A multiple model approach for predictive controlof nonlinear hybrid systems

Naresh N. Nandola, Sharad Bhartiya *

Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India

Abstract

This paper presents modeling and control of nonlinear hybrid systems using multiple linearized models. Each linearized model is alocal representation of all locations of the hybrid system. These models are then combined using Bayes theorem to describe the nonlinearhybrid system. The multiple models, which consist of continuous as well as discrete variables, are used for synthesis of a model predictivecontrol (MPC) law. The discrete-time equivalent of the model predicts the hybrid system behavior over the prediction horizon. The MPCformulation takes on a similar form as that used for control of a continuous variable system. Although implementation of the control lawrequires solution of an online mixed integer nonlinear program, the optimization problem has a fixed structure with certain computa-tional advantages. We demonstrate performance and computational efficiency of the modeling and control scheme using simulationson a benchmark three-spherical tank system and a hydraulic process plant.

Keywords: Nonlinear hybrid system; Multiple model control; Model predictive control

1. Introduction

Hybrid systems are characterized by interactionsbetween continuous and discrete dynamics. The termhybrid has also been used to describe processes that involvecontinuous dynamics and discrete (logical) decisions [1–3].Applications of hybrid systems have arisen in manufactur-ing systems, automobile control, and computer disk drivecontrol among others. Although the use of a hybrid systemframework in modeling and control of chemical processeshas emerged only recently, large continuous plants havealways used logic controllers to implement safety featuressuch as the triggering of a coolant pump and the varioussafety interlocks. However, current trends in the chemicalprocess industry emphasize the need for flexible processing,which invariably necessitates a greater degree of logical

decision-making along with the continuous control laws.Similarly, batch processes, which are naturally character-ized by a sequence of event or time-driven operations,can be described through mixed continuous and logicalvariables. The logic component may be realized throughthe use of on/off valves, a speed selector switch and if-then-else rules. The control of such hybrid systems is oftenbased on heuristics resulting from plant operation experi-ence. Hybrid systems and their applications have attractedresearchers from control and computer science communityto focus on various aspects such as stability [4], simulation[5], verification [6–8], identification [9], control [1,2,10,11],modeling and analysis [12,13], and optimization [14].

A number of modeling formalisms that represent hybridsystems have been proposed in the literature. These formal-isms can be broadly assigned to the following three catego-ries [15]: (i) a discrete formalism, such as finite automata,that can be extended with continuous variables resultingin hybrid frameworks such as timed automata [16] andhybrid Petri Nets [17], (ii) a continuous formalism that

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can accommodate discrete variables or logical conditionsby appropriately switching between system dynamics, and(iii) an approach that directly combines the continuoussubsystem with its discrete counterpart through an inter-face. In the current work, we adopt the second modelingstrategy. In particular, we present a framework that repre-sents nonlinear hybrid systems by combining multiple, par-tially linearized models. Thus, the framework can representarbitrary nonlinear hybrid dynamic systems while retaininga fixed model structure. Subsequently, a model predictivecontrol (MPC) scheme is presented that uses the multiple,partially linearized models for predicting future plantbehavior. The utility of the framework is demonstratedby modeling and control of two different applications,namely a three-spherical tank system and a hydraulic pro-cess plant. The results confirm that the multiple linearizedmodel approach provides superior predictions for a nonlin-ear hybrid system relative to a single linearized model. Fur-ther, the approach also exhibits computational advantagesover the mixed logical dynamic model [2] due to its com-pact formulation. The current paper is organized as fol-lows: Section 2 develops the multiple, partially linearizedmodeling framework. Development of the model predictivecontrol law based on the above mentioned framework ispresented in Section 3. Section 4 documents the modelingand control of the two applications. Finally, the work isconcluded in Section 5.

2. Multiple linearized modeling framework for nonlinear

hybrid systems

Hybrid systems may involve both continuous and dis-crete states as well as continuous and discrete inputs. Typ-ically, the flow-field describing the evolution of continuousstates is dependent on discrete phenomena characterized bydiscrete state events as well as control events due to discreteinputs. A formal description of hybrid systems may be rep-resented by a hybrid automaton (HA) [18,19] as the follow-ing septuple (L,X,A,W,E,Inv,Act). The continuous statesx 2 X � Rn of the hybrid automaton HA evolve accordingto the continuous dynamic flow-field and external inputsW � Rq. The status of the discrete state and input variablesdetermine the location l 2 L of HA. Each location maps tothe flow-field via the map Act. E is a finite set of eventsdescribed by a five-tuple ðl; a;Guardll0 ; Jumpll0 ; l

0Þ, wherel 0 2 L represents the new location upon occurrence of theevent and a 2 A serves as a label of the event. Prior tothe transition, the continuous state X must lie in the guardset denoted by Guardll0 . The transition from location l tolocation l 0 represents a jump in the continuous states x

and their values are reset by Jumpll0 . The invariant Inv

maps the locations to the set of subset of X. Thus, if anevent occurs, Inv evaluates to false and the discrete statetransits to a new discrete location.

The choice of the specific model structure of the hybridautomaton is dependent on the application at hand. Forexample, in formal verification, a simpler model, usually

an over-approximation of the original model, is employed.Timed automata [16] or rectangular automata [15] are com-mon choices. Similarly, representation of hybrid systemsfor simulations has resulted in development of various plat-forms such as Simulink/Stateflow and Modelica that usestateflow chart formalism. On the other hand, controlapplications as well as fault detection and diagnosis studieshave led to development of various linear modeling para-digms such as piecewise affine [20] (PWA) and mixed logi-cal dynamic [2] (MLD) models. The equivalence betweenMLD and a variety of other linear hybrid models has alsobeen proven [21]. In the area of nonlinear hybrid systems,Buss et al. [22] presented the Hybrid State Model (HSM),which models the nonlinear dynamics of the system withdiscrete states as well as discrete control inputs and formsthe starting point for development of the multiple partiallylinearized modeling framework. The HSM model assumesthat the system is driven by continuous and discrete manip-ulated inputs, uc(t) and ud(t), respectively. The continuousstates xc(t) evolve based on the flow-field fl, which is depen-dent on the location l of the system. To enable identifica-tion of the different locations and the transitions betweenthem, suitable event generating functions sj, j = 1, . . . ,ns

are defined. When one or more of these functions take ona value of zero, that is sj = 0, an event is said to occur.We assume that if a function equals zero at t then it willresult in a sign change, that is a positive value at t� willbecome negative at t+ after equaling zero and vice-versa.Thus, as long as the scalar functions sj do not cross zero(that is sj 6 0 or sj P 0), no new event occurs and the stateevolves according to the flow-field fl. For example, considerthe level (h) of a tank being filled whose maximum level forsafe operation is hmax. We may then define a functions1 = h � hmax, which is non-positive if the level of the tankis below the threshold (location l) and non-negative ifabove (location l 0). A value of zero (s1 = 0) identifies occur-rence of the event that the tank has achieved its maximumsafe level. A number of such functions sj, j = 1, . . . ,ns maybe necessary to uniquely identify all locations of the hybridsystem. The identification of the event may be representedby _sj = 0, j = 1, . . . ,ns, that is an event is said to occur ifone or more event generating functions become zero. Whenan event occurs, the system transits to a new locationaccompanied by a reset in both the continuous statesxc(t) and discrete states xd(t), using a prescribed mapb(xc,uc,xd,ud,t). Thus, the Hybrid State Model may be rep-resented as,

_xc ¼ flðxc; xd ; uc; ud ; tÞ_xd ¼ 0

�if sjðxc; uc; xd ; ud ; tÞ 6¼ 0;

j ¼ 1; . . . ; ns ð1ÞxcðtþÞxdðtþÞ

� �¼ bðxc; uc; xd ; ud ; tÞ if _j¼1...ns ½sjðxc; uc; xd ; ud ; tÞ ¼ 0�

ð2ÞycðtÞydðtÞ

� �¼ hðxc; uc; xd ; ud ; tÞ ð3Þ

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where yc(t) and yd(t) represent continuous and discrete out-puts, respectively. In the above equation, fl represents theflow-field corresponding to location l. The discrete statestake on a fixed value until they are reset upon occurrenceof an event. For example, the discrete state variable FILL_TANK(t) = 1 may indicate permission to commence fillingof the tank. The value of this state continues to be 1 until astate event occurs (such as s1 = h � hmax = 0) at whichpoint the discrete state may be reset to 0, that is FILL_TANK(t+) = 0. To identify occurrence of an event andhence the new location and the corresponding flow-field,one may define appropriate binary indicator variablesdj 2 {0,1} corresponding to each event generating functionsj as follows:

sj 6 0$ ðdj ¼ 1Þ; j ¼ 1; . . . ; ns ð4Þ

The status of the indicator variables uniquely defines thelocation l and hence the flow-field fl. Thus, the continuousstate evolution map of the HSM (Eq. (1)) may be rewrittenas,

_xc ¼ fgðxc; uc; dÞ ð5Þ

where fg is a global flow-field that subsumes all location-dependent flow-fields fl and is parameterized by the indica-tor vector d, whose values are implied by Eq. (4). Note thatthe vector d with ns binary elements can describe 2ns loca-tions. A change in the status of one or more elements ofd corresponds to an event that may be triggered by discon-tinuity in states (that is, a State Event (SE)) and/or discon-tinuity in inputs (that is, a Control Event (CE)). This factenables one to relate the status of the discrete states to bin-ary variables d as follows:

xdðtÞ ¼ bdðdðtÞÞ ð6Þ

where the elements of bd depend on the status of the vectord(t). In most chemical processes, the continuous states rep-resent physical quantities and typically exhibit a switch-likebehavior upon occurrence of an event rather than a suddenjump. Thus, the continuous states are reset after an eventas follows:

xcðtþÞ ¼ xcðt�Þ ð7Þ

Eqs. (6) and (7) represent the reset map b in the HSM pre-sented in Eq. (2).

The HSM allows nonlinear functions to trigger an event.In most chemical processes, however, events are generatedby (i) crossing of some threshold by the state of the systemresulting in a SE, and (ii) activation and deactivation ofinputs resulting in a CE. Therefore, in many instances,the condition sj 6 0 used in Eq. (4) can be specified in termsof linear constraints such as xi 6 c1 or axi + bui 6 c2. Thus,one may simplify Eq. (4) as follows:

ða0jucðtÞ þ b0jxðtÞ � cj 6 0Þ $ ðdj ¼ 1Þ; j ¼ 1; . . . ; ns ð8Þ

where aj and bj are coefficient vectors that relate appropri-ate elements of input and state vectors and cj is some scalarvalue. The implication in Eq. (8) may be reformulated as

linear inequalities using equivalence with propositional lo-gic expressions (such as Big-M constraints; see, for exam-ple, [23,24]) followed by a matrix representation for thens equations in Eq. (8) as follows:

E1ucðtÞ þ E2dðtÞ þ E3xðtÞ 6 E4 ð9Þ

where Ei (i = 1,2,3,4) are the corresponding coefficientmatrices of appropriate size and d represents the vectorof binary variables comprising all indicator variables dj.Thus, the modified Hybrid State Model can be written asfollows:

_xc ¼ fgðxc; uc; dÞ ð10ÞxdðtÞ ¼ bdðdðtÞÞ ð11ÞE1ucðtÞ þ E2dðtÞ þ E3xðtÞ 6 E4 ð12ÞxcðtþÞ ¼ xcðt�Þ ð13ÞxdðtþÞ ¼ bdðdðtþÞÞ ð14Þ

The HSM represented in Eqs. (10)–(14) may be used formodeling and control of nonlinear hybrid systems. Typi-cally, models derived from real systems will be well-posedand solution of Eqs. (10)–(14) will result in a unique trajec-tory x(t), for a given initial condition and control input tra-jectory [2]. However, use of the above HSM for a modelbased receding horizon control scheme needs solution ofa MINLP optimization problem online. Further, the futureprediction will require numerous integrations of Eq. (10).The following route is adopted to simplify the controlproblem formulation as well as computations,

(i) obtain multiple linear models by linearizing the origi-nal HSM Eq. (10) at multiple operating points char-acterized by continuous variables (xc, uc). Thus ateach operating point, we obtain linear models corre-sponding to all locations;

(ii) discretize the continuous time linearized multiplemodels to obtain a discrete-time representation;

(iii) obtain the nonlinear hybrid model prediction byweighting the multiple linearized models using aBayesian criterion.

The remainder of this section discusses each of theabove three steps.

2.1. Model linearization

Performing a Taylor series expansion on Eq. (10)around the point (xc, uc) and retaining the binary vectord as a parameter, one may obtain a linearized model asfollows:

_xc ¼ AðdÞxc þ BðdÞuc þ f ðdÞ ð15ÞxdðtÞ ¼ bdðdÞ ð16ÞE1ucðtÞ þ E2dðtÞ þ E3xðtÞ 6 E4 ð17Þ

Note that a fixed value of the parameter vector d

corresponds to a unique location of the hybrid system.

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The deviation form of variables has not been used as thisallows representation of non-equilibrium operation, a com-mon feature in hybrid systems applications, resulting in theaffine representation in Eq. (15).

Remark 1. The continuous time model in Eqs. (15)–(17) isnonlinear since the system matrices are a function of theindicator variables, which in turn are based on state andcontrol events.

Next, the above model is discretized in the time domain,which in turn will enable writing the prediction equationsneeded in MPC. However, the presence of discrete vari-ables makes time-discretization a non-trivial task.

2.2. Discretization of linearized hybrid model

One may use a numerical integration technique like theexplicit Euler method to obtain a discrete-time representa-tion of the linear model in Eq. (15). This approach has beenused in the MLD framework [2]. However, the stability andaccuracy of the explicit Euler method necessitate use of asmall step size. To circumvent the inaccuracies due to theapproximate discretization one could discretize the systemusing the analytical solution to the integration of continu-ous time linear systems using a zero order hold. We call thismethod exact discretization and present details below.

Exact discretization: The method begins by fixing thevalues of binary variables d in Eqs. (15) and (16), therebyobtaining a model for a system at a fixed location [25].Since ns binary variables result in 2ns possible locations,we obtain 2ns linear models from Eqs. (15) and (16), respec-tively as follows:

_xc ¼ AixcðtÞ þ Biu

cðtÞ þ f i; i ¼ 1; 2; . . . ; 2ns ð18Þxd ¼ bdi; i ¼ 1; 2; . . . 2ns ð19Þ

Models for continuous states (Eq. (18)) are then discretizedusing the standard technique based on integration of lineartime invariant systems [26]. Thus, the hybrid model takesthe form,

xckþ1

xdkþ1

� �¼ ½Ui� ½0�

½0� ½0�

" #xc

k

xdk

� �þ ½Ci�½0�

" #uc

k þ�f di

bdi

" #;

i ¼ 1; 2; . . . ; 2ns ð20Þ

The above equation may be rewritten in a compact form asfollows:

xkþ1 ¼ Uixk þ Ciuck þ f di; i ¼ 1; 2; . . . ; 2ns ð21Þ

The 2ns models obtained above may then be combinedusing a corresponding scalar logical multiplier [25] ‘i toproduce a discrete-time representation of Eqs. (15) and(16), which accounts for all locations. Note that the con-straints in Eq. (17) can be rewritten in discrete-time formby merely replacing the continuous time variable t by asampling time index k. Thus, the discrete-time representa-tion of Eqs. (15)–(17) is as follows:

xkþ1 ¼X2ns

i¼1

‘i;kUi

!xk þ

X2ns

i¼1

‘i;kCi

!uc

k þX2ns

i¼1

‘i;kf di

!

ð22ÞE1uc

k þ E2dk þ E3xk 6 E4 ð23Þ

The logical multiplier ‘i is based on the indicator variables,and is designed to take on a value 1 if and only if the ithcombination of the binary variables is encountered andzero, otherwise. This is demonstrated using the followingexample.

Example 1. Let us assume a hybrid system with two binaryvariables d1 and d2 (that is d ¼ ½d1 d2�T ) whose valuescharacterize the locations of the hybrid system. In this case,we have four possible locations and the same number oflinearized models, which take the form shown in Eq. (21).The four logical multipliers, ‘1, ‘2, ‘3 and ‘4 correspond tothe cases ½0 0�T , ½0 1�T , ½1 0�T and ½1 1�T , respectively andare defined as,

‘1 ¼ ð1� d1Þð1� d2Þ‘2 ¼ ð1� d1Þðd2Þ‘3 ¼ ðd1Þð1� d2Þ‘4 ¼ ðd1Þðd2Þ

Eqs. (22) and (23) can be recast into the MLD model [2].This is accomplished by noting that the RHS of Eq. (22)has nonlinear terms due to multiplication between the indi-cator variables dk, state variables xc

k and inputs uck. In the

MLD models, these multiplicative terms are masked byintroducing auxiliary binary and auxiliary continuous vari-ables and their associated constraints. The increased size ofthe MLD model typically imposes a large computationalburden in its use. On the other hand, our experience hasshown that retaining the model in the nonlinear form asshown by Eqs. (22) and (23) is computationally efficientsince it involves fewer numbers of variables and con-straints. The linearized discrete-time model (22) and (23)can be expressed in a compact form by defining thefollowing,

Lk ¼ ð‘1;kÞI ð‘2;kÞI � � � ð‘2ns ;kÞI½ � ð24Þ

U ¼ UT1 UT

2 � � � UT2ns

� �T ð25Þ

C ¼ CT1 CT

2 � � � CT2ns

� �T ð26Þ

�f ¼ f Td1

f Td2� � � f T

d2ns

� �T ð27Þ

I is an identity matrix of appropriate size. Thus, the modelin Eqs. (22) and (23) takes the form,

xkþ1 ¼ ðLkUÞxk þ ðLkCÞuck þ Lk

�f ð28ÞE1uc

k þ E2dk þ E3xk 6 E4 ð29Þ

The outputs of the linearized model may be written asfollows:

yk ¼ Cxk ð30Þ

Fig. 1. Bayesian weighting based multiple model scheme [30] for obtainingmultiple, partially linearized frameworks.

135

Eqs. (28)–(30) represent the final form of the linearizedmodel. Note that this model describes all locations of thenonlinear hybrid system in the vicinity of a single operatingpoint. We may obtain similar linearized discrete-time mod-els at different operating points [27] characterized by thecontinuous states and continuous inputs (xc, uc) and finallycombine them to reconstitute the nonlinear model. Methodsfor obtaining multiple model predictions typically consist ofusing past measurements to determine the validity of eachmodel. The validity of the models may be characterizedusing various criteria such as Bayes’ rule [28] and member-ship functions in a fuzzy logic approach [29]. The Bayesianweighting method has been employed for multiple modelpredictive control [30] as well as H1 control of nonlinearprocesses [31]. The current work uses the Bayesian methodfor obtaining multiple model predictions of nonlinear hy-brid systems. The next section briefly reviews this method.

2.3. Model predictions using Bayesian weighting of multiple

linearized models

Nonlinear systems can be decomposed into several localoperating regimes [32] and linear models built for each ofthese regimes. Determining the number of local operatingregimes and identification of the operating points is anon-trivial task. Criteria for selection of local operatingpoints and their numbers depend on insights andknowledge of the system [28,30,32]. Recently, theoreticapproaches to multiple modeling have emerged which arebased on a variable model structure [33]. Here the localmodels are treated as random models, which take on spe-cific values based on adaptive design methods. In our work,we follow the former approach and assume that nl localregimes have been identified that can adequately describethe overall operation. Each of these regimes may bedescribed by a local model of the form of Eqs. (28)–(30).The outputs of these individual models are then used toobtain the final weighted model. By defining new coefficientmatrices appropriately the weighted model takes the sameform as a single model described in Eqs. (28)–(30). A typ-ical approach involves weighting the different linear modelsto reconstitute the overall model. Thus, the overallweighted model may be written as,

xkþ1 ¼ ðLkUavgÞxk þ ðLkCavgÞuck þ Lk

�f avg ð31ÞE1uc

k þ E2dk þ E3xk 6 E4 ð32Þyk ¼ Cxk ð33Þwhere the blended system matrices depend on weighting ofthe different models as follows:

Uavg ¼Xnl

i¼1

wiUi ð34Þ

Cavg ¼Xnl

i¼1

wiCi ð35Þ

�f avg ¼Xnl

i¼1

wi�f i ð36Þ

wi represents the weight of the model for the ith regime.One popular approach to determining the weight wi isbased on a Bayesian interpretation of the plant model mis-match [30]. Here, wi has a value between 0 and 1 and sum-mation of all weights equals unity. The weights arecalculated online as shown in Fig. 1. The model bank con-tains models developed around multiple operating pointsas discussed previously. The Bayesian weight calculatorcalculates the weights, wi, using the past history of residualsand a probability is assigned to each model. The recursiveBayes theorem for the ith model at the kth time instant canbe used to evaluate the posterior probability as follows:

pri;k ¼exp � 1

2eI

i;kKei;k

� �pri;k�1Pnl

j¼1 exp � 12eI

j;kKej;k

� �prj;k�1

ð37Þ

where ei;k ¼ yk � yi;k represents the residual between themeasurement yk and the output prediction by the ith modelyi,k at the kth instant. pri,k is the posterior probability of themeasurement being closest to the prediction of the ith lin-earized model at kth instant. K is a time invariant weight-ing matrix known as convergence matrix and typicallychosen to be diagonal. In view of the similarity with thenormal distribution, K can be interpreted as the inversematrix of the residual covariance. Higher values of diago-nal elements of K indicate a small residual variance andthus greater confidence in the residual of each model. Thushigher the values of the elements of K, faster is the rejectionof models with large residuals. The user-defined K allowsstrategies ranging from a winner-take-all approach (largeK) to a non-discriminating averaging approach (small K).Finally, the weights wi corresponding to each model maybe obtained as follows:

Fig. 2. Schematic representation of the multiple, partially linearized(MPL) model. The MPL model combines all locations (x-axis) of thehybrid system with local information at different operating points (y-axis)thus representing a nonlinear hybrid system.

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wi;k ¼pri;kPnlj¼1prj;k

ð38Þ

Note that the model priors should take on values greaterthan zero to ensure their participation in the overall modelprediction at all k. Eqs. (31)–(38) may thus be used formodeling and control of a nonlinear hybrid system. Sincethe model is linear only at a fixed location and nonlinearotherwise, we will refer to this model as the multiple, par-tially linearized (MPL) model.

Remark 2. The method of obtaining the MPL model may berepresented schematically as in Fig. 2. The linearized hybridmodel in Eqs. (28)–(30) is shown as the dashed horizontalline obtained by linearizing the nonlinear hybrid state modelin Eqs. (10)–(14) at points P1, P2 and so on. Although thesemodels are obtained by linearization at a point, each modelaccounts for all locations l1, l2, lns and is therefore nonlinear.However, at any given location, the model becomes lineartime invariant with fixed system matrices. Finally, thesemodels are combined using the Bayesian weightingapproach. Thus, Eqs. (31)–(33), approximate the nonlinearoperating range as well as all locations of the hybrid system.

For comparison purposes, the discrete-time linearhybrid model using the explicit Euler method was alsodeveloped. This model takes on a similar form as in Eqs.(28)–(30) and is provided in Appendix I. We will use theterm MPL-Exact model when using exact discretizationwith the MPL framework and MPL-Euler model whenusing discretization by explicit Euler along with the MPLframework. In the next section we develop a model predic-tive control scheme for control of nonlinear hybrid systemsusing the MPL model.

3. MPC formulation

Model predictive control (MPC) is a form of feedbackcontrol, where the current value of the manipulated vari-

ables is determined online as the solution of an optimalcontrol problem over a horizon of given length. The behav-ior of the system over the horizon is predicted with a modeland the current plant state estimate assumed as the initialstate for this prediction. While a possibly large set of con-trol moves is computed, only the first one is implemented.When information about the plant state is available at thenext sampling instant, the model is updated and optimiza-tion repeated over a shifted horizon. The ability to system-atically include constraints and the capability to handleplants with multiple inputs and outputs has made MPCan attractive technique in the process industry. One ofthe main issues in implementation of nonlinear MPC iscomputational complexity and the time required for predic-tion. This is further exacerbated in case of hybrid systemsdue to the presence of binary variables. In this work, weuse a quadratic cost function of the form

minuc

k;...;uc

kþm�1dk;...;dkþp�1

J ¼Xp

i¼1

ykþi � yref

2

KyþXm�1

i¼0

uckþi � uc

kþi�1

2

Ku

ð39Þsubject to mixed integer constraints of Eq. (32) and variousprocess and safety constraints,

ymax 6 y 6 ymin ð40Þuc

max 6 u 6 ucmin ð41Þ

Ky, Ku represents error penalty and move suppression,respectively and p,m are the prediction and control hori-zons, respectively.

Remark 3. Eq. (39) shows that the binary decisionvariables d are propagated up to p-1 samples in future.However, these binary variables consist of those related tocontrol events and others related to state events. Althoughno distinction is made for ease of notation, the binaryvariables indicating control events (such as status of asolenoid valve) need be propagated only over the controlhorizon or m instants in future.

The future predictions may be obtained by propagatingthe Eqs. (31) and (33) for p steps in future, which results inthe following prediction equations for the state andoutput

vk ¼ H1kxk þH2klck þH3k ð42Þ

wk ¼ H1kxk þH2klck þH3k ð43Þ

The different vectors are defined as follows:

vk ¼ ½xTkþ1 xT

kþ2 . . . xTkþp�1 xT

kþp�T

wk ¼ ½yTkþ1 yT

kþ2 . . . yTkþp�1 yT

kþp�T

lck ¼ ½ucT

k ucT

kþ1 . . . ucT

kþm�1�T

�dk ¼ ½dTk dT

kþ1 dTkþ2 . . . dT

kþp�1�T

Similarly, the constraints resulting from propositional logicin Eq. (32) as well as the process and safety limits in Eqs.(40) and (41) may be written as,

137

E1lck þ E2

�dk þ E3vk�1 6 E4 ð44Þlc

min 6 lck 6 lc

max ð45Þwmin 6 wk 6 wmax ð46Þ

H1k ¼

ðLkUavgÞðLkþ1UavgÞðLkUavgÞ

ðLkþ2UavgÞðLkþ1UavgÞ . . . ðLkUavgÞ:

:

:

ðLkþp�1UavgÞðLkþp�2UavgÞ . . . ðLkUavgÞ

2666666666664

3777777777775

H2k ¼

ðLkCavgÞ ½0�ðLkþ1UavgÞðLkCavgÞ ðLkþ1CavgÞ

ðLkþ2UavgÞðLkþ1UavgÞðLkCavgÞ ðLkþ2UavgÞðLkþ1C

: :

: :

: :

: :

: :

ðLkþp�1UavgÞðLkþp�2UavgÞ . . . ðLkCavgÞ ðLkþp�1UavgÞðLkþp�2UavgÞ .

266666666666666664

½0� ½0�½0� ½0�: ½0�: :

ðLkþm�2CavgÞ :

ðLkþm�1UavgÞðLkþm�2CavgÞ ðLkþm�1CavgÞ: ðLkþmUavgÞðLkþm�1CavgÞ þ: :

ðLkþp�1UavgÞðLkþp�2UavgÞ . . . ðLkþm�2CavgÞ ðLkþp�1UavgÞðLkþp�2UavgÞ . . .

þðLkþp�1UavgÞðLkþp�2UavgÞ . .

. . .þ ðLkþp�1UavgÞðLkþp�2CavgÞ

H3k ¼

ðLk�f avgÞ

½ðLkþ1UavgÞðLk�f avgÞ� þ ½ðLkþ1

�f avgÞ�½ðLkþ2UavgÞðLkþ1UavgÞðLk

�f avgÞ� þ ½ðLkþ2UavgÞðLkþ1�f avgÞ�

:

:

:

:

½ðLkþp�1UavgÞðLkþp�2UavgÞ . . . ðLk�f avgÞ� þ ½ðLkþp�1UavgÞ . . . ðLkþ1

�f av

266666666666664

E1 ¼

E1 0 0 :: m times

0 E1 : : 0

: 0 : : :

: : : : :

0 0 0 :: E1

0 : : :: E1

: : : :: :

p times 0 0 :: E1

266666666666664

377777777777775

Ei ¼ diag Ei Ei :: :: p time½

E4 ¼

E4

E4

:

:

p times

26666664

37777775

Coefficient matrices H1k, H2k, H3k and E1, E2, E3, E4 aredefined as follows:

. . .

. . .

avgÞ . . .

. . .

. . .

. . .

. . .

. . .

. . ðLkþ1CavgÞ . . .

ðLkþmCavgÞ

ðLkþm�1CavgÞ. ðLkþmCavgÞþþ ðLkþp�1CavgÞ

37777777777777777777775

þ ½ðLkþ2�f avgÞ�

gÞ� þ . . .þ ½ðLkþp�1�f avgÞ�

377777777777775

s �; i ¼ 2; 3

where Ei(i = 1,2,3,4) are coefficient matrices in Eq. (32).

138

Coefficient matrices H1k, H2k and H3k may be obtainedby pre-multiplying each element block of H1k,H2k and H3k

by matrix C, respectively. Substituting Eq. (43) in Eq. (39)results in simplification of objective function,

minlk;

�dk

J ¼ ðwk � wref ÞT

Wyðwk � wref Þ

þ ðRlck � R0uc

k�1ÞT

WuðRlck � R0uc

k�1Þ ð47Þ

where Wy,Wu are error penalty and move suppressionweight matrices of appropriate size, respectively. MatricesR and R0 are defined as follows:

R ¼

I 0 0 :: 0 0

�I I 0 :: 0 0

0 �I I :: 0 0

: : : :: :: ::

0 0 0 :: I 0

0 0 0 :: �I I

2666666666664

3777777777775

and R0 ¼

I

0

0

:

:

m times

2666666666664

3777777777775

where I is an identity matrix whose size is determined bythe number of manipulated inputs. Minimization of theabove objective function subject to mixed integer con-straints in Eq. (44) and the process and safety constraintsin Eqs. (45) and (46) constitute the MPC law.

Note that the above formulation can accommodate sin-gle partially linearized model (Eqs. (28)–(30)) or the MPLmodel (Eqs. (31)–(33)) since both these models have a sim-ilar form. Also, the MPL-Euler model discussed in Appen-dix I can be accommodated in the above formulation.Further, if we fix the binary variables, the above MPC for-mulation is identical to that used for discrete-time model forthe conventional non-hybrid system (see for example Ricker[34]). Although, the control law shown in Eq. (47) necessi-tates an online solution of an MINLP, the MPL modeloffers an advantage of a fixed structure of the objectivefunction and constraints. Thus, one may derive an efficient

Fig. 3. Schematic of the 3-Ta

solution scheme by making use of analytical gradient andHessian to speedup the solution of online optimization.

4. Application

We evaluate the MPL modeling and control strategyusing simulation of two different applications, namely athree-tank benchmark system, which has been modifiedto enhance nonlinearity, and a PLC based operation of ahydraulic process plant.

4.1. Spherical three-tank system

The three-tank system is based on the three-tank bench-mark problem that has been used by a number of research-ers [35,36]. In order to enhance the nonlinear behavior, wemodify the benchmark problem to three spherical tanksinstead of the cylindrical tanks, as shown in Fig. 3. The sys-tem consists of two independent pumps that deliver theliquid flows Q1 and Q2 to Tank-1 and Tank-2, respectivelythrough the two control valves. Six independent solenoid(on/off) valves (V1, V2,V13,V23,VL1 and VN3) can be manip-ulated to interrupt the flows into or out of the three-tanks.Tank-1 and Tank-3 as well as Tank-2 and Tank-3 are con-nected through upper and lower pipes. The lower pipes arelocated at the bottom of the tanks while the upper pipes arelocated at a height hv(0.3 m). This system exhibits typicalcharacteristics of a hybrid dynamical system. The systemtransits between its locations due to the logic inputs (thesolenoid valves) and continuous variables (if h1 > hv thenthe outflow dynamics in Tank-1 changes). The nonlinearityresults from the spherical shape of the tanks as well as theconstitutive relationship between the exit flows and thelevel in each tank.

The six solenoid valves may be assigned binary indicatorvariables, whose value equals 1 when the correspondingvalve is open and 0 otherwise. Thus, the opening and clos-ing of valves may be classified as control events (CE),

nk benchmark problem.

139

which may be captured in the dynamic model by the use ofthe indicator variables as follows:

ph1ðhmax � h1Þdh1

dt¼ ðQ1 � V 13Q13V 13

� V 1Q13V 1� V L1QL1

Þð48Þ

ph2ðhmax � h2Þdh2

dt¼ ðQ2 � V 23Q23V 23

� V 2Q23V 2Þ ð49Þ

ph3ðhmax � h3Þdh3

dt¼ ðV 13Q13V 13

þ V 23Q23V 23þ V 1Q13V 1

þ V 2Q23V 2� V N3QN Þ

ð50Þ

where V1,V2,V13,V23,VL1 and VN3 represents binary indica-tor variables for corresponding valves, and hmax representstank diameter (0.6 m). Variables Qi represent flowratesthrough valves Vi and may be evaluated using the followingconstitutive equations,

Qi3V i3¼ azSi3signðhi � h3Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2gðhi � h3Þj

p; i ¼ 1; 2 ð51Þ

QL1 ¼ azSL1

ffiffiffiffiffiffiffiffiffi2gh1

pð52Þ

QN ¼ azSN

ffiffiffiffiffiffiffiffiffi2gh3

pð53Þ

The expressions for Qi3V idepends on whether the heights hj

(j = 1,2,3) are greater or less than hv = 0.3 m. Thus,

Qi3V i¼ V iazSisignðmaxfhi; hvg �maxfh3; hvgÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gjðmaxfhi; hvg �maxfh3; hvgÞj

p; i ¼ 1; 2

ð54Þ

Note that Eq. (54) represents all state events and thus weneed not define indicator variables for discrete states. Thus,

xc ¼ x ¼ y ¼ h1 h2 h3½ �Tuc ¼ Q1 Q2½ �Tud ¼ d ¼ V 13 V 23 V 1 V 2 V L1 V N3½ �T

where Si, Si3, SL1 and SN are cross sectional areas of valvesand assumed identical for all valves (0.95 cm2) and az is thedischarge coefficient, which is assumed to be unity. Next,we develop three linearized models of the form (28)–(30)for the three-spherical tank system. The points of lineariza-tion are listed below

(i) Model-I: h1 = h2 = 0.15 (or 25%), h3 = 0.14 (or 23%)and Q1 = Q2 = 0

(ii) Model-II: h1 = h2 = 0.25 (or 42%), h3 = 0.24 (or 40%)and Q1 = Q2 = 0

(iii) Model-III: h1 = h2 = 0.35 (or 58%), h3 = 0.34 (or57%) and Q1 = Q2 = 0

Model-I and Model-II correspond to levels below theupper pipe connections while Model-III corresponds to alevel above the upper pipe. The three points correspondto low, medium and high levels in the three tanks. Also,the points of linearization were chosen such that the conti-nuity of the max function is maintained. Alternatively,smooth approximation of the max function may be used.Each of the three models describes all locations of thehybrid system. The values of the coefficient matrices

Uavg;Cavg; �f avg;Lk have not been shown owing to theirextensiveness.

In order to compare the accuracy of a single partiallylinearized model with our MPL framework, we simulatethe plant model of (Eqs. (48)–(54)) by increasing the flowrates Q1 and Q2 to 0.7% of their maximum value(Qi,max = 0.0015 m3/s) at t = 0. Prior to the change, thethree-tanks were nearly empty and the two control valvesas well as the six solenoid valves were completely closed.During the course of the simulation, a number of arbitrarychanges were made in the status of the six solenoid valves.The levels in the three-tanks are shown in Fig. 4a–c, respec-tively. The nonlinear ‘‘plant’’ (solid line) based on Eqs.(48)–(54) is compared with the single Model-II (dotted line)and the MPL model (dashed-dotted). It can be noted thatthe MPL model consistently provides accurate predictionsrelative to the single partially linearized model throughoutthe course of the simulation. Extensive simulations withdifferent inputs consistently indicate the superior perfor-mance of the MPL framework over the entire level rangein the tanks. This can be explained by the fact that theMPL framework is a composite of all three models. As dis-cussed earlier, weights for each model is calculated onlinebased on the model residuals and are shown in Fig. 4d.The solid line with squares, dashed-dotted line withinverted triangles and dotted line with circles correspondto Model-I, Model-II and Model-III, respectively. It isobserved that only one of the three models is selected atany given time by the Bayesian approach of Eqs. (37)and (38). This winner-take-all results from the large val-ues assigned to K (diag[1000 1000 1000]). The large Khas been used to reflect the fact that the linearized modelsaccurately describe the plant (nonlinear model) locally.Fig. 4d shows that MPL framework uses Model-I untilapproximately t = 10,000 s after which it switches toModel-III.

We use the MPL model based MPC for a setpoint track-ing control problem, which involves filling of empty tanksto desired levels, followed by multiple setpoint changes. Todemonstrate the computational advantages of the multiplepartially linearized model over the MLD model, we com-pare our results with a multiple MLD (MMLD) modelbased MPC. Three MLD models are obtained at the sameoperating points by introducing auxiliary variables andassociated constraints. These MLD models are then usedin the multiple model scheme shown in Fig. 1 with Bayes-ian weighting. The MPC formulation using MLD model isprovided in Online Electronic Annex-I. We also considertwo different levels of complexities in terms of independentmanipulation of solenoid valves (or number of binaryinputs) to study the scalability of our formulation in termsof the computation time. In the first case, we reduce thenumber of binary inputs to 3 by assuming that each pairof valves V1 and V13, V2 and V23,VL1 and VN3 are switched(on/off) simultaneously. In the second case, all solenoidvalves are manipulated independently resulting in sixbinary inputs. Both cases use a sampling time ts of 3 s,

0

25

50

75

h 1 (%

)

h 2 (%

)

h 3 (%

)

0

25

50

75

0 5000 10000 150000

25

50

75

Time (sec)0 5000 10000 15000

0

0.5

1

Time (sec)

Wei

ght

Fig. 4. (a)–(c) Comparison of predictions of multiple model MPL framework (dashed-dotted line) and a single linearized model, Model-II (dotted line)with the first principles model (solid line) for various changes in inputs. (d) weights of the three models during the simulation. Model-I (solid line withsquare), Model-II (dashed-dotted line with inverted triangle) and Model-III (dotted line with circle) for MPL case.

140

prediction horizon, p = 5 (15 s) and control horizon, m = 2(6 s). We consider nonlinear dynamic Eqs. (48)–(54) as the‘‘plant’’ model and MPL/MMLD framework as the predic-tion model. Model prediction is corrected by adding a con-stant plant model mismatch over a prediction horizon.

Note that the MPL model based MPC results in anonline MINLP optimization problem while the MMLDmodel based MPC results in an online MIQP optimizationproblem. The branch and bound strategy is used to solvethe mixed integer programs. The relaxed NLP (for MINLPin MPL model based MPC) and QP (for MIQP in MMLDmodel based MPC) problems are solved using fmincon andquadprog in MATLAB 6.5 (Mathworks Inc, Natick, MA,USA). All simulations have been performed on a3.0 GHz P-IV machine with 1 GB RAM.

Case 1. Simultaneous switching of V1 and V13,V2 andV23,VL1 and VN3.

Fig. 5 documents filling and control of the levels in thethree-tanks at different levels using the MPL-Exact model(solid line) as well as the corresponding MMLD-Exactmodel (dotted line). Various characteristics of the 3-tankbenchmark problem are reflected in these results. Forexample, Fig. 5b shows that the filling of Tank-2 is fasterthan emptying. This can be explained by noting thatTank-2 does not have an exit at its bottom and can be emp-tied only by discharging to Tank-3. Similarly it takes longerto fill Tank-3 than either Tank-1 or Tank-2. Manipulatedvariables (Q1,Q2 and the six solenoid valves) for the MPL(solid line), and MMLD (dotted line) are shown in

Fig. 6. In order to convert the partially linearized modelto MLD framework, we introduced 4 auxiliary binary vari-ables, 27 auxiliary continuous variables and 121 linearmixed integer constraints. The resulting MPC problem isan MIQP with 14 binary variables, 139 continuous vari-ables and 655 constraints (for p = 5 and m = 2) (see OnlineElectronic Annex-I). On the other hand, MPC using MPL-Exact approach requires an online solution of an MINLPwith only 6 binary variables, 4 continuous variables and50 constraints. Note that in this example, the binary vari-ables represent control events and hence they need to bepropagated only up to the control horizon, m = 2 (seeRemark 3). Since the MPL and the MMLD frameworksare equivalent, the two solutions are nearly identical andthe minor differences can be attributed to numerical meth-ods. However, the main advantage of the MPL frameworkover the MMLD framework lies in computational effi-ciency. The mean and standard deviation of the computa-tion times needed to calculate the control moves using theMPL-Exact model are 0.6777 s and 0.8014 s, respectively.On the other hand, the mean and standard deviation ofusing MMLD-Exact model based control were 438.84 sand 220.49 s for the same operation. This clearly demon-strates the efficiency of the MPL model based MPC overthe MMLD model based MPC.

We also tested the MPL-Euler model and the MMLD-Euler model with the same filling followed by level controlproblem. Unlike the exact discretization approach, wheremodels corresponding to each location are obtained andthen recombined using logical multipliers, the use of

0

25

50

75

h 1 (%

)h 2 (

%)

h 3 (%

)

0

25

50

75

0 100 200 300 400 500 600 7000

25

50

75

Time (sec)

Fig. 5. Case 1: 3 binary manipulated inputs. Model predictive control of levels h1, h2, and h3 in the 3-Spherical Tank system using the MPL-Exact model(solid line) and the corresponding MMLD-Exact model (dotted line). 100% corresponds to 0.6 m height.

0

15

x 10 −4

Q1 (

m3 /s

)

Q2 (

m3 /s

)

0

15

x 10 −4

0

1

0 200 400 600 7500

1

0 200 400 600 7500

1

Time (sec)

VL

1 and

VN

3V

13 a

nd V

1

V23

and

V2

Time (sec)

Fig. 6. Case 1: 3 binary manipulated inputs. Control moves for the level control problem in the 3-Spherical Tank system using the MPL-Exact model(solid line) and MMLD-Exact model (dotted line).

141

explicit Euler simplifies the various expressions in MPLmodel (see Appendix I) and also reduces the number of

auxiliary variables needed in the MLD model. TheMMLD-Euler based MPC results in an online optimiza-

142

tion problem with 6 binary variables, 34 continuous vari-ables and 170 constraints, which is significantly smallerthan the MMLD-Exact model based MPC problem. Thesimulation profiles for these are not presented here due tospace limitation. However, the mean and standard devia-tion of the time required to compute each online solutionduring control are provided in Table 1. It is observed thatthe MPL-Euler model has advantages over the MPL-Exactmodel in terms of computation efficiency. However, theMPL-Euler framework results in higher overshoots andlarger settling times as compared to MPL-Exact frame-work, which is attributed to inadequacies associated withthe explicit Euler discretization method. The mean andstandard deviation of computation time per samplinginstant for MPL-Euler model are 0.65 s and 0.72 s, respec-tively and for MMLD-Euler model they are 8.1 s and4.54 s, respectively. These differences can be explained bythe size of optimization problem discussed above.

The 3-tank application represents an integrating system.Ideally, once the three-tanks have achieved the desired setpoints, the controller should turn off all the valves until anew set point is implemented. If such an approach is notpursued, the controller attempts to maintain the desiredlevel by continual opening and closing the solenoid valvesdue to integrating nature of the three-tank system andhence an oscillatory behavior around the vicinity of set-points is observed in outputs. To avoid such oscillatorybehavior, one could penalize excessive movement of thesolenoid valves directly in the objective function in Eq.(39). In the present work, we used the simple heuristic thatall valves are closed after the outputs attain their setpoints.For example, at t = 260 s all tanks achieve their prescribedlevel and therefore the heuristic deactivates the controllerand closes all valves in order to maintain level at that par-ticular setpoint (see Fig. 5). At time, t = 363 s the controlleris reactivated as a new setpoint is implemented. To achievethe higher desired level h1, the controller increases Q1 andalso opens valves V13, V1 and V23 V2 in order to achievelower level in Tank-2. Further Q2 remains closed in orderto decrease level in Tank-2. Again at time, t = 450 s thecontroller is deactivated as the desired setpoints areachieved.

Table 1Mean and Standard deviation of computation time for various case ofMPC simulation of 750 s for three-spherical tank system using samplingtime ts = 3 s

Exact Explicit Euler

Case 1a Case 2b Case 1a Case 2b

MPL Mean(s) 0.6777 1.9295 0.6461 0.8219SD(s) 0.8014 2.8310 0.7194 1.3660

MMLD Mean(s) 438.84 * 8.0989 196.88SD(s) 220.4904 * 4.5365 87.5231

a Case 1: 3 binary inputs.b Case 2: 6 binary inputs.* Could not solved a single optimization problem in entire day.

Case 2. Independent manipulation of all valves.

In Case 1, pairs of solenoid valves were simultaneouslyoperated to reduce the size of the optimization problem.Ideally, however, it is desirable to manipulate the solenoidvalves independently. This results in an optimization prob-lem for MPL-Exact with 12 binary and 4 continuous vari-ables and 62 constraints. The corresponding MMLDformalism resulted in an online optimization problem with70 binary variables, 209 continuous variables and 1457constraints. Simulation results using the MPL-Exact for-mulation are shown in Fig. 7 as the solid line. As expected,the current case has higher degrees of freedom and there-fore performs better than Case 1. For example, implemen-tation of the set point at t = 150 s necessitates draining ofTank 1 through VL1, while VN3 must remain closed toincrease h3. However, as noted in Fig. 5c, simultaneousopening of VL1 and VN3 results in draining of Tank-3 asin Case 1. Thus, in Case 2, a simultaneous decrease in h1

and increase in h3 at t = 150 is better handled than in Case1. The higher degrees of freedom in Case 2 also results in asuperior dynamic performance. For example, the settlingtimes of Tank-3 for Cases 1 and 2 following setpointchanges at 150 s are noted as 120 s and 80 s, respectively.However, the mean and standard deviation of the compu-tation time for Case 2 when using MPL-Exact base MPCare 1.9 s and 2.8 s, which are higher than in Case 1 (seeTable 1). The MMLD-Exact model for Case 2 resulted inan intractable online optimization problem, where not asingle optimization problem could be solved in an entireday. The manipulated variable profile (Q1, Q2 and the sixsolenoid valves) for the MPL-Exact (solid line) is shownin Fig. 8. We also studied the control problem of Case 2using MPL-Euler formulation discussed previously. Thesimulation results are documented in Fig. 7 as the dottedline. As expected, control with MPL-Exact discretizationperforms better than with the MPL-Euler model since itprovides relatively accurate approximation of the nonlinearplant model. Control with the MPL-Euler model results inhigher overshoots and larger settling times as compared tocontrol using the MPL-Exact model. This behavior may beexplained by the inaccurate integration of the nonlinearplant by the explicit Euler technique. However, the MPL-Euler model requires lesser time to solve the online MINLPthan when using the MPL-Exact model. The mean andstandard deviation of computation time needed per onlinesolution of the control problem with the MPL-Euler modelare 0.8219 s and 1.3660 s, respectively (see Table 1). Thismay be explained by the fact that although the numberof decision variables and constraints remain same in bothMPL-Exact and MPL-Euler models, the computationaloverheads in evaluation of LN

k in Eq. A-10 is small as com-pared to Lk in Eq. (31). The manipulated variables (Q1,Q2

and the six solenoid valves) profile for the MPL-Euler isshown as the dotted line in Fig. 8.

Table 1 also documents the statistics of computationalefficiency when using MMLD-Exact and MMLD-Euler

0

25

50

75

0

25

50

75

0 100 200 300 400 500 600 7000

25

50

75

Time (sec)

h 3 (%

)h 2

(%)

h 1 (%

)

Fig. 7. Case 2: 6 binary manipulated inputs. Model predictive control of levels h1, h2, and h3 in the 3-Spherical Tank system using the MPL-Exact model(solid line) and the corresponding MPL-Euler model (dotted line). 100% corresponds to 0.6 m height.

0

15

x 10 −4

Q1

(m3 /s

)

Q2

(m2 /s

)

0

15

x 10 −4

0

1

V13

0

1

V23

0

1

V1

0

1

V2

0 200 400 600 7500

1

Time (sec)

VL

1

0 200 400 600 7500

1

Time (sec)

VN

3

Fig. 8. Case 2: 6 binary manipulated inputs. Control moves for the level control problem in the 3-Spherical Tank system using the MPL-Exact model(solid line) and MPL-Euler model (dotted line).

143

as the controller models. Upon comparing the computationtimes between the above two cases, it is observed that con-

trol with MPL models is scalable (doubling the number ofbinary variables increases computational time 3 times with

144

MPL-Exact and 1.25 times with MPL-Euler). The compu-tation time required for control with the MMLD-Exactmodel is significantly higher (�50 times) than its corre-sponding MMLD-Euler model. On the other hand, forthe MPL models this difference is not significant. This isdue to the fact that in MPL models the number of decisionvariables and number of constraints remain same in eithercase and the difference arises due to the simpler form of LN

k

in Eq. A-10 for MPL-Euler model than Lk in Eq. (31) forMPL-Exact model. On the other hand, MMLD-Exactmodel introduces far larger number of auxiliary variablesand constraints than the corresponding MMLD-Eulermodel, thus making it computationally expensive. Thus,the MPL model provides a computationally attractiveframework for control of nonlinear hybrid systems.

Fig. 9. Schematic diagram of the hydraulic process plant for temperatureand level control in Tank-1 and Tank-3 (adapted from [25]).

4.2. Hydraulic process

In this example we demonstrate modeling and control ofa hydraulic process reported in the literature. The probleminvolves discrete as well as continuous states. In practicediscrete decisions are typically implemented using a pro-grammable logic controller (PLC). However, control of ahybrid systems using MPC can integrate the PLC-like deci-sion-making and manipulation of continuous inputs into asingle control law formulation. A schematic of the hydrau-lic process plant is shown in Fig. 9, whose modeling andcontrol has been reported in the literature [25]. In thehydraulic plant, water is heated in Tank-21 and Tank-22by two separate on/off electric heaters. The warm wateris then supplied to Tank-3 by means of two separate on/off pumps. Finally, the warm water from Tank-3 exits asthe final product by an on/off pump, a part of which isrecycled by a speed-regulated pump to Tank-1. Tank-1supplies water to Tank-21 and Tank-22. Tank-1 andTank-3 also receive cold water by means of an on/off pumpand a speed-regulated pump, respectively. A first principlesmodel has been provided by Colmenares et al. [25] and isreproduced below,

dh1

dt¼ 0:1154ðq31þq0�q11�q12Þ ð55Þ

dh3

dt¼ 0:1154ðq21þq22þq23�q31�q32Þ ð56Þ

h1dT 1

dt¼ 0:1154ðq31ðT 3�T 1Þþq0ð10�T 1ÞÞ ð57Þ

h3

dT 3

dt¼ 0:1154ðq21ðT 21�T 3Þþq22ðT 22�T 3Þþq23ð10�T 3ÞÞ

ð58Þdh21

dt¼ 0:1346ðq11�q21Þ ð59Þ

h21

dT 21

dt¼ 0:1346ð7:102QH1þq11ðT 1�T 21ÞÞ ð60Þ

dh22

dt¼ 0:1346ðq12�q22Þ ð61Þ

h22

dT 22

dt¼ 0:1346ð7:102QH2þq12ðT 1�T 22ÞÞ ð62Þ

where q0, q11, q12, q21, q22, q32, represent flows that havebeen normalized to take a value of 1 when they are activeand 0 otherwise (that is, they are discrete inputs). Flowrates q23 and q31 represent continuous normalized flowsand the rate of heat addition by the heaters namely QH1

and QH2 take discrete values of 0 or 1. The interpretationof all other variables is available from Fig. 9.

The control objective of this process is to obtain warmwater while maintaining levels in Tank-1 and Tank-3within prescribed upper and lower limits. The operationof Tank-21 and Tank-22 is governed by discrete decisions,which may be modeled using discrete states. Each of thetwo tanks must fill up to a height of 15 cm and heated toa temperature of 25 �C starting from 10 �C before theycan discharge to Tank-3. Once the tanks are ready to dis-charge the controller decides to discharge or the tankscan merely wait to discharge. However, once a particulartank begins discharging it cannot be interrupted until thewater level drops to 5 cm. Simultaneous filling and dis-charging is also not permitted. Further, during dischargingof a particular tank, the corresponding heater is turned off.

145

Also, Tank-21 and Tank-22 cannot be simultaneously dis-charged. However, they may be filled simultaneously.

Colmenares et al. [25] used the MLD framework tomodel this system. However, a large numbers of auxiliaryvariable were arbitrarily dropped from the MLD formula-tion to enable reasonable computation times. We model thehydraulic process using the proposed partially linearizedmodel by linearizing the nonlinear Eqs. (55)–(62) arounda fixed operating point characterized by continuous vari-ables. The operating point considered here is identical tothat used by Colmenares et al. [25] at q23 = q31 =0.5, T3 = 20 �C, T1 = 15 �C, T21 = T22 = 25 �C, andh1 = h21 = h22 = h3 = 15 cm. The model consists of eightcontinuous states (h1,h3,T1,T3,h21,T21,h22,T22), two discretestates OFFLOAD_T21 and OFFLOAD_T22, two continu-ous inputs (q23 and q31), and seven discrete inputs(q0,q11,q12,q21,q22,QH1,QH2). We introduce 7 extra logical(binary) variables (d1,d2,d3,d4,d5,e1,e2) during the problemformulation. Of these logical variables, d1 and d3 indicatethe maximum levels in Tank-21 and Tank-22, respectively.While d2 and d4 represent the corresponding minimum lev-els, d5 represents the permission to begin offloading eitherTank-21 or Tank-22. A particular tank offloads only whend5 equals 1. As d5 is a common variable for both tanks, wehave assigned a higher priority to offload Tank-21 in caseof a tie. Logical variables e1 and e2 keep information aboutthe status of loading/offloading of Tank-21 and Tank-22,respectively and hence are directly related to the discretestates. The logical expressions are then converted to linearinequalities. Fifty-six linear inequality constraints havebeen introduced to convert the decision-making into

12

18

24

T3 (

deg.

C)

10

15

20

h 3 (cm

)

0 1 2 3

10

15

20

Ti

h 1 (cm

)

Fig. 10. Control of temperature (T3) and level (h3) in Tank-3 and level (h

inequalities. Online Electronic Annex-II presents all logicalexpressions necessary to implement the logical decision-making.

Since the current application was based around a singleoperating point, we used only a single partially linearizedmodel. The original nonlinear dynamic Eqs. (55)–(62) areconsidered as the ‘‘plant’’ model and the partially linear-ized model as the controller model. Note that the PLCdecision-making scheme is dependent on the states of thesystems as discussed above. This necessitates immediatechanges in the status of discrete inputs as soon as the statesreach their threshold, which in turn compels selection ofprediction horizon (p) equal to control horizon (m) in theMPC problem. We use a sampling time, ts = 30 s, predic-tion horizon and control horizon of unity (p = m = 1),which result in an online optimization problem with 14 bin-ary variables, 2 continuous variables and 90 constraints.Model predictions are corrected using constant outputfeedback. We consider setpoint changes in the temperatureof the exit flow as 17 �C during first hour, 20 �C during sec-ond hour, 15 �C for the next two hours, 20 �C for fifth hour22 �C for the sixth hour and 18 �C for last hour as well as15 cm as the levels in Tank-3 and Tank-1 throughout thesimulation. The simulation results of MPC are shown inFig. 10. The oscillatory nature of all variables is due to cyc-lic filling and emptying of Tank-21 and Tank-22. The pro-files of heights in Tank-1 and Tank-3 are also closer to theirsetpoints. However, we could not achieve the setpoint of22 �C in Tank-3. The reason for this is that the logical deci-sions are designed such that we cannot get water warmerthan 25 �C and we require nonstop constant out flow of

4 5 6 7

me (hour)

1) in Tank-1 of the hydraulic process plant during setpoint tracking.

146

water from Tank-3. During the same time, level of Tank-3also should be maintained around 15 cm, which necessi-tates the supply of cold water to Tank-3 from sump via aside stream. This issue is related to reachability of hybrid

Fig. 11. Manipulated inputs for control of temperature (T3) and level (h3) insimulation results corresponding to 1–3 h are shown from the total simulation

0

10

20

h 21 (

cm)

10

20

30

T21

(deg

.C)

0

10

20

h 22 (c

m)

1 1.5 210

20

30

Time

T22

(de

g.C

)

Fig. 12. Levels (h21 and h22) and temperature (T21 and T22) in Tank 21 and TanTank-3 and level in Tank-1. (For clarity, simulation results corresponding to

systems and is not addressed here. The simulation resultsfor 2 continuous and 7 binary manipulated variables areshown in Fig. 11. Profile of temperatures and heightsin Tank-21 and Tank-22 are shown in Fig. 12, which

Tank-3 and level (h1) in Tank-1 of hydraulic process plant. (For clarity,duration of 7 hours.)

2.5 3 3.5 4

(hour)

k 22 of the hydraulic process plant for control of temperature and level in1–4 h are shown from the total simulation duration of 7 h.)

147

demonstrate that the hybrid MPC controller indeed imple-ments the PLC-like decision-making in addition to themanipulation of the continuous inputs. The mean and stan-dard deviation for the computation time of the controlmoves are 8.1614 s and 5.0215 s, respectively.

5. Conclusion

Applications of hybrid systems are becoming increas-ingly common in the process industry. In this work, we pre-sented a framework for modeling of nonlinear hybridprocesses. The framework uses multiple partially linearizedmodel representation with blending of models using Bayes’theorem. This enables us to write equations for synthesis ofan MPC controller, which is similar in form to those com-monly used in the literature for control of conventionalcontinuous variables processes. Further, we demonstratemodeling and control on two applications namely thethree-spherical tank systems and the hydraulic process.The MPL model refrains from masking mixed and nonlin-ear terms using auxiliary variables, resulting in a controllaw that requires solution of an MINLP. If we were to pur-sue an MLD model the resulting control law would requiresolution of an MIQP, which would have a significantly lar-ger number of binary and mixed variables as well as con-straints relative to the MPL model. The increased size ofthe MIQP makes it computationally unattractive relativeto the MINLP with the MPL model.

The main hurdle in the optimal control of hybrid sys-tems is the requirement of an online solution to an MIN-LP. In the multiple model framework, the nonlinearbehavior is described by models with a defined structure.Thus, the MPC formulation results in an objective functionand constraints of a fixed structure regardless of the under-lying nonlinearity. One may therefore design optimizationalgorithms that exploit this feature. The computational effi-ciency already demonstrated with the current formulationpresents further opportunities to enable online control ofnonlinear hybrid systems. The multi-parametric optimiza-tion for control of hybrid systems is a promising approachthat significantly reduces the online computational expense[37,38]. Another area that needs to be investigated is han-dling noisy measurements and unmeasured disturbances.For example, the heuristic used in the 3-tank system toavoid the oscillatory behavior may not be effective in pres-ence of noise. This issue may be partly addressed throughdevelopment of state estimators such as in [39] and devel-opment of disturbance models.

Appendix I. Discretization of continuous time linear hybrid

dynamic system using the Explicit Euler method

Consider linearized hybrid system of the form (15)–(17).Eq. (15) may be discretized using explicit Euler integrationtechnique as follows:

xckþ1 ¼ xc

k þ tsðAðdkÞxck þ BðdkÞuc

k þ f ðdkÞÞ ðA-1Þ

From Eqs. (16) and (A-1), discrete-time equation for boththe continuous and the discrete states may be written asfollows:

xckþ1

xdkþ1

� �¼½Ics� ½0�½0� ½0�

� �xc

k

xdk

� �

þ ts½AðdkÞ� ½0�½0� ½0�

� �xc

k

xdk

� �

þ ts½BðdkÞ�½0�

� �uc

k þ ts½f ðdkÞ�

t�1s ½bdðdkÞ�

� �ðA-2Þ

where Ics is an identity matrix of size A. Eq. (A-2) may thenbe rewritten as

xkþ1 ¼ ðHN þ tsANðdkÞÞxk þ tsBNðdkÞuck þ tsf NðdkÞ ðA-3Þ

where AN(dk),BN(dk) and fN(dk) are time varying coefficientmatrices whose elements are polynomial of binary variables(element of vector dk). These matrices can be decomposedas follows

tsANðdkÞ ¼ An0 þ pn

1An1 þ pn

2An2 þ � � � þ pn

s Ans ðA-4Þ

tsBNðdkÞ ¼ Bn0 þ pn

1Bn1 þ pn

2Bn2 þ � � � þ pn

s Bns ðA-5Þ

tsf NðdkÞ ¼ f n0 þ pn

1f n1 þ pn

2f n2 þ � � � þ pn

s f ns ðA-6Þ

where pni ði ¼ 1; 2; 3; . . . ; sÞ is the ith term of representative

polynomial of binary variables. Eqs. (A-4, A-5,A-6) maythen be rewritten as follows

tsANðdkÞ ¼ LNk UN ðA-7Þ

tsBNðdÞ ¼ LNk CN ðA-8Þ

tsf NðdkÞ ¼ LNk

�f N ðA-9Þ

where,

LNk ¼ IN pn

1IN pn2IN . . . pn

s IN� �

;

is a function of discrete variables ði:e: dkÞ ðA-10Þ

UN ¼ AnT

0 AnT

1 AnT

2 . . . AnT

s

h iT

ðA-11Þ

CN ¼ BnT

0 BnT

1 BnT

2 . . . BnT

s

h iT

ðA-12Þ

�f N ¼ f nT

0 f nT

1 f nT

2 . . . f nT

s

� �T ðA-13Þ

IN is an identity matrix of size of Ani .

Thus, discrete-time equation for state can be written asfollows

xkþ1 ¼ HN þ LNk UN

� �xk þ LN

k CN

� �uc

k þ LNk

�f N ðA-14Þ

while the equation for output and mixed constraints re-main same as in Eqs. (29) and (30). Note that structurally,both Eq. (28) (exact discretization) and Eq. (A-14) (explicitEuler discretization) are same, the only difference appear-ing in the definition of Lk and LN

k . The structural definitionof Lk in case of exact discretization remains same for anysystem while in numerical integration techniques like expli-cit Euler structure of LN

k vary system to system. However,size of LN

k may be smaller than the size of Lk for the samesystem. The main draw back of numerical techniques (e.g.

148

Explicit Euler) is that a large step size (sampling time) mayresult in unstable behavior.

Appendix II. Supplementary data

Supplementary data associated with this article can befound, in the online version, at doi:10.1016/j.jprocont.2007.07.003.

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