ISA Transactions 48 (2009) 273–282

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ISA Transactions

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Design of nonlinear PID controller and nonlinear model predictive controller fora continuous stirred tank reactorJ. Prakash ∗, K. SrinivasanDepartment of Instrumentation Engineering, Madras Institute of Technology, Anna University, Chennai-44, India

a r t i c l e i n f o

Article history:Received 7 September 2008Received in revised form4 February 2009Accepted 9 February 2009Available online 19 March 2009

Keywords:CSTRPID controller and model predictivecontroller

a b s t r a c t

In this paper, the authors have represented the nonlinear system as a family of local linear state spacemodels, local PID controllers have been designed on the basis of linear models, and the weighted sum ofthe output from the local PID controllers (Nonlinear PID controller) has been used to control the nonlinearprocess. Further, NonlinearModel Predictive Controller using the family of local linear state spacemodels(F-NMPC) has been developed. The effectiveness of the proposed control schemes has been demonstratedon a CSTR process, which exhibits dynamic nonlinearity.

Crown Copyright© 2009 Published by Elsevier Ltd on behalf of ISA. All rights reserved.

1. Introduction

PID controller and linear model predictive controller arethe two most popular control schemes that have been widelyimplemented throughout the chemical process industries for thepast two decades. However, control of nonlinear system usingabove linear control schemes don’t give satisfactory performanceat all operating points, the reason being that the processparameters of the nonlinear process will vary with the operatingconditions. Moreover, the PID controller tuned at one operatingcondition may not provide satisfactory servo and regulatoryperformances at shifted operating points. It should be noted that,to achieve improved closed loop performance a different set ofcontroller settings for each operating condition have to be used.In the case of model based control schemes, the accuracy of the

model will have a significant effect on the closed loop performanceof the control system. Themultiple-linearmodels concept has beenused in the recent years for modeling of nonlinear systems [1].In addition, multiple-linear model based approaches for controllerdesign [2–5] have attracted the process control community. Aplethora of multiple-model adaptive control schemes have beenproposed in the control literature [6–9]. Gao et al. [10] hasproposed a nonlinear PID controller for CSTR using local modelnetworks. Omar Galan et al. [11] have reported the real-timeimplementation of multi-linear model based control strategies onthe laboratory scale process.

∗ Corresponding author.E-mail address: prakaiit@rediffmail.com (J. Prakash).

0019-0578/$ – see front matter Crown Copyright© 2009 Published by Elsevier Ltd ondoi:10.1016/j.isatra.2009.02.001

A simple way to describe a nonlinear dynamic system usingmultiple linear models has been proposed by Takagi–Sugeno [12]and it is being used in this paper to develop Nonlinear PID con-troller and Nonlinear Model Predictive Controller. The proposedcontrol scheme consists of a family of controllers (Local Con-trollers) and a scheduler. As suggested by Kuipers and Astrom [13],either local PID controller outputs or the local PID controller pa-rameters can be interpolated. In the case of interpolation of con-troller parameters, the controllers’ structure have to be assumedas homogeneous, whereas interpolation of controllers output doesnot impose any such constraints. At each sampling instant, thescheduler will assign weights for each controller and the weightedsum of the outputs will be applied as an input to the plant in thecase of interpolation of local controller outputs.As suggested, one can also apply operating regime approaches

to develop an operating regime based model that can be appliedin a model-based controller [14,15]. Since global information canbe applied to determine the control input at each sampling instant,the nonlinear model based controller is expected to achieve bettercontrol performance. Recently, stability analysis of a multi-modelpredictive control algorithm with an application to the controlof chemical reactors has been reported by Leyla, Özkan andKothare, [16].The key unit operation in chemical plants namely the continu-

ous stirred tank reactor (CSTR) exhibits highly nonlinear dynamicbehavior. Hence, there arises a need to develop computationallynon-intensive control schemes in order to achieve tighter con-trol of strong nonlinear processes. A plethora of advanced con-trol schemes such as neural adaptive controller [17], nonlinearinternal model control scheme [18] and fuzzy model predictive

behalf of ISA. All rights reserved.

274 J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282

Nomenclature

CA Concentration (mol/l)T Temperature (K)qc Coolant flow rate (l/min)q Feed flow rate (l/min)CA0 Feed concentration (mol/l)T0 Feed temperature (K)Tc0 Inlet coolant temperature (K)V CSTR volume (l)hA Heat transfer term (cal/(min K))k0 Reaction rate constant (min−1)E/R Activation energy term (K)−1H Heat of reaction (cal/mol)ρ, ρc Liquid density (g/l)Cp, Cpc Specific heats (cal/(g K))x(k) True state variabley(k) Measured variablesu(k) Process inputsA State transition matrix (continuous domain)B Input matrix (continuous domain)C Measurement matrixKi Steady State gain of the ith process modelKc,i Proportional gain of ith PID controllerTr,i Integral time of ith PID controllerTd,i Derivative time of ith PID controllerNP Prediction horizonNc Control horizonWE Error weighting matrix (N-MPC)WU Controller weighting matrix (N-MPC)

Greek letter words

Φ State transition matrix (Discrete domain)Γ Input coupling matrix (Discrete domain)ξ Damping factorωn Un-damped natural frequencyλ Tuning parameter (IMC-PID controller)

control scheme [19] have been already attempted on the CSTR pro-cess which is considered for the simulation study in this paper.Even with the introduction of powerful nonlinear control strate-gies such as nonlinear internal model control schemes and neuraladaptive control, the proposed control schemes remain an attrac-tive control strategy, because it offer advantages such as simple de-sign and low computational complexity.The main contributions of the paper are as follows: firstly, the

nonlinear system is represented as a family of local linear statespace models. Secondly, local PID controllers have been designedon the basis of local linear models, the weighted sum of theoutput from local PID controllers has been used to control thenonlinear process, and finally a nonlinear model predictive controlscheme using the family of local linear state spacemodels has beenproposed to control nonlinear process.The organization of the paper is as follows. Section 2 discusses

the T–S fuzzymodel. Section 3 presents the design of nonlinear PIDcontroller. Section 4 deals with nonlinear model predictive controlschemes formulation using local linear models. Section 5 dealswith analytical (first principle) model based predictive controlformulation. The process considered for simulation study hasbeen discussed in Section 6. Simulation results are presented inSection 7 and the conclusions drawn from the simulation studiesin Section 8.

2. Takagi–Sugeno (T–S) fuzzy model

Consider a nonlinear system represented by the followingnonlinear differential equations:

x =−

f (x, u, d) (1)

y =−

g(x, u, d). (2)Eq. (1) describes a deterministic system evolution and can be

obtained from the material and energy balances of the processunder consideration. Eq. (2) describes the relationships betweenthe measurements and the state variables. In order to describe adiscrete nonlinear system, Eqs. (1) and (2) can also be functionallyrepresented in discrete form asx(k) = f [x(k− 1), u(k− 1), d(k− 1)] (3)y(k) = g [x(k− 1), u(k− 1)] (4)where, x(k) is the system state vector (x(k) ∈ Rn), u(k) is thesystem input/known deterministic input (u(k) ∈ Rm), d(k) theunmeasured disturbance/unknown input (d(k) ∈ Rq), and y(k) isthe measured variable (y(k) ∈ Rr ). The parameters k representsthe sampling instant and the symbol f and g represent an n-dimensional function vectors. We assume that measurements aremade at discrete sampling instants with sampling period T . Notethat the d(t) term described in Eq. (1) is assumed to be piecewiseconstant for kT ≤ t < (k+ 1)TA T–S fuzzy model has been proposed to represent a nonlinear

system using locally linearized models [12]. Two differentmethods for developing a T–S fuzzy model have been suggestedin the literature, namely (i) the black box identification viafuzzy clustering technique [20] and (ii) Linearization of anexisting nonlinear system around the centers of the fuzzy regionpartitioning the state space. The T–S fuzzy model is nothingbut a piecewise interpolation of local linear models throughmembership function. The T–S fuzzy model is described by IF-THEN rules, which represent local linear relations of the nonlinearsystem. The rule to describe the nonlinear system around anoperating point is as follows:Rule i (i = 1 : N)If z1(k) isMi,1 and . . . and zg(k) isMi,g then

xi(k) = Φi(x(k− 1)− xi)+ Γi(u(k− 1)− ui) (5)yi(k) = Cixi(k) (6)where, zj(k) are the premise variables andMij(k) are the fuzzy sets.Φi, Γi, and Ci are known time invariant matrices of appropriatedimensions. In this work it is assumed that such a model of theprocess can be developed from the first principles by linearizingthem around different operating steady state values (xi and ui). Theglobal system behavior is described by a fuzzy fusion of all linearmodel outputs. For a given input vector, u(k), the global state andoutput of fuzzy model are inferred as follows:

x(k) =N∑i=1

hi(z(k))[Φi(x(k− 1)− xi)

+Γi(u(k− 1)− ui)+ xi] (7)

y(k) = Cx(k) (8)where the membership grades hi(z(k)) are defined as

hi(z(k)) =µi(z(k))µ(k)

(9)

µi(z(k)) =g∏j=1

Mij (10)

µ(k) =N∑i=1

µi(z(k)). (11)

J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282 275

It should be noted that the grade of membership should be

hi(z(k)) ∈ [0, 1] andN∑i=1

hi(z(k)) = 1. (12)

3. Nonlinear PID controller (N-PID) design using local linearmodels

In this section, the design of local PID controllers on the basisof local linear models, which were described in the previoussection, is discussed. Further, the method to combine the localPID controller outputs yielding a global controller output has beenoutlined. The global controller output u(k) has been determined bythe following rules:Rule i (i = 1 : N)If z1(k) isMi,1 and . . . and zg(k) isM1,g then

ui(k) = Kc,i(e(k)− e(k− 1))+Kc,iTr,iT e(k)+

(Kc,i ∗ Td,i)T

× (e(k)− 2 ∗ e(k− 1)+ e(k− 2))+ ui(k− 1) (13)

where, T is the sampling time. It should be noted that PIDcontroller could be designed to satisfy the stability, performance,and robustness criteria for each local linear model. Kc,i, Tr,i andTd,i are the proportional gain, integral time and derivative timevalues of the ith PID controllers determined using standard optimalPID tuning methods. The global controller output is described by afusion of all linear PID controller outputs.

4. Nonlinear model predictive controller using local linearmodels (F-NMPC)

In the proposed N-MPC formulation, at every sampling instantthe fuzzy dynamic model (Refer Eqs. (7) and (8)) is used forpredicting the future behavior of the plant over a finite numberof future time steps, say Np which is called prediction horizon.A set of Nc future manipulated input moves {u(k/k), u(k +1/k) . . . u(k + Nc − 1/k)}(where Nc is called the control horizon)are determined by constrained optimization with the objective ofminimizing the predicted deviation of the process output fromthe target over the prediction horizon as well as minimizing theexpenditure of control effort in driving the process output to target,subject to pre-specified operating constraints. The proposed F-NMPC is implemented in a moving horizon framework, that is,only u(k/k) is implemented at each sampling instant and theoptimization is repeated at each sampling instant based on theupdated information from the plant.The fuzzy dynamic model developed in the Section 2 can

be used recursively to obtain multi-step prediction. Given asequence of future control moves {u(k/k) · · · u(k + 1/k) · · · u(k+ Nc − 1/k)}, a Np step ahead output prediction can be writtenas follows:

x(k+ j+ 1/k) =N∑i=1

hiz(k+ j)[Φi [(x(k+ j/k))− xi]

+Γu,i (u(k+ j/k)− ui)+ xi] for j = 0, . . . ,Nc − 1 (14)

x(k+ j+ 1/k) =N∑i=1

hiz(k+ j)[Φi [[x(k+ j/k)]− xi]

+Γu,i(u(k+ Nc − 1/k)− ui )+ xi]

for j = Nc, . . . ,NP − 1 (15)y(k+ j/k) = C x(k+ j/k); for j = 1, . . . ,NP . (16)

To account for plant model mismatch and unmeasureddisturbances, a simple unmeasured disturbance estimator similarto the dynamic matrix control scheme is incorporated as follows:

yc(k+ j/k) = y(k+ j/k)+ d(k+ j/k) (17)

where

d(k+ j/k) = d(k/k) = (ym(k)− y(k)) for j = 1, . . . ,Np. (18)

In the above Eq. (18), ym(k) represents the measured outputat the kth instant and y(k) represents the model output at thekth instant. Given a future setpoint trajectory yr(k + j/k), (j =1, . . . ,Np), the nonlinear model predictive controller designproblem can be formulated as:

u(k/k) min. . . u(k+ Nc − 1/k) J (19)

where,

J =Np∑j=1

[E(k+ j/k)]TWE [E(k+ j/k)]

+

Nc−1∑j=0

[1u(k+ j/k)]TWu [1u(k+ j/k)]

E(k+ j/k) = yr(k+ j/k)− yc(k+ j/k) (20)

1u(k+ j/k) = u(k+ j/k)− u(k+ j− 1/k). (21)

Subject to the following constraints

uL ≤ u(k+ j/k) ≤ uH for j = 0, . . . ,Nc − 1 (22)

yL ≤ yc(k+ j/k) ≤ yH for j = 1, . . . ,Np (23)1u(k+ Nc/k) = 1u(k+ Nc + 1/k)

= · · ·1u(k+ Np − 1/k) = 0. (24)

The resulting constrained optimization problem can be solvedusing any standard optimization technique.

5. Nonlinear model predictive controller using first principle(analytical) model–A-NMPC

The objective of the A-NMPC is to calculate a set of futurecontrolmoves (Control horizon) byminimization of a cost functionon a moving finite horizon (Prediction horizon). The optimizationproblem is solved on-line, based on the predictions obtainedfrom a nonlinear model. It is possible to use different empiricalnonlinear models for predictions in the controller, but the mostattractive approach is to use the first principle models [21]). In theanalytical model based N-MPC formulation, given a sequence offuture control moves {u(k/k) · · · u(k+ 1/k) · · · u(k+ Nc − 1/k)},a Np step ahead output prediction using the first principle modelcould be written as follows:

x(k+ j+ 1|k) = x(k+ j|k)

+

∫ (k+j+1)T

(k+j)TF[x(τ ),u(k+ j|k), d

]dτ ;

j = 0, 1, . . . .Np − 1 (25)

y(k+ j+ 1/k) = Cx(k+ j+ 1/k); for j = 0, 1, . . . ,NP − 1.(26)

To account for plant model mismatch and unmeasured distur-bances, a simple unmeasured disturbance estimator similar to thedynamic matrix control scheme is incorporated as follows:

yc(k+ j/k) = y(k+ j/k)+ d(k+ j/k) (27)

where

d(k+ j/k) = d(k/k) = (ym(k)− y(k)) for j = 1, . . . ,Np. (28)

276 J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282

Table 1Operating data for CSTR process.

Process variable Normal operating condition

Process flow rate (q) 100.0 l/minFeed concentration (CA0) 1 mol/lFeed temperature (T0) 350 KInlet coolant temperature (Tc0) 350 KCSTR volume (V ) 100 lHeat transfer term (hA) 7× 105 cal/(min K)Reaction rate constant (k0) 7.2× 1010 min−1

Activation energy term (E/R) 1× 104 KHeat of reaction (−1H) −2× 105 cal/molLiquid density (ρ, ρc ) 1× 103 g/lSpecific heats (Cp , Cpc ) 1 cal/(g K)

Given a future setpoint trajectory yr(k + j/k), (j = 1, . . . ,Np),the nonlinear model predictive controller design problem can beformulated as:

u(k/k) min. . . u(k+ Nc − 1/k)J (29)where,

J =Np∑j=1

[E(k+ j/k)]T WE [E(k+ j/k)]

+

Nc−1∑j=0

[1u(k+ j/k)]T Wu [1u(k+ j/k)]

E(k+ j/k) = yr(k+ j/k)− yc(k+ j/k) (30)

1u(k+ j/k) = u(k+ j/k)− u(k+ j− 1/k). (31)Subject to the following constraints

uL ≤ u(k+ j/k) ≤ uH for j = 0, . . . ,Nc − 1 (32)

yL ≤ yc(k+ j/k) ≤ yH for j = 1, . . . ,Np (33)1u(k+ Nc/k) = 1u(k+ Nc + 1/k)

= · · ·1u(k+ Np − 1/k) = 0. (34)The resulting constrained optimization problem can be solvedusing any standard optimization technique.

6. Continuous stirred tank reactor (CSTR)

The first principle model of the continuous stirred tank systemand the operating point data (Refer Table 1) as specified in thepaper titled Fuzzy Model Predictive Control by Huang et al. [19]have been used in this simulation study. In the process consideredfor simulation study, an irreversible, exothermic reaction A → Boccurs in constant volume reactor that is cooled by a single coolantstream. The process is modeled by the following equations:

dCA(t)dt=q(t)V

(CA0(t)− CA(t))− k0CA (t) exp(−ERT (t)

)(35)

dT (t)dt=q(t)V(T0(t)− T (t))−

(−1H)k0CA(t)ρCp

exp(−ERT (t)

)+ρcCpcρCpV

qc(t){1− exp

(−hAqc(t)ρCp

)}(Tc0(t)− T (t)) . (36)

The state x(t) and input u(t) vectors are given by x(t) = [CA; T ]and u(t) = [qc ].

7. Simulation studies

In all the simulation runs, the process is simulated using thenonlinear first principle model (Eqs. (35) and (36)). The truestate variables are computed by solving the nonlinear differentialequations using differential equation solver in Matlab 6.5.

7.1. Fuzzy dynamic model for the CSTR process

For fuzzy dynamic model design for the CSTR process, thecoolant flow rate (qc) has been chosen as the premise variableand triangular membership functions have been used to partitionthe input space qc . The universe of discourse is divided into fiveintervals which are defined by the linguistic variables, very low[97 100], low [97 100 103], medium [100 103 106], high [103 106109] and very high [106 109] respectively. Further, local modelparameters (consequent part of T–S fuzzy model) are determinedby linearizing the nonlinear differential equations ((35) and (36))around the centers of the fuzzy region partitioning the operatingspace of the system. The detailed design procedure of the fuzzydynamic model for the CSTR process has been reported in Senthilet al. [22]. The linear time invariant discrete state space models(Refer Eqs. (5) and (6)) for five different operating points of CSTRare:

Operating point: 1 (qc = 97CA = 0.0795 and T = 443.4566)

Φ1 =

[1.2040e−001 −3.1008e−0031.5350e+002 1.4438

]Γ1 =

[1.2927e−004−9.6293e−002

].

Operating point: 2 (qc = 100, CA = 0.0885 and T = 441.1475)

Φ2 =

[1.7133e−001 −3.2672e−0031.4362e+002 1.4733

]Γ2 =

[1.3035e−004−9.4559e−002

].

Operating point: 3 (qc = 103, CA = 0.0989 and T = 438.7763)

Φ3 =

[2.2479e−001 −3.4252e−0031.3333e+002 1.5012

]Γ3 =

[1.3074e−004−9.2643e−002

].

Operating point: 4 (qc = 106, CA = 0.1110 and T = 436.3091)

Φ4 =

[2.8071e−001 −3.5731e−0031.2254e+002 1.5270

]Γ4 =

[1.3038e−004−9.0506e−002

].

Operating point: 5 (qc = 109, CA = 0.1254 and T = 433.6921)

Φ5 =

[3.3941e−001 −3.7084e−0031.1123e+002 1.5504

]Γ5 =

[1.2913e−004−8.8085e−002

].

C =[1 00 1

].

For the CSTR process considered for the simulation study Senthilet al. [13] has shown that the linear dynamic model is not ableto capture the dynamic behavior of the CSTR process, whereas thefuzzy dynamic model is able to capture the dynamic nonlinearityadequately.

7.2. Nonlinear PID controller (N-PID) design for CSTR process

In this work, we have intended to interpolate five PID controlleroutputs. That is, for each local linear model described in theprevious subsection, a PID controller has been designed. In orderto prevent a sharp spike in the controller output, at the timeof step change in the setpoint, the derivative of the measured

J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282 277

Fig. 1. Servo response of CSTR with F-NMPC, N-PID, and A-NMPC (a) Process output (b) Controller output.

Fig. 2. Histogram of computation time per sampling instant.

output has been used in the control law instead of a derivativeof the error. That is, PV derivative type PID controller form hasbeen implemented in this work (P and I on setpoint error andD on Process Variable). Further, the tuning parameters of eachPID controller have been determined using the IMC [23] tuningrules proposed by Morari and Zafiriou, [24]. The process transferfunction relating the reactor concentration to the coolant flow rateat all operating points has been found to be of the form:

Gi(s) =Ki

s2 + 2ξiωn,is+ ω2n,i∀ i = 1 : 5.

Table 2 provides the values of the second-order transferfunction model parameters such as process gain, damping factorand un-damped natural frequency at different operating points.The IMC based PID tuning procedure will yield the followingcontroller parameters:

Kc,i =2ξi

ωn,iKiλ; Tr,i =

2ξiωn,i; Td,i =

12ξiωn,i

.

It should be noted that using the model parameters reported inTable 2, we have obtained the controller parameters of each localPID controller. The PID controllers’ parameters at five different

278 J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282

Table 2Damping factor and un-damped natural frequency at different operating points.

Operating point Damping factor Freq. (rad/s) Process gain

At qc = 97, CA = 0.0795, T = 443.4566 0.661 3.93 0.0028At qc = 100, CA = 0.0885, T = 441.1475 0.540 3.64 0.0032At qc = 103, CA = 0.0989, T = 438.7763 0.416 3.34 0.0037At qc = 106, CA = 0.1110, T = 436.3091 0.285 3.03 0.0043At qc = 109, CA = 0.1254, T = 433.6921 0.141 2.71 0.0052

Table 3PID controllers’ parameters at different operating points.

Operating point Kc,i Tr,i Td,i

At qc = 97, CA = 0.0795, T = 443.4566 119.4321λ

0.3367 0.1926

At qc = 100, CA = 0.0885, T = 441.1475 92.6928λ

0.2973 0.2546

At qc = 103, CA = 0.0989, T = 438.7763 67.4294λ

0.2491 0.3601

At qc = 106, CA = 0.1110, T = 436.3091 43.2812λ

0.1876 0.5792

At qc = 109, CA = 0.1254, T = 433.6921 19.1813λ

0.1037 1.3124

Table 4ISE values of F-NMPC, N-PID, and A-NMPC for setpoint tracking.

Sampling instants interval A-NMPC F-NMPC N-PID

10 ≤ k ≤ 49 2.52e–05 3.68e–05 5.07e–0550 ≤ k ≤ 79 8.84e–05 9.42e–05 1.47e–0480 ≤ k ≤ 120 1.55e–05 2.67e–05 3.10e–05

Table 5ISE values of A-NMPC, F-NMPC and N-PID in the presence of setpoint change andload change.

Sampling instants interval A-NMPC F-NMPC N-PID

10 ≤ k ≤ 69 1.16e–05 1.18e–05 1.66e–0570 ≤ k ≤ 119 1.17e–04 1.43e–04 2.03e–04120 ≤ k ≤ 175 5.51-06 2.77e–06 1.11e–05

Fig. 3. Variation in feed temperature of CSTR.

operating points have been reported in Table 3. It should be notedthat the controller gain has been found to be the function of thefilter time constant lamda (λ).

7.3. Nonlinear model predictive controller for CSTR process

A simple model predictive control for CSTR has been developedusing the local linear models (Refer Section 7.1). F-NMPC and A-

NMPC schemes for CSTR have been developed with the samplingtime of 0.083 min, prediction horizon of NP = 5, and controlhorizon of Nc = 1.The error weighting matrix and the controllerweighting matrix used in the N-MPC formulation are WE =1e4 and WU = 0. The following constraints on the manipulatedinput (coolant flow rate) are imposed 95 < qc < 108.

7.4. Servo performance

The setpoint variations as shown in Fig. 1(a) have beenintroduced for assessing the tracking capability of the proposed (i)F-NMPC formulation using the local linear models, (ii) proposedN-PID control scheme using multiple-linear PID controllers and(iii) analytical/first principle model based NMPC(A-NMPC). Fromthe response, it can be inferred that, the F-NMPC formulationbased on local linearmodels, N-PID control schemeusingmultiple-linear PID controllers and A-NMPC scheme are able to maintainthe reactor concentration at the setpoint. The ISE values of F-NMPC, N-PID and A-NMPC are reported in Table 4. From Table 4,it can be inferred that the ISE values of A-NMPC and F-NMPC havebeen found to be considerably less than N-PID. The variation inthe controller outputs is presented in Fig. 1(b). The observations(qualitative) of the above simulation study are as follows:Both the proposed controllers and A-NMPC are able tomaintain

the setpoint at the desired value. However, the performances ofF-NMPC and A-NMPC at all the operating points are found to bebetter than N-PID, as there is less overshoot and settles to thesetpoint faster. The F-NMPC provides performance comparable tothat of A-NMPC.Fig. 2 presents histograms of computation time at each

sampling instant obtained using A-NMPC and F-MPC. It can beconcluded that the computation time per iteration (Matlab 7.0,Intel Core 2 Duo Processor-2.13 GHz) of the proposed F-NMPCalgorithm is in the range 0.01–0.7 s, whereas, for the A-NMPCalgorithm, the value is in the range of 0.25–1.6 s. For the N-PID thecomputation time per iteration has been found to be in the range0.001–0.0016 s.The proposed F-NMPC helps to reduce the number of compu-

tations needed, compared to the rigorous model based NMPC (A-NMPC). Also, in the A-NMPC, the nonlinear differential equationshave to be numerically integrated to obtain the predicted estimatesof the output variables. On the other hand, in the F-NMPC, althoughmore matrices are needed, all of them have constant values, whichlimit the calculation to (i) the determination of weights, whichwill

J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282 279

Fig. 4. Servo and regulatory responses of CSTR with F-NMPC, A-NMPC and N-PID (a) Process output (b) Controller output.

Fig. 5. Servo response of CSTR with N-PID for various values of filter time constant (a) Process output (b) Controller output.

be provided by the operating region membership functions, and(ii) state propagation calculations of each model using the appro-priate matrices and a weighted average of the local linear modeloutputs. Since, it is not necessary to carry out numerical integra-tion of nonlinear differential equations; the proposed F-NMPC ap-proach has better implementation capabilities than the A-NMPCapproach. Note that the computation time of the F-NMPC for evenhigher-order problems will be always less demanding, in compar-ison to that of the rigorous model based NMPC(A-NMPC).

7.5. Servo-regulatory performance

Simulation studies have been carried out to demonstratethe disturbance rejection capability of the proposed F-NMPCformulation based on local linear models, analytical model basedNMPC (A-NMPC) and N-PID Controller at nominal and at shiftedoperated points.A step change in the feed temperature of magnitude 2 ◦K (from

350 ◦K to 352 ◦K) has been introduced at the 10th sampling

280 J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282

Fig. 6. Servo response of CSTR with F-NMPC for various values of prediction horizon (a) Process output (b) Controller output.

Fig. 7. Performance of N-PID in the presence of measurement noise (a) Process output (b) Controller output.

instants and the value has been maintained upto the 110thsampling instants and is then brought back to 350 ◦K. (Refer Fig. 3).The ISE values are computed for A-NMPC, F-NMPC and N-PID andare reported in Table 5. The ISE value of F-NMPC and A-NMPC havebeen found to be considerably less than that of N-PID for the servo-regulatory performance case also. The following observation canbedrawn from the simulation studies• From 10th sampling instants to 70th sampling instants ofFig. 4(a), it can be inferred that the controllers (A-NMPC, N-

PID and F-NMPC) are able to reject the disturbance quickly andbring the reactor concentration back to the nominal value ofthe setpoint. This part of the simulation demonstrates that thecontrollers are able to reject the disturbance at the nominaloperating point.

• With the disturbance being persistent, a step change in thesetpoint has been introduced at 70th sampling instant and itcan be noted that both the controllers are able to maintain the

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Fig. 8. Performance of F-NMPC in the presence of measurement noise (a) Process output (b) Controller output.

Table 6Mean and standard deviation of the true value of the controlled variable for various values of alpha.

Alpha Sampling instants interval F-NMPC µ (σ) N-PID µ (σ)

0.25 51 ≤ k ≤ 100 0.0989(3.28e–04) 0.0989(2.32e–04)251 ≤ k ≤ 300 0.1107(7.15e–04) 0.1108(4.24e–04)451 ≤ k ≤ 500 0.0989(3.57e–04) 0.0989(2.64e–04)

0.5 51 ≤ k ≤ 100 0.0988(4.31e–04) 0.0989(2.93e–04)251 ≤ k ≤ 300 0.1107(8.09e–04) 0.1108(4.31e–04)451 ≤ k ≤ 500 0.0988(4.48e–04) 0.0989(2.61e–04)

0.75 51 ≤ k ≤ 100 0.0988(4.58e–04) 0.0989(3.09e–04)251 ≤ k ≤ 300 0.1107(7.15e–04) 0.1108(4.23e–04)451 ≤ k ≤ 500 0.0988(3.59e–04) 0.0989(2.72e–04)

concentration at the setpoint, as evident from 70th samplinginstants to 110th sampling instants of Fig. 4(a).• At 110th sampling instants a simultaneous step change in thesetpoint (Refer Fig. 4(a)) as well as a step change in the feedtemperature (Fig. 3) has been introduced and it can be inferredthat the performance of the controllers has been found to besatisfactory. This part of the simulation demonstrates that thecontrollers are able to reject the disturbance aswell asmaintainthe process variable at the setpoint. It should be noted that theperformance of F-NMPC is found to be better than A-NMPC andN-PID.

7.6. Performance of N-PID for various values of filter time constant

In order to show the tradeoff between performance androbustness of the proposed nonlinear PID control scheme, weperformed simulation studies for various values of filter timeconstant (lamda). The closed loop responses for step changes inthe setpoint for various values of lamda are shown in Fig. 5(a).The manipulated variable profiles for various values of filter timeconstants are shown in Fig. 5(b). It should be noted that we haveused single tuning parameter (lamda) for all the local controllers.However, for each local controller appropriate values of filterconstant can be chosen. As we increase the value of lamda, it wasobserved that the responses have been found to be of over-damped

type (lamda 0.5 and 1.5). For lamda value equal to 0.25, it canbe observed that controller output was found to be aggressive, ascompared to other values of lamda (Refer Fig. 5(b)).

7.7. Performance of F-NMPC for various values of prediction horizon

In order to assess the effect of the prediction horizon, wehave performed simulation studies for various values of predictionhorizon. The closed loop responses to step changes in the setpointand for various values of prediction horizon are shown in Fig. 6.In all the simulation runs, a control horizon of 1 is used. For theprocess considered for simulation study the prediction horizonseems not to have appreciable effect as shown in Fig. 6. Thesetpoint tracking performance has been found to be almost thesame for all the values of prediction horizon.

7.8. Performance of N-PID and F-NMPC in the presence of measure-ment noise

The performances of the proposed N-PID and F-NMPC controlschemes in the presence of measurement noise are shown inFigs. 7 and 8 respectively. In both the control schemes, Gaussianwhite noise of mean zero and standard deviation of 0.0012 mol/lhas been added to the true value of the process variable (reactorconcentration). A digital first order filter has been used to filterthe noisy process measurement and the control calculations are

282 J. Prakash, K. Srinivasan / ISA Transactions 48 (2009) 273–282

performed based on the filtered value of the process variable, incase of N-PID and F-NMPC. From the Figs. 7 and 8, the performanceof both the control schemes has been found to be satisfactory.The mean and the standard deviation of the true value of themeasured variable (concentration) for various values of alpha havebeen reported in Table 6. As we reduce the value of alpha, thecontroller action of both the control schemes has been found to besmooth and the standard deviation of the controlled variable hasbeen found to be less.

8. Conclusions

In this paper, the authors have proposed a simple andstraightforward procedure for designing a Nonlinear PID control(N-PID) scheme and Nonlinear Model Predictive Control scheme(F-NMPC) using local linear models for the CSTR process, whichexhibits significant variation in the damping factor and un-damped natural frequency. From the extensive simulation studies,it can be concluded that the proposed controllers have goodsetpoint tracking, disturbance rejection capabilities at nominaland shifted operated points and robustness properties. Further,the performance of the proposed nonlinear model predictivecontrol scheme using local linear models, has been comparedwith nonlinearmodel predictive control using an analytical model.From the extensive simulation study, it can be concluded that theproposed F-NMPC helps to reduce the number of computationsneeded, compared to the analytical model based NMPC. Theproposedmodel based control scheme (F-NMPC) canbe consideredas an alternative to analytical model based control scheme (A-NMPC).

References

[1] Johansen T, Murray-Smith R. Multiple model approaches to modeling andcontrol. London: Taylor and Francis Limited; 1997.

[2] Xue ZK, Li SY. Multi-model modeling and predictive control based on Localmodel networks. Control and Intelligent Systems 2006;34(2):105–12.

[3] Arslan E, CamurdanMC, Palazoglu A, Arkun Y. Multi-model scheduling controlof nonlinear systems using gap metric. Industrial & Engineering ChemistryResearch 2004;43:8275–83.

[4] Boling Jari M, Seborg Dale E, Hespanha JP. Multi-model adaptive control of asimulated pH neutralization process. Control Engineering Practice 2007;15:663–72.

[5] Prakash J, Senthil R. Design of observer based nonlinear model predictivecontroller for a continuous stirred tank reactor. Journal of Process Control2008;18:504–14.

[6] Dougherty Danielle, Cooper Doug. A practical multiple model adaptivestrategy for multivariable model predictive control. Control EngineeringPractice 2003;11:649–64.

[7] Narendra KS, Balakrishnan J. Adaptive control using multiple models. IEEETransactions on Automatic Control 1997;4(2):171–87.

[8] Wojsznis WK, Blevins TL. Evaluating PID adaptive techniques for industrialimplementation. In: Proceedings of the 2002 American control conference.2002. p. 1151–5.

[9] Tan W, Marquez HJ, Chen T, Liu J. Multi-model analysis and controllerdesign for nonlinear processes. Computers & Chemical Engineering 2004;28:2667–75.

[10] Gao Ruiyao, O’dywer Aidan, Coyle Eugene. A Nonlinear PID control forCSTR using local model networks. In: Proceedings of 4th world congress onintelligent control and automation. 2002. p. 3278–82.

[11] GalanOmar, Romagnoli, PalazogluAhmet. Real-time implementation ofmulti-linear model-based control strategies-An application to a bench-scale pHneutralization reactor. Journal of Process Control 2004;14:571–9.

[12] Takagi T, Sugeno M. Fuzzy identification of systems and its applications tomodeling and control. IEEE Transactions on Systems Man Cybernetics 1985;15:116–32.

[13] Kuipers B, Astrom K. The composition and validation of heterogeneous controllaws. Automatica 1994;30(2):233–49.

[14] Aufderheide B, Bequette BW. Extension of dynamic matrix control to multiplemodels. Computers and Chemical Engineering 2003;27:1079–96.

[15] Tian Z, Hoo KA. Multiple model-based control of the Tennessee–Eastmanprocess. Industrial & Engineering Chemistry Research 2005;44:3187–202.

[16] Özkan Leyla, KothareMV. Stability analysis of amulti-model predictive controlalgorithm with application to control of chemical reactors. Journal of ProcessControl 2006;16:81–90.

[17] Krishnapura VG, Jutan Arthur. A neural adaptive controller. ChemicalEngineering Science 2000;55:3803–12.

[18] Nahas EP, Henson MA, Seborg DE. Nonlinear internal model control strategyfor neural network models. Computers and Chemical Engineering 1992;16(12):1039–57.

[19] Huang YL, Lou HH, Gong JP, Edgar Thomas. Fuzzy model predictive control.IEEE Transaction on Fuzzy Systems 2000;8:5–77.

[20] Babuska R, Verbruggen H. Fuzzy set methods for local modeling andidentification in multi-model approaches to modeling and control. London:Taylor and Francis Limited; 1997. p. 75–100.

[21] Minh Tran, Dimitrios K. Varvarezos Mohamad Nasir The importance of first-principles model-based steady-state gain calculations in model predictivecontrol—a refinery case study. Control Engineering Practice 2005;13(8):1369–82.

[22] Senthil R, Janarthanan K, Prakash J. Nonlinear state estimation usingfuzzy kalman filter. Industrial Engineering Chemistry Research 2006;45:8678–8688.

[23] Panda Rames C, Yu Cheng-Ching, Huang Hsiao-Ping. PID tuning rules forSOPDT system: Review and some new results. ISA Transactions 2004;43(2):283–295.

[24] Morari, Zafiriou. Robust process control. Englewood Cliffs (NJ): Prentice Hall;1989.