stability analysis of a multi-model predictive control algorithm with application to control of...

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Journal of Process Control 16 (2006) 81–90

Stability analysis of a multi-model predictive control algorithmwith application to control of chemical reactors

Leyla Ozkan a, Mayuresh V. Kothare b,*

a Center for Process Analytics and Control Technology, Department of Chemical Engineering and Advanced Materials,

University of Newcastle, Merz Court, Newcastle upon Tyne, NE17RU, UKb Chemical Process Modeling and Control Research Center, Department of Chemical Engineering, Lehigh University,

111 Research Drive Bethlehem, PA 18015, USA

Received 17 March 2005; received in revised form 14 June 2005; accepted 14 June 2005

Abstract

We study a stabilizing multi-model predictive control strategy for controlling nonlinear process at different operating conditions.The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function that has finite and infinite hori-zon cost components. The finite horizon cost consists of free input variables that direct the system towards a terminal region whichcontains the desired operating point. The infinite horizon cost has an upper bound and steers the system to the desired operatingpoint. The system is represented by a sequence of piecewise linear models. Based on the condition of the system states, the sequenceof piecewise linear models is updated and the controller�s objective function switches form quasi-infinite to infinite horizon objectivefunction. This results in a hybrid control structure. A recent approach in the analysis of hybrid systems that uses multiple Lyapunovfunctions is employed in the stability analysis of the closed-loop system. The stabilizing hybrid control strategy is illustrated on twoexamples and their closed-loop stability properties are studied.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Model predictive control; Multiple models; Hybrid systems; Linear matrix inequalities; Stability

1. Introduction

Model predictive control (MPC), also known asreceding horizon control, has become the control strat-egy of choice in industrial applications that typicallyinvolve linear systems subject to linear inequality con-straints. However, chemical processes are in generalinherently nonlinear and operate over a wide range ofoperating conditions. The use of multiple model/controllers is a common strategy in dealing with thecomplexity of nonlinear systems and has led to thedevelopment of various multiple model/controller

0959-1524/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.jprocont.2005.06.013

* Corresponding author. Tel.: +1 610 758 6654; fax: +1 610 7585057.

E-mail address: [email protected] (M.V. Kothare).

approaches. Considerable research has been focusedon the development and utilization of multiple model/controller banks within the MPC framework [21,2,5,13,14,8,7,19,20] in order to cope with nonlinear systems.Other closely related approaches that fill in this generalcategory include gain scheduling [9,24,27,28], multi-model adaptive control [29,25], model reference adap-tive control [17,18,11], supervisory control [16,23]. Thebasis of these approaches is the decomposition of thesystem�s full range of operation into a number of oper-ating regimes in which a simpler local model and/or con-troller is applied. The local models and controllers arethen incorporated to give a global model and/orcontroller.

In any control strategy, the question of closed-loopstability is of great importance. The general approach

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Notation

C concentration [kmol/m3]F mass flow rate [kg/h]Gpi polymer production rate [kg/h]I identity matrixJ cost functionK state feedback controller gainM number of piecewise linear modelsMpw weight average molecular weight [kg/kmol]m summation indexP positive definite matrixQ positive definite matrixQI weighting matrixq dimensionless feed flow rateR weighting matrixT temperature [K]t time [h]U control moved in infinite horizon cost func-

tionu input vectorX partitionX symmetric matrixx(kjk) state vector at time instant kx(k + ijk) ith step predicted state vector at time in-

stant kx1 dimensionless reactor concentration

x2 dimensionless reactor temperatureV Lyapunov functiony output vectorYAp polymer composition

Greek symbols

b dimensionless heat of reactionc Lyapunov functionk dimensionless activation energyd dimensionless heat transfer coefficientn intermediate variable in S-procedures dimensionless time/ Damkohler number

Superscripts/subscripts

A monomer AB monomer BC chain transfer agentf feedi region, indexj cooling jacket, indexn tuning parameters switching instantT transposet terminal region

82 L. Ozkan, M.V. Kothare / Journal of Process Control 16 (2006) 81–90

in stability analysis of MPC is that the performanceindex is interpreted as a Lyapunov function and shownto be monotonically decreasing. Various predictive con-trol strategies with stability guarantee have been pro-posed. A detailed review can be found in [1,15]. Theseapproaches either impose contractive constraints on thesystem states or drive the system states to zero or insidea region where the control law is obtained from a stabi-lizing linear controller. The stability results of these for-mulations depend on the feasibility of the control law.This issue has recently been addressed by [10]. In theirwork, a controller strategy that combines the boundedcontrol approach with MPC for constrained linear sys-tems was proposed. The main idea is the utilization ofbounded control whenever MPC results in infeasibilityand hence guarantee that the system will evolve in thestability region defined by the bounded controller.

Closed-loop stability in multiple model/control ap-proaches has also been studied [18,11,26] since designinglocal controllers that stabilize each individual model maynot result in a stable global closed-loop system. In gen-eral, the use of multiple models in a control structurenecessitates a means of switching among the availablemodels to the one that best describes the current operat-ing condition. The switching from one model/controller

to another based on a logical argument (supervisoryscheme) results in a hybrid system. An approach thathas found wide utility in stability analysis of hybrid sys-tems is multiple Lyapunov functions [3,4]. The idea orig-inates from the fact that it may not be possible to find acommon Lyapunov function that ensures stability for allthe subsystems and the global system. A closely relatedwork is the stability analysis of piecewise linear systemsby [22] in which piecewise quadratic Lyapunov functionswere constructed using convex optimization in terms oflinear matrix inequalities (LMIs) as an alternative to aglobally quadratic Lyapunov function.

In this paper, we extend the multi-model predictivecontrol strategy formulated in our previous work [19]to incorporate a stabilizing contractive constraint. Wethen analyze stability of the resulting closed-loop systemusing the multiple Lyapunov function approach pro-posed by [3,4]. The use of multiple Lyapunov functionsis well suited for our problem since the overall system in[19] is described by a sequence of piecewise linear modelsand based on the system state, the sequence of models isupdated. In addition, the objective function switchesfrom a quasi-infinite horizon to an infinite horizon costfunction when the system states enter the terminal re-gion that contains the origin.

L. Ozkan, M.V. Kothare / Journal of Process Control 16 (2006) 81–90 83

The paper is organized as follows: Section 2 providesthe background information on the multi-model predic-tive control strategy introduced in [19] and the multipleLyapunov functions approach in the stability analysis ofhybrid systems. In Section 3, the multi-model predictivecontrol strategy is extended to incorporate a contractiveconstraint and the stability of the resulting closed-loopis analysed using the multiple Lyapunov functions ap-proach. In Section 4, the proposed control strategy isimplemented on two examples. Finally Section 5 pre-sents a summary and discussion of the results.

2. Background

2.1. Multi-model predictive control

Consider a nonlinear system represented by a se-quence of piecewise linear equations as in (1) along atransition trajectory between two operating points.

xðk þ 1Þ ¼ AixðkÞ þ BiuðkÞ þ bi

yðkÞ ¼ CixðkÞ þDiuðkÞ þ di

�i ¼ 1; . . . ;M ð1Þ

Assume that each piecewise linear model is assigned toan ellipsoid region. The desired operating point is de-fined as the origin of the system and the ellipsoid regionthat contains the origin is called the terminal region. It isalso assumed that the number of moves required for thesystem to transition from one point to another is pre-specified. The algorithm is illustrated in Fig. 1.

We consider the following optimization problem atany instant k depending on the system state (see [19,Eq. (20)]):

minu

J1ðkÞ ¼ minu

Xn

m¼0

½xTðk þ mjkÞQIxðk þ mjkÞ(

þ uTðk þ mjkÞRuðk þ mjkÞ�

þX1

m¼nþ1

½xTðk þ mjkÞQIxðk þ mjkÞ

þ uTðk þ mjkÞRuðk þ mjkÞ�)

ð2Þ

x1

x2

OP1

OP2

(x,u)=(0,0)

u( |k),kx ( |k)k

Fig. 1. Illustration of multi-model predictive control algorithm.

s.t.

V ðxðk þ mþ 1jkÞÞ � V ðxðk þ mjkÞÞ

6 �kxðk þ mjkÞk2QI� kuðk þ mjkÞk2R

and

kEtxðk þ mjkÞ þ etjj 6 1; m ¼ nþ 1; . . . ;1

for x(k) 62 {xjkEtx + etk 6 1}. Here, kEtx + etk 6 1 isthe equation that defines the terminal region as an ellip-soid; V(x) = xTPx, P > 0 is a quadratic function of thestate x. This problem has a quasi-infinite horizon objec-tive function with free input variables (u(k + mjk),m = 0, . . . ,n) for the finite part.

For x(k) 2 {xjkEtx + etk 6 1} the optimization prob-lem is given by

minu

JðkÞ ¼ minu

X1m¼0

kxðk þ mjkÞk2QIþ kuðk þ mjkÞk2R

( )

ð3Þ

s.t.

V ðxðk þ mþ 1jkÞÞ � V ðxðk þ mjkÞÞ

6 �kxðk þ mjkÞk2QI� kuðk þ mjkÞk2R ð4Þ

kEtxðk þ mjkÞ þ etk 6 1; m ¼ 0; . . . ;1

In this optimization problem, input variables (u(k +mjk), m = n + 1, . . . ,1) are calculated from a state feed-back control law (u(k + mjk) = Ktx(k + mjk)).

Using standard techniques in convex optimizationutilizing LMIs, the following control recipe is obtained(see [19])

minc;nQ;Yt

c ð5Þ

s.t.

1 xTðkjkÞ

xðkjkÞ Q

" #P 0 ð6Þ

and

Q � � � �

ðAtQþ BtYtÞ Qþ nbtbTt bte

Tt n 0 0

EtQ netbTt �nðI� ete

Tt Þ 0 0

Q1=2I Q 0 0 cI 0

R1=2Yt 0 0 0 cI

2666666664

3777777775P 0

ð7Þ

for x(k) satisfying kEtx(k) + etk 6 1 and

minuðkþmjkÞ;m¼0;1;...;n; n;Q;Yt

c ð8Þ


s.t.

1 � � � � � � � �Q0.5

I xðkjkÞ cI 0 0 0 0 0

R0.5uðkjkÞ 0 cI 0 0 0 0

Q0.5I xðk þ 1jkÞ 0 0 cI 0 0 0

R0.5uðk þ 1jkÞ 0 0 0 cI 0 0

..

. . ..

xðk þ nþ 1jkÞ 0 0 0 0 0 Q

26666666666664

37777777777775P 0 ð9Þ

and

Q � � � �ðAtQþ BtYtÞ Qþ nbtb

Tt bte

Tt n 0 0

EtQ netbTt �nðI� ete

Tt Þ 0 0

Q1=2I Q 0 0 cI 0

R1=2Yt 0 0 0 cI

26666664

37777775P 0

ð10Þfor x(k) 62 {xjkEtx(k) + etk 6 1}. The detailed derivationof this algorithm can be found in [19].

It is also possible to incorporate input constraints.We consider bounds on input at time k such as

jujðkÞj 6 uj;max; k P 0 and j ¼ 1; 2; . . . ; nu

Inputs can be split into sequences: {u(kjk),u(k +1jk),u(k + 2jk), . . . ,u(k + njk),Ut} where u(kjk),u(k +1jk),u(k + 2jk), . . . ,u(k + njk) are free variables and Ut

are future control moves in the terminal region givenby the state feedback law.

Ut : uðk þ mjkÞ ¼ Ktxðk þ mjkÞ; nþ 1 6 m 6 1Kt ¼ YtQ

�1

Since u(kjk),u(k + 1jk),u(k + 2jk), . . . ,u(k + njk) are freedecision variables, constraints on these variables can beimposed directly:

jujðk þ mjkÞj 6 uj;max;

j ¼ 1; 2; . . . ; nu and m ¼ 0; 1; . . . ; n ð11Þ

For the remaining future manipulated variables, theexistence of a symmetric matrix X satisfying the follow-ing inequality:

X Yt

YTt Q

� �P 0 with Xjj 6 u2j;max; j ¼ 1; 2; . . . ; nu

ð12Þguarantees that juj(k + mjk)j 6 uj,max, j = 1,2, . . . ,nu,m = n + 1, n + 2, . . . ,1. More detailed derivation ofthe control strategy can be found in [19].

2.2. Stability analysis of hybrid systems

In analyzing the stability characteristics of hybrid sys-tems, an approach that has received substantial atten-

tion is multiple Lyapunov functions [4]. Consider aswitched system in the following from:

_xðtÞ ¼ fiðxðtÞÞ; i 2 Q ’ f1; . . . ;Ng ð13Þ

where xðtÞ 2 Rn. It is suggested that if each subsystem_xðtÞ ¼ fiðxðtÞÞ has a Lyapunov function establishing sta-bility individually, then there should be restrictions onswitching in order to guarantee global stability since itis well known that switching between the stable systemswill not necessarily result in a global stable system. In [4]the following notation is used for analyzing switchedand hybrid systems. Assume a switching sequence

S ¼ fx0; ði0; t0Þ; ði1; t1Þ; . . . ; ðiN ; tNÞ; . . .g

where x0 is the initial condition and (ik, tk) means thatthe system evolves according to _xðtÞ ¼ f ik ðxðtÞ; tÞ fortk 6 t 6 tk+1. This trajectory is denoted by xS(t). As-sume also that the switching sequence is minimal inthe sense that ij 5 ij+1, j 2 Z+. Denote the end pointsof the times that the system i is active by Sji. The intervalcompletion IðT Þ of a strictly increasing sequence oftimes T = t0, t1, t2, . . . , tN, . . . is the set[j2Zþ

½t2j; t2jþ1� ð14Þ

Hence, IðSjiÞ denotes the set of times that the �i �th sys-tem is active. Lastly, let e(T) denote the even sequenceof T.

eðT Þ : t0; t2; t4; . . .

Definition 2.1 [4]. Let V be a function that is continu-ous positive definite about the origin. Given a strictlyincreasing sequence of time T 2 R, we say that V isLyapunov-like for function f and trajectory x(t) over Tif:

• _V ðxðtÞÞ < 0; 8t 2 IðT Þ,• V is monotonically non-increasing on e(T).

Theorem 1. Consider candidate Lyapunov functions Vi,

i = 1, . . . ,N and vector fields _x ¼ f iðxÞ with fi(0) = 0 for

all i. LetS be the set of all switching sequences associated

with the system. If for every possible S 2 S we have that

for all i, Vi is Lyapunov-like for fi and xS(Æ) over Sji thenthe system is stable in the sense of Lyapunov.

Proof. See [4]. h

The main result of this theorem is that it is not re-quired for the individual Lyapunov functions to de-crease monotonically at instances other than the evensequence of switching times and is interpreted in Fig. 2for N = 2 where the solid line shows which Lyapunovfunction is active.

t0 t1 t5 t6t3t2 t4

V1 (t)

V2 (t)

Fig. 2. An interpretation of Theorem 1 for N = 2.


Although Theorem 1 presented here is for continuoussystems it is also applicable to discrete systems and theproof can be found in [4]. The idea in Theorem 1 is easyto understand and there is no limitation in the selectionof the Lyapunov function. However, it is only applicableto a system comprised of stable subsystems or subsys-tems that have been stabilized by appropriate regional(local) controllers.

3. Stabilizing multi-model predictive control

The multi-model predictive control algorithm in Sec-tion 2.1 not only consists of a receding horizon controlformulation but also a hybrid structure due to the dy-namic scheduling of the sequence of models along thetransition trajectory. In addition, the control strategyswitches to an infinite horizon receding controller, inwhich stability is guaranteed, once the states are in theterminal region. Therefore, in the stability analysis ofthe closed-loop system, the use of multiple Lyapunovfunctions is appropriate. Assuming that the systembehaves exactly according to the sequence of piecewiselinear models during the transition, closed-loop stabilitycan be guaranteed by showing the monotonicity ofthe objective function

PNi¼0kxðk þ ijkÞk2QI

þ kuðkþijkÞk2R þ kxðk þ N þ 1jkÞk2P for x(k) outside the terminalregion and xTPx for x(k) in the terminal region. How-ever, in the implementation of the multi-model predic-tive control strategy, the linear models are only usedto approximate the nonlinear system behavior and it israther probable that the resulting response can be quitedifferent than what has been predicted. Thus, a condi-tion is required explicitly in the control algorithm thatwill guarantee closed-loop stability.

Consider c(k) as a Lyapunov function at time k forthe system x(k) 62 the terminal region and impose themonotonically decreasing condition as an explicit con-

tractive constraint (15) in the multi-model predictivecontrol algorithm in (8) after the initialization stepk = 0.

cðkÞ < cðk � 1Þ ð15Þ

Then, the multi-model predictive control algorithm withstability guarantee is provided by the following theorem.

Theorem 2. Let x(kjk) be the state of the systemmeasured at time k in region i. Suppose the operating

regions are approximated as ellipsoids. Suppose we know

a priori the sequence of regions that the system transitions

through and the number of steps required for the system to

switch from one region to another. Then the multi-model

predictive control algorithm with stability guarantee is

defined as the optimization problem in (8) subject to the

constraints (9)–(12), and (15) if x(k) is not in the terminalregion. If x(kjk) is in the terminal region then the

minimization problem in (5) is solved subject to the

constraints (6), (7), and (12). The closed-loop system is

stable if the feasible solutions of the control strategy

defined above are implemented in a receding horizon

fashion.

Proof. Consider a Lyapunov function of the followingform at time k for the closed-loop system

V ðxðkÞÞ ¼ V aðxðkÞÞ ð16Þ

where

aðxðkÞÞ ¼ 1 if xðkÞ 62 terminal region ð17Þ

aðxðkÞÞ ¼ 2 if xðkÞ 2 terminal region ð18Þ

with two candidate Lyapunov functions

V 1ðkÞ ¼ cðkÞ ð19Þ

V 2ðkÞ ¼ xTðkÞPxðkÞ ð20Þ

Show that

• V(x(k)) > 0 for x(k)5 0 and V(x(k)) = 0 if and onlyif x(k) = x(kjk) = 0. For x(k)5 0.1. If (x(k)) 62 the terminal region, V(x(k)) = c(k)

which is an upper bound on the quadratic quasi-finite horizon objective function where QI, R, andP are positive definite. Therefore, V(x(k)) > 0,"x(k) 5 0 and x(k) 62 the terminal region.

2. If x(k) 2 the terminal region V(x(k)) =xT(k)Px(k). Since P is positive definite V(x(k)) >0, "x(k)5 0 and x(k) 2 the terminal region.

If x(k) = 0, then x(k) 2 the terminal region, u(k) = 0since u(k) = Kx(k), and V(x(k)) = xT(k)Px(k) = 0.Conversely, if V(x(k)) = 0, then c(k) = 0 for x(k) 62the terminal region and xT(k)Px(k) = 0 otherwise.In the former case, c(k) satisfies the inequality in (21):


Xn

m¼0

½xTðk þ mjkÞQIxðk þ mjkÞ

þ uTðk þ mjkÞRuðk þ mjkÞ�þ xTðk þ nþ 1jkÞPxðk þ nþ 1jkÞ 6 cðkÞ ð21Þ

Therefore, the quasi-infinite horizon performanceindex is also equal to 0. Since QI, R, and P in (21)are positive definite matrices, x(k + ijk),u(k + ijk)must be all zero. Consequently, the system is in theterminal region and x(k) = 0.

• V(x(k)) < V(x(k � 1)).Suppose that the multi-model predictive control algo-rithm is feasible after the initialization step (k = 0).Then, Va(x(k)) < Va(x(k�1)) holds due to the contractiveconstraint (15) for 1 6 k 6 ts � 1 and the Lyapunovinequality constraint in (3) for ts + 1 6 k < 1 wherets denotes the switching instant of the control algo-rithm from quasi-infinite horizon to infinite horizon,i.e., the first instant the states are in the terminalregion. h

Remark 1. The stability results depend on the feasibilityof the control algorithm and measurability of the states.

Remark 2. At k = ts it is not required for the Lyapunovfunction V(x(k)) to satisfy the monotonically decreasingcondition.

V ðxðtsÞÞ � V ðxðts � 1ÞÞ < 0

This result is deduced from Theorem 1.

Fig. 3. Performance of multi-model predictive control with stabilityguarantee during a transition from OP3 to OP2 (CSTR).

4. Illustrative examples

The illustration of the proposed multi-model predic-tive algorithm with stability guarantee is carried outon two examples. The first example is a nonlinear CSTRwith an irreversible reaction. The second example is anindustrial scale solution copolymerization reactor. Inboth examples, the tuning parameters QI and R areequal to identity matrices.

4.1. Nonlinear CSTR

Consider a standard two-state CSTR with an exo-thermic irreversible first-order reaction A ! B. Thedynamics of the system can be described by a set oftwo nonlinear ordinary differential equations obtainedfrom material and energy balances with the assumptionof constant volume, perfect mixing, and constant phys-ical properties.

dx1ds

¼ �/x1jðx2Þ þ qðx1f � x1Þdx2ds

¼ b/x1jðx2Þ � ðqþ dÞx2 þ duþ qx2f

y ¼ x2

ð22Þ

where x1 is the concentration, x2 is the reactor tempera-ture, and u is the cooling jacket temperature. The

parameters are jðx2Þ ¼ exp x21þx2

k

� �/ ¼ 0.072, q = 1.0,

b = 8.0, d = 0.3, k = 20.0, x1f = 1.0, x2f = 0.0. Underthe nominal operating conditions u = 0, the reactorexhibits three steady states; (x1,x2) = (0.2353,4.7050),(0.5528,2.7517), (0.856,0.889). The state space of thesystem is divided into multiple regions and piecewise lin-ear models are obtained by the linearization of the non-linear model in each region. The regions are first definedas slabs by bounding the state variable x such as

Ri ¼ x 2 Rnj � a 6 x 6 bf gWhen Ri is a slab, a degenerated ellipsoid of the formkEix + ei 6 1k can be found that approximates Ri [12].Suppose that Ri = {xjd1 6 cTx 6 d2}, then we can take

Ei;j ¼2cj

d2;j � d1;j; ei;j ¼

ðd2;j þ d1;jÞðd2;j � d1;jÞ

ð23Þ

where i specifies the region and j shows the correspond-ing column number. This approximation can only bedone if the regions are in the form of slabs and we cantake d1 and d2 larger than the physical limitations ofthe system. In this case, the slabs for each correspondingregion are

R1 ¼ x 2 R2 0.3

1.5

� �� 6 x 60.77

4.0

� ��

R2 ¼ x 2 R2 0.15

3.5

� �6 x 6

0.45

4.5

� ��

R3 ¼ x 2 R2 0.0

4.0

� �6 x 6

0.33

6.0

� �� ð24Þ

In this example, we consider a transition from OP3(x,u) = ([0.2353,4.7050]T,0) to OP2 (x,u) = ([0.5528,2.7517]T,0). The performance of the controller strategyis shown in Fig. 3.

In Fig. 4 the global Lyapunov function is plotted.In this simulation, the switching time ts = 1.9. This is

the instant at which the controller switches from the

Table 1Operating conditions of MMVA solution copolymerization reactor

OP1 OP2

Inputs

Monomer A feed rate (kg/h) 18.0 22.5Monomer B feed rate (kg/h) 90.0 90.0Initiator feed rate (kg/h) 0.18 0.18Solvent feed rate (kg/h) 36.0 36.0Chain transfer agent feed rate (kg/h) 2.7 2.7Inhibitor feed rate (kg/h) 0.0 0.0Reactor jacket temperature (K) 336 336Reactor feed temperature (K) 353 353

Outputs

Polymer production rate (kg/h) 23.35 24.9Mole fraction of A in dead polymer 0.56 0.64Weight average molecular weight (kg/kmol) (105) 0.34 0.39Reactor temperature (K) 353.06 353.3Fig. 4. Va vs. time for a transition from OP3 to OP2 (CSTR).


quasi-infinite to the infinite horizon objective and thestates enter the terminal region for the first time. An in-crease in the Lyapunov function value is detected at thisinstant. This can be attributed to the fact that, when thestates that are in the terminal region the infinite horizonobjective function becomes effective and it does notcontain free input variables u(k + ijk), i = 0, . . . ,n in itsformulation. However, this does not violate the require-ments of stability and the closed-loop system is guaran-teed to be stable as was mentioned in Remark 2.

4.2. Methylmethacrylate–vinylacetate solutioncopolymerization reactor

The modified control strategy has also been tested onan industrial scale solution copolymerization reactormodel. The nonlinear model was taken from [6]. This

Fig. 5. Closed-loop response for a transition from OP1 to OP2 (MMVA

model is based on a free radical mechanism with 27 sep-arate reactions and has 12 states. The dynamic behaviorof the process largely depends on the monomer feedratio. In this study, the reactor is represented with fourlinear models obtained by the Jacobian linearization ofthe nonlinear model when it experiences a transitionfrom OP1 to OP2 shown in Table 1. Only the terminalregion is approximated as an ellipsoid which can beobtained by calculating an ellipsoid with the maximumvolume for the symmetric polytope defined for the ter-minal region. This also has been formulated as anLMI problem.

Figs. 5–7 show the performance of the control algo-rithm and the course of the Lyapunov function Va(x(k))

during the transition from OP1 to OP2, respectively.

solution copolymerization reactor, four piecewise linear models).

Fig. 6. Manipulated variable profiles for transition from OP1 to OP2 (MMVA solution copolymerization reactor, four piecewise linear models).

Fig. 7. Va vs. time for a transition from OP1 to OP2 (MMVA solutioncopolymerization reactor, four piecewise linear models).


In this example, it is observed that there are two timeinstants (t = 7 h and t = 7.75 h) at which the states enterthe terminal region. The solution copolymerization reac-tor is highly nonlinear and the implementation of themulti-model controller causes the states to move outof the terminal region at the next sampling time. Thiscan be avoided if the nonlinear system is representedas a convex hull of linear models in the terminal regioninstead of a piecewise linear model or if the terminal re-gion is smaller, and the number of free input variablesare increased. The control algorithm is still stable dueto the contractive constraint and forces the states to-wards the terminal region.

The results presented in the previous examples as-sume that the states are measurable. The same study

could be done by using estimated state values. Then,the Lyapunov function V aðxðkÞÞ would be a function ofxðkÞ. However, this does not guarantee that estimatedstates will converge to the real states.

5. Summary and conclusion

This work presented a stabilizing multi-model predic-tive control algorithm which has a contractive constraintto guarantee closed-loop stability. Furthermore, the sta-bility of the closed-loop is analyzed by employing themultiple Lyapunov functions approach originally pro-posed in [3,4]. Two different Lyapunov functions areproposed. Depending on the system state, (in the termi-nal region or outside) the corresponding Lyapunovfunctions are assigned. The use of multiple Lyapunovfunctions has enabled us to relax the monotonicallydecreasing condition of the Lyapunov function whenthe control algorithm switches from a quasi-infinitehorizon to an infinite horizon strategy. This is concludedfrom the description of Lyapunov-like function in Defi-nition 2.1. In this definition, the individual Lyapunovfunctions are only required to decrease monotonicallyduring the time interval they are active and over theset of even sequence of switching times. The stability re-sults depend on the feasibility of the control algorithm.Therefore, in the illustrative studies the tuning parame-ter n is modified accordingly.

One should note that the stability results are onlyvalid provided that the real system is described by afamily of piecewise linear systems. Therefore, stability


of the switched system using multiple Lyapunov func-tions does not necessarily guarantee stability of thenonlinear system, a limitation common to many multi-model control approaches applied to nonlinear systems.However, if the nonlinear system is described by a rea-sonable number of piecewise linear models, as done inthe illustrative examples, the proposed approach pro-vides an acceptable approach for controlling nonlinearchemical systems while guaranteeing stability.

The work presented here has only dealt with thedevelopment of a stabilizing control strategy and stabil-ity analysis of the closed-loop system. A further studythat will enhance the performance of the control strat-egy will be to consider the modeling aspects. Through-out the study it is assumed that the piecewise affinemodels describe the system dynamics exactly. Thisassumption is inadequate for processes with high nonlin-earity. Therefore, it is necessary to modify the modelstructure for capturing the nonlinearity of the process.A possible way is to incorporate uncertainty into theformulation by utilizing a polytopic uncertainty descrip-tion in which the nonlinear system is assumed to lie in apolytope. The polytope is defined as the convex hull ofmultiple models obtained from Jacobian linearization.

Furthermore, in the current formulation, the controlalgorithm assumes fixed piecewise linear models and afixed sequence of models along the transition trajectory.It is more likely that the system may not follow the pre-defined sequence, especially if the process is as nonlinearas the solution copolymerization reactor example. Inthat case, we either increase the number of linear modelsused in the formulation or we update model definitionsand the sequence of models if the models are obtainedby Jacobian linearization along a new dynamic trajec-tory. A new dynamic trajectory that will describe thesystem behavior starting from the current conditionsto the final region can be calculated by the implementa-tion of the input sequence, which is the solution of theoptimization problem in Eq. (8) at that instant, onthe real system. A relevant and a worthwhile study,especially for an industrial perspective, would be thecomparison of the developed control algorithm withnonlinear MPC techniques on a industrial scale process.Such a study should consist of not only performancecomparisons, disturbance rejection, but discuss feasibil-ity, stability and computational features as well.

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