½1πΌπΌοΏ½. (A.63)
The fluidβs properties are evaluated at the mean temperature πποΏ½οΏ½,οΏ½,
πποΏ½οΏ½,οΏ½1 = 1.066 Γ 10β14πποΏ½οΏ½,οΏ½ 3 β 3.6432 Γ 10β11πποΏ½οΏ½,οΏ½
2 + 6.6706 Γ 10β8πποΏ½οΏ½,οΏ½ + 1.433 Γ 10β6,
(A.64)
89
πποΏ½οΏ½,οΏ½1 = 3.4288 Γ 10β11πποΏ½οΏ½,οΏ½ 3 β 9.1803 Γ 10β8πποΏ½οΏ½,οΏ½
2 + 1.29410β4πποΏ½οΏ½,οΏ½ β 5.2076 Γ 10β3,
(A.65)
Similarly the convective heat transfer coefficient between the gas and the internal outlet surface of the pipe and the ambient air is given by
and the Nusseltβs Number by
πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½2 = πποΏ½οΏ½2. (A.67)
The thermal conductivity is evaluated at the ambient temperature,
πποΏ½οΏ½,οΏ½2 = 3.4288 Γ 10β11πποΏ½οΏ½οΏ½ 3 β 9.1803 Γ 10β8πποΏ½οΏ½οΏ½
2 + 1.29410β4πποΏ½οΏ½οΏ½ β 5.2076 Γ 10β3.
(A.68)
The energy balance in the solid phase of the connecting pipe body reduces to
πποΏ½οΏ½πποΏ½οΏ½πππποΏ½οΏ½
ππππ= βοΏ½οΏ½,οΏ½οΏ½π΄π΄οΏ½οΏ½1οΏ½πποΏ½οΏ½,οΏ½ β πποΏ½οΏ½οΏ½ + βοΏ½οΏ½,οΏ½οΏ½π΄π΄οΏ½οΏ½2οΏ½πποΏ½οΏ½οΏ½ β πποΏ½οΏ½οΏ½ (A.69)
βοΏ½οΏ½,οΏ½οΏ½ =πποΏ½οΏ½,οΏ½2
π·π·οΏ½οΏ½2πππ’π’οΏ½οΏ½,οΏ½οΏ½οΏ½2
, (A.66)
91
Appendix B
B. Three-way catalytic converter model
The three-way-catalytic converter (TWC) model is formulated on the basis that heat and mass transfer in both gaseous and solid phases are the most dominant phenomena. Table B.1 and Table B.2 contain the parameters of the model. This model is formulated by the following system of partial differential equations:
πππποΏ½οΏ½οΏ½πποΏ½
π π πποΏ½
πππποΏ½
ππππ= β
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½πποΏ½
π΄π΄οΏ½
πππποΏ½
πππ₯π₯+ ππβοΏ½οΏ½οΏ½πποΏ½ β πποΏ½οΏ½ (B.1)
πποΏ½πποΏ½(1 β ππ) πππποΏ½ππππ
= ππβοΏ½οΏ½οΏ½πποΏ½ β πποΏ½οΏ½ + πποΏ½(1 β ππ) ππ2πποΏ½πππ₯π₯2 β πποΏ½πΌπΌοΏ½ οΏ½π π οΏ½,οΏ½π₯π₯βοΏ½,οΏ½
οΏ½οΏ½
οΏ½=1 (B.2)
πππποΏ½πππ΅π΅οΏ½,οΏ½
ππππ= β
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
π΄π΄οΏ½
πππ΅π΅οΏ½,οΏ½
πππ₯π₯+ πππποΏ½βοΏ½,οΏ½οΏ½,οΏ½οΏ½π΅π΅οΏ½,οΏ½ β π΅π΅οΏ½,οΏ½οΏ½ (B.3)
πππποΏ½πππ΅π΅οΏ½,οΏ½
πππ₯π₯= ππ
πποΏ½πποΏ½βοΏ½,οΏ½οΏ½,οΏ½οΏ½π΅π΅οΏ½,οΏ½ β π΅π΅οΏ½,οΏ½οΏ½ β πΌπΌοΏ½πποΏ½π π οΏ½,οΏ½ (B.4)
Name Value Description π΄π΄οΏ½ 0.0119 Cross-sectional area of the substrate [ππ2] πποΏ½ 1500 Specific heat capacity of the substrate [π½π½/(πππππΎπΎ)]
π·π·οΏ½ 0.001105 Hydraulic diameter of a channel [ππ]
πποΏ½ 3.0 Thermal conductivity of the substrate [W/(mK)]
πΏπΏ 0.1435 Reactor length [ππ]
ππ 2740 Geometric surface area per unit reactor volume [m2/m3]
ππ 0.757 Reactor void fraction [β] πποΏ½ 2240 Density of the substrate [ππππ/ππ3]
πποΏ½ 1.98 Γ 10β5 Washcoat thickness [m]
πΌπΌοΏ½ 273 Catalytic surface area per unit reactor volume [m2/m3]
πΉπΉ 600 Oxygen storage capacity [πππ π ππ/ππ3]
ππβ 3.66 Sherwood number [β]
πππ’π’ 3.66 Nusselt number [β]
πΌπΌοΏ½ 10 Number of reactions in the TWC
Table B.1: Parameters for the catalyst model
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Name Type Description
πποΏ½,οΏ½οΏ½ Input TWC feed gas temperature [πΎπΎ] πΈπΈοΏ½οΏ½ Input Engine-out emissions [πππ π ππ/πππ π ππ],πΈπΈοΏ½οΏ½ β β5 οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ Input Total mass flow rate of the mixture [ππππ/π π ] πποΏ½οΏ½οΏ½ Input Vehicle speed [ππππ/β] πποΏ½οΏ½ Output NOx mass flow rate in [ππππ/ππππ] πποΏ½οΏ½ Output CO mass flow rate in [ππππ/ππππ] πποΏ½οΏ½ Output HC mass flow rate in [ππππ/ππππ]
Table B.2: Inputs and output of the TWC model
ππ Pre β exponential factors [πππ π ππ πΎπΎ/ππ3π π ] π΄π΄ctivation energy [π½π½ ]
For the rate of reactions*
1 π΄π΄οΏ½,1 = 1.064380 Γ 1020 πΈπΈοΏ½,1 = 90000
2 π΄π΄οΏ½,2 = 1.064380 Γ 1020 πΈπΈοΏ½,2 = 90000
3 π΄π΄οΏ½,3 = 5.687341 Γ 1018 πΈπΈοΏ½,3 = 95000
4 π΄π΄οΏ½,4 = 1.625393 Γ 1017 πΈπΈοΏ½,4 = 120000
5 π΄π΄οΏ½,5 = 5.610389 Γ 1012 πΈπΈοΏ½,5 = 90000
6 π΄π΄οΏ½,6 = 1.819701 Γ 1013 πΈπΈοΏ½,6 = 90000
7 π΄π΄οΏ½,7 = 5.175123 Γ 1013 πΈπΈοΏ½,7 = 90000
8 π΄π΄οΏ½,8 = 2.610655 Γ 1011 πΈπΈοΏ½,8 = 85000
9 π΄π΄οΏ½,9 = 2.972991 Γ 1011 πΈπΈοΏ½,9 = 85000
10 π΄π΄οΏ½,10 = 2.009992 Γ 1013 πΈπΈοΏ½,10 = 85000
For the reaction inhibition term**
1 π΄π΄1 = 65.5 πΈπΈ1 = β7990
2 π΄π΄2 = 2080 πΈπΈ2 = β3000
3 π΄π΄3 = 3.98 πΈπΈ3 = β96534
4 π΄π΄4 = 4.79 Γ 105 πΈπΈ4 = 31036
Table B.3: Pre-exponential factors and activation energy for the reaction model of the three-way catalyst.* Parameters obtained from experiments [56]. ** Data taken from [59].
This set of partial differential equations is commonly used in many 1-dimensional three way catalytic converter models to describe the energy exchange and mass transport phenomena between the gas and the washcoat layer along the direction of flow (βπ₯π₯-axesβ). Note that, equation (B.1) indicates that the gas temperature πποΏ½ is changing along the channel due to the convective heat transfer with the washcoat surface. The conductive heat transfer in the gas phase is
93
normally neglected in this type of model. However, according to (B.2), the temperature of the solid phase πποΏ½ is governed by the convective heat transfer with the surrounding gas and the heat conduction inside the substrate. This explains the presence of the second order partial derivative of the temperature with respect to π₯π₯ in the right hand side of the equation. The last term accounts for the heat generated during the chemical reaction that take place in the wash coat surface.
Based on the mass balance equation for the gas, the concentration of each species π΅π΅οΏ½,οΏ½ changes along the π₯π₯ direction as a result of the convective mass transfer that takes place within the washcoat. In the solid phase the species concentration π΅π΅οΏ½,οΏ½ is governed by this convective mass transfer and also the consumption of species during the reactions that occur in the solid substrate. The mass transfer process, in conjunction with those chemical reactions, is the key process for converting the engine-out pollutants into less harmful gases.
B.1 Reaction kinetics of the three-way catalyst
Table B.4 contains the reaction mechanism of the three way catalytic converter. The pre-exponential factors and the activation energy are given in Table B.3. The reaction mechanism can be used to derive the consumption rate of the species π π οΏ½,οΏ½.
B.1.1 Consumption rates
Consider a vector ππ β β11 containing the especies involved in the reaction mechanism shown in Table B.4 and its stoichiometric matrix ππ β β10Γ11,
ππ = οΏ½π΅π΅ππ, ππ2,π΅π΅ππ2,π»π»2,π»π»2ππ,π΅π΅π»π»1.8οΏ½οΏ½οΏ½οΏ½, π΅π΅π»π»1.8
οΏ½οΏ½οΏ½οΏ½,ππππ, ππ2,π΅π΅ππ2ππ3,π΅π΅ππππ2οΏ½β€, (B.5)
ππ =
β
βββββββββββββ
β1 β0.5 1 0 0 0 0 0 0 0 00 β0.5 0 β1 1 0 0 0 0 0 00 β1.45 1 0 0.9 β1 0 0 0 0 00 β1.45 1 0 0.9 0 β1 0 0 0 0
β2 0 2 0 0 0 0 β2 1 0 00 β0.5 0 0 0 0 0 0 0 β1 20 0 0 0 0 0 0 β1 0.5 β1 2
β1 0 1 0 0 0 0 0 0 1 β20 0 1 0 0.9 β1 0 0 0 1.9 β3.80 0 1 0 0.9 0 β1 0 0 1.9 β3.8
β
βββββββββββββ
. (B.6)
This matrix satisfies the following mass conserversation equation,
ππππ = 0, (B.7)
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ππ Reaction Reaction rate [πππ π ππ/π π ππ3]
Oxidation reactions
1 π΅π΅ππ + 0.5ππ2 β π΅π΅ππ2 π π οΏ½,1 = πΊπΊβ1π΄π΄οΏ½,1πππ₯π₯ππ(βπΈπΈοΏ½,1/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½π΅π΅οΏ½,οΏ½2
2 π»π»2 + 0.5ππ2 β π»π»2ππ π π οΏ½,2 = πΊπΊβ1π΄π΄οΏ½,2πππ₯π₯ππ(βπΈπΈοΏ½,2/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½2π΅π΅οΏ½,οΏ½2
3 π΅π΅π»π»1.8οΏ½οΏ½οΏ½οΏ½ + 1.45ππ2 β π΅π΅ππ2 + 0.9π»π»2ππ π π οΏ½,3 = πΊπΊβ1π΄π΄οΏ½,3πππ₯π₯ππ(βπΈπΈοΏ½,3/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
π΅π΅οΏ½,οΏ½2
4 π΅π΅π»π»1.8οΏ½οΏ½οΏ½οΏ½ + 1.45ππ2 β π΅π΅ππ2 + 0.9π»π»2ππ π π οΏ½,4 = πΊπΊβ1π΄π΄οΏ½,4πππ₯π₯ππ(βπΈπΈοΏ½,4/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
π΅π΅οΏ½,οΏ½2
ππππ reduction reaction
5 2ππππ + 2π΅π΅ππ β 2π΅π΅ππ2 + ππ2 π π οΏ½,5 = π΄π΄οΏ½,5πππ₯π₯ππ(βπΈπΈοΏ½,5/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½π΅π΅οΏ½,οΏ½οΏ½
Oxygen storage reactions
6 π΅π΅ππ2ππ3 + 0.5ππ2 β 2π΅π΅ππππ2 π π οΏ½,6 = π΄π΄οΏ½,6πππ₯π₯ππ(βπΈπΈοΏ½,6/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½2(1 β ππ)
7 π΅π΅ππ2ππ3 + ππππ β 2π΅π΅ππππ2 + 0.5ππ2 π π οΏ½,7 = π΄π΄οΏ½,7πππ₯π₯ππ(βπΈπΈοΏ½,7/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½(1 β ππ)
8 2π΅π΅ππππ2 + π΅π΅ππ β π΅π΅ππ2ππ3 + π΅π΅ππ2 π π οΏ½,8 = π΄π΄οΏ½,8πππ₯π₯ππ(βπΈπΈοΏ½,8/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½ππ
9 π΅π΅π»π»1.8οΏ½οΏ½οΏ½οΏ½ + 3.8π΅π΅ππππ2 β 1.9π΅π΅ππ2ππ3 + π΅π΅ππ2 + 0.9π»π»2ππ π π οΏ½,9 = π΄π΄οΏ½,9πππ₯π₯ππ(βπΈπΈοΏ½,9/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½ππ
10 π΅π΅π»π»1.8οΏ½οΏ½οΏ½οΏ½ + 3.8π΅π΅ππππ2 β 1.9π΅π΅ππ2ππ3 + π΅π΅ππ2 + 0.9π»π»2ππ π π οΏ½,10 = π΄π΄οΏ½,10πππ₯π₯ππ(βπΈπΈοΏ½,10/π π οΏ½πποΏ½)π΅π΅οΏ½,οΏ½οΏ½ππ
Table B.4: Reaction mechanism in the three-way catalytic converter [59].
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The production rate π π οΏ½ is calculated according to
π π οΏ½ = ππβ€π π , (B.8)
where π π οΏ½ = οΏ½π π οΏ½,1, π π οΏ½,2, β― , π π οΏ½,οΏ½οΏ½οΏ½β€. The consumption rate π π οΏ½,οΏ½ for each species ππ
in the vector ππ is simply the negative of π π οΏ½,οΏ½. Therefore, the consumption rates for the gases of interest are calculated as follow:
π π οΏ½,οΏ½2 = 0.5π π οΏ½,1 + 0.5π π οΏ½,2 + 1.45π π οΏ½,3 + 1.45π π οΏ½,4 + 0.5π π οΏ½,6, (B.9)
π π οΏ½,οΏ½οΏ½ = π π ππ,1 + 2π π ππ,5 + π π ππ,8, (B.10)
π π οΏ½,οΏ½2 = π π ππ,2, (B.11)
π π οΏ½,οΏ½οΏ½ = 2π π ππ,5 + π π ππ,7, (B.12)
π π οΏ½,οΏ½οΏ½1.8οΏ½οΏ½οΏ½οΏ½ = π π ππ,3 + π π ππ,9, (B.13)
π π οΏ½,οΏ½οΏ½1.8οΏ½οΏ½οΏ½οΏ½ = π π ππ,4 + π π ππ,10. (B.14)
Table B.4 contain an inhibition term πΊπΊ, which is calculated according to:
πΊπΊ = πποΏ½οΏ½1 + πΎπΎ1π΅π΅οΏ½,οΏ½οΏ½ + πΎπΎ2π΅π΅οΏ½,οΏ½οΏ½οΏ½2οΏ½1 + πΎπΎ3π΅π΅οΏ½,οΏ½οΏ½ 2 π΅π΅οΏ½,οΏ½οΏ½
2 οΏ½οΏ½1 + πΎπΎ4π΅π΅οΏ½,οΏ½οΏ½ 0.7 οΏ½, (B.15)
where
π΅π΅οΏ½,οΏ½οΏ½ = π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½+ π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
, (B.16)
and
πΎπΎοΏ½ = π΄π΄οΏ½ππβ οΏ½οΏ½
οΏ½οΏ½οΏ½οΏ½, (B.17)
where the parameters πΎπΎοΏ½ and πΈπΈοΏ½ are given in Table B.3. The oxygen storage reactions are used to model the oxygen storage capacity of the TWC. Ceria in the form of π΅π΅ππππ2 is considered to be in the oxygen enriched state, while π΅π΅ππ2ππ3 is in the oxygen depleted state. The extent of oxygen stored ππ is therefore defined as the instantaneous proportion of π΅π΅ππππ2 in the total ceria:
ππ =πΌπΌοΏ½οΏ½οΏ½2
πΌπΌοΏ½οΏ½οΏ½2+ 2πΌπΌοΏ½οΏ½2οΏ½3
, (B.18)
The oxygen storage level is governed by the following differential equation:
ππ Μ = 1πΉπΉ
οΏ½2π π οΏ½,6 + 2π π οΏ½,7 β 2π π οΏ½,8 β 3.8π π οΏ½,9 β 3.8π π οΏ½,10οΏ½, (B.19)
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B.2 Enthalpy of reaction
In a reacting system, the molar enthalpy of the compound ππ in [π½π½/πππ π ππ] is given by
βΜ οΏ½ = ΞβΜ οΏ½,οΏ½οΏ½ + ΞβΜ οΏ½, (B.20)
where ΞβΜ οΏ½,οΏ½οΏ½ in the molar enthalpy of formation of the species ππ, which are
specified in Table B.6. By assuming constant pressure condition inside the TWC, the term ΞβΜ οΏ½ can be computed from,
ΞβΜ οΏ½(ππ) = βΜ οΏ½ β βΜ οΏ½οΏ½οΏ½,οΏ½ = οΏ½ πποΏ½Μ ,οΏ½πππποΏ½
οΏ½οΏ½οΏ½οΏ½
, (B.21)
where πποΏ½οΏ½οΏ½ is the reference temperature taken as 298 πΎπΎ. ππ is approximated as the substrate temperature πποΏ½. Since the compounds π΅π΅ππππ2 and π΅π΅ππ2ππ3 are in solid state inside the TWC, theirs specific heat are constant values given by 61.6 and 114.6 [π½π½/πππ π πππΎπΎ] respectively. On the other hand, the specific heat of the reacting gases πποΏ½Μ ,οΏ½, is approximated by modelling the species as ideal gases and using the following polynomials,
πποΏ½Μ ,οΏ½ = πποΏ½ + πποΏ½πποΏ½ + πποΏ½πποΏ½ 2 + πποΏ½πποΏ½
3, (B.22)
where πποΏ½Μ ,οΏ½ is given in [π½π½/πππ π πππΎπΎ]. The coefficients (πποΏ½, πποΏ½, πποΏ½, πποΏ½) are specified in Table B.5. After computing the molar enthalpy for all the species ππ, it is possible to construct the following vector:
βΜ (πποΏ½) =
οΏ½βΜ οΏ½οΏ½, βΜ οΏ½2, βΜ οΏ½οΏ½2
, βΜ οΏ½2, βΜ οΏ½2οΏ½, βΜ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
, βΜ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½, βΜ οΏ½οΏ½, βΜ οΏ½2
, βΜ οΏ½οΏ½2οΏ½3, βΜ οΏ½οΏ½οΏ½2
οΏ½β€
ββββοΏ½οΏ½
. (B.23)
The vector of the enthalpies of reaction is obtained from
π₯π₯βοΏ½ = ππβΜ (πποΏ½), (B.24)
where π₯π₯βοΏ½ contains 10 enthalpies of reactions required by the last term in (B.2); that is:
ΞβοΏ½ = οΏ½ΞβοΏ½,1,π₯π₯βοΏ½,2, β― ,π₯π₯βοΏ½,10οΏ½β€ (B.25)
97
ππ πποΏ½ πποΏ½ πποΏ½ πποΏ½ π΅π΅ππ 2.8160 Γ 101 1.6750 Γ 10β3 5.3720 Γ 10β6 β2.2220 Γ 10β9
ππ2 25.48 1.52 Γ 10β2 β7.1550 Γ 10β6 1.3120 Γ 10β9
π΅π΅ππ2 2.2260 Γ 101 5.9810 Γ 10β2 β3.5010 Γ 10β5 7.4690 Γ 10β9
π»π»2 2.9110 Γ 101 β1.9160 Γ 10β3 4.0030 Γ 10β6 β8.7040 Γ 10β10
π»π»2ππ 3.2240 Γ 101 1.9230 Γ 10β3 1.0550 Γ 10β5 β3.5950 Γ 10β9
π΅π΅π»π»οΏ½οΏ½οΏ½οΏ½ 3.15 2.3830 Γ 10β1 β1.2180 Γ 10β4 2.4620 Γ 10β8
π΅π΅π»π»οΏ½οΏ½οΏ½οΏ½ 1.9890 Γ 101 5.0240 Γ 10β2 1.2690 Γ 10β5 β1.1010 Γ 10β8
ππππ 2.9340 Γ 101 β9.3950 Γ 10β4 9.7470 Γ 10β6 β4.1870 Γ 10β9
ππ2 2.8900 Γ 101 β1.5710 Γ 10β3 8.0810 Γ 10β6 β2.8730 Γ 10β9
Table B.5: Ideal-gas specific heat coefficients of various gases [60]
ππ ΞβΜ Μ οΏ½,οΏ½οΏ½ [π½π½/πππ π ππ]
π΅π΅ππ β1.1050 Γ 105 ππ2 0
CO2 β3.9350 Γ 105 π»π»2 0
π»π»2ππ β2.4180 Γ 105 π΅π΅π»π»fast 20000 π΅π΅π»π»slow β7.4600 Γ 104
ππππ 9.1300 Γ 104 ππ2 0
π΅π΅ππ2ππ3 β17.96200 Γ 105 π΅π΅ππππ2 β10.88700 Γ 105
Table B.6: Standard molar enthalpy of formation of the given species [60]
B.3 Heat and mass transfer coefficients
Heat and mass transfer coefficient are given by
βοΏ½οΏ½ = οΏ½οΏ½οΏ½οΏ½
πππ’π’οΏ½οΏ½, (B.26)
βοΏ½,οΏ½οΏ½,οΏ½ =π·π·οΏ½οΏ½2
π·π·οΏ½ππβοΏ½οΏ½,οΏ½, (B.27)
Nusselt (πππ’π’οΏ½οΏ½) and Sherwood (βοΏ½,οΏ½οΏ½,οΏ½) numbers are calculated according to the correlations:
πππ’π’οΏ½οΏ½ = 3.66οΏ½1 + π·π·οΏ½πΏπΏ
ππππβ,οΏ½οΏ½οΏ½
0.45
, (B.28)
98
ππβοΏ½οΏ½,οΏ½ = 3.66οΏ½1 + π·π·οΏ½πΏπΏ
πππποΏ½,οΏ½οΏ½,οΏ½οΏ½0.45
, (B.29)
Peclet (Pe) numbers are expressed in terms of the Prandtl (Pr), Schmidt (Sc) and Reynolds (Re) numbers as follows:
ππππβ,οΏ½οΏ½= π π πποΏ½οΏ½
ππππ, (B.30)
πππποΏ½,οΏ½οΏ½,οΏ½ = π π πποΏ½οΏ½πππποΏ½, (B.31)
where
π π πποΏ½οΏ½ =οΏ½ΜοΏ½ποΏ½οΏ½οΏ½π·π·οΏ½
πππ΄π΄οΏ½πποΏ½, (B.32)
ππππ =πποΏ½πποΏ½
πποΏ½, (B.33)
πππποΏ½ =πποΏ½
πποΏ½π·π·οΏ½,οΏ½2
. (B.34)
By treating the exhaust gas as a binary mixture of ππ2 and the compound ππ under atmospheric pressure, the gas phase diffusion coefficients,π·π·οΏ½,οΏ½2
, are obtained from
π·π·οΏ½,οΏ½2
=10β7πποΏ½
1.75οΏ½1/πποΏ½οΏ½οΏ½οΏ½οΏ½ + 1/πποΏ½οΏ½οΏ½οΏ½οΏ½2οΏ½
12
οΏ½πποΏ½ 13 + πποΏ½2
13 οΏ½2 , (B.35)
Where πποΏ½οΏ½οΏ½οΏ½οΏ½ is the molar mass of the compound ππ in [ππ/πππ π ππ], πποΏ½ in [πΎπΎ] and the diffusion volumes πποΏ½, are taken from:
ππ πποΏ½[-] π΅π΅ππ 18.9 ππ2 16.6 π»π»2 7.07
π΅π΅3π»π»6 61.38 π΅π΅π»π»4 24.42 ππ2 17.9
Table B.7: Diffusion volumes
In Table B.7: Diffusion volumes for the fast and slow oxidising fuel are approximated by those of π΅π΅3π»π»6 and π΅π΅π»π»4 respectively.
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B.4 Solution scheme
Before considering the solution of (B.1)-(B.4) it is important to specify some auxiliary conditions to complete the PDE problem. π₯π₯ is referred to as the boundary-value variable and the number of the boundary conditions is determined by the higher-order derivative in π₯π₯. Therefore in (B.1), πποΏ½ is first order in π₯π₯, thus the boundary condition is set as
πποΏ½(0, ππ) = πποΏ½,οΏ½οΏ½, (B.36)
where πποΏ½,οΏ½οΏ½ is the feed gas temperature in πΎπΎ of the TWC, which in turn is the outlet temperature of the connecting pipe, πποΏ½οΏ½,οΏ½.
Equation (B.2) is second order in π₯π₯, thus it requires two boundaries conditions. These are determined by assuming non-conducting boundary conditions in the front and the back of the monolith (TWCβs substrate)
πππποΏ½(0, ππ)πππ₯π₯
= 0 (B.37)
πππποΏ½(πΏπΏ, ππ)πππ₯π₯
= 0. (B.38)
The equation (B.3), which describe the mass transfer of the compound ππ, require only one boundary condition for each chemical specie,
π΅π΅οΏ½,οΏ½2(0, ππ) = π΅π΅οΏ½,οΏ½2,οΏ½οΏ½, (B.39)
π΅π΅οΏ½,οΏ½οΏ½(0, ππ) = π΅π΅οΏ½,οΏ½οΏ½,οΏ½οΏ½, (B.40)
π΅π΅οΏ½,οΏ½2(0, ππ) = π΅π΅οΏ½,οΏ½2,οΏ½οΏ½, (B.41)
π΅π΅οΏ½,οΏ½οΏ½(0, ππ) = π΅π΅οΏ½,οΏ½οΏ½,οΏ½οΏ½, (B.42)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½(0, ππ) = 0.85π΅π΅οΏ½,οΏ½οΏ½,οΏ½οΏ½, (B.43)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½(0, ππ) = 0.15π΅π΅οΏ½,οΏ½οΏ½,οΏ½οΏ½. (B.44)
This concentration at the front of the TWC are those of the engine-out emission concatenated in the vector πΈπΈοΏ½οΏ½ (Section Engine-out emissions). The PDE problem is first order in ππ and therefore it requires one initial condition (IC) for each equation in the model.
100
Figure B.1: Discretisation along the catalytic converterβs length, ππ = 1,2,3, β¦ ,60
The technique used in thesis for solving the PDE system is the method of lines (MOL). This method essentially uses finite difference relationships to the spatial derivatives such that a system of ODEs approximates the original PDE. In this method, the spatial domain is divided into small segments Ξπ₯π₯. In [56], four nodes (Ξπ₯π₯ = 35.875 ππππ) were used since they provide comparative results to that of the 51 nodes,. The Figure B.1 shows the scheme of the discretisation along the spatial coordinate in the TWC. The exhaust gas mass flow rate οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ and the inputs over nodes 1, 9, 13, 17, 21, 25, 29 drive the rest of 53 nodes by a set of 53 ODEs (one per node).
ππππ (ππ) β’ 1in β’ 2 β’ 3 β’ 4
ππππ (ππ) β’ 5 β’ 6 β’ 7 β’ 8
π΅π΅ππ,ππ2
(ππ) β’ 9 in β’ 10 β’ 11 β’ 12
π΅π΅ππ,π΅π΅ππ (ππ) β’ 13 in β’ 14 β’ 15 β’ 16 out
π΅π΅ππ,π»π»2
(ππ) β’ 17 in β’ 18 β’ 19 β’ 20
π΅π΅ππ,ππππ (ππ) β’ 21 in β’ 22 β’ 23 β’ 24 out
π΅π΅π»π»ππ,πππππ π ππ (ππ) β’ 25 in β’ 26 β’ 27 β’ 28 out
π΅π΅π»π»ππ,π π πππ π π€π€ (ππ) β’ 29 in β’ 30 β’ 31 β’ 32 out
π΅π΅ππ,ππ2
(ππ) β’ 33 β’ 34 β’ 35 β’ 36
π΅π΅ππ,π΅π΅ππ (ππ) β’ 37 β’ 38 β’ 39 β’ 40
π΅π΅ππ,π»π»2
(ππ) β’ 41 β’ 42 β’ 43 β’ 44
π΅π΅ ππ,ππππ(ππ) β’ 45 β’ 46 β’ 47 β’ 48
π΅π΅π»π»ππ,πππππ π ππ (ππ) β’ 49 β’ 50 β’ 51 β’ 52
π΅π΅π»π»ππ,π π πππ π π€π€ (ππ) β’ 53 β’ 54 β’ 55 β’ 56
ππ (ππ) β’ 57 β’ 58 β’ 59 β’ 60
Gas and substratetemperature
Species concentrationsin the gas
Species in the Substrate
Oxygen storage level
Nodes
οΏ½ΜοΏ½ποΏ½οΏ½οΏ½
101
After solving these differential equations, it is possible to calculate the mass flow rate of the legislated tail pipe emission according to
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½πποΏ½οΏ½πποΏ½
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½), (B.45)
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½πποΏ½οΏ½πποΏ½
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½), (B.46)
οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ = οΏ½ΜοΏ½ποΏ½οΏ½οΏ½ οΏ½πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πποΏ½π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) +
πποΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
πποΏ½π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½)οΏ½, (B.47)
where emissions at the end of the TWC are approximated with the value at the nodes 16, 24, 28, 32, in Figure B.1,
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½ (16) (ππ), (B.48)
π΅π΅οΏ½,οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½ (24) (ππ), (B.49)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
(28) (ππ), (B.50)
π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½(οΏ½,οΏ½) = π΅π΅οΏ½,οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
(32) (ππ). (B.51)
Finally, οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½, οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ and οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½ are converted into [ππππ/ππππ], by taking into account the vehicle speed, πποΏ½οΏ½οΏ½ in [ππππ/β],
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½,οΏ½οΏ½οΏ½, (B.52)
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½,οΏ½οΏ½οΏ½, (B.53)
πποΏ½οΏ½ = 3.6 Γ 109 οΏ½ΜοΏ½ποΏ½οΏ½,οΏ½οΏ½οΏ½
πποΏ½,οΏ½οΏ½οΏ½, (B.54)
where πποΏ½,οΏ½οΏ½οΏ½ = max(10, πποΏ½οΏ½οΏ½).
103
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Minerva Access is the Institutional Repository of The University of Melbourne
Author/s:
Ramos Herrera, Miguel Antonio
Title:
Extremum seeking for spark advance calibration under tailpipe emissions constraints
Date:
2016
Persistent Link:
http://hdl.handle.net/11343/118621
File Description:
Extremum Seeking for Spark Advance Calibration under Tailpipe Emissions Constraints
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