Extremum Seeking for Spark Advance Calibration under ...

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 engine’s 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 𝑆𝑆� . ................................ 24

Figure 3.2: Lyapunov functions with critical points 𝑢𝑢�∗ on the optimal

solution 𝑢𝑢�∗. (a) 𝑢𝑢�

∗ ∈ 𝑆𝑆�. (b) 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆�. ................................................... 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 𝛼𝛼. ....................................................................................................... 28

Figure 3.5: Convergence results in example 1 with different values of 𝛼𝛼. .......... 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 𝛼𝛼. ...... 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/Simulink™. ........................ 45

Figure 4.2: Discretization of the catalytic converter. 𝑗𝑗 = 1,2,3, … ,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 𝜆𝜆 ................................................ 52

Figure 4.5: Steady-state maps, 𝑁𝑁 = 1395 rpm, 𝑇𝑇�,�� = 53.2 Nm,𝑉𝑉�� =30, 𝜆𝜆 = 1.01............................................................................................. 53

Figure 4.6: 𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 and 𝑁𝑁𝑁𝑁 maps, 𝑁𝑁 = 1395 rpm, 𝑇𝑇�,�� = 53.2 Nm, 𝑉𝑉�� = 30, 𝜆𝜆 = 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 4.12: Tailpipe 𝑁𝑁𝑁𝑁 emission and conversion efficiency with UES. ............ 60

Figure 4.13: Constrained extremum seeking scheme to minimise fuel consumption subject to 𝑁𝑁𝑁𝑁 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 𝑁𝑁𝑁𝑁 emission and conversion efficiency with CES ............. 63

Figure 4.16: CES result for different smoothing parameter 𝛼𝛼 ............................. 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 𝛼𝛼 = 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 converter’s length, 𝑗𝑗 = 1,2,3, … ,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 𝑁𝑁𝑁𝑁 emission and spark timing for different 𝛼𝛼 values at 200 s. ...................................................................................................... 64

Table 4.3: Average 𝑁𝑁𝑁𝑁 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 engine’s model .................................................... 81

Table A.7: Inputs, outputs and states for the engine’s 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 (𝐵𝐵2𝐻𝐻6), propane (𝐵𝐵3𝐻𝐻8) and inert gases such as carbon dioxide (𝐵𝐵𝑁𝑁2) and molecular nitrogen (𝑁𝑁2), [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:

𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢), (2.1)

𝑦𝑦 = ℎ(𝑥𝑥). (2.2)

Suppose there exists some state 𝑥𝑥∗, such that, 𝑦𝑦∗ = ℎ(𝑥𝑥∗) is an extremum of the mapping ℎ(⋅). The main objective in extremum-seeking control is to force the solutions of the system (2.1) to eventually converge to a neighbourhood of 𝑥𝑥∗, 𝑦𝑦∗. And more important, to do so without the knowledge of the functional relationship of the map ℎ(⋅).

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 plant’s input, and subsequently the plant’s 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:

𝑢𝑢 = 𝛼𝛼(𝑥𝑥, 𝜃𝜃). (2.3)

The closed-loop system (2.1)-(2.2) and (2.3) is then

𝑥𝑥̇ = 𝑓𝑓�𝑥𝑥, 𝛼𝛼(𝑥𝑥, 𝜃𝜃)�, (2.4)

Assumption 1 [15]. There is a smooth function 𝑙𝑙: ℝ → ℝ� such that

𝑓𝑓(𝑥𝑥, 𝛼𝛼(𝑥𝑥, 𝜃𝜃)) = 0, if and only if 𝑥𝑥 = 𝑙𝑙(𝜃𝜃), (2.5)

Assumption 2 [15]. For each 𝜃𝜃 ∈ ℝ, the equilibrium 𝑥𝑥 = 𝑙𝑙(𝜃𝜃) of the system (2.4) is locally exponentially stable with decay and overshoot constants uniform in 𝜃𝜃.

Assumption 3 [15]. There exists 𝜃𝜃∗ ∈ ℝ such that

(ℎ ∘ 𝑙𝑙)′(𝜃𝜃∗) = 0, (2.6)

(ℎ ∘ 𝑙𝑙)′′(𝜃𝜃∗) < 0. (2.7)

This means that the steady-state input-output map 𝑦𝑦 = ℎ(𝑙𝑙(𝜃𝜃)) has a maximum at 𝜃𝜃 = 𝜃𝜃∗. Tuning parameters of the ES are 𝑘𝑘, 𝜔𝜔ℎ, 𝜔𝜔�, 𝜔𝜔 and 𝑎𝑎. Under these assumptions, local exponential stability was demonstrated. That is, if the initial condition is chosen sufficiently close to the extremum and ‘small’ values for the controller parameters are chosen, then 𝜃𝜃 converges to a small neighbourhood of 𝜃𝜃∗.

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 controller’s tuning parameters

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are only 𝑘𝑘, 𝑎𝑎, and 𝜔𝜔. The stability analysis was based on stronger assumptions compared to those in [15]:

Assumption 2 [16]. For each constant 𝜃𝜃 ∈ ℝ, the corresponding equilibrium of the system (2.4) is globally asymptotically stable, uniformly in 𝜃𝜃.

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 Δ, 𝜈𝜈, it is possible to adjust the controller parameter 𝑘𝑘, 𝑎𝑎, 𝜔𝜔 such that all solutions starting in 𝐵𝐵∆={𝜃𝜃 ∈ ℝ||𝜃𝜃 − 𝜃𝜃∗| ≤ Δ} converge to the ball 𝐵𝐵� = {𝜃𝜃 ∈ ℝ||𝜃𝜃 − 𝜃𝜃∗| ≤ ν}.

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, 𝑓𝑓(𝑥𝑥, 𝑢𝑢), possesses an asymptotically stable equilibrium surface 𝑥𝑥 = 𝑙𝑙(𝑥𝑥, 𝑢𝑢), uniformly in 𝑢𝑢. In addition, the steady-state input-output map 𝑄𝑄(𝑢𝑢) = ℎ(𝑙𝑙(𝑢𝑢)) is a 𝑁𝑁 times continuously differentiable function and has a global extremum (maximum or minimum). The derivatives are denoted by

𝐷𝐷�(𝑄𝑄) = �𝑑𝑑𝑄𝑄𝑑𝑑𝑢𝑢

, ⋯ , 𝑑𝑑�𝑄𝑄

𝑑𝑑𝑢𝑢� ��

. (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 (𝐷𝐷�(⋅)� ) 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 controller’s 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 optimiser’s 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 systems’ 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 plant’s 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 ‘barrier-projections’. 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 plant’s 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 ‘hemstitching’ or ‘boundary-following method’ 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 engine’s 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 engine’s 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 map’s 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

𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢) (3.1)

with the following two outputs

𝑦𝑦� = ℎ�(𝑥𝑥), 𝑦𝑦1 = ℎ1(𝑥𝑥),

(3.2)

where 𝑓𝑓: ℝ� × ℝ → ℝ�, ℎ�: ℝ� → ℝ and ℎ1: ℝ� → ℝ. Let 𝐷𝐷�1 (⋅) and 𝐷𝐷�

2(⋅) be the first and second derivative operators with respect to 𝑢𝑢.

Assumption 1: 𝑓𝑓(⋅), ℎ�(⋅), and ℎ1(⋅) are smooth.

Assumption 2: There exists a differentiable function 𝑙𝑙: ℝ → ℝ�, such that 𝑓𝑓(𝑥𝑥, 𝑢𝑢) = 0 if and only if 𝑥𝑥 = 𝑙𝑙(𝑢𝑢).

Assumption 3: For each constant 𝑢𝑢 ∈ ℝ, the equilibrium 𝑥𝑥 = 𝑙𝑙(𝑢𝑢) is asymptotically stable, uniformly in 𝑢𝑢.

These assumptions imply two reference-to-output maps at the equilibrium,

𝑦𝑦�,�� = ℎ�(𝑙𝑙(𝑢𝑢)) = ℎ� ∘ 𝑙𝑙(𝑢𝑢) = 𝑓𝑓�(𝑢𝑢), 𝑦𝑦1,�� = ℎ1(𝑙𝑙(𝑢𝑢)) = ℎ1 ∘ 𝑙𝑙(𝑢𝑢) = 𝑓𝑓1(𝑢𝑢),

(3.3)

Assumption 4: 𝑓𝑓�(⋅) and 𝑓𝑓1(⋅) are twice continuously differentiable. Moreover, 𝐷𝐷�

2𝑓𝑓�(𝑢𝑢) > 0, 𝐷𝐷�2𝑓𝑓1(𝑢𝑢) > 0 ∀ 𝑢𝑢 ∈ ℝ. Thus, 𝑓𝑓�(⋅) and 𝑓𝑓1(⋅) are strictly convex

functions.

Assumption 5: 𝑓𝑓�(⋅) has a strict local minimum, 𝑢𝑢∗ ∈ ℝ, such that 𝐷𝐷�1𝑓𝑓�(𝑢𝑢∗) = 0.

Let 𝑆𝑆 be the constraint set defined by

𝑆𝑆 = {𝑢𝑢 ∈ ℝ|𝑓𝑓1(𝑢𝑢) ≤ 0}. (3.4)

The set 𝑆𝑆�� = {𝑢𝑢 ∈ ℝ|𝑓𝑓1(𝑢𝑢) < 0} is the interior of 𝑆𝑆 and its boundary is denoted by 𝜕𝜕𝑆𝑆 = {𝑢𝑢 ∈ ℝ|𝑓𝑓1(𝑢𝑢) = 0}.

Assumption 6: The set 𝑆𝑆 is not empty.

The objective is to steer the system (3.1) to the equilibrium 𝑥𝑥�∗ and 𝑢𝑢�

∗ which solves the inequality constrained optimisation problem given by

min�∈�

𝑓𝑓�(𝑢𝑢), (3.5)

and to do so without any explicit knowledge about 𝑥𝑥�∗, 𝑢𝑢�

∗, 𝑙𝑙, ℎ�, ℎ1. In addition, the initial condition for 𝑢𝑢 is not required to be close to 𝑢𝑢�

∗. 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:

min�∈��

𝑓𝑓�(𝑢𝑢), (3.6)

where

𝑆𝑆� = �𝑢𝑢 ∈ ℝ| lim�→∞

1𝑇𝑇

� 𝑓𝑓1(𝑢𝑢)𝑑𝑑𝑑𝑑�

�−�≤ 0�. (3.7)

Here 𝑇𝑇 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

𝑦𝑦� = 𝑓𝑓�(𝑢𝑢), 𝑦𝑦1 = 𝑓𝑓1(𝑢𝑢),

(3.8)

where 𝑓𝑓�: ℝ → ℝ, 𝑓𝑓1: ℝ → ℝ.

The goal is to solve the following constrained optimisation problem:

min�∈��

𝑓𝑓�(𝑢𝑢), (3.9)

𝑆𝑆� = {𝑢𝑢 ∈ ℝ|𝑓𝑓1(𝑢𝑢) ≤ 0}. (3.10)

Suppose 𝑓𝑓�(⋅), 𝑓𝑓1(⋅), and 𝑆𝑆� satisfy Assumptions 4-6. Moreover, 𝑓𝑓�(⋅), 𝑓𝑓1(⋅), and their respective derivatives 𝐷𝐷�

1𝑓𝑓�(⋅) and 𝐷𝐷�1𝑓𝑓1(⋅) 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 𝑢𝑢�

∗, is unique.

The objective of this section is to propose a continuous-time optimisation algorithm whose solutions asymptotically converge to 𝑢𝑢�

∗, regardless of whether it resides in the interior of the constraint set 𝑆𝑆�,�� or on its boundary 𝜕𝜕𝑆𝑆�. In addition to this, the initial condition for 𝑢𝑢 could be specified in any of these sets.

The following update law is proposed for the input to the static maps.

�̇�𝑢 = �−𝐷𝐷�1𝑓𝑓�(𝑢𝑢) 𝑖𝑖𝑓𝑓 𝑓𝑓1(𝑢𝑢) < 0

−𝐷𝐷�1𝑓𝑓1(𝑢𝑢) 𝑖𝑖𝑓𝑓 𝑓𝑓1(𝑢𝑢) ≥ 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

�̇�𝑢 = −[1 − 𝑠𝑠�𝑓𝑓1(𝑢𝑢)�] ⋅ 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) − 𝑠𝑠�𝑓𝑓1(𝑢𝑢)� ⋅ 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢), (3.12)

where 𝑠𝑠 : ℝ → ℝ , and 𝑝𝑝 ∈ ℝ, such that

𝑠𝑠(𝑝𝑝) = �0 𝑖𝑖𝑓𝑓 𝑝𝑝 < 01 𝑖𝑖𝑓𝑓 𝑝𝑝 ≥ 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 𝑢𝑢. There exists a significant research body that addresses the stability analysis of such systems that are not locally Lipchitz on 𝑢𝑢, [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 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) to 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢) as 𝑢𝑢 approaches 𝜕𝜕𝑆𝑆� can be eliminated by approximating the step function in (3.13) with a smooth function. There are many possibilities to achieve this. Let 𝛼𝛼 ∈ ℝ>0, 𝜀𝜀 ∈ ℝ and 𝜎𝜎: ℝ × ℝ>0 × ℝ → ℝ. Some smooth approximations for (3.13) are listed below.

1) The sigmoid function:

𝜎𝜎(𝑝𝑝, 𝛼𝛼, 𝜀𝜀) = 1

1 + 𝑒𝑒−(�−�)�

(3.14)

2) The twice differentiable spline function:

𝜎𝜎(𝑝𝑝, 𝛼𝛼, 𝜀𝜀) =

⎩⎪⎨

⎪⎧ 0 𝑖𝑖𝑓𝑓 𝑝𝑝 − 𝜀𝜀 ≤ −𝛼𝛼316

�𝑝𝑝 − 𝜀𝜀𝛼𝛼

�5− 5

8�𝑝𝑝 − 𝜀𝜀

𝛼𝛼 �

3+ 15

16�𝑝𝑝 − 𝜀𝜀

𝛼𝛼� + 1

2 𝑖𝑖𝑓𝑓 − 𝛼𝛼 ≤ 𝑝𝑝 − 𝜀𝜀 ≤ 𝛼𝛼

1 𝑖𝑖𝑓𝑓 𝛼𝛼 ≤ 𝑝𝑝 − 𝜀𝜀

(3.15)

3) The hyperbolic tangent function:

𝜎𝜎(𝑝𝑝, 𝛼𝛼, 𝜀𝜀) = 0.5 + 0.5 tanh�𝑝𝑝 − 𝜀𝜀𝛼𝛼

� (3.16)

Note that for 𝜀𝜀 = 0 all these approximations are centred with respect to 𝑝𝑝 = 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:

�̇�𝑢 = −[1 − 𝜎𝜎(𝑓𝑓1(𝑢𝑢), 𝛼𝛼, 0)]𝐷𝐷�1𝑓𝑓�(𝑢𝑢) − 𝜎𝜎(𝑓𝑓1(𝑢𝑢), 𝛼𝛼, 0)𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). (3.17)

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For simplicity and where the context allows, the function’s argument of 𝜎𝜎 will be dropped, so that

�̇�𝑢 = −(1 − 𝜎𝜎)𝐷𝐷�1𝑓𝑓�(𝑢𝑢) − 𝜎𝜎𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). (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 𝑢𝑢�

∗. Before establishing this, it is necessary to show that the system possesses an equilibrium 𝑢𝑢� = 𝑢𝑢�

∗. Note that depending on the particular problem, 𝑢𝑢�

∗ can be an element of 𝜕𝜕𝑆𝑆� or 𝑆𝑆�,��.

First, consider the case for 𝑢𝑢�∗ ∈ 𝑆𝑆�,��. In the interior of the constraint set 𝑆𝑆�,

the smooth function 𝜎𝜎 → 0, thereby, the equation (3.18) reduces to following dynamic system:

�̇�𝑢 = −𝐷𝐷�1𝑓𝑓�(𝑢𝑢). (3.19)

According to Assumption 5, 𝑓𝑓�(⋅) has a strict local minimum 𝑢𝑢∗ and 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢∗) = 0. Furthermore, by Assumption 4 𝑓𝑓�(⋅) is strictly convex, then 𝑢𝑢∗ = 𝑢𝑢�

∗ and the system (3.19) has an equilibrium 𝑢𝑢� = 𝑢𝑢�∗.

Now consider the case for 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆�. The system in (3.18) is designed to

smooth the gradients as 𝑢𝑢 approaches 𝜕𝜕𝑆𝑆� 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 𝑢𝑢� and 𝑢𝑢�, 𝑢𝑢� < 𝑢𝑢�, such that �̇�𝑢(𝑢𝑢�) and �̇�𝑢(𝑢𝑢�) are of opposite signs. Then there exists a 𝑢𝑢� ∈ [𝑢𝑢�, 𝑢𝑢�] with �̇�𝑢(𝑢𝑢�) = 0 (Bolzano’s Theorem). Thus when 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆� and 𝐷𝐷�1𝑓𝑓1(𝑢𝑢) ⋅ 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) < 0 in a region of 𝑢𝑢�∗, the

system (3.19) has an equilibrium 𝑢𝑢�.

In order to show that 𝑢𝑢� = 𝑢𝑢�∗, some conditions must be satisfied. Consider the

equation (3.18). Since 𝑓𝑓1(𝑢𝑢) = 0 for 𝑢𝑢 ∈ 𝜕𝜕𝑆𝑆�, then 𝜎𝜎 = 0.5 and (3.18) reduces to

�̇�𝑢 = −0.5𝐷𝐷�1𝑓𝑓�(𝑢𝑢)−0.5𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). (3.20)

It follows that for 𝑢𝑢 ∈ 𝜕𝜕𝑆𝑆� the gradients must satisfy 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) = −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢) for the system to have an equilibrium 𝑢𝑢� = 𝑢𝑢�

∗. This is a strong condition and in general difficult to meet since the derivatives of 𝑓𝑓1(⋅) and 𝑓𝑓�(⋅) 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 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) ≠ −𝐷𝐷�1𝑓𝑓1(𝑢𝑢) on 𝜕𝜕𝑆𝑆�.

24

Figure 3.1: (a) Discontinuous gradient system, equation (3.11). (b) Smooth gradient at the boundary of the constraint set 𝑆𝑆�

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 𝑢𝑢�.

The system (3.18) can be expressed in the following form

�̇�𝑢 = −𝐵𝐵(𝑢𝑢), (3.21)

where

𝐵𝐵(𝑢𝑢) = (1 − 𝜎𝜎)𝐷𝐷�1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). (3.22)

Let 𝐷𝐷 ⊂ ℝ be a domain containing the equilibrium point of (3.21). Let 𝑉𝑉:𝐷𝐷 → ℝ be an antiderivative of 𝐵𝐵(𝑢𝑢), so that

𝐷𝐷�1𝑉𝑉(𝑢𝑢) = 𝐵𝐵(𝑢𝑢). (3.23)

Then (3.21) can be written as follows:

�̇�𝑢 = −𝐷𝐷�1𝑉𝑉(𝑢𝑢). (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 𝑉𝑉(⋅) is a natural Lyapunov function for (3.21) [53].

The time derivative of 𝑉𝑉(𝑢𝑢) along the trajectories of (3.24) is given by:

𝑉𝑉(̇𝑢𝑢) = 𝐷𝐷�1𝑉𝑉(𝑢𝑢)𝑢𝑢,̇ (3.25)

𝑉𝑉(̇𝑢𝑢) = −[𝐷𝐷�1𝑉𝑉(𝑢𝑢)]2. (3.26)

0

u$c

_u

0

u$c

_u

−𝐷𝐷�1𝑓𝑓1(𝑢𝑢)

−𝐷𝐷�1𝑓𝑓�(𝑢𝑢)

𝑓𝑓1(𝑢𝑢) < 0

𝑓𝑓1(𝑢𝑢) > 0

(𝑎𝑎) (𝑏𝑏)

25

It is evident from (3.26) that 𝑉𝑉(̇𝑢𝑢) is zero at 𝑢𝑢� and negative definite in 𝐷𝐷 − {𝑢𝑢�}. Then, 𝑢𝑢� is an asymtotically stable equilibrium point of (3.21). Furthermore, according to Section 3.2.1, if 𝑢𝑢�

∗ ∈ 𝑆𝑆�,��, or 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆� and

𝐷𝐷�1𝑓𝑓�(𝑢𝑢�

∗) = −𝐷𝐷�1𝑓𝑓1(𝑢𝑢�

∗), then 𝑢𝑢� = 𝑢𝑢�∗. Consequently, 𝑢𝑢�

∗ is an asymtotically stable equilibrium point of (3.21) under these conditions.

Figure 3.2: Lyapunov functions with critical points 𝑢𝑢�

∗ on the optimal solution 𝑢𝑢�∗. (a)

𝑢𝑢�∗ ∈ 𝑆𝑆�. (b) 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆�

Note that the critical point of 𝑉𝑉(𝑢𝑢), namely 𝑢𝑢�∗, is the equilibrium point of

(3.24). Therefore, 𝑢𝑢� = 𝑢𝑢�∗. It follows that, under the aforementioned conditions,

𝑢𝑢�∗ = 𝑢𝑢�

∗. 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:

min�

𝑉𝑉(𝑢𝑢), (3.27)

which in turn, it is the minimisation of the Lyapunov function of the system (3.21).

0_u 0_u

V V

𝑢𝑢�∗

𝑢𝑢�∗ 𝑢𝑢�

∗

𝑢𝑢�∗

𝑠𝑠𝑒𝑒𝑑𝑑 𝑆𝑆� 𝑆𝑆�

(𝑎𝑎) (𝑏𝑏)

26

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 𝑢𝑢�

∗ such that all solutions of (3.21) asymptotically converge to 𝑢𝑢�

∗ if 𝑢𝑢�∗ ∈ 𝑆𝑆�,��. A similar stability result was

concluded when 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆�, however 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢�∗) = −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�∗) must hold for the

solution to converge to 𝑢𝑢�∗.

Now, consider the case when 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) ≠ −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢) on 𝜕𝜕𝑆𝑆�. If 𝑢𝑢�∗ ∈ 𝑆𝑆�,��, then

the smooth function 𝜎𝜎 → 0, therefore, the equation (3.18) reduces to (3.19) and follows the same analysis as above to conclude that 𝑢𝑢� = 𝑢𝑢�

∗. On the other hand, if 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆�, then 𝑓𝑓1(𝑢𝑢�∗) = 0 and, 𝜎𝜎 = 0.5, whereby (3.18) reduces to

�̇�𝑢 = −0.5𝐷𝐷�1𝑓𝑓�(𝑢𝑢)−0.5𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). (3.28)

However, since 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) ≠ −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢) and 𝐷𝐷�1𝑓𝑓1(𝑢𝑢)𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) < 0 on 𝜕𝜕𝑆𝑆�, (3.28) does not possess and equilibrium point on 𝜕𝜕𝑆𝑆�. That is, 𝑢𝑢� ≠ 𝑢𝑢�

∗.

Despite this, it is possible to have an equilibrium point in a neighbourhood of 𝑢𝑢�

∗ by choosing a suitable value for the parameter 𝛼𝛼 in the smooth function (3.14). To demonstrate this, consider the equilibria of (3.18):

0 = −(1 − 𝜎𝜎�)𝐷𝐷�1𝑓𝑓�(𝑢𝑢�) − 𝜎𝜎�𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�), (3.29)

solving for 𝜎𝜎� yields

𝜎𝜎� = 𝐷𝐷�1𝑓𝑓�(𝑢𝑢�)

(𝐷𝐷�1𝑓𝑓�(𝑢𝑢�) − 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�)), (3.30)

substituting (3.14) in (3.30) gives:

1

1 + 𝑒𝑒−�1(��)�

= 𝐷𝐷�1𝑓𝑓�(𝑢𝑢�)

(𝐷𝐷�1𝑓𝑓�(𝑢𝑢�) − 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�)). (3.31)

Now solving for 𝑓𝑓1(𝑢𝑢�):

𝑓𝑓1(𝑢𝑢�) = −𝛼𝛼𝑙𝑙𝛼𝛼�−𝐷𝐷�1𝑓𝑓1(𝑢𝑢�)

𝐷𝐷�1𝑓𝑓�(𝑢𝑢�)

�. (3.32)

Consider the following two cases for the equation (3.32). If |𝐷𝐷�1𝑓𝑓�(𝑢𝑢�)| >

|𝐷𝐷�1𝑓𝑓1(𝑢𝑢�)|, then 𝑓𝑓1(𝑢𝑢�)>0 and the equilibrium point of (3.18) lies outside the

constrained set. That is 𝑢𝑢� ∉ 𝑆𝑆�. Now, if |𝐷𝐷�1𝑓𝑓�(𝑢𝑢�)| < |𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�)| then 𝑓𝑓1(𝑢𝑢�) < 0 and 𝑢𝑢� ∈ 𝑆𝑆�,��. As a consequence, if 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆� and 𝐷𝐷�1𝑓𝑓�(𝑢𝑢�

∗) ≠−𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�∗), then the proposed algorithm might converge to an equilibrium point

that either violates the constraint or resides in 𝑆𝑆��. However, by reducing the parameter 𝛼𝛼 in (3.32), it is possible to construct a 𝑁𝑁(𝛼𝛼)-sized neighbourhood centred at 𝑓𝑓1(𝑢𝑢�

∗) = 0, such that

|𝑓𝑓1(𝑢𝑢�) − 𝑓𝑓1(𝑢𝑢�∗)| ≤ 𝑘𝑘𝛼𝛼, (3.33)

27

for some 𝑘𝑘 > 0. Then, by continuity:

|𝑢𝑢� − 𝑢𝑢�∗| ≤ 𝜇𝜇. (3.34)

It is concluded that the practitioner may choose a sufficiently small 𝛼𝛼 such that 𝑢𝑢� is arbitrarily close to 𝑢𝑢�

∗, and consequently, 𝑓𝑓1(𝑢𝑢�) resides in a small region of 𝑓𝑓1(𝑢𝑢�

∗).

3.2.3 Numerical examples

Example 1. Consider the following problem:

min�

(𝑢𝑢 + 1)2

s. t. 2(𝑢𝑢 − 2)2 − 2 ≤ 0,

(3.35)

Let 𝑓𝑓�(𝑢𝑢) = (𝑢𝑢 + 1)2 and 𝑓𝑓1(𝑢𝑢) = 2(𝑢𝑢 − 2)2 − 2. The constraint set is 𝑆𝑆� = [1,3]. To solve this problem, the following input update law for 𝑢𝑢 is used.

�̇�𝑢 = −[(1 − 𝜎𝜎)𝐷𝐷�1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎𝐷𝐷�

1𝑓𝑓1(𝑢𝑢)], (3.36)

where

𝜎𝜎 = 1

1 + 𝑒𝑒−�1(�)�

. (3.37)

Assumptions 4 to 6 are satisfied. From Figure 3.3 it is apparent that the unique constrained optimum occurs at 𝑢𝑢�

∗ = 1 and resides on the boundary of 𝑆𝑆�, that is, 𝑓𝑓1(𝑢𝑢�

∗ ) = 0. Figure 3.4 shows the proposed dynamical system as a function of 𝑢𝑢 for three different values of the parameter 𝛼𝛼.

Note that the upper diagonal line corresponds to −𝐷𝐷�1𝑓𝑓1(𝑢𝑢), while −𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) is represented by the lower diagonal. If 𝑢𝑢 was only driven by −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢), then for all 𝑢𝑢(0) ∈ ℝ, the solutions would asymptotically converge to 𝑢𝑢 = 2. Conversely, if 𝑢𝑢 was driven by −𝐷𝐷�

1𝑓𝑓�(𝑢𝑢), then the solutions would asymptotically converge to 𝑢𝑢 = −1. None of these is the constrained optimum. However, since 𝐷𝐷�

1𝑓𝑓1(1) =−𝐷𝐷�

1𝑓𝑓o(1), then for any 𝛼𝛼 ∈ ℝ>0 the proposed dynamical system (3.36) possesses an equilibrium point at 𝑢𝑢� = 1, which is the optimal solution of (3.35).

The convergence to the optimal solution for 𝑢𝑢(0) = 4 and different 𝛼𝛼 values is shown in Figure 3.5. Note that this condition implies an initialisation outside 𝑆𝑆� and violates the constraint. With this initialisation, it is possible to demonstrate the efficacy of the proposed optimisation algorithm to drive 𝑢𝑢 towards the feasible set [1,3]. To do so, 𝑢𝑢 is initially driven by the gradient of the constraint, −𝐷𝐷�

1𝑓𝑓1(𝑢𝑢). Once the trajectory approaches 𝑢𝑢 = 3 the gradient transition to −𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) and 𝑢𝑢 enters into the feasible set [1,3]. Note that 𝑢𝑢 = 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 𝑢𝑢 = 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 𝛼𝛼

-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 𝛼𝛼

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 𝛼𝛼 in the speed of convergence to the constrained optimum is also shown in Figure 3.5. Decreasing 𝛼𝛼 increases the rate of convergence of the input to 𝑢𝑢 = 1, and consequently the rate of convergence to 𝑓𝑓�(1) = 4 and 𝑓𝑓1(1) = 0. To explain this, consider again Figure 3.4. With 𝛼𝛼 = 0.1, the gradient transition occurs when 𝑢𝑢 is close to either 𝑢𝑢 = 1 or 𝑢𝑢 = 3. In the interval (1,3), 𝜎𝜎 → 0 and the dynamic system (3.36) rapidly approximates the gradient descent −𝐷𝐷�

1𝑓𝑓�(𝑢𝑢), which is represented by the lower diagonal in the same figure. Conversely, for 𝛼𝛼 = {1,2}, the algorithm provides a smoother gradient transition, and this occurs before 𝑢𝑢 approaches the boundaries of the constraint set. However, in the interior of [1,3], 𝜎𝜎 ≇ 0, then, a lower |�̇�𝑢| is obtained compared to the case with 𝛼𝛼 = 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 𝛼𝛼. This function is obtained by solving the following antiderivative:

𝑉𝑉(𝑢𝑢) = �𝐵𝐵(𝑢𝑢) 𝑑𝑑𝑢𝑢, (3.38)

𝑉𝑉(𝑢𝑢) = �{[1 − 𝜎𝜎]𝐷𝐷�1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎𝐷𝐷�

1𝑓𝑓1(𝑢𝑢)} 𝑑𝑑𝑢𝑢, (3.39)

𝑉𝑉(𝑢𝑢) = ��1 − 1

1 + 𝑒𝑒−�1(�)�

�𝐷𝐷�1𝑓𝑓�(𝑢𝑢) + 1

1 + 𝑒𝑒−�1(�)�

𝐷𝐷�1𝑓𝑓1(𝑢𝑢)�𝑑𝑑𝑢𝑢. (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 𝑉𝑉. As expected, the optimal solution of minimising 𝑓𝑓�(𝑢𝑢) over 𝑆𝑆 is the critical point of 𝑉𝑉(⋅), that is, 𝑢𝑢�

∗ = 𝑢𝑢�∗.

Example 2, There are situations where 𝐷𝐷�1𝑓𝑓�(𝑢𝑢�

∗) ≠ −𝐷𝐷�1𝑓𝑓1(𝑢𝑢�

∗). In such cases 𝑢𝑢� ≠ 𝑢𝑢�

∗ and the solution of the dynamical system (3.36) will not converge to the constraint optimum, but arbitrarily close to it if an appropriate 𝛼𝛼 is chosen. To illustrate this, consider a modification of the cost function used in example 1. The optimisation problem is stated as:

min�

2(𝑢𝑢 + 1)2

s. t. 2(𝑢𝑢 − 2)2 − 2 ≤ 0,

(3.41)

Let 𝑓𝑓�(𝑢𝑢) = 2(𝑢𝑢 + 1)2, 𝑓𝑓1(𝑢𝑢) = 2(𝑢𝑢 − 2)2 − 2. The constraint set is 𝑆𝑆� = [1,3]. Note that the optimal solution is still 𝑢𝑢�

∗ = 1 but now 𝐷𝐷�1𝑓𝑓�(𝑢𝑢�

∗) > −𝐷𝐷�1𝑓𝑓1(𝑢𝑢�

∗).

The convergence of 𝑢𝑢, 𝑓𝑓�(𝑢𝑢), and 𝑓𝑓1(𝑢𝑢) for 𝑢𝑢(0) = 4 is shown in Figure 3.7 Clearly, all solutions converge to an equilibrium point such that 𝑢𝑢� < 𝑢𝑢�

∗, and the constraint is not satisfied, that is, 𝑓𝑓1(𝑢𝑢�) > 0. However, by decreasing 𝛼𝛼, 𝑢𝑢� might reside in a small neighbourhood |𝑢𝑢� − 𝑢𝑢�

∗| ≤ 𝜇𝜇(𝛼𝛼), thus 𝑢𝑢� will approach 𝑢𝑢�∗ from

below.

31

Although the 𝑁𝑁(𝛼𝛼)-neighbourhood of 𝑢𝑢�

∗ could be arbitrarily small, decreasing 𝛼𝛼 also results in a fast gradient transition. This is observed in the response of �̇�𝑢 in Figure 3.7. The sharp transition might not be of any concern for this ‘toy’ example, however, it may introduce issues in real systems. For example, if 𝑢𝑢 represents an actuator, this would require a large controller effort to cope with the high rate of change of 𝑢𝑢 as it approaches the boundary of 𝑆𝑆�.

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 plant’s 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 𝛼𝛼

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:

�̇�𝑢 = 𝐵𝐵�� 𝑢𝑢,𝐷𝐷�1𝑉𝑉(𝑢𝑢)�. (3.42)

Without loss of generality, consider the case for minimization, and the optimiser:

𝐵𝐵�� 𝑢𝑢,𝐷𝐷�1𝑉𝑉(𝑢𝑢)� = − 𝐷𝐷�

1𝑉𝑉(𝑢𝑢), (3.43)

where

𝐷𝐷�1𝑉𝑉(𝑢𝑢) = (1 − 𝜎𝜎) ⋅ 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎 ⋅ 𝐷𝐷�1𝑓𝑓1(𝑢𝑢). (3.44)

3.3.1 Singular perturbation and averaging analysis

The controller parameters in Figure 3.8 are 𝑎𝑎,𝑤𝑤�,𝛼𝛼, 𝜖𝜖, Throughout this chapter 𝜖𝜖 = 𝛿𝛿𝑤𝑤𝑤𝑤� for some 𝛿𝛿 > 0. Let the smooth function be:

𝜎𝜎 = 1

1 + 𝑒𝑒−ℎ1(�)�

, (3.45)

and 𝜁𝜁(ℎ�(𝑥𝑥), ℎ1(𝑥𝑥), 𝑑𝑑) = [ℎ�(𝑥𝑥)𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑), ℎ1(𝑥𝑥)𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑)]⊤, 𝜂𝜂 = [𝜂𝜂�, 𝜂𝜂1]⊤. Let 𝑔𝑔(⋅) be given by:

𝑔𝑔(𝜎𝜎, 𝑎𝑎, 𝜂𝜂) = −(1 − 𝜎𝜎) ⋅ 𝑔𝑔�(𝑎𝑎, 𝜂𝜂) − 𝜎𝜎 ⋅ 𝑔𝑔1(𝑎𝑎, 𝜂𝜂), (3.46)

where 𝑔𝑔�(𝑎𝑎, 𝜂𝜂) = 2𝑎𝑎−1𝜂𝜂�, and 𝑔𝑔1(𝑎𝑎, 𝜂𝜂) = 2𝑎𝑎−1𝜂𝜂1, which on steady-state aproximate 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) and 𝐷𝐷�1𝑓𝑓1(𝑢𝑢) [54]. Consequently, 𝑔𝑔(𝜎𝜎, 𝑎𝑎, 𝜂𝜂) approximates −𝐷𝐷�

1𝑉𝑉(𝑢𝑢).

The closed-loop equations of the system in Figure 3.8 with the gradient estimator 𝑔𝑔(⋅) are given by

𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢) 𝑦𝑦� = ℎ�(𝑥𝑥) 𝑦𝑦1 = ℎ1(𝑥𝑥)

𝑤𝑤�𝑤𝑤𝑠𝑠 + 𝑤𝑤�𝑤𝑤

𝑢𝑢��̇ = 𝜖𝜖𝐵𝐵��(𝑢𝑢��, 𝑔𝑔(𝜎𝜎, 𝑎𝑎, 𝜂𝜂))

𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

𝑢𝑢� +

𝑢𝑢 = 𝑢𝑢�� + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

𝑦𝑦1

𝜂𝜂�

𝜂𝜂1

𝑦𝑦�


33

𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢�� + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)), (3.47)

𝑢𝑢��̇ = 𝛿𝛿𝑤𝑤𝑤𝑤�𝐵𝐵��𝑢𝑢��, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜂𝜂)�, (3.48)

𝜂𝜂̇ = −𝜔𝜔𝜔𝜔�[𝜂𝜂 − 𝜁𝜁(ℎ�(𝑥𝑥), ℎ1(𝑥𝑥),𝑤𝑤𝑑𝑑)]. (3.49)

Let �̃�𝑢 = 𝑢𝑢�� − 𝑢𝑢�∗ and 𝑑𝑑′ = 𝜔𝜔𝑑𝑑. The system (3.47)-(3.49) is brought into the

standard singular perturbation form

𝑤𝑤 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑′

= 𝑓𝑓(𝑥𝑥, �̃�𝑢 + 𝑢𝑢�∗ + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′)), (3.50)

𝑑𝑑�̃�𝑢𝑑𝑑𝑑𝑑′

= 𝛿𝛿𝑤𝑤�𝐵𝐵��̃�𝑢 + 𝑢𝑢�∗, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜂𝜂)�, (3.51)

𝑑𝑑𝜂𝜂𝑑𝑑𝑑𝑑′

= −𝜔𝜔�[𝜂𝜂 − 𝜁𝜁(ℎ�(𝑥𝑥), ℎ1(𝑥𝑥), 𝑑𝑑′)]. (3.52)

Setting 𝑤𝑤 = 0 ‘freezes’ 𝑥𝑥 at its equilibrium 𝑥𝑥= 𝑙𝑙(�̃�𝑢 + 𝑢𝑢�∗ + 𝑎𝑎𝑠𝑠𝑖𝑖 𝛼𝛼(𝑑𝑑′)) to obtain

the following reduced system:

𝑑𝑑𝑢𝑢�𝑑𝑑𝑑𝑑′

= 𝛿𝛿𝑤𝑤�𝐵𝐵��𝑢𝑢� + 𝑢𝑢�∗, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜂𝜂�)�, (3.53)

𝑑𝑑𝜂𝜂�𝑑𝑑𝑑𝑑′

= −𝜔𝜔�[𝜂𝜂� − 𝜁𝜁(𝑓𝑓�(�̃�𝑢 + 𝑢𝑢�∗ + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′)), 𝑓𝑓1(�̃�𝑢 + 𝑢𝑢�

∗ + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′), 𝑑𝑑′)], (3.54)

and its average system is

𝑑𝑑𝑢𝑢��𝑑𝑑𝑑𝑑′

= 𝛿𝛿𝑤𝑤�𝐵𝐵��𝑢𝑢�� + 𝑢𝑢�∗, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜂𝜂��)�, (3.55)

𝑑𝑑𝜂𝜂��𝑑𝑑𝑑𝑑′

= −𝜔𝜔�[𝜂𝜂�� − 𝜇𝜇(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎))], (3.56)

where

𝜇𝜇(𝑢𝑢�� + 𝑢𝑢�∗ , 𝑎𝑎) = [𝜇𝜇�(𝑢𝑢�� + 𝑢𝑢�

∗, 𝑎𝑎), 𝜇𝜇1(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎)]⊤, (3.57)

and

𝜇𝜇�(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎) = 1

2𝜋𝜋� ℎ�(𝑢𝑢�� + 𝑢𝑢�

∗ + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼(𝑑𝑑′)), 𝑑𝑑′)2�

0𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′)𝑑𝑑𝑑𝑑′ (3.58)

𝜇𝜇1(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎) = 1

2𝜋𝜋� ℎ1(𝑢𝑢�� + 𝑢𝑢�

∗ + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼(𝑑𝑑′)), 𝑑𝑑′)2�

0𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′)𝑑𝑑𝑑𝑑′. (3.59)

By introducing the new time scale 𝑠𝑠 = 𝛿𝛿𝜔𝜔�𝑑𝑑, the system (3.55)-(3.56) is transformed into standard singular perturbation form

𝑑𝑑𝑢𝑢��𝑑𝑑𝑠𝑠

= 𝐵𝐵��𝑢𝑢�� + 𝑢𝑢�∗, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜂𝜂��)�, (3.60)

𝛿𝛿 𝑑𝑑𝜂𝜂��𝑑𝑑𝑠𝑠

= −[𝜂𝜂�� − 𝜇𝜇(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎))]. (3.61)

Setting 𝛿𝛿 = 0 ‘freezes’ 𝜂𝜂�� at its equilibrium: 𝜂𝜂��= 𝜇𝜇(𝑢𝑢�� + 𝑢𝑢�∗, 𝑎𝑎), and the

following reduced system is obtained:

34

𝑑𝑑𝑢𝑢�𝑑𝑑𝑠𝑠

= 𝐵𝐵��𝑢𝑢� + 𝑢𝑢�∗, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜇𝜇(𝑢𝑢� + 𝑢𝑢�

∗ , 𝑎𝑎))�, (3.62)

which can be written in the new coordinates 𝑢𝑢�� = 𝑢𝑢� + 𝑢𝑢�∗ as follow.


= 𝐵𝐵��𝑢𝑢��, 𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜇𝜇(𝑢𝑢��, 𝑎𝑎))�. (3.63)

Note that

𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜇𝜇(𝑢𝑢��, 𝑎𝑎)) = −(1 − 𝜎𝜎) ⋅ 𝑔𝑔��𝑎𝑎, 𝜇𝜇�(𝑢𝑢��, 𝑎𝑎)� − 𝜎𝜎 ⋅ 𝑔𝑔1�𝑎𝑎,𝜇𝜇1(𝑢𝑢��, 𝑎𝑎)�, (3.64)

where 𝑔𝑔�(𝑎𝑎,𝜇𝜇�(𝑢𝑢��, 𝑎𝑎))=𝐷𝐷��1 𝑓𝑓�(𝑢𝑢��) , and 𝑔𝑔1(𝑎𝑎, 𝜇𝜇1(𝑢𝑢��, 𝑎𝑎))=𝐷𝐷��

1 𝑓𝑓1(𝑢𝑢��). The difference

⏐⏐⏐𝐷𝐷��1 𝑓𝑓�(𝑢𝑢��) − 𝐷𝐷��

1 𝑓𝑓�(𝑢𝑢��)⏐⏐⏐ can be made arbitrarily small by reducing the parameter 𝑎𝑎, [54]. As a result

𝑔𝑔(𝑎𝑎, 𝜎𝜎, 𝜇𝜇(𝑢𝑢��, 𝑎𝑎)) = −𝐷𝐷��1𝑉𝑉(𝑢𝑢��). (3.65)

Therefore equation (3.63) can be written as


= 𝐵𝐵�� 𝑢𝑢��,𝐷𝐷��1𝑉𝑉(𝑢𝑢��)�. (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 𝑉𝑉(𝑢𝑢).

3.3.2 Stability

Since the proposed extremum seeking scheme solves the equivalent unconstrained optimisation problem of minimising 𝑉𝑉(𝑢𝑢), 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 𝑦𝑦 = 𝑄𝑄(𝑢𝑢), in general 𝑦𝑦 ∈ ℝ, 𝑢𝑢 ∈ ℝ�. Suppose 𝑄𝑄(⋅) has a strict local minimum, 𝑢𝑢∗ ∈ ℝ�, such that 𝐷𝐷�

1𝑄𝑄(𝑢𝑢∗) = 0, 𝐷𝐷�2𝑄𝑄(𝑢𝑢∗) > 0. Moreover, 𝐷𝐷�

1𝑄𝑄(⋅) and 𝐷𝐷�2𝑄𝑄(⋅) are continuous.

Suppose the map 𝑄𝑄 is 𝑁𝑁 times continuously differentiable.

Assumption 2 [13]. A large class of off-line optimisation methods can be formulated in the following form

𝜉𝜉 ̇ = 𝐵𝐵�𝜉𝜉, 𝐷𝐷�𝑄𝑄(𝜉𝜉)�, (3.67)

35

where 𝜉𝜉 ∈ ℝ� and 𝐷𝐷�(⋅) is the vector of the iterated derivatives of 𝑄𝑄 with respect to its argument. The system (3.67) possesses an equilibrium 𝜉𝜉∗ = 𝑢𝑢∗ and 𝑢𝑢∗ is the strict local minimum of 𝑄𝑄 defined in Assumption 1.

The continuous-time optimisation algorithm typically must satisfy:

Property 2 [13]. There exists 𝛽𝛽� ∈ 𝐾𝐾ℒ, and Δ > 0, such that the following holds: for any 𝜈𝜈 > 0, there exists 𝜖𝜖∗ > 0 such that for any 𝜖𝜖 ∈ (0, 𝜖𝜖∗) and |𝑤𝑤|∞ <𝜖𝜖, the solutions of the following system:

𝜉𝜉 ̇ = 𝐵𝐵(𝜉𝜉, 𝐷𝐷�𝑄𝑄(𝜉𝜉) + 𝑤𝑤), (3.68)

satisfy

|𝜉𝜉(𝑑𝑑) − 𝜉𝜉∗| ≤ 𝛽𝛽�(|𝜉𝜉(0) − 𝜉𝜉∗|, 𝑑𝑑) + 𝜈𝜈, (3.69)

for all |𝜉𝜉(0) − 𝜉𝜉∗| ≤ Δ.

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 𝑉𝑉(𝑢𝑢) has a strict local minimum 𝑢𝑢�

∗, such that 𝐷𝐷�1𝑉𝑉(𝑢𝑢�

∗) = 0, and 𝐷𝐷�2𝑉𝑉(𝑢𝑢�

∗) > 0, thus Assumption 1 holds.

The proposed update law for 𝑢𝑢 given by (3.42) has the form of (3.67) for 𝑁𝑁 = 1. Moreover, the Section 3.2.2 showed that this system has an asymptotically stable equilibrium at 𝑢𝑢�

∗. 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 𝛿𝛿, 𝜔𝜔� and 𝜔𝜔, such that 𝑢𝑢 will converge to a small neighbourhood of 𝑢𝑢�

∗. Let 𝑧𝑧1 = 𝜂𝜂 − 𝜇𝜇(𝑢𝑢��, 𝑎𝑎), and 𝑧𝑧2 = 𝑥𝑥 − 𝑙𝑙(𝑢𝑢�� + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑑𝑑′)), Theorem 2 in [13] is stated as follows:

Theorem 2 [13]: Suppose that Property 2 and Assumption 4, 5 hold. Then there exist and 𝛽𝛽�, 𝛽𝛽� and 𝛽𝛽� ∈ 𝐾𝐾ℒ and Δ > 0 such that the following holds: for any positive 𝜈𝜈, there exist 𝑎𝑎∗ > 0 and 𝜔𝜔�

∗ > 0, such that for any 𝑎𝑎 ∈ (0, 𝑎𝑎∗) and 𝜔𝜔� ∈ (0, 𝜔𝜔�

∗ ), there exists 𝛿𝛿∗(𝑎𝑎) > 0 such that for any 𝛿𝛿 ∈ (0, 𝛿𝛿∗(𝑎𝑎)), there exists 𝑤𝑤∗(𝑎𝑎,𝑤𝑤�, 𝛿𝛿) > 0 such that for any 𝑤𝑤 ∈ (0,𝑤𝑤∗(𝑎𝑎,𝑤𝑤�, 𝛿𝛿)), the solutions of the system (3.47)-(3.49) satisfy

|�̃�𝑢(𝑑𝑑)| ≤ 𝛽𝛽��|�̃�𝑢(𝑑𝑑0)|, 𝛿𝛿𝜔𝜔�𝜔𝜔(𝑑𝑑 − 𝑑𝑑0)� + 𝜈𝜈, (3.70)

|𝑧𝑧1(𝑑𝑑)| ≤ 𝛽𝛽��|𝑧𝑧1(𝑑𝑑0)|, 𝜔𝜔�𝜔𝜔(𝑑𝑑 − 𝑑𝑑0)� + 𝜈𝜈, (3.71)

|𝑧𝑧2(𝑑𝑑)| ≤ 𝛽𝛽��|𝑧𝑧2(𝑑𝑑0)|, (𝑑𝑑 − 𝑑𝑑0)� + 𝜈𝜈, (3.72)

for all |(�̃�𝑢(𝑑𝑑0), 𝑧𝑧1(𝑑𝑑0), 𝑧𝑧2(𝑑𝑑0))| < Δ , and 𝑑𝑑 ≥ 𝑑𝑑0 ≥ 0.

36

Thus, for any 𝜈𝜈 > 0, it is possible to tune 𝛿𝛿, 𝜔𝜔�, 𝜔𝜔 such that 𝑢𝑢�� will converge to a 𝜈𝜈-sized ball centred on 𝑢𝑢�

∗. Since in general 𝑢𝑢�∗ will be in a small region of

optimal solution 𝑢𝑢�∗, it is necessary to discuss the convergence of 𝑢𝑢�� with respect

to 𝑢𝑢�∗.

With this in mind, consider the case for 𝑢𝑢�∗ ∈ 𝑆𝑆��. As mentioned in Section

3.2.1, the proposed optimisation algorithm possesses the property that 𝑢𝑢�∗ = 𝑢𝑢�

∗. According to (3.70), it follows that 𝑢𝑢�� converges to the ball ℬ�� = {𝑢𝑢�� ∈ ℝ|𝑢𝑢�� − 𝑢𝑢�

∗| ≤𝜈𝜈}, and ℬ�� ⊂ 𝑆𝑆��, provided a sufficiently small 𝜈𝜈.

The average of the constraint output is given by

𝑓𝑓1,��(𝑢𝑢) = 1𝑇𝑇

� 𝑓𝑓1(𝑢𝑢�� + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑))𝑑𝑑𝑑𝑑,�

�−� (3.73)

where 𝑇𝑇 = 2𝜋𝜋/𝑤𝑤. Using a Taylor series expansion, the integrand in (3.73) can be written in the following form:

𝑓𝑓1(𝑢𝑢�� + 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)) = 𝑓𝑓1(𝑢𝑢��) + 𝑎𝑎𝐷𝐷�1𝑓𝑓1(𝑢𝑢��)𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑) + �2

2 𝐷𝐷�2𝑓𝑓1(𝑢𝑢��) 𝑠𝑠𝑖𝑖𝛼𝛼2(𝑤𝑤𝑑𝑑) + Ο(𝑎𝑎3), (3.74)

thereby, for a sufficiently small 𝑎𝑎, 𝑓𝑓1,��(𝑢𝑢) in (3.73) can be approximated by

𝑓𝑓1,��(𝑢𝑢) ≅ 𝑓𝑓1(𝑢𝑢��), (3.75)

Since 𝑢𝑢 converges to the ball ℬ�� ⊂ 𝑆𝑆��, then, the constrained is satisfied on average.

On the other hand, in the case of 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆, Section 3.2.1, showed that 𝑢𝑢�

∗ is in the ball ℬ��

∗ = {𝑢𝑢�∗ ∈ ℝ|𝑢𝑢�

∗ − 𝑢𝑢�∗| ≤ 𝜇𝜇(𝛼𝛼)}. According to (3.70), 𝑢𝑢�� converges to the

set ℬ�� = {𝑢𝑢�� ∈ ℝ|𝑢𝑢�� − 𝑢𝑢�∗ | ≤ 𝜈𝜈}. Thus, 𝑢𝑢�� converges to an 𝑁𝑁(𝛼𝛼 + 𝜈𝜈)-sized

neighbourhood of 𝑢𝑢�∗ and 𝑓𝑓1,��(𝑢𝑢) will be in an 𝑁𝑁(𝛼𝛼 + 𝜈𝜈)-sized neighourhood of

𝑓𝑓1(𝑢𝑢�∗) = 0. Hence, if 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆, 𝑢𝑢�� may converge to an equilibrium point that resides outside the constraint set 𝑆𝑆 and consequently 𝑓𝑓1,��(𝑢𝑢) > 0. Although this constraint violation could still be ‘small’ if the parameters 𝑎𝑎, 𝛿𝛿, 𝜔𝜔�, 𝜔𝜔, 𝛼𝛼 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 𝑢𝑢�� close to the optimal solution of the problem. However, when 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆, 𝑢𝑢�� 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 𝑢𝑢�

∗ ∈ 𝜕𝜕𝑆𝑆, then 𝑢𝑢�

∗ = 𝑢𝑢�∗. This eliminates the need of having the ball ℬ��

∗ in the first place.

37

To incorporate integral action in the controller, consider the proposed update law for 𝑢𝑢 with a modification in the smooth function (3.45) given by

�̇�𝑢 = −𝐷𝐷�1𝑉𝑉(𝑢𝑢), (3.76)

where

𝐷𝐷�1𝑉𝑉(𝑢𝑢) = (1 − 𝜎𝜎) ⋅ 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎 ⋅ 𝐷𝐷�1𝑓𝑓1(𝑢𝑢), (3.77)

and

𝜎𝜎 = 1

1 + 𝑒𝑒−[�1(�)−�]�

. (3.78)

Here 𝜀𝜀 ∈ ℝ. The benefits of this parameter can be appreciated by considering the equilibrium point of (3.76),

0 = (1 − 𝜎𝜎) ⋅ 𝐷𝐷�1𝑓𝑓�(𝑢𝑢) + 𝜎𝜎 ⋅ 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢, ), (3.79)

solving for 𝜎𝜎:

𝜎𝜎 = 𝐷𝐷�1𝑓𝑓�(𝑢𝑢)

(𝐷𝐷�1𝑓𝑓�(𝑢𝑢) − 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢)). (3.80)

Since 𝑢𝑢�∗ ∈ 𝜕𝜕𝑆𝑆 , then, the goal is to have an equilibrium point on 𝜕𝜕𝑆𝑆. Note that

𝑓𝑓1(𝑢𝑢�∗) = 0. Suppose in general that 𝐷𝐷�

1𝑓𝑓1(𝑢𝑢�∗)𝐷𝐷�

1𝑓𝑓�(𝑢𝑢�∗) < 0, Then by subtituting

(3.78) in (3.80):

11 + 𝑒𝑒�

�= 𝐷𝐷�

1𝑓𝑓�(𝑢𝑢�∗)

(𝐷𝐷�1𝑓𝑓�(𝑢𝑢�

∗) − 𝐷𝐷�1𝑓𝑓1(𝑢𝑢�

∗)), (3.81)

solving for 𝜀𝜀:

𝜀𝜀 = 𝛼𝛼𝑙𝑙𝛼𝛼 �−𝐷𝐷�1𝑓𝑓1(𝑢𝑢�

∗)𝐷𝐷�

1𝑓𝑓�(𝑢𝑢�∗)

�. (3.82)

Then, for any 𝛼𝛼 > 0, there is an offset 𝜀𝜀 such that (3.76) possesses an equilibrium point 𝑢𝑢� = 𝑢𝑢�

∗. Moreover, as discussed in Section 3.2.2, one of the properties of the gradient system (3.76) is that 𝑉𝑉(⋅) has a strict local minimum 𝑢𝑢�

∗ = 𝑢𝑢�, and therefore, 𝑢𝑢�∗ = 𝑢𝑢�

∗. 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 𝜀𝜀 such that (3.79) has an equilibrium point on the boundary of the constraint.

The potential benefit of the offset 𝜀𝜀 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

𝜎𝜎 = 1

1 + 𝑒𝑒−[�1(�)−�]�

(3.83)

38

𝜙𝜙 ̇ = 𝑟𝑟 − 𝑦𝑦1(𝑥𝑥) (3.84)

𝜀𝜀 = 𝑘𝑘�𝜙𝜙. (3.85)

The reference is set to 𝑟𝑟 = 0 since the integral action aims to regulate the system output at 𝑦𝑦1 = 0. For the sake of simplicity, the feedback control law is chosen to be 𝜀𝜀 = 𝑘𝑘�𝜙𝜙, where 𝑘𝑘� 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:

𝑥𝑥̇ = 100(𝑢𝑢 − 𝑥𝑥), (3.86)

𝑦𝑦0 = (𝑥𝑥 + 1)2, (3.87)

𝑦𝑦1 = 2(𝑥𝑥 − 4)2 − 8. (3.88)

The steady state input-output maps are

𝑓𝑓�(𝑢𝑢) = (𝑢𝑢 + 1)2, (3.89)

𝑓𝑓1(𝑢𝑢) = 2(𝑢𝑢 − 4)2 − 8. (3.90)

The objective is to solve the following problem:

min�

𝑓𝑓�(𝑢𝑢(𝑑𝑑))

s. t. lim�→∞

1𝑇𝑇

� 𝑓𝑓1(𝑢𝑢(𝑑𝑑))𝑑𝑑𝑑𝑑�

�−�≤ 0.

(3.91)

The solution of this constrained optimisation is 𝑢𝑢�∗ = 2, 𝑓𝑓�(2) = 9, 𝑓𝑓1(2) = 0.

This problem is firstly solved by using the proposed dynamical system (3.47)-(3.49) with no integral action. The controller’s parameter are chosen to be 𝑎𝑎 = 0.1, 𝜔𝜔 = 3𝜋𝜋𝑠𝑠−1, 𝜔𝜔� = 0.005, 𝜖𝜖 = 0.1, and 𝛼𝛼 = 2. The initial conditions are 𝑢𝑢��(0) = 7, 𝑥𝑥(0) = 7, 𝜂𝜂�(0) = 0, 𝜂𝜂1(0) = 0, and the period for averaging is 𝑇𝑇 = 2𝜋𝜋/𝜔𝜔.

Figure 3.9 shows the simulation results. Notice that 𝑢𝑢�� converges within a small region around 𝑢𝑢�

∗. 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 𝑢𝑢�� to converge closer to 𝑢𝑢�

∗. This is accomplished by updating the parameter 𝜀𝜀. The Figure 3.10 illustrates the effect of the integral action. 𝑢𝑢�� is still cycling but converges on average to 𝑢𝑢�

∗. Moreover, the constraint is met with an average value of -0.0013, and 𝜀𝜀 = 0.562.

39

𝑑𝑑(𝑠𝑠) 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 plant’s 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 vehicle’s 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 engine’s 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 engine’s 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 model’s equations can be found in the Appendix A.

The engine model is simulated using the software Matlab/Simulink™. Inside this environment, the engine’s inputs are the normalized air-fuel ratio 𝜆𝜆, spark timing 𝜃𝜃, intake valve closing 𝑉𝑉��, valve overlap 𝑉𝑉��, engine speed 𝑁𝑁, desired load 𝑇𝑇�,��, and vehicle speed 𝑉𝑉��.

45

Figure 4.1: Engine-aftertreatment model in Matlab/Simulink™

Connecting pipe

λ

V int (CAD-ABDC)

Vovlap (CAD)

Vspd (km/h)

NO (mg/km)

Tb,ref (Nm)

θ (CAD-BTDC)

N(RPM)

BSFC (g/kWh)

-C-

[Vspd]

Goto

u

αt

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

η%

TWC

Tb,ref

Tb

α 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. 𝑗𝑗 = 1,2,3, … ,60

A PI controller is used to adjust the engine throttle position (𝛼𝛼�) to track 𝑇𝑇�,��

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 𝑝𝑝��, exhaust gas temperature 𝑇𝑇��, and engine-out emission 𝐸𝐸��. The engine out emission is a vector that contains the concentration of oxygen 𝑁𝑁2, nitric oxide 𝑁𝑁𝑁𝑁, carbon monoxide 𝐵𝐵𝑁𝑁, hydrogen 𝐻𝐻2, and hydrocarbon 𝐻𝐻𝐵𝐵.

𝑇𝑇𝒈𝒈 (𝑗𝑗) • 1in • 2 • 3 • 4

𝑇𝑇𝒔𝒔 (𝑗𝑗) • 5 • 6 • 7 • 8

𝐵𝐵𝒈𝒈,𝑁𝑁2

(𝑗𝑗) • 9 in • 10 • 11 • 12

𝐵𝐵𝒈𝒈,𝐵𝐵𝑁𝑁 (𝑗𝑗) • 13 in • 14 • 15 • 16 out

𝐵𝐵𝒈𝒈,𝐻𝐻2

(𝑗𝑗) • 17 in • 18 • 19 • 20

𝐵𝐵𝒈𝒈,𝑁𝑁𝑁𝑁 (𝑗𝑗) • 21 in • 22 • 23 • 24 out

𝐻𝐻𝐵𝐵𝒈𝒈,𝑓𝑓𝑎𝑎𝑠𝑠𝑑𝑑 (𝑗𝑗) • 25 in • 26 • 27 • 28 out

𝐻𝐻𝐵𝐵𝒈𝒈,𝑠𝑠𝑙𝑙𝑠𝑠𝑤𝑤 (𝑗𝑗) • 29 in • 30 • 31 • 32 out

𝐵𝐵𝒔𝒔,𝑁𝑁2

(𝑗𝑗) • 33 • 34 • 35 • 36

𝐵𝐵𝒔𝒔,𝐵𝐵𝑁𝑁 (𝑗𝑗) • 37 • 38 • 39 • 40

𝐵𝐵𝒔𝒔,𝐻𝐻2

(𝑗𝑗) • 41 • 42 • 43 • 44

𝐵𝐵 𝒔𝒔,𝑁𝑁𝑁𝑁(𝑗𝑗) • 45 • 46 • 47 • 48

𝐻𝐻𝐵𝐵𝒔𝒔,𝑓𝑓𝑎𝑎𝑠𝑠𝑑𝑑 (𝑗𝑗) • 49 • 50 • 51 • 52

𝐻𝐻𝐵𝐵𝒔𝒔,𝑠𝑠𝑙𝑙𝑠𝑠𝑤𝑤 (𝑗𝑗) • 53 • 54 • 55 • 56

𝜓𝜓 (𝑗𝑗) • 57 • 58 • 59 • 60

Gas and substratetemperature

Species concentrationsin the gas

Species in the Substrate

Oxygen storage level

Nodes

𝐿𝐿=143.5mm

�̇�𝑚��

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 TWC’s length 𝐿𝐿 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, 𝑇𝑇� and 𝑇𝑇� respectively. This group of temperatures encompasses the nodes 1-8. Similarly, the concentrations of the species in the gas 𝐵𝐵�,�, and in the TWC’s washcoat (substrate) 𝐵𝐵�,� are located along nodes 9-32, and 33-56, respectively. Note that the hydrocarbons (𝐻𝐻𝐵𝐵) are modelled as slow-oxidising 𝐻𝐻𝐵𝐵�,��, 𝐻𝐻𝐵𝐵�,�� and fast-oxidising fuel, 𝐻𝐻𝐵𝐵�,��, 𝐻𝐻𝐵𝐵�,�� 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 𝐵𝐵3𝐻𝐻6 and 𝐵𝐵𝐻𝐻4 respectively [59]. Finally, the oxygen storage level 𝜓𝜓 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 �̇�𝑚�� at the temperature 𝑇𝑇�, 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 × 109 �̇�𝑚��

𝑇𝑇��𝑁𝑁, (4.1)

where �̇�𝑚�� denotes the fuel mass flow rate in kg/s, 𝑁𝑁 is the engine speed in rad/s and 𝑇𝑇�� is the produced engine torque in Nm. The high level goal is to adjust the engine’s parameters to minimise (4.1) whilst taking into account the legislated tailpipe emissions. These pollutants are the outputs of the TWC: 𝑁𝑁𝑁𝑁,𝐵𝐵𝑁𝑁, 𝐻𝐻𝐵𝐵. 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:

�̇�𝑚��,�� = �̇�𝑚��𝑀𝑀��𝑀𝑀�

𝐵𝐵�,��(�,�), (4.2)

�̇�𝑚��,�� = �̇�𝑚��𝑀𝑀��𝑀𝑀�

𝐵𝐵�,��(�,�), (4.3)

�̇�𝑚��,�� = �̇�𝑚�� 𝑀𝑀��

𝑀𝑀�𝐵𝐵�,��(�,�) +

𝑀𝑀��

𝑀𝑀�𝐵𝐵�,��(�,�)�, (4.4)

where the exhaust gas mass flow rate �̇�𝑚�� is given in kg/s, 𝑀𝑀� is its molar mass, and 𝑀𝑀��,𝑀𝑀��,𝑀𝑀��,��,𝑀𝑀��,�� are the molar masses of 𝐵𝐵𝑁𝑁, 𝑁𝑁𝑁𝑁, 𝐵𝐵3𝐻𝐻6 and 𝐵𝐵𝐻𝐻4 respectively. The emissions at the end of the TWC are calculated from the concentration at the nodes 16, 24, 28, 32:

𝐵𝐵�,��(�,�) = 𝐵𝐵�,�� (16) , (4.5)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,�� (24) , (4.6)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,��

(28) , (4.7)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,��

(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 (𝑚𝑚𝑔𝑔/𝑘𝑘𝑚𝑚)

𝑁𝑁𝑁𝑁 150

𝐵𝐵𝑁𝑁 2300 𝐻𝐻𝐵𝐵 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. �̇�𝑚��,��, �̇�𝑚��,�� and �̇�𝑚��,�� can be converted into mg/km by taking into account the vehicle speed, 𝑉𝑉��,

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉��, (4.9)

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉��, (4.10)

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉��, (4.11)

where 𝑉𝑉�� ≠ 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 𝑚𝑚�� 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 𝜃𝜃 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

𝜃𝜃∗ = arg min �∈��

BSFC (𝜃𝜃), (4.12)

where

𝐵𝐵�� = �𝜃𝜃 ∈ Θ ⊂ ℝ|𝑚𝑚�� ≤ 𝑚𝑚��,��. (4.13)

Here Θ is the physical range for 𝜃𝜃 and 𝑚𝑚��,�� 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 plant’s 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

𝐵𝐵��,� = �𝜃𝜃 ∈ Θ ⊂ ℝ| lim�→∞

1𝑇𝑇

� 𝑚𝑚�� 𝑑𝑑𝑑𝑑�

�−�≤ 𝑚𝑚��,��, (4.14)

where 𝑇𝑇 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 𝜃𝜃, but also other engine inputs such as the intake valve closing 𝑉𝑉��. 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 𝑁𝑁 = 1395 rpm, brake torque 𝑇𝑇�,�� = 53.2 Nm and vehicle speed 𝑉𝑉�� = 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 (𝜆𝜆) 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 𝜆𝜆 with the corresponding effect in the conversion efficiency of the engine aftertreatment system. As discussed in the previous section, the 𝑁𝑁𝑁𝑁 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 𝜆𝜆 set point was chosen to be 𝜆𝜆 = 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 𝜆𝜆

A second test was conducted in order to select the intake valve closing

position (𝑉𝑉��). In this case, values for spark timing 𝜃𝜃 and 𝑉𝑉�� 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

𝑆𝑆�� = �(𝜃𝜃, 𝑉𝑉��)|𝑚𝑚�� ≤ 𝑚𝑚��,��, (4.15)

where 𝑚𝑚��,�� is 150 mg/km for 𝑁𝑁𝑁𝑁 emission. As a result, by fixing the intake valve timing 𝑉𝑉�� at any value in the interval [93.7, 98], the set 𝐵𝐵�� 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 𝑁𝑁𝑁𝑁 emission (𝑚𝑚��), Figure 4.6.

Figure 4.5: Steady-state maps, 𝑁𝑁 = 1395 rpm, 𝑇𝑇�,�� = 53.2 Nm, 𝑉𝑉�� = 30, 𝜆𝜆 = 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: 𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 and 𝑁𝑁𝑁𝑁 maps, 𝑁𝑁 = 1395 rpm, 𝑇𝑇�,�� = 53.2 Nm, 𝑉𝑉�� = 30, 𝜆𝜆 = 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 engine’s parameters vary. For instance, the three different intake valve closing positions {94, 96, 98} displace the 𝑚𝑚�� 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 𝜃𝜃 and 𝑉𝑉�� 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 𝑆𝑆� = {(𝜃𝜃, 𝑉𝑉��)|𝜃𝜃 ∈[2,42], 𝑉𝑉�� ∈ [46,96]}, is needed to close and bound 𝑆𝑆�� 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 𝛼𝛼� to effectively emulate a driver pushing the pedal such that the engine’s 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,

𝑥𝑥̇ = 𝑓𝑓�𝑥𝑥, 𝜃𝜃 ̅+ 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼(𝑤𝑤𝑑𝑑) , 𝛼𝛼��, (4.16)

𝜃𝜃 ̅̇ = 𝜖𝜖𝐵𝐵�� 𝜃𝜃,̅𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)�, (4.17)

𝜂𝜂�̇ = −𝜔𝜔𝜔𝜔�[𝜂𝜂� − 𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 ⋅ 𝑠𝑠𝑖𝑖𝛼𝛼(𝜔𝜔𝑑𝑑)], (4.18)

where 𝜖𝜖 = 𝛿𝛿𝜔𝜔𝜔𝜔� and,

𝐵𝐵�� 𝜃𝜃,̅𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)� = −𝐷𝐷� ̅

1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵), (4.19)

𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵) = 2𝑎𝑎−1𝜂𝜂�. (4.20)

The extremum seeking parameters 𝑎𝑎, 𝛿𝛿, 𝜔𝜔, 𝜔𝜔� 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

𝜃𝜃 ̅̇ = 𝜖𝜖𝐵𝐵�� 𝜃𝜃,̅𝐷𝐷�̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)�

𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼(𝑤𝑤𝑑𝑑)

𝜃𝜃 = 𝜃𝜃 ̅+ 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼(𝑤𝑤𝑑𝑑)

𝑇𝑇�,��

𝑇𝑇𝑏𝑏𝑟𝑟𝑎𝑎𝑘𝑘𝑒𝑒

PI TWC Plant

Engine/

𝜃𝜃 ̅ +

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵

𝛼𝛼�


2𝑎𝑎

𝑠𝑠𝑖𝑖𝛼𝛼(𝑤𝑤𝑑𝑑)

Derivative Estimator

Optimiser

𝐷𝐷�̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)

𝜂𝜂�

56

time scale), the derivate estimator (medium time scale) and the optimiser (slowest time scale) is obtained. To do so, the extremum seeking controller’s parameters are set to 𝑎𝑎 = 1.5, 𝜔𝜔 = 2𝜋𝜋𝑠𝑠−1, 𝜔𝜔� = 0.01, 𝜖𝜖 = 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 𝑉𝑉�� = 94 CAD-ABDC, valve overlap 𝑉𝑉�� = 30 CAD, and air-fuel ratio with 𝜆𝜆 = 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 𝑁𝑁𝑁𝑁 emission is well above the Euro-3 limit.

To explain this tailpipe violation, it is necessary to further investigate the convergence of the plant’s 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 TWC’s 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

(%)


0 50 100 150 200 250 300 350 4000

2

4

x 10-3

Cg;

H2( %

)

t (s)


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 𝐻𝐻𝐵𝐵 and 𝐵𝐵𝑁𝑁 are reduced significantly but the tailpipe 𝑁𝑁𝑁𝑁 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 𝜓𝜓 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

)


0 50 100 150 200 250 300 350 4000

100

200

300

HC

g;f

ast

(ppm

)


0 50 100 150 200 250 300 350 4000

100

200

300

HC

g;s

low

(ppm

)


0 50 100 150 200 250 300 350 4000.15

0.2

0.25

Cs;O

2(%

)


0 50 100 150 200 250 300 350 4000

0.5

1x 10

-3

Cs;

CO

(%)

t (s)


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 𝑁𝑁𝑁𝑁 emission is above the Euro-3 limit. Next section proposes an approach to handle the 𝑁𝑁𝑁𝑁 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)

0 50 100 150 200 250 300 350 4000

0.5

1x 10

-4

Cs;

H2(%

)


0 50 100 150 200 250 300 350 4000

100

200

300

400

500

Cs;

NO

(ppm

)


0 50 100 150 200 250 300 350 4000

50

100

150

200

HC

s;fast

(ppm

)


0 50 100 150 200 250 300 350 4000

50

100

150

200

HC

s;sl

ow(p

pm

)


0 50 100 150 200 250 300 350 4000

50

100

A(%

)

t (s)


60

Figure 4.12: Tailpipe 𝑁𝑁𝑁𝑁 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 plant’s output of 𝑁𝑁𝑁𝑁 emission (𝑚𝑚��) 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 Δ𝑚𝑚�� = 𝑚𝑚�� − 𝑚𝑚��,��.

The equations of the CES for the dynamic system are given by


𝜃𝜃 ̅̇ = 𝜖𝜖𝐵𝐵�� 𝜃𝜃,̅ 𝜎𝜎(𝛼𝛼,∆𝑚𝑚��),𝐷𝐷� ̅1� (∆𝑚𝑚��),𝐷𝐷� ̅

1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)�, (4.22)


𝜂𝜂1̇ = −𝜔𝜔𝜔𝜔�[𝜂𝜂1 − Δ𝑚𝑚�� ⋅ 𝑠𝑠𝑖𝑖𝛼𝛼(𝜔𝜔𝑑𝑑)], (4.24)

where 𝜖𝜖 = 𝛿𝛿𝜔𝜔𝜔𝜔� and

𝐵𝐵�� = −[1 − 𝜎𝜎(Δ𝑚𝑚��,𝛼𝛼)] ⋅ 𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵) − 𝜎𝜎(Δ𝑚𝑚��,𝛼𝛼) ⋅ 𝐷𝐷� ̅

1� (∆𝑚𝑚��), (4.25)

𝜎𝜎(Δ𝑚𝑚��,𝛼𝛼) = 1

1 + 𝑒𝑒−Δ��, (4.26)

0 50 100 150 200 250

200

400

600

800

Cg;

NO

(ppm

)

pre-catalystwith UES

post-catalyst withUES

0 50 100 150 200 25020

25

30

35

40

E/

cien

cy(%

)

t (s)

61

Figure 4.13: Constrained extremum seeking scheme to minimise fuel consumption subject to 𝑁𝑁𝑁𝑁 emission constraint

where

𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵) = 2𝑎𝑎−1𝜂𝜂�, (4.27)

𝐷𝐷� ̅1� (∆𝑚𝑚��) = 2𝑎𝑎−1𝜂𝜂1. (4.28)

The extremum seeking parameters 𝑎𝑎, 𝛿𝛿, 𝜔𝜔, 𝜔𝜔� and the smoothing parameter 𝛼𝛼 > 0 need to be adjusted to obtain a desirable convergence on the closed loop. The procedure to tune the parameters 𝑎𝑎, 𝛿𝛿, 𝜔𝜔, 𝜔𝜔� is the same as that in UES to provide the scale separation. The parameter 𝛼𝛼 is used to control the transition between the gradient estimates given in (4.27) and (4.28) as the spark timing 𝜃𝜃 evolves and approaches the constraint boundary, that is, Δ𝑚𝑚�� = 0. In addition, 𝛼𝛼 is also used to reduce the steady-state error between the post-catalyst average 𝑁𝑁𝑁𝑁 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 (𝑚𝑚�� = 150 mg/km). This can be observed in the 𝑚𝑚�� 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 𝜃𝜃 ̅evolves until it asymptotically converges to 28.55 CAD. At this point 𝑚𝑚�� =

Engine/ TWC Plant


𝜃𝜃 ̅̇ = 𝜖𝜖𝐵𝐵�� 𝜃𝜃,̅ 𝜎𝜎(𝛼𝛼, ∆𝑚𝑚��),𝐷𝐷�̅1� (∆𝑚𝑚��), 𝐷𝐷�̅

1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)�

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵


𝑚𝑚��

𝑚𝑚��,��

+

−

𝐷𝐷�̅1� (∆𝑚𝑚��)

𝐷𝐷�̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)

𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

𝜃𝜃� +

𝜃𝜃 = 𝜃𝜃 ̅+ 𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

𝑎𝑎𝑠𝑠𝑖𝑖𝛼𝛼 (𝑤𝑤𝑑𝑑)

PI 𝑇𝑇�,��

𝑇𝑇��

𝛼𝛼� ∆𝑚𝑚��

2𝑎𝑎 2

𝑎𝑎

𝜂𝜂0

𝜂𝜂1

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 𝑁𝑁𝑁𝑁 concentration lower than that obtained with the UES. Although the engine-out 𝑁𝑁𝑁𝑁 conversion efficiency is around 35% for both of the controllers, the CES retards the spark timing such that TWC converts 𝑁𝑁𝑁𝑁 from a lower inlet concentration and the resulting post-catalyst emission is within the Euro-3 limit.

0 20 40 60 80 100 120 140 160 180 20010

20

30

40

503(C

AD

BT

DC

)

0 20 40 60 80 100 120 140 160 180 200400

425

450

475

500

525

550

BS

FC

(g=kW

h)

0 20 40 60 80 100 120 140 160 180 200

150

200

250

300

350

mN

O(m

g=km

)

t (s)

Average NOx (CES)

Average NOx (UES)

Euro-3 limit

180 190 200148

150

152

7mN

O

63

Figure 4.16: CES result for different smoothing parameter 𝛼𝛼

0 20 40 60 80 100 120 140 160 180 20015

17.520

22.525

27.530

32.5

7 3(C

AD

BT

DC

)

, = 1, = 5, = 10

0 20 40 60 80 100 120 140 160 180 200

-10

0

10

"7m

NO

(mg=

km

)

t (s)

Figure 4.15: Tailpipe 𝑁𝑁𝑁𝑁 emission and conversion efficiency with CES

0 50 100 150 200 250

200

400

600

800

Cg;

NO

(ppm

)

pre-catalystwith UES

post-catalyst withUES

0 50 100 150 200 250

200

400

600

800

Cg;

NO

(ppm

)

pre-catalystwith CES

post-catalyst withCES

0 50 100 150 200 25020

25

30

35

40

E/

cien

cy(%

)

t (s)

NO-conversionwith UESNO-conversionwith CES

64

Smooth parameter

𝛼𝛼

Average emission Δ𝑚𝑚�� (mg/km)

Average emission 𝑚𝑚�� (mg/km)

Spark timing 𝜃𝜃 ̅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 𝑁𝑁𝑁𝑁 emission and spark timing for different 𝛼𝛼 values at 200 s

To study the effect of the parameter 𝛼𝛼 in the convergence properties of the proposed CES, consider the Figure 4.16 . For a small 𝛼𝛼 a faster respond with a greater overshoot in the output response is observed. Table 4.2 summarizes the average 𝑁𝑁𝑁𝑁 emission (Δ𝑚𝑚��) and optimal spark timings for different 𝛼𝛼. The average 𝑁𝑁𝑁𝑁 emission approaches to 150 mg/km from below if a small 𝛼𝛼 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 𝐷𝐷� ̅

1(Δ𝑚𝑚��) is, in absolute value, greater than the gradient of the cost function 𝐷𝐷� ̅

1(𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵). This can be appreciated by observing the steady-state values of these gradients in Figure 4.17. As a consequence, for any 𝛼𝛼 > 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 𝛼𝛼.

Figure 4.17: Gradient convergence with CES

0 50 100 150 200 250-8

-4

0

4

8

gra

d.e

stim

ate

of(B

SF

C)

0 50 100 150 200 250-8

-4

0

4

8

gra

d.e

stim

ate

of("

mN

O)

D̂73(BSFC)) (g=kWhCAD!1)

D̂73("mNO)) (mg=kmCAD!1)

0 50 100 150 200 250-4

-2

0

2

4

6

8

Fop

t (s)

, = 0:1, = 5

65

Note that a choice of a small 𝛼𝛼 may potentially introduce undesirable effects. Consider the update law 𝐵𝐵�� in (4.25). The Figure 4.17 shows the average of this gradient-based system for an 𝛼𝛼 value of 0.1 and 5. With the former, 𝐵𝐵�� clearly exhibits several sudden rates of change during the first 50 seconds of the transient response. With this 𝛼𝛼, the algorithm will introduce a sharp transition between the gradients at the vicinity of the boundary of the constraint where ∆𝑚𝑚�� = 0. Depending on the application at hand, this might require considerable actuator effort. To alleviate this, the CES should be tuned to a higher 𝛼𝛼 value. For instance, consider the test with 𝛼𝛼 = 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 𝛼𝛼 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 ∆𝑚𝑚��. To do so, the smoothing function is slightly modified and the closed- loop equations (4.25)-(4.28) are augmented with an integrator state 𝜙𝜙. The equations of the CES equipped with integral action are given by


𝜃𝜃 ̅̇ = 𝜖𝜖𝐵𝐵�� 𝜃𝜃,̅ 𝜎𝜎(𝛼𝛼,∆𝑚𝑚��, 𝜀𝜀),𝐷𝐷� ̅1� (∆𝑚𝑚��),𝐷𝐷� ̅

1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵)�, (4.30)

𝜙𝜙 ̇ = −𝜖𝜖Δ𝑚𝑚��, (4.31)


𝜂𝜂1̇ = −𝜔𝜔𝜔𝜔�[𝜂𝜂1 − Δ𝑚𝑚�� ⋅ 𝑠𝑠𝑖𝑖𝛼𝛼(𝜔𝜔𝑑𝑑)], (4.33)

where 𝜖𝜖 = 𝛿𝛿𝜔𝜔𝜔𝜔� and

𝐵𝐵�� = −[1 − 𝜎𝜎(𝛼𝛼, ∆𝑚𝑚��, 𝜀𝜀)] ⋅ 𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵) − 𝜎𝜎(𝛼𝛼, ∆𝑚𝑚��, 𝜀𝜀) ⋅ 𝐷𝐷� ̅

1� (∆𝑚𝑚��), (4.34)

𝜎𝜎(Δ𝑚𝑚��,𝛼𝛼, 𝜀𝜀) = 1

1 + 𝑒𝑒−Δ��−��

, (4.35)

where

𝐷𝐷� ̅1� (𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵) = 2𝑎𝑎−1𝜂𝜂�, (4.36)

𝐷𝐷� ̅1� (∆𝑚𝑚��) = 2𝑎𝑎−1𝜂𝜂1, (4.37)

66

𝜀𝜀 = 𝑘𝑘�𝜙𝜙. (4.38)

Once again, 𝛿𝛿, 𝜔𝜔, 𝜔𝜔�, 𝛼𝛼 and 𝑘𝑘� are design parameters. By properly choosing the integral gain, the feedback control law (4.38) updates the offset parameter 𝜀𝜀. 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 𝑘𝑘� = 0.25. The controller parameters

0 20 40 60 80 100 120 140 160 180 20010

20

30

40

50

3(C

AD

BT

DC

)

0 20 40 60 80 100 120 140 160 180 200400

425

450

475

500

525

550

BS

FC

(g=kW

h)

0 20 40 60 80 100 120 140 160 180 200135

140

145

150

155

160

165

mN

O(m

g=km

)

Euro-3 limit

0 20 40 60 80 100 120 140 160 180 200-6

-4

-2

0

2

4

"(!

)

t (s)

67

𝛿𝛿, 𝜔𝜔, 𝜔𝜔� and the engine operating variables are the same as those used in the previous section. An 𝛼𝛼 = 5 is selected as this ensured a smooth behavior of 𝐵𝐵��. This may represent a practical situation where reducing 𝛼𝛼 is not possible and the average output is required to be on the boundary of the constraint, that is, Δ𝑚𝑚�� = 0. Figure 4.18 shows the dynamic respond of spark timing 𝜃𝜃,̅ average emission 𝑚𝑚��, and offset parameter 𝜀𝜀 with the CES. The average 𝑁𝑁𝑁𝑁 emission 𝑚𝑚�� 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 𝑘𝑘�

Average emission Δ𝑚𝑚�� (mg/km)

Average emission 𝑚𝑚�� (mg/km)

Offset parameter

𝜀𝜀

Spark timing 𝜃𝜃(̅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 𝑁𝑁𝑁𝑁 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 (𝑘𝑘� = 0.85)

produces a faster convergence to the optimal spark timing where Δ𝑚𝑚�� = 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 𝜃𝜃 ̅ converges to the constrained optimum. To illustrate it, consider the modified sigmoid function defined in (4.35) and note that 𝜀𝜀 converges to a positive value (0.868) with 𝑘𝑘� = 0.85. With no integral action the offset is 𝜀𝜀 = 0 and the constrained spark timing converges into a feasible point such that Δ𝑚𝑚�� = −0.57 mg/km at steady-state (Table 4.2 for 𝛼𝛼 = 5), as a result, 𝜎𝜎(−0.57,5,0) = 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 𝜀𝜀 = 0.868 translates the sigmoid curve to the right where 𝜎𝜎 = 0.456, such that the dynamic system (4.34) has an equilibrium at the constrained optimal spark timing and consequently Δ𝑚𝑚�� = 0.

68

Figure 4.19: Sigmoid function with smoothing parameter 𝛼𝛼 = 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 𝑁𝑁𝑁𝑁 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 controller’s tuning parameters are chosen to be 𝑎𝑎 = 1.5, 𝜔𝜔 = 2𝜋𝜋𝑠𝑠−1, 𝜔𝜔� = 0.01, 𝜖𝜖 = 0.09, 𝛼𝛼 = 5, and 𝑘𝑘� = 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 𝑚𝑚��=150.00 mg/km, and 𝐵𝐵𝐵𝐵𝑆𝑆𝐵𝐵 =575.73 g/kWh.

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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 engines’ 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

0 50 100 150 200 2505

15

25

35

45

3(C

AD

BT

DC

)

0 50 100 150 200 250500

600

700

800

BS

FC

(g=kW

h)

0 50 100 150 200 250100

125

150

175

200

mN

O(m

g=km

)

0 50 100 150 200 250-5

0

5

10

15

20

"(!

)

t (s)

70

timing was achieved with the simplest version of the proposed scheme, that is, without updating the offset parameter 𝜀𝜀. 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 plant’s 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 𝑁𝑁𝑁𝑁 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 𝑁𝑁𝑁𝑁 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 𝑁𝑁𝑁𝑁 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 engine’s basic specifications are given in Table A.1.

Parameter Description Manufacturer Ford Australia

Cylinders In line 6 Capacity 3984 𝑐𝑐𝑚𝑚3

Bore 92.25 𝑚𝑚𝑚𝑚 Stroke 99.31 𝑚𝑚𝑚𝑚

Conrod length 153.85 𝑚𝑚𝑚𝑚

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

λ

V int (CAD-ABDC)

Vovlap (CAD)

Vspd (km/h)

NO (mg/km)

Tb,ref (Nm)

( )

N(RPM)

BSFC (g/kWh)

-C-

[Vspd]

Goto

u

αt

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

η%

TWC

Tb,ref

Tb

α 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 𝑀𝑀� [𝑘𝑘𝑔𝑔/𝑚𝑚𝑠𝑠𝑙𝑙] 𝐴𝐴𝑖𝑖𝑟𝑟 28.96 × 10−3 𝑁𝑁𝑁𝑁 30.01 × 10−3 𝑁𝑁2 32 × 10−3 𝐵𝐵𝑁𝑁 28.01 × 10−3 𝐻𝐻2 2.016 × 10−3 𝑁𝑁2 28.0134 × 10−3

𝐵𝐵3𝐻𝐻6 42.0797 × 10−3 𝐵𝐵𝐻𝐻4 16.043 × 10−3 𝑀𝑀�� 13.82 × 10−3

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 system’s states are given in Table A.4.

Name Value Description 𝛼𝛼� 6 Number of cylinders [−] 𝑉𝑉� 6.637 × 10−4 Cylinder swept volume [𝑚𝑚3] 𝑉𝑉�� 0.004 Intake manifold volume [𝑚𝑚3] 𝑎𝑎� 1.3381 × 10−3 Throttle parameter [−] 𝑏𝑏� 4.6367 × 10−4 Throttle parameter [−] 𝑐𝑐� 9.5868 × 10−6 Throttle parameter [−]

𝜂𝜂��(⋅) Variable Volumetric efficiency [−] 𝑅𝑅� 8.31 Universal gas constant [𝐽𝐽/𝑚𝑚𝑠𝑠𝑙𝑙𝐾𝐾] 𝛾𝛾 1.44 Ratio of specific heats 𝑐𝑐�/𝑐𝑐�

𝑝𝑝�� 101000 Ambient pressure [𝑃𝑃𝑎𝑎] 𝑇𝑇�� 298 Ambient temperature [𝐾𝐾] 𝐴𝐴𝐵𝐵𝑅𝑅� 14.50 Stoichiometric air fuel ratio for gasoline

Table A.3: Parameters defining the throttle and intake manifold model

78

Name Type Description

𝜆𝜆 Input

Normalized air/fuel ratio [−] 𝑁𝑁 Input

Engine speed [𝑟𝑟𝑎𝑎𝑑𝑑/𝑠𝑠]

𝑉𝑉�� Input Intake valve closing [𝐵𝐵𝐴𝐴𝐷𝐷 𝐴𝐴𝐵𝐵𝑇𝑇𝐷𝐷𝐵𝐵 ] 𝑉𝑉�� Input Valve overlap [𝐵𝐵𝐴𝐴𝐷𝐷] 𝛼𝛼� Input Throttle position [𝑑𝑑𝑒𝑒𝑔𝑔] 𝑝𝑝�� State Intake manifold pressure [𝑃𝑃𝑎𝑎]

�̇�𝑚�� Output Fuel mass flow rate [𝑘𝑘𝑔𝑔/𝑠𝑠] �̇�𝑚�� Output Total mas flow rate of the mixture [𝑘𝑘𝑔𝑔/𝑠𝑠]

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,

�̇�𝑚�� = 𝐵𝐵�𝐴𝐴�(𝛼𝛼�)

�𝑅𝑅𝑇𝑇�� 𝑝𝑝��

𝑝𝑝��

1�� 2𝛾𝛾

𝛾𝛾 − 1�1 − � 𝑝𝑝��

𝑝𝑝��

(�−1)/�

��

12

, (A.1)

for the pressure ratios 𝑝𝑝��/𝑝𝑝�� greater than the critical value, which is defined as

𝑝𝑝�� = � 2𝛾𝛾 + 1

��

�−1, (A.2)

and

�̇�𝑚�� = 𝐵𝐵�𝐴𝐴�(𝛼𝛼�)

�𝑅𝑅𝑇𝑇��𝛾𝛾� 2

𝛾𝛾 + 1�

�+12(�−1)

, (A.3)

when the pressure ratio is less than 𝑝𝑝��. The product of the throttle open area and the discharge coefficient is determined by

𝐵𝐵�𝐴𝐴�(𝛼𝛼�) = 10−3 × (𝑎𝑎�𝛼𝛼�2 + 𝑏𝑏�𝛼𝛼� + 𝑐𝑐�), (A.4)

whose coefficients {𝑎𝑎�, 𝑏𝑏�, 𝑐𝑐�} 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:

�̇�𝑚�� = 𝑝𝑝��𝛼𝛼�𝑉𝑉�𝑁𝑁4𝜋𝜋𝑅𝑅𝑇𝑇��

𝜂𝜂��𝑝𝑝��,𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁�, (A.5)

where the volumetric efficiency for the Ford Falcon engine was approximated by the polynomial:

79

where

𝑃𝑃�� = 10−3𝑝𝑝��, (A.7)

𝑉𝑉�� = 𝑉𝑉�� − 58.5, (A.8)

𝑁𝑁�� = 𝑁𝑁 602𝜋𝜋

. (A.9)

Here, the manifold pressure is expressed in (kPa), intake valve closing position 𝑉𝑉�� relative to 58.5 CAD-ABDC, and engine speed 𝑁𝑁 in (rpm). Based on this change of variables, the coefficients 𝑎𝑎��,�,�,�,�,� were experimentally obtained and are given in Table A.5.

𝜂𝜂�� parameters 𝑎𝑎𝑎𝑎00000 0.35689 𝑎𝑎𝑎𝑎01001 −1.67540 × 10−5 𝑎𝑎𝑎𝑎00001 1.8431 × 10−4 𝑎𝑎𝑎𝑎01010 −9.5532 × 10−4 𝑎𝑎𝑎𝑎00002 −3.6965 × 10−8 𝑎𝑎𝑎𝑎01100 1.12060 × 10−3 𝑎𝑎𝑎𝑎00010 1.6316 × 10−3 𝑎𝑎𝑎𝑎02000 1.12920 × 10−1 𝑎𝑎𝑎𝑎00011 1.78080 × 10−6 𝑎𝑎𝑎𝑎10000 8.8101 × 10−3 𝑎𝑎𝑎𝑎00020 −9.6684 × 10−5 𝑎𝑎𝑎𝑎10001 −8.2443 × 10−7 𝑎𝑎𝑎𝑎00100 −2.7419 × 10−4 𝑎𝑎𝑎𝑎10010 2.9296 × 10−5 𝑎𝑎𝑎𝑎00101 3.48180 × 10−7 𝑎𝑎𝑎𝑎10100 1.4431 × 10−5 𝑎𝑎𝑎𝑎00110 −1.3453 × 10−4 𝑎𝑎𝑎𝑎11000 1.5496 × 10−4 𝑎𝑎𝑎𝑎00200 −1.1835 × 10−4 𝑎𝑎𝑎𝑎20000 −5.2594 × 10−5 𝑎𝑎𝑎𝑎01000 −0.20586

Table A.5: Volumetric efficiency parameters

(A.6) can be expanded as:

𝜂𝜂�� = (

𝑎𝑎𝑎𝑎02000𝜆𝜆2 + 𝑎𝑎𝑎𝑎01100𝜆𝜆𝑉𝑉�� + 𝑎𝑎𝑎𝑎01001𝜆𝜆𝑁𝑁�� + 𝑎𝑎𝑎𝑎11000𝜆𝜆𝑃𝑃�� + 𝑎𝑎𝑎𝑎01010 𝜆𝜆𝑉𝑉�� + 𝑎𝑎𝑎𝑎01000 𝜆𝜆 + 𝑎𝑎𝑎𝑎00200𝑉𝑉��

2 + 𝑎𝑎𝑎𝑎00101𝑉𝑉��𝑁𝑁�� + 𝑎𝑎𝑎𝑎10100𝑉𝑉��𝑃𝑃�� + 𝑎𝑎𝑎𝑎00110𝑉𝑉��𝑉𝑉�� + 𝑎𝑎𝑎𝑎00100𝑉𝑉�� + 𝑎𝑎𝑎𝑎00002𝑁𝑁��

2 + 𝑎𝑎𝑎𝑎10001 𝑁𝑁��𝑃𝑃�� + 𝑎𝑎𝑎𝑎00011𝑁𝑁��𝑉𝑉�� + 𝑎𝑎𝑎𝑎00001𝑁𝑁�� + 𝑎𝑎𝑎𝑎20000 𝑃𝑃��

2 + 𝑎𝑎𝑎𝑎10010 𝑃𝑃��𝑉𝑉�� + 𝑎𝑎𝑎𝑎10000𝑃𝑃�� + 𝑎𝑎𝑎𝑎00020𝑉𝑉��

2 + 𝑎𝑎𝑎𝑎00010𝑉𝑉�� + 𝑎𝑎𝑎𝑎00000),

(A.10)

The fuel mass flow rate is given by

𝜂𝜂�� = �� 𝑎𝑎��,�,�,�,�,�

2

�=0

2

�=0

2

�=0

2

�=0

2

�=0𝑃𝑃��

� 𝜆𝜆�𝑉𝑉�� 𝑉𝑉��

� 𝑁𝑁�� , (A.6)

80

�̇�𝑚�� =�̇�𝑚��

𝜆𝜆(𝐴𝐴𝐵𝐵𝑅𝑅�), (A.11)

and the total mass flow rate of the mixture by

�̇�𝑚�� = �̇�𝑚�� + �̇�𝑚��. (A.12)

Finally the differential equation that governs the intake manifold pressure is given by

𝑑𝑑𝑝𝑝��𝑑𝑑𝑑𝑑

= 𝛾𝛾𝑅𝑅�𝑇𝑇��𝑀𝑀��𝑉𝑉��

(�̇�𝑚�� − �̇�𝑚��) (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,

�̇�𝑄�� − �̇�𝑊�� = �̇�𝑚��𝛥𝛥ℎ�, (A.14)

the heat due to the fuel combustion and heat exchanged with the walls is given by,

�̇�𝑄�� = �̇�𝑚��𝑄𝑄�� + 𝐴𝐴��ℎ��𝑇𝑇�� − 𝑇𝑇��, (A.15)

and the work output by

�̇�𝑊�� = �̇�𝑚��𝑄𝑄�� 𝜂𝜂��,��𝑝𝑝��, 𝑆𝑆𝐾𝐾, 𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁�. (A.16)

Then (A.14) can be written as

�̇�𝑚��𝑄𝑄��1 − 𝜂𝜂��,��𝑝𝑝��, 𝑆𝑆𝐾𝐾, 𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁�� +𝐴𝐴��ℎ��𝑇𝑇�� − 𝑇𝑇�� = �̇�𝑚�� 𝛥𝛥ℎ�.

(A.17)

In (A.17), three variables need to be specified: Δℎ�, ℎ�� and 𝜂𝜂��,��(⋅). The term 𝛥𝛥ℎ� is determined by approximating the gases’ properties in the combustion chamber as those of air modelled as ideal gas,

𝛥𝛥ℎ� = 28.1(𝑇𝑇�� − 𝑇𝑇��) + 1.9670 × 10−3�𝑇𝑇��

2 − 𝑇𝑇�� 2 �

2+

4.8020 × 10−6�𝑇𝑇�� 3 − 𝑇𝑇��

3 �3

−1.9 × 10−9�𝑇𝑇��

4 − 𝑇𝑇�� 4 �

4,

(A.18)

81

Name Value Description 𝐽𝐽�� 0.15 Crankshaft moment of inertia [𝑘𝑘𝑔𝑔 𝑚𝑚2] 𝐴𝐴�� 0.04215 Total combustion chamber area [𝑚𝑚2] 𝐵𝐵 0.9225 Cylinder bore diameter [𝑚𝑚]

𝑄𝑄�� 44 × 106 Lower heating value of the fuel [𝐽𝐽/𝑘𝑘𝑔𝑔] 𝑎𝑎� 69 Cylinder wall heat transfer parameter 𝑏𝑏� 0.2 Cylinder wall heat transfer parameter

𝑚𝑚��𝑐𝑐�� 115000 Thermal mass of the engine [𝐽𝐽/𝐾𝐾] 𝐾𝐾�� 3.5 Equilibirum constant of water-gas shift reaction 𝑟𝑟�� 1.8 Fuel 𝐻𝐻 to 𝐵𝐵 ratio 𝑟𝑟�2 0.2095 Proportion of 𝑁𝑁2 in air [𝑚𝑚𝑠𝑠𝑙𝑙/𝑚𝑚𝑠𝑠𝑙𝑙]

𝜂𝜂��,��(⋅) Variable Net indiacated efficiency [−] 𝑒𝑒��(⋅) Variable 𝐵𝐵𝑁𝑁 emissions [𝑚𝑚𝑠𝑠𝑙𝑙/(𝑘𝑘𝑔𝑔 𝑓𝑓𝑢𝑢𝑒𝑒𝑙𝑙)] 𝑒𝑒��(⋅) Variable 𝑁𝑁𝑁𝑁 emissions [𝑚𝑚𝑠𝑠𝑙𝑙/(𝑘𝑘𝑔𝑔 𝑓𝑓𝑢𝑢𝑒𝑒𝑙𝑙)] 𝑒𝑒��(⋅) Variable 𝐻𝐻𝐵𝐵 emissions [𝑚𝑚𝑠𝑠𝑙𝑙/(𝑘𝑘𝑔𝑔 𝑓𝑓𝑢𝑢𝑒𝑒𝑙𝑙)]

Table A.6: Parameters for the engine’s model


𝜆𝜆 Input

Normalized air/fuel ratio [−] 𝑁𝑁 Input

Engine speed [𝑟𝑟𝑎𝑎𝑑𝑑/𝑠𝑠]

�̇�𝑁 Input

Engine acceleration [𝑟𝑟𝑎𝑎𝑑𝑑/𝑠𝑠2] 𝑉𝑉�� Input Intake valve closing [𝐵𝐵𝐴𝐴𝐷𝐷 − 𝐴𝐴𝐵𝐵𝑇𝑇𝐷𝐷𝐵𝐵 ] 𝑉𝑉�� Input Valve overlap [𝐵𝐵𝐴𝐴𝐷𝐷] 𝑆𝑆𝐾𝐾 Input Spark timing [𝐵𝐵𝐴𝐴𝐷𝐷 − 𝐵𝐵𝑇𝑇𝐷𝐷𝐵𝐵] 𝑝𝑝�� Input Intake manifold pressure [𝑃𝑃𝑎𝑎]

�̇�𝑚�� Input Fuel mass flow rate [𝑘𝑘𝑔𝑔/𝑠𝑠] �̇�𝑚�� Input Total mas flow rate of the mixture [𝑘𝑘𝑔𝑔/𝑠𝑠] 𝑇𝑇�� State Lumped engine temperature [𝐾𝐾] 𝑇𝑇�� Output Exhaust temperature [𝐾𝐾]

𝑇𝑇�� Output Engine brake torque [𝑁𝑁𝑚𝑚] 𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 Output Brake specific fuel consumption [𝑔𝑔/𝑘𝑘𝑊𝑊ℎ] 𝐸𝐸�� Output Engine-out emissions ∈ ℝ5 [𝑚𝑚𝑠𝑠𝑙𝑙/𝑚𝑚𝑠𝑠𝑙𝑙]

Table A.7: Inputs, outputs and states for the engine’s model

82

The heat transfer coefficient, ℎ�� is obtained by using,

ℎ�� = 𝑎𝑎�𝑘𝑘��

𝐵𝐵 �

4�̇�𝑚��

𝛼𝛼� 𝜋𝜋𝐵𝐵𝜇𝜇��

��

, (A.19)

where the thermal conductivity is given by

𝑘𝑘�� = 3.4288 × 10−11𝑇𝑇�� 3 − 9.1803 × 10−8𝑇𝑇��

2 +

1.294 × 10−4𝑇𝑇�� − 5.2076 × 10−3, (A.20)

and the dynamic viscosity is by

𝜇𝜇�� = 1.066 × 10−14𝑇𝑇�� 3 − 3.6432 × 10−11𝑇𝑇��

2 +

6.6706 × 10−8𝑇𝑇�� + 1.433 × 10−6. (A.21)

The indicated thermal efficiency for the Ford Falcon engine was approximated by the polynomial,

𝜂𝜂�� =

�� 𝑎𝑎�,�,�,�,�,�𝑃𝑃�� 𝜆𝜆�𝑆𝑆𝐾𝐾 �𝑉𝑉��

� 𝑉𝑉�� 𝑁𝑁��

�2

�=0

2

�=0

2

�=0

2

�=0

2

�=0

2

�=0, (A.22)

where the variables 𝑃𝑃��, 𝑉𝑉�� and 𝑁𝑁��, were given in (A.7)-(A.9). The coefficients 𝑎𝑎�,�,�,�,�,� were experimentally obtained and are reported in Table A.8.

𝜂𝜂��,�� parameters

𝑎𝑎000000 −8.4001 × 10−1 𝑎𝑎002000 −1.2642 × 10−4 𝑎𝑎000001 3.4125 × 10−5 𝑎𝑎010000 1.4883 𝑎𝑎000002 −6.2929 × 10−9 𝑎𝑎010001 −4.1641 × 10−5 𝑎𝑎000010 1.536 × 10−3 𝑎𝑎010010 −2.7417 × 10−3 𝑎𝑎000011 6.8762 × 10−7 𝑎𝑎010100 −3.6511 × 10−3 𝑎𝑎000020 −5.1696 × 10−5 𝑎𝑎011000 8.8654 × 10−3 𝑎𝑎000100 1.211 × 10−3 𝑎𝑎020000 −9.4752 × 10−1 𝑎𝑎000101 5.2989 × 10−7 𝑎𝑎100000 8.2314 × 10−3 𝑎𝑎000110 −8.6462 × 10−5 𝑎𝑎100001 −4.9836 × 10−7 𝑎𝑎000200 −6.1361 × 10−5 𝑎𝑎100010 4.4968 × 10−5 𝑎𝑎001000 8.6231 × 10−4 𝑎𝑎100100 6.3069 × 10−5 𝑎𝑎001001 1.2589 × 10−6 𝑎𝑎101000 −8.5707 × 10−5 𝑎𝑎001010 4.4078 × 10−5 𝑎𝑎110000 8.5054 × 10−3 𝑎𝑎001100 5.3959 × 10−5 𝑎𝑎200000 −1.1767 × 10−4

Table A.8: The net indicated efficiency coefficients

The net indicated efficiency is then expanded as

83

𝜂𝜂��,�� = ( 𝑎𝑎000200 𝑉𝑉��

2 + 𝑎𝑎000101𝑉𝑉��𝑁𝑁�� + 𝑎𝑎010100𝑉𝑉�� +𝑎𝑎100100𝑉𝑉��𝑃𝑃�� + 𝑎𝑎001100𝑉𝑉��𝑆𝑆𝐾𝐾 + 𝑎𝑎000110𝑉𝑉��𝑉𝑉�� +𝑎𝑎000100𝑉𝑉�� + 𝑎𝑎000002𝑁𝑁��

2 + 𝑎𝑎010001𝑁𝑁��𝜆𝜆 + 𝑎𝑎100001𝑁𝑁��𝑃𝑃�� +𝑎𝑎001001𝑁𝑁��𝑆𝑆𝐾𝐾 + 𝑎𝑎000011𝑁𝑁��𝑉𝑉�� + 𝑎𝑎000001𝑁𝑁�� + 𝑎𝑎020000𝜆𝜆2 + 𝑎𝑎110000𝜆𝜆𝑃𝑃�� + 𝑎𝑎011000𝜆𝜆(𝑆𝑆𝐾𝐾) + 𝑎𝑎010010𝜆𝜆𝑉𝑉�� + 𝑎𝑎010000𝜆𝜆

+𝑎𝑎200000𝑃𝑃�� 2 + 𝑎𝑎101000 𝑃𝑃��𝑆𝑆𝐾𝐾 + 𝑎𝑎100010𝑃𝑃��𝑉𝑉�� + 𝑎𝑎100000 𝑃𝑃��

+𝑎𝑎002000 𝑆𝑆𝐾𝐾 2 + 𝑎𝑎001010 𝑆𝑆𝐾𝐾�𝑉𝑉�� + 𝑎𝑎001000𝑆𝑆𝐾𝐾 + 𝑎𝑎000020 𝑉𝑉�� 2

+ 𝑎𝑎000010𝑉𝑉�� + 𝑎𝑎000000),

(A.23)

A.2.1 Torque production and friction

Newton’s second law applied to the engine crankshaft yields,

𝑇𝑇�� = 𝑇𝑇�� − 𝑇𝑇�� − 𝐽𝐽��𝑑𝑑𝑁𝑁𝑑𝑑𝑑𝑑

, (A.24)

where

𝑇𝑇�� =�̇�𝑚��𝑄𝑄��

𝑁𝑁𝜂𝜂��,��𝑝𝑝��, 𝑆𝑆𝐾𝐾,𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁�, (A.25)

and

𝑇𝑇�� =𝛼𝛼��𝑉𝑉�

4𝜋𝜋𝑝𝑝��𝑁𝑁, 𝑇𝑇��. (A.26)

The frictional effective pressure, 𝑝𝑝��, is approximated by

𝑝𝑝�� = 𝑎𝑎�1𝑇𝑇�� + 𝑎𝑎�2𝑇𝑇�� 2 + 𝑎𝑎�3𝑁𝑁�� + 𝑎𝑎�4, (A.27)

where 𝑇𝑇�� is the lumped engine temperature 𝑇𝑇�� in ℃ and,

𝑎𝑎�1 = −7881 (A.28)

𝑎𝑎�2 = 50.596 (A.29)

𝑎𝑎�3 = 39.107 (A.30)

𝑎𝑎�4 = 324910.6 (A.31)

The brake specific fuel consumption in [𝑔𝑔/𝑘𝑘𝑊𝑊ℎ] is given by

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 = 3.6 × 109 �̇�𝑚��

𝑇𝑇��𝑁𝑁, (A.32)

and the energy balance applied to the engine yields,

𝑚𝑚��𝑐𝑐�� 𝑑𝑑𝑇𝑇��

𝑑𝑑𝑑𝑑= 𝐴𝐴��ℎ��𝑇𝑇�� − 𝑇𝑇�� + 𝑁𝑁𝑇𝑇��. (A.33)

84

A.2.2 Engine-out emissions

Engine-out emissions such as 𝐵𝐵𝑁𝑁, 𝑁𝑁𝑁𝑁 and 𝐻𝐻𝐵𝐵 were obtained as static maps, which depend on several engine parameters,

𝑒𝑒��(𝜆𝜆) (A.34)

𝑒𝑒��(𝑝𝑝��, 𝑆𝑆𝐾𝐾, 𝜆𝜆,𝑁𝑁) (A.35)

𝑒𝑒��(𝑝𝑝��, 𝑆𝑆𝐾𝐾,𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁) (A.36)

where 𝑒𝑒� is the normalized emission of the compound 𝑋𝑋 in [𝑚𝑚𝑠𝑠𝑙𝑙�/𝑘𝑘𝑔𝑔 𝑓𝑓𝑢𝑢𝑒𝑒𝑙𝑙]. By introducing the change of variables (A.7)-(A.9), the normalized emission are given by the following calibrated polynomials,

(A.37)

𝑒𝑒��𝑃𝑃��, 𝑆𝑆𝐾𝐾, 𝜆𝜆,𝑁𝑁�� =

(A.38)

𝑒𝑒��(𝑃𝑃��, 𝑆𝑆𝐾𝐾,𝜆𝜆, 𝑉𝑉��, 𝑉𝑉��,𝑁𝑁��) =

𝑒𝑒𝐵𝐵𝑁𝑁(𝜆𝜆 ) = (15430.2129𝜆𝜆5 − 82708.4092𝜆𝜆4 + 175681.3381𝜆𝜆3 − 184575.6623𝜆𝜆2 + 95730.3456𝜆𝜆 − 19554.9757)

(0.00026607 (𝑃𝑃𝑖𝑖𝑚𝑚 )2 + 0.043891 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆 − 0.00071808 𝑃𝑃𝑖𝑖𝑚𝑚 𝑆𝑆𝐾𝐾

+ 2.3068 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.094552 𝑃𝑃𝑖𝑖𝑚𝑚 + 7.2269 𝜆𝜆2

+ 0.071725 𝜆𝜆𝑆𝑆𝐾𝐾 − 0.0018909 𝜆𝜆𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 17.9685 𝜆𝜆− 0.00044268 (𝑆𝑆𝐾𝐾)2 − 3.1327 × 10−6𝑆𝑆𝐾𝐾 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.0067427𝑆𝑆𝐾𝐾

+ 8.0627 × 10−7�𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 − 0.0023642 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 15.5957)

�−8.6207 × 10−6(𝑃𝑃𝑖𝑖𝑚𝑚 )3 + 0.0015661 (𝑃𝑃𝑖𝑖𝑚𝑚 )2𝜆𝜆 − 2.5081 × 10−5(𝑃𝑃𝑖𝑖𝑚𝑚 )2𝑆𝑆𝐾𝐾

+ 1.0709 × 10−5(𝑃𝑃𝑖𝑖𝑚𝑚 )2𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 + 2.0362 × 10−5(𝑃𝑃𝑖𝑖𝑚𝑚 )2𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 − 1.3387 × 10−7(𝑃𝑃𝑖𝑖𝑚𝑚 )2𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.00019104 (𝑃𝑃𝑖𝑖𝑚𝑚 )2 − 0.40518 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆2

+ 0.006661 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆𝑆𝑆𝐾𝐾 − 0.0027099 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 − 0.0021044 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝

− 2.1798 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.72644 𝑃𝑃𝑖𝑖𝑚𝑚 𝜆𝜆 − 5.9171 × 10−6𝑃𝑃𝑖𝑖𝑚𝑚 (𝑆𝑆𝐾𝐾)2

+ 1.0742 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝑆𝑆𝐾𝐾 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 + 2.299 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝑆𝑆𝐾𝐾 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 3.1168 × 10−8𝑃𝑃𝑖𝑖𝑚𝑚 𝑆𝑆𝐾𝐾 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.0034456 𝑃𝑃𝑖𝑖𝑚𝑚 𝑆𝑆𝐾𝐾 − 1.6157 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 (𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑 𝑆𝑆 )2

− 3.9845 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 4.1223 × 10−7𝑃𝑃𝑖𝑖𝑚𝑚 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚

+ 0.0017333 𝑃𝑃𝑖𝑖𝑚𝑚 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 − 1.495 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 �𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2 + 2.5882 × 10−7𝑃𝑃𝑖𝑖𝑚𝑚 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.00020505 𝑃𝑃𝑖𝑖𝑚𝑚 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 1.6978

× 10−9𝑃𝑃𝑖𝑖𝑚𝑚 �𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 + 1.3202 × 10−5𝑃𝑃𝑖𝑖𝑚𝑚 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.35554 𝑃𝑃𝑖𝑖𝑚𝑚

− 36.5564 𝜆𝜆3 − 0.24804 𝜆𝜆2𝑆𝑆𝐾𝐾 + 0.24286 𝜆𝜆2𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆+ 0.2056 𝜆𝜆2𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 − 0.0038141 𝜆𝜆2𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 121.9179 𝜆𝜆2

+ 0.0019239 𝜆𝜆(𝑆𝑆𝐾𝐾)2 − 0.0035712 𝜆𝜆𝑆𝑆𝐾𝐾 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 − 0.0026637 𝜆𝜆𝑆𝑆𝐾𝐾 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝

+ 5.3709 × 10−5𝜆𝜆𝑆𝑆𝐾𝐾 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.20893 𝜆𝜆𝑆𝑆𝐾𝐾 − 0.0019114 𝜆𝜆(𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 )2

85

(A.39)

The overall reaction of the combustion of gasoline is given by

𝛼𝛼��𝐵𝐵𝐻𝐻�� + 𝛼𝛼��𝑟𝑟�2𝑁𝑁2 + �1 − 𝑟𝑟�2

�𝑁𝑁2�

→ 𝛼𝛼��𝐵𝐵𝐻𝐻��+ 𝛼𝛼��𝐵𝐵𝑁𝑁 + 𝛼𝛼��𝑁𝑁𝑁𝑁 + 𝛼𝛼��2

𝐵𝐵𝑁𝑁2

+ 𝛼𝛼�2𝑁𝑁2 + 𝛼𝛼�2

𝑁𝑁2 + 𝛼𝛼�2𝐻𝐻2 + 𝛼𝛼�2�𝐻𝐻2𝑁𝑁,

(A.40)

and the water-gas shift reaction by

𝐵𝐵𝑁𝑁2 + 𝐻𝐻2 → 𝐵𝐵𝑁𝑁 + 𝐻𝐻2𝑁𝑁. (A.41)

Thus, it is possible to estimate the other products of the reaction (A.40), [56].

𝑒𝑒��2= 1

𝑀𝑀��− 𝑒𝑒�� − 𝑒𝑒��, (A.42)

𝑒𝑒�2� = 0.5𝑟𝑟��

1𝑀𝑀��

− 𝑒𝑒��

1 + 𝑒𝑒��𝐾𝐾��𝑒𝑒��2

, (A.43)

𝑒𝑒�2=

𝑟𝑟�2

𝑀𝑀��𝐴𝐴𝐵𝐵𝑅𝑅�𝜆𝜆 − 0.5𝑒𝑒�� − 0.5𝑒𝑒�� − 𝑒𝑒��2

− 0.5𝑒𝑒�2� , (A.44)

𝑒𝑒�2=

𝑒𝑒��𝑒𝑒�2�

𝐾𝐾��𝑒𝑒��2

, (A.45)

where

− 0.0024392 𝜆𝜆𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 2.8594 × 10−5𝜆𝜆𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.25166 𝜆𝜆𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆

− 0.0021685 𝜆𝜆�𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2 + 7.3965 × 10−5𝜆𝜆𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.25842 𝜆𝜆𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝

− 4.1835 × 10−7𝜆𝜆�𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 + 0.0061968 𝜆𝜆𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 131.0003 𝜆𝜆 − 9.5166 × 10−6(𝑆𝑆𝐾𝐾)3 + 7.8065 × 10−6(𝑆𝑆𝐾𝐾)2𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 + 1.5802 × 10−5(𝑆𝑆𝐾𝐾)2𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝

+ 2.0864 × 10−7(𝑆𝑆𝐾𝐾)2𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 − 0.0013541 (𝑆𝑆𝐾𝐾)2 − 2.8959 × 10−5𝑆𝑆𝐾𝐾 (𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 )2 − 5.2006 × 10−5𝑆𝑆𝐾𝐾 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 1.0303 × 10−7𝑆𝑆𝐾𝐾 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.0039259 𝑆𝑆𝐾𝐾 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 − 2.5525

× 10−5𝑆𝑆𝐾𝐾 �𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2 + 1.8598 × 10−7𝑆𝑆𝐾𝐾 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.002035 𝑆𝑆𝐾𝐾 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝

− 4.1617 × 10−9𝑆𝑆𝐾𝐾 �𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 − 5.2757 × 10−5𝑆𝑆𝐾𝐾 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚

− 0.039895 𝑆𝑆𝐾𝐾 + 2.0968 × 10−5(𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 )3 + 4.3798 × 10−5(𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 )2𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 − 1.5891 × 10−7(𝑉𝑉𝑖𝑖𝛼𝛼 𝑑𝑑𝑆𝑆 )2𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚

+ 0.0011484 (𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 )2 + 4.6537 × 10−5𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 �𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2 − 4.1651 × 10−7𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.0021636 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 − 1.0669

× 10−8𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 �𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 + 1.6185 × 10−6𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.021872 𝑉𝑉𝑖𝑖𝛼𝛼𝑑𝑑𝑆𝑆

+ 1.4185 × 10−5�𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙 𝑝𝑝 �3 − 8.0137 × 10−8�𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚

+ 0.0013281 �𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �2 − 2.1102 × 10−8𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 �𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 − 7.0696

× 10−6𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 0.078225 𝑉𝑉𝑠𝑠𝑎𝑎𝑙𝑙𝑝𝑝 + 1.5873 × 10−10 �𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �3 + 2.0081

× 10−7�𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 �2 − 0.0026787 𝑁𝑁𝑟𝑟𝑝𝑝𝑚𝑚 + 46.1231�

86

𝐴𝐴𝐵𝐵𝑅𝑅� = 1 + 0.25𝑟𝑟��𝑟𝑟�2

� 𝑀𝑀��𝑀𝑀��

�. (A.46)

The engine-out gases needed for the three-way catalytic converter model are 𝑁𝑁2,𝐵𝐵𝑁𝑁,𝐻𝐻2,𝑁𝑁𝑁𝑁 and 𝐻𝐻𝐵𝐵. Their concentration are given by

𝐵𝐵�,� = 𝑒𝑒�𝑀𝑀��

1 + 𝜆𝜆𝐴𝐴𝐵𝐵𝑅𝑅�, (A.47)

where 𝐵𝐵�,� is the exhaust concentration of the species X in [𝑚𝑚𝑠𝑠𝑙𝑙/𝑚𝑚𝑠𝑠𝑙𝑙]. Then the engine-out emissions of interest for the TWC can be represented by

𝐸𝐸�� = �𝐵𝐵�,�2,𝐵𝐵�,��,𝐵𝐵�,�2

,𝐵𝐵�,��,𝐵𝐵�,��⊤, (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 𝑎𝑎��1 0.0179 Nusselt paramter inner convection 𝑏𝑏��1 0.95 Nusselt paramter inner convection 𝑎𝑎��2 0 Nusselt paramter outer convection 𝐷𝐷�� 0.060 Inner diameter at outlet [𝑚𝑚] 𝐴𝐴��1 0.181 Inner surface area [𝑚𝑚2] 𝐴𝐴��2 0.199 Outer surface area [𝑚𝑚2]

𝑚𝑚��𝑐𝑐�� 4140 Exhasut manifold thermal mass [𝐽𝐽/𝐾𝐾]

Table A.9: Parameters for the exhaust manifold model


𝑇𝑇��,� Input Exhaust manifold inlet temperature [𝐾𝐾] �̇�𝑚�� Input Total mass flow rate of the mixture [𝑘𝑘𝑔𝑔/𝑠𝑠] 𝑇𝑇�� State Lumped exhaust manifold temperature [𝐾𝐾] 𝑇𝑇��,� Output Exhaust manifold outlet temperature [𝐾𝐾]

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

�̇�𝑚��𝛥𝛥ℎ��,� = ℎ��,��𝐴𝐴��1�𝑇𝑇�� − 𝑇𝑇��,��, (A.49)

where

87

𝑇𝑇��,� ≈𝑇𝑇��,� + 𝑇𝑇��,�

2, (A.50)

and

𝛥𝛥ℎ��,� = 28.1(𝑇𝑇��,� − 𝑇𝑇��,�) + 1.9670 × 10−3�𝑇𝑇��,�

2 − 𝑇𝑇��,� 2 �

2+

4.8020 × 10−6�𝑇𝑇��,�

3 − 𝑇𝑇��,� 3 �

3−

1.9 × 10−9�𝑇𝑇��,� 4 − 𝑇𝑇��,�

4 �4

. (A.51)

In order to compute ℎ��,��, the Nusselt correlation is firstly approximated by

𝑁𝑁𝑢𝑢��,��= 𝑎𝑎��1�𝑅𝑅𝑒𝑒��,��

��1, (A.52)

where

𝑅𝑅𝑒𝑒��,��=

4�̇�𝑚��

𝜋𝜋𝐷𝐷��𝜇𝜇��,�𝛼𝛼�. (A.53)

Finally

ℎ��,�� =𝑘𝑘��,�

𝐷𝐷��1𝑁𝑁𝑢𝑢��,��

. (A.54)

The conductivity and dynamic viscosity are calculated by,

𝑘𝑘��,� = 3.4288 × 10−11�𝑇𝑇��,��3 − 9.1803 × 10−8�𝑇𝑇��,��

2 +1.29410−4�𝑇𝑇��,�� − 5.2076 × 10−3, (A.55)

𝜇𝜇��,� = 1.066 × 10−14�𝑇𝑇��,��3 − 3.6432 × 10−11�𝑇𝑇��,��

2

+ 6.6706 × 10−8�𝑇𝑇��,�� + 1.433 × 10−6, (A.56)

The energy balance in the solid phase of the exhaust manifold body reduces to

𝑚𝑚��𝑐𝑐��𝑑𝑑𝑇𝑇��𝑑𝑑𝑑𝑑

= ℎ��,��𝐴𝐴��1�𝑇𝑇��,� − 𝑇𝑇��. (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

�̇�𝑚��𝛥𝛥ℎ��,� = ℎ��,��𝐴𝐴��1�𝑇𝑇�� − 𝑇𝑇��,��, (A.58)

Where

𝑇𝑇��,� ≈𝑇𝑇��,� + 𝑇𝑇��,�

2 (A.59)

88

Name Value Description 𝑎𝑎��1 0.0641 Nusselt paramter inner convection 𝑏𝑏��1 0.789 Nusselt paramter inner convection 𝑎𝑎��2 529 Nusselt paramter outer convection 𝐷𝐷��1 0.060 Inner diameter [𝑚𝑚] 𝐷𝐷��2 0.064 Outer diameter [𝑚𝑚] 𝐴𝐴��1 0.0826 Inner surface area [𝑚𝑚2] 𝐴𝐴��2 0.0882 Outer surface area [𝑚𝑚2]

Table A.11: Parameters for the exhaust manifold model


𝑇𝑇��,� Input Connecting pipe inlet temperature [𝐾𝐾] �̇�𝑚�� Input Total mass flow rate of the mixture [𝑘𝑘𝑔𝑔/𝑠𝑠] 𝑇𝑇�� State Lumped pipe temperature [𝐾𝐾] 𝑇𝑇��,� Output Connecting pipe outlet temperature [𝐾𝐾]

Table A.12: Inputs, states and outputs for the connecting pipe

and

𝛥𝛥ℎ��,� = 28.1(𝑇𝑇��,� − 𝑇𝑇��,�) +1.9670 × 10−3�𝑇𝑇��,�

2 − 𝑇𝑇��,� 2 �

2+

4.8020 × 10−6�𝑇𝑇��,�

3 − 𝑇𝑇cp,� 3 �

3−

1.9 × 10−9�𝑇𝑇��,� 4 − 𝑇𝑇��,�

4 �4

. (A.60)

The convective heat transfer coefficient between the gas and the internal wall of the pipe is calculated by

ℎ��,�� =𝑘𝑘��,�1

𝐷𝐷��1𝑁𝑁𝑢𝑢��,��1

, (A.61)

where, the Nusselt’s number is approximated by

𝑁𝑁𝑢𝑢��,��1= 𝑎𝑎��1 �𝑅𝑅𝑒𝑒��,��1

��1

, (A.62)

and the Reynold’s number by

𝑅𝑅𝑒𝑒��,��1=

4�̇�𝑚��

𝜋𝜋𝐷𝐷��1𝜇𝜇��,�1𝛼𝛼�. (A.63)

The fluid’s properties are evaluated at the mean temperature 𝑇𝑇��,�,

𝜇𝜇��,�1 = 1.066 × 10−14𝑇𝑇��,� 3 − 3.6432 × 10−11𝑇𝑇��,�

2 + 6.6706 × 10−8𝑇𝑇��,� + 1.433 × 10−6,

(A.64)

89

𝑘𝑘��,�1 = 3.4288 × 10−11𝑇𝑇��,� 3 − 9.1803 × 10−8𝑇𝑇��,�

2 + 1.29410−4𝑇𝑇��,� − 5.2076 × 10−3,

(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 Nusselt’s Number by

𝑁𝑁𝑢𝑢��,��2 = 𝑎𝑎��2. (A.67)

The thermal conductivity is evaluated at the ambient temperature,

𝑘𝑘��,�2 = 3.4288 × 10−11𝑇𝑇�� 3 − 9.1803 × 10−8𝑇𝑇��

2 + 1.29410−4𝑇𝑇�� − 5.2076 × 10−3.

(A.68)

The energy balance in the solid phase of the connecting pipe body reduces to

𝑚𝑚��𝑐𝑐��𝑑𝑑𝑇𝑇��

𝑑𝑑𝑑𝑑= ℎ��,��𝐴𝐴��1�𝑇𝑇��,� − 𝑇𝑇�� + ℎ��,��𝐴𝐴��2�𝑇𝑇�� − 𝑇𝑇�� (A.69)

ℎ��,�� =𝑘𝑘��,�2

𝐷𝐷��2𝑁𝑁𝑢𝑢��,��2

, (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:

𝜖𝜖𝑝𝑝��𝑐𝑐�

𝑅𝑅𝑇𝑇�

𝜕𝜕𝑇𝑇�

𝜕𝜕𝑑𝑑= −

�̇�𝑚��𝑐𝑐�

𝐴𝐴�

𝜕𝜕𝑇𝑇�

𝜕𝜕𝑥𝑥+ 𝑆𝑆ℎ��𝑇𝑇� − 𝑇𝑇�� (B.1)

𝜌𝜌�𝑐𝑐�(1 − 𝜖𝜖) 𝜕𝜕𝑇𝑇�𝜕𝜕𝑑𝑑

= 𝑆𝑆ℎ��𝑇𝑇� − 𝑇𝑇�� + 𝑘𝑘�(1 − 𝜖𝜖) 𝜕𝜕2𝑇𝑇�𝜕𝜕𝑥𝑥2 − 𝑙𝑙�𝛼𝛼� �𝑅𝑅�,�𝛥𝛥ℎ�,�

��

�=1 (B.2)

𝜖𝜖𝜌𝜌�𝜕𝜕𝐵𝐵�,�

𝜕𝜕𝑑𝑑= −

�̇�𝑚��

𝐴𝐴�

𝜕𝜕𝐵𝐵�,�

𝜕𝜕𝑥𝑥+ 𝑆𝑆𝜌𝜌�ℎ�,��,��𝐵𝐵�,� − 𝐵𝐵�,�� (B.3)

𝑆𝑆𝜌𝜌�𝜕𝜕𝐵𝐵�,�

𝜕𝜕𝑥𝑥= 𝑆𝑆

𝑙𝑙�𝜌𝜌�ℎ�,��,��𝐵𝐵�,� − 𝐵𝐵�,�� − 𝛼𝛼�𝑀𝑀�𝑅𝑅�,� (B.4)

Name Value Description 𝐴𝐴� 0.0119 Cross-sectional area of the substrate [𝑚𝑚2] 𝑐𝑐� 1500 Specific heat capacity of the substrate [𝐽𝐽/(𝑘𝑘𝑔𝑔𝐾𝐾)]

𝐷𝐷� 0.001105 Hydraulic diameter of a channel [𝑚𝑚]

𝑘𝑘� 3.0 Thermal conductivity of the substrate [W/(mK)]

𝐿𝐿 0.1435 Reactor length [𝑚𝑚]

𝑆𝑆 2740 Geometric surface area per unit reactor volume [m2/m3]

𝜖𝜖 0.757 Reactor void fraction [−] 𝜌𝜌� 2240 Density of the substrate [𝑘𝑘𝑔𝑔/𝑚𝑚3]

𝑙𝑙� 1.98 × 10−5 Washcoat thickness [m]

𝛼𝛼� 273 Catalytic surface area per unit reactor volume [m2/m3]

𝛹𝛹 600 Oxygen storage capacity [𝑚𝑚𝑠𝑠𝑙𝑙/𝑚𝑚3]

𝑆𝑆ℎ 3.66 Sherwood number [−]

𝑁𝑁𝑢𝑢 3.66 Nusselt number [−]

𝛼𝛼� 10 Number of reactions in the TWC

Table B.1: Parameters for the catalyst model

92


𝑇𝑇�,�� Input TWC feed gas temperature [𝐾𝐾] 𝐸𝐸�� Input Engine-out emissions [𝑚𝑚𝑠𝑠𝑙𝑙/𝑚𝑚𝑠𝑠𝑙𝑙],𝐸𝐸�� ∈ ℝ5 �̇�𝑚�� Input Total mass flow rate of the mixture [𝑘𝑘𝑔𝑔/𝑠𝑠] 𝑉𝑉�� Input Vehicle speed [𝑘𝑘𝑚𝑚/ℎ] 𝑚𝑚�� Output NOx mass flow rate in [𝑚𝑚𝑔𝑔/𝑘𝑘𝑚𝑚] 𝑚𝑚�� Output CO mass flow rate in [𝑚𝑚𝑔𝑔/𝑘𝑘𝑚𝑚] 𝑚𝑚�� Output HC mass flow rate in [𝑚𝑚𝑔𝑔/𝑘𝑘𝑚𝑚]

Table B.2: Inputs and output of the TWC model

𝑖𝑖 Pre − exponential factors [𝑚𝑚𝑠𝑠𝑙𝑙 𝐾𝐾/𝑚𝑚3𝑠𝑠] 𝐴𝐴ctivation energy [𝐽𝐽 ]

For the rate of reactions*

1 𝐴𝐴�,1 = 1.064380 × 1020 𝐸𝐸�,1 = 90000

2 𝐴𝐴�,2 = 1.064380 × 1020 𝐸𝐸�,2 = 90000

3 𝐴𝐴�,3 = 5.687341 × 1018 𝐸𝐸�,3 = 95000

4 𝐴𝐴�,4 = 1.625393 × 1017 𝐸𝐸�,4 = 120000

5 𝐴𝐴�,5 = 5.610389 × 1012 𝐸𝐸�,5 = 90000

6 𝐴𝐴�,6 = 1.819701 × 1013 𝐸𝐸�,6 = 90000

7 𝐴𝐴�,7 = 5.175123 × 1013 𝐸𝐸�,7 = 90000

8 𝐴𝐴�,8 = 2.610655 × 1011 𝐸𝐸�,8 = 85000

9 𝐴𝐴�,9 = 2.972991 × 1011 𝐸𝐸�,9 = 85000

10 𝐴𝐴�,10 = 2.009992 × 1013 𝐸𝐸�,10 = 85000

For the reaction inhibition term**

1 𝐴𝐴1 = 65.5 𝐸𝐸1 = −7990

2 𝐴𝐴2 = 2080 𝐸𝐸2 = −3000

3 𝐴𝐴3 = 3.98 𝐸𝐸3 = −96534

4 𝐴𝐴4 = 4.79 × 105 𝐸𝐸4 = 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 (“𝑥𝑥-axes”). Note that, equation (B.1) indicates that the gas temperature 𝑇𝑇� 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 𝑇𝑇� 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 𝑥𝑥 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 𝐵𝐵�,� changes along the 𝑥𝑥 direction as a result of the convective mass transfer that takes place within the washcoat. In the solid phase the species concentration 𝐵𝐵�,� 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 𝑅𝑅�,�.

B.1.1 Consumption rates

Consider a vector 𝑞𝑞 ∈ ℝ11 containing the especies involved in the reaction mechanism shown in Table B.4 and its stoichiometric matrix 𝜈𝜈 ∈ ℝ10×11,

𝑞𝑞 = �𝐵𝐵𝑁𝑁, 𝑁𝑁2,𝐵𝐵𝑁𝑁2,𝐻𝐻2,𝐻𝐻2𝑁𝑁,𝐵𝐵𝐻𝐻1.8��, 𝐵𝐵𝐻𝐻1.8

��,𝑁𝑁𝑁𝑁, 𝑁𝑁2,𝐵𝐵𝑒𝑒2𝑁𝑁3,𝐵𝐵𝑒𝑒𝑁𝑁2�⊤, (B.5)

𝜈𝜈 =

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛

−1 −0.5 1 0 0 0 0 0 0 0 00 −0.5 0 −1 1 0 0 0 0 0 00 −1.45 1 0 0.9 −1 0 0 0 0 00 −1.45 1 0 0.9 0 −1 0 0 0 0

−2 0 2 0 0 0 0 −2 1 0 00 −0.5 0 0 0 0 0 0 0 −1 20 0 0 0 0 0 0 −1 0.5 −1 2

−1 0 1 0 0 0 0 0 0 1 −20 0 1 0 0.9 −1 0 0 0 1.9 −3.80 0 1 0 0.9 0 −1 0 0 1.9 −3.8

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞

. (B.6)

This matrix satisfies the following mass conserversation equation,

𝜈𝜈𝑞𝑞 = 0, (B.7)

94

𝑖𝑖 Reaction Reaction rate [𝑚𝑚𝑠𝑠𝑙𝑙/𝑠𝑠 𝑚𝑚3]

Oxidation reactions

1 𝐵𝐵𝑁𝑁 + 0.5𝑁𝑁2 → 𝐵𝐵𝑁𝑁2 𝑅𝑅�,1 = 𝐺𝐺−1𝐴𝐴�,1𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,1/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��𝐵𝐵�,�2

2 𝐻𝐻2 + 0.5𝑁𝑁2 → 𝐻𝐻2𝑁𝑁 𝑅𝑅�,2 = 𝐺𝐺−1𝐴𝐴�,2𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,2/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,�2𝐵𝐵�,�2

3 𝐵𝐵𝐻𝐻1.8�� + 1.45𝑁𝑁2 → 𝐵𝐵𝑁𝑁2 + 0.9𝐻𝐻2𝑁𝑁 𝑅𝑅�,3 = 𝐺𝐺−1𝐴𝐴�,3𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,3/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��

𝐵𝐵�,�2

4 𝐵𝐵𝐻𝐻1.8�� + 1.45𝑁𝑁2 → 𝐵𝐵𝑁𝑁2 + 0.9𝐻𝐻2𝑁𝑁 𝑅𝑅�,4 = 𝐺𝐺−1𝐴𝐴�,4𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,4/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��

𝐵𝐵�,�2

𝐍𝐍𝐍𝐍 reduction reaction

5 2𝑁𝑁𝑁𝑁 + 2𝐵𝐵𝑁𝑁 → 2𝐵𝐵𝑁𝑁2 + 𝑁𝑁2 𝑅𝑅�,5 = 𝐴𝐴�,5𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,5/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��𝐵𝐵�,��

Oxygen storage reactions

6 𝐵𝐵𝑒𝑒2𝑁𝑁3 + 0.5𝑁𝑁2 → 2𝐵𝐵𝑒𝑒𝑁𝑁2 𝑅𝑅�,6 = 𝐴𝐴�,6𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,6/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,�2(1 − 𝜓𝜓)

7 𝐵𝐵𝑒𝑒2𝑁𝑁3 + 𝑁𝑁𝑁𝑁 → 2𝐵𝐵𝑒𝑒𝑁𝑁2 + 0.5𝑁𝑁2 𝑅𝑅�,7 = 𝐴𝐴�,7𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,7/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��(1 − 𝜓𝜓)

8 2𝐵𝐵𝑒𝑒𝑁𝑁2 + 𝐵𝐵𝑁𝑁 → 𝐵𝐵𝑒𝑒2𝑁𝑁3 + 𝐵𝐵𝑁𝑁2 𝑅𝑅�,8 = 𝐴𝐴�,8𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,8/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��𝜓𝜓

9 𝐵𝐵𝐻𝐻1.8�� + 3.8𝐵𝐵𝑒𝑒𝑁𝑁2 → 1.9𝐵𝐵𝑒𝑒2𝑁𝑁3 + 𝐵𝐵𝑁𝑁2 + 0.9𝐻𝐻2𝑁𝑁 𝑅𝑅�,9 = 𝐴𝐴�,9𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,9/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��𝜓𝜓

10 𝐵𝐵𝐻𝐻1.8�� + 3.8𝐵𝐵𝑒𝑒𝑁𝑁2 → 1.9𝐵𝐵𝑒𝑒2𝑁𝑁3 + 𝐵𝐵𝑁𝑁2 + 0.9𝐻𝐻2𝑁𝑁 𝑅𝑅�,10 = 𝐴𝐴�,10𝑒𝑒𝑥𝑥𝑝𝑝(−𝐸𝐸�,10/𝑅𝑅�𝑇𝑇�)𝐵𝐵�,��𝜓𝜓

Table B.4: Reaction mechanism in the three-way catalytic converter [59].

95

The production rate 𝑅𝑅� is calculated according to

𝑅𝑅� = 𝜈𝜈⊤𝑅𝑅, (B.8)

where 𝑅𝑅� = �𝑅𝑅�,1, 𝑅𝑅�,2, ⋯ , 𝑅𝑅�,��⊤. The consumption rate 𝑅𝑅�,� for each species 𝑖𝑖

in the vector 𝑞𝑞 is simply the negative of 𝑅𝑅�,�. Therefore, the consumption rates for the gases of interest are calculated as follow:

𝑅𝑅�,�2 = 0.5𝑅𝑅�,1 + 0.5𝑅𝑅�,2 + 1.45𝑅𝑅�,3 + 1.45𝑅𝑅�,4 + 0.5𝑅𝑅�,6, (B.9)

𝑅𝑅�,�� = 𝑅𝑅𝑟𝑟,1 + 2𝑅𝑅𝑟𝑟,5 + 𝑅𝑅𝑟𝑟,8, (B.10)

𝑅𝑅�,�2 = 𝑅𝑅𝑟𝑟,2, (B.11)

𝑅𝑅�,�� = 2𝑅𝑅𝑟𝑟,5 + 𝑅𝑅𝑟𝑟,7, (B.12)

𝑅𝑅�,��1.8�� = 𝑅𝑅𝑟𝑟,3 + 𝑅𝑅𝑟𝑟,9, (B.13)

𝑅𝑅�,��1.8�� = 𝑅𝑅𝑟𝑟,4 + 𝑅𝑅𝑟𝑟,10. (B.14)

Table B.4 contain an inhibition term 𝐺𝐺, which is calculated according to:

𝐺𝐺 = 𝑇𝑇��1 + 𝐾𝐾1𝐵𝐵�,�� + 𝐾𝐾2𝐵𝐵�,��2�1 + 𝐾𝐾3𝐵𝐵�,�� 2 𝐵𝐵�,��

2 ��1 + 𝐾𝐾4𝐵𝐵�,�� 0.7 �, (B.15)

where

𝐵𝐵�,�� = 𝐵𝐵�,��+ 𝐵𝐵�,��

, (B.16)

and

𝐾𝐾� = 𝐴𝐴�𝑒𝑒− ��

��, (B.17)

where the parameters 𝐾𝐾� and 𝐸𝐸� 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 𝐵𝐵𝑒𝑒𝑁𝑁2 is considered to be in the oxygen enriched state, while 𝐵𝐵𝑒𝑒2𝑁𝑁3 is in the oxygen depleted state. The extent of oxygen stored 𝜓𝜓 is therefore defined as the instantaneous proportion of 𝐵𝐵𝑒𝑒𝑁𝑁2 in the total ceria:

𝜓𝜓 =𝛼𝛼��2

𝛼𝛼��2+ 2𝛼𝛼��2�3

, (B.18)

The oxygen storage level is governed by the following differential equation:

𝜓𝜓 ̇ = 1𝛹𝛹

�2𝑅𝑅�,6 + 2𝑅𝑅�,7 − 2𝑅𝑅�,8 − 3.8𝑅𝑅�,9 − 3.8𝑅𝑅�,10�, (B.19)

96

B.2 Enthalpy of reaction

In a reacting system, the molar enthalpy of the compound 𝑖𝑖 in [𝐽𝐽/𝑚𝑚𝑠𝑠𝑙𝑙] is given by

ℎ̅� = Δℎ̅�,�� + Δℎ̅�, (B.20)

where Δℎ̅�,�� in the molar enthalpy of formation of the species 𝑖𝑖, which are

specified in Table B.6. By assuming constant pressure condition inside the TWC, the term Δℎ̅� can be computed from,

Δℎ̅�(𝑇𝑇) = ℎ̅� − ℎ̅��,� = � 𝑐𝑐�̅,�𝑑𝑑𝑇𝑇�

��

, (B.21)

where 𝑇𝑇�� is the reference temperature taken as 298 𝐾𝐾. 𝑇𝑇 is approximated as the substrate temperature 𝑇𝑇�. Since the compounds 𝐵𝐵𝑒𝑒𝑁𝑁2 and 𝐵𝐵𝑒𝑒2𝑁𝑁3 are in solid state inside the TWC, theirs specific heat are constant values given by 61.6 and 114.6 [𝐽𝐽/𝑚𝑚𝑠𝑠𝑙𝑙𝐾𝐾] respectively. On the other hand, the specific heat of the reacting gases 𝑐𝑐�̅,�, is approximated by modelling the species as ideal gases and using the following polynomials,

𝑐𝑐�̅,� = 𝑎𝑎� + 𝑏𝑏�𝑇𝑇� + 𝑐𝑐�𝑇𝑇� 2 + 𝑑𝑑�𝑇𝑇�

3, (B.22)

where 𝑐𝑐�̅,� is given in [𝐽𝐽/𝑚𝑚𝑠𝑠𝑙𝑙𝐾𝐾]. The coefficients (𝑎𝑎�, 𝑏𝑏�, 𝑐𝑐�, 𝑑𝑑�) are specified in Table B.5. After computing the molar enthalpy for all the species 𝑖𝑖, it is possible to construct the following vector:

ℎ̅(𝑇𝑇�) =

�ℎ̅��, ℎ̅�2, ℎ̅��2

, ℎ̅�2, ℎ̅�2�, ℎ̅��

, ℎ̅��, ℎ̅��, ℎ̅�2

, ℎ̅��2�3, ℎ̅��2

�⊤

⏐⏐⏐⏐��

. (B.23)

The vector of the enthalpies of reaction is obtained from

𝛥𝛥ℎ� = 𝜈𝜈ℎ̅(𝑇𝑇�), (B.24)

where 𝛥𝛥ℎ� contains 10 enthalpies of reactions required by the last term in (B.2); that is:

Δℎ� = �Δℎ�,1,𝛥𝛥ℎ�,2, ⋯ ,𝛥𝛥ℎ�,10�⊤ (B.25)

97

𝑖𝑖 𝑎𝑎� 𝑏𝑏� 𝑐𝑐� 𝑑𝑑� 𝐵𝐵𝑁𝑁 2.8160 × 101 1.6750 × 10−3 5.3720 × 10−6 −2.2220 × 10−9

𝑁𝑁2 25.48 1.52 × 10−2 −7.1550 × 10−6 1.3120 × 10−9

𝐵𝐵𝑁𝑁2 2.2260 × 101 5.9810 × 10−2 −3.5010 × 10−5 7.4690 × 10−9

𝐻𝐻2 2.9110 × 101 −1.9160 × 10−3 4.0030 × 10−6 −8.7040 × 10−10

𝐻𝐻2𝑁𝑁 3.2240 × 101 1.9230 × 10−3 1.0550 × 10−5 −3.5950 × 10−9

𝐵𝐵𝐻𝐻�� 3.15 2.3830 × 10−1 −1.2180 × 10−4 2.4620 × 10−8

𝐵𝐵𝐻𝐻�� 1.9890 × 101 5.0240 × 10−2 1.2690 × 10−5 −1.1010 × 10−8

𝑁𝑁𝑁𝑁 2.9340 × 101 −9.3950 × 10−4 9.7470 × 10−6 −4.1870 × 10−9

𝑁𝑁2 2.8900 × 101 −1.5710 × 10−3 8.0810 × 10−6 −2.8730 × 10−9

Table B.5: Ideal-gas specific heat coefficients of various gases [60]

𝑖𝑖 Δℎ̅̅�,�� [𝐽𝐽/𝑚𝑚𝑠𝑠𝑙𝑙]

𝐵𝐵𝑁𝑁 −1.1050 × 105 𝑁𝑁2 0

CO2 −3.9350 × 105 𝐻𝐻2 0

𝐻𝐻2𝑁𝑁 −2.4180 × 105 𝐵𝐵𝐻𝐻fast 20000 𝐵𝐵𝐻𝐻slow −7.4600 × 104

𝑁𝑁𝑁𝑁 9.1300 × 104 𝑁𝑁2 0

𝐵𝐵𝑒𝑒2𝑁𝑁3 −17.96200 × 105 𝐵𝐵𝑒𝑒𝑁𝑁2 −10.88700 × 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

ℎ�� = ��

𝑁𝑁𝑢𝑢��, (B.26)

ℎ�,��,� =𝐷𝐷��2

𝐷𝐷�𝑆𝑆ℎ��,�, (B.27)

Nusselt (𝑁𝑁𝑢𝑢��) and Sherwood (ℎ�,��,�) numbers are calculated according to the correlations:

𝑁𝑁𝑢𝑢�� = 3.66�1 + 𝐷𝐷�𝐿𝐿

𝑃𝑃𝑒𝑒ℎ,��

0.45

, (B.28)

98

𝑆𝑆ℎ��,� = 3.66�1 + 𝐷𝐷�𝐿𝐿

𝑃𝑃𝑒𝑒�,��,��0.45

, (B.29)

Peclet (Pe) numbers are expressed in terms of the Prandtl (Pr), Schmidt (Sc) and Reynolds (Re) numbers as follows:

𝑃𝑃𝑒𝑒ℎ,��= 𝑅𝑅𝑒𝑒��

𝑃𝑃𝑟𝑟, (B.30)

𝑃𝑃𝑒𝑒�,��,� = 𝑅𝑅𝑒𝑒��𝑆𝑆𝑐𝑐�, (B.31)

where

𝑅𝑅𝑒𝑒�� =�̇�𝑚��𝐷𝐷�

𝜖𝜖𝐴𝐴�𝜇𝜇�, (B.32)

𝑃𝑃𝑟𝑟 =𝑐𝑐�𝜇𝜇�

𝑘𝑘�, (B.33)

𝑆𝑆𝑐𝑐� =𝜇𝜇�

𝜌𝜌�𝐷𝐷�,�2

. (B.34)

By treating the exhaust gas as a binary mixture of 𝑁𝑁2 and the compound 𝑖𝑖 under atmospheric pressure, the gas phase diffusion coefficients,𝐷𝐷�,�2

, are obtained from

𝐷𝐷�,�2

=10−7𝑇𝑇�

1.75�1/𝑀𝑀�� + 1/𝑀𝑀��2�

12

�𝑉𝑉� 13 + 𝑉𝑉�2

13 �2 , (B.35)

Where 𝑀𝑀�� is the molar mass of the compound 𝑖𝑖 in [𝑔𝑔/𝑚𝑚𝑠𝑠𝑙𝑙], 𝑇𝑇� in [𝐾𝐾] and the diffusion volumes 𝑉𝑉�, are taken from:

𝑖𝑖 𝑉𝑉�[-] 𝐵𝐵𝑁𝑁 18.9 𝑁𝑁2 16.6 𝐻𝐻2 7.07

𝐵𝐵3𝐻𝐻6 61.38 𝐵𝐵𝐻𝐻4 24.42 𝑁𝑁2 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 𝐵𝐵3𝐻𝐻6 and 𝐵𝐵𝐻𝐻4 respectively.

99

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. 𝑥𝑥 is referred to as the boundary-value variable and the number of the boundary conditions is determined by the higher-order derivative in 𝑥𝑥. Therefore in (B.1), 𝑇𝑇� is first order in 𝑥𝑥, thus the boundary condition is set as

𝑇𝑇�(0, 𝑑𝑑) = 𝑇𝑇�,��, (B.36)

where 𝑇𝑇�,�� is the feed gas temperature in 𝐾𝐾 of the TWC, which in turn is the outlet temperature of the connecting pipe, 𝑇𝑇��,�.

Equation (B.2) is second order in 𝑥𝑥, thus it requires two boundaries conditions. These are determined by assuming non-conducting boundary conditions in the front and the back of the monolith (TWC’s substrate)

𝜕𝜕𝑇𝑇�(0, 𝑑𝑑)𝜕𝜕𝑥𝑥

= 0 (B.37)

𝜕𝜕𝑇𝑇�(𝐿𝐿, 𝑑𝑑)𝜕𝜕𝑥𝑥

= 0. (B.38)

The equation (B.3), which describe the mass transfer of the compound 𝑖𝑖, require only one boundary condition for each chemical specie,

𝐵𝐵�,�2(0, 𝑑𝑑) = 𝐵𝐵�,�2,��, (B.39)

𝐵𝐵�,��(0, 𝑑𝑑) = 𝐵𝐵�,��,��, (B.40)

𝐵𝐵�,�2(0, 𝑑𝑑) = 𝐵𝐵�,�2,��, (B.41)

𝐵𝐵�,��(0, 𝑑𝑑) = 𝐵𝐵�,��,��, (B.42)

𝐵𝐵�,��(0, 𝑑𝑑) = 0.85𝐵𝐵�,��,��, (B.43)

𝐵𝐵�,��(0, 𝑑𝑑) = 0.15𝐵𝐵�,��,��. (B.44)

This concentration at the front of the TWC are those of the engine-out emission concatenated in the vector 𝐸𝐸�� (Section Engine-out emissions). The PDE problem is first order in 𝑑𝑑 and therefore it requires one initial condition (IC) for each equation in the model.

100

Figure B.1: Discretisation along the catalytic converter’s length, 𝑗𝑗 = 1,2,3, … ,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 Δ𝑥𝑥. In [56], four nodes (Δ𝑥𝑥 = 35.875 𝑚𝑚𝑚𝑚) 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 �̇�𝑚�� 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).

𝑇𝑇𝒈𝒈 (𝑗𝑗) • 1in • 2 • 3 • 4

𝑇𝑇𝒔𝒔 (𝑗𝑗) • 5 • 6 • 7 • 8

𝐵𝐵𝒈𝒈,𝑁𝑁2

(𝑗𝑗) • 9 in • 10 • 11 • 12

𝐵𝐵𝒈𝒈,𝐵𝐵𝑁𝑁 (𝑗𝑗) • 13 in • 14 • 15 • 16 out

𝐵𝐵𝒈𝒈,𝐻𝐻2

(𝑗𝑗) • 17 in • 18 • 19 • 20

𝐵𝐵𝒈𝒈,𝑁𝑁𝑁𝑁 (𝑗𝑗) • 21 in • 22 • 23 • 24 out

𝐵𝐵𝐻𝐻𝒈𝒈,𝑓𝑓𝑎𝑎𝑠𝑠𝑑𝑑 (𝑗𝑗) • 25 in • 26 • 27 • 28 out

𝐵𝐵𝐻𝐻𝒈𝒈,𝑠𝑠𝑙𝑙𝑠𝑠𝑤𝑤 (𝑗𝑗) • 29 in • 30 • 31 • 32 out

𝐵𝐵𝒔𝒔,𝑁𝑁2

(𝑗𝑗) • 33 • 34 • 35 • 36

𝐵𝐵𝒔𝒔,𝐵𝐵𝑁𝑁 (𝑗𝑗) • 37 • 38 • 39 • 40

𝐵𝐵𝒔𝒔,𝐻𝐻2

(𝑗𝑗) • 41 • 42 • 43 • 44

𝐵𝐵 𝒔𝒔,𝑁𝑁𝑁𝑁(𝑗𝑗) • 45 • 46 • 47 • 48

𝐵𝐵𝐻𝐻𝒔𝒔,𝑓𝑓𝑎𝑎𝑠𝑠𝑑𝑑 (𝑗𝑗) • 49 • 50 • 51 • 52

𝐵𝐵𝐻𝐻𝒔𝒔,𝑠𝑠𝑙𝑙𝑠𝑠𝑤𝑤 (𝑗𝑗) • 53 • 54 • 55 • 56

𝜓𝜓 (𝑗𝑗) • 57 • 58 • 59 • 60

Gas and substratetemperature

Species concentrationsin the gas

Species in the Substrate

Oxygen storage level

Nodes

�̇�𝑚��

101

After solving these differential equations, it is possible to calculate the mass flow rate of the legislated tail pipe emission according to

�̇�𝑚��,�� = �̇�𝑚��𝑀𝑀��𝑀𝑀�

𝐵𝐵�,��(�,�), (B.45)

�̇�𝑚��,�� = �̇�𝑚��𝑀𝑀��𝑀𝑀�

𝐵𝐵�,��(�,�), (B.46)

�̇�𝑚��,�� = �̇�𝑚�� 𝑀𝑀��

𝑀𝑀�𝐵𝐵�,��(�,�) +

𝑀𝑀��

𝑀𝑀�𝐵𝐵�,��(�,�)�, (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,

𝐵𝐵�,��(�,�) = 𝐵𝐵�,�� (16) (𝑑𝑑), (B.48)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,�� (24) (𝑑𝑑), (B.49)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,��

(28) (𝑑𝑑), (B.50)

𝐵𝐵�,��(�,�) = 𝐵𝐵�,��

(32) (𝑑𝑑). (B.51)

Finally, �̇�𝑚��,��, �̇�𝑚��,�� and �̇�𝑚��,�� are converted into [𝑚𝑚𝑔𝑔/𝑘𝑘𝑚𝑚], by taking into account the vehicle speed, 𝑉𝑉�� in [𝑘𝑘𝑚𝑚/ℎ],

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉�,��, (B.52)

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉�,��, (B.53)

𝑚𝑚�� = 3.6 × 109 �̇�𝑚��,��

𝑉𝑉�,��, (B.54)

where 𝑉𝑉�,�� = max(10, 𝑉𝑉��).

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

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http://hdl.handle.net/11343/118621

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Extremum Seeking for Spark Advance Calibration under Tailpipe Emissions Constraints

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