Model Predictive Control of MLD Models with Integrators J.L. Villa, Member IEEE M. Duque, A. Gauthier and N. Rakoto-Ravalontsalama, Member IEEE

Abstract— Model Predictive Control (MPC) based on MixedLogical Dynamical (MLD) Systems was proposed as a suitable approach to the control problem of Hybrid Systems. In this paper we study some extensions of the basic structure in orderto gain precision in the regulation problem. We study therobustness of the resulting controlled system using simulationsof a three-tank benchmark.

I. INTRODUCTION

Continuous dynamical systems interacting with discrete-event systems are called Hybrid Systems. This class of

systems has been studied in several ways, and it is an actual research subject in control and computer science communities, c.f. [10] and [7].

Recently, Bemporad and Morari in [1] proposed MixedLogical Dynamical (MLD) systems as an alternative representation of Hybrid Systems which is suitable forsolving control problems using Model Predictive Control(MPC) ideas. On the other hand, MLD models areequivalent to PWA models as has been proved in [6]. Thisequivalence is useful because of several analysis tools developed for PWA systems. Several algorithms have been proposed in the literature in order to translating MLDsystems into PWA systems efficiently, cf. [2] and [9].

Using MPC, the output of the controlled systemconverges to the reference value if this one is a steady-statepoint of the closed-loop system. For linear constrainedsystems, every point in the reachable space of the systemcan be a steady-state point, and the convergence of thecontrolled system is assured under certain conditions, see [4] and references therein. In hybrid systems, every reachable point in the state-space is not necessarily a steady-state point and the error between the reference value and the output ofthe system is not always zero.

For this reason we study the introduction of integrators inthe formulation problem, in particular their explicitintroduction in the MLD model, and we apply the results ina three-tank benchmark .

The paper is organized as follows. In Section II wepresent the general MLD model and the formulation of theMPC problem, then in Section III, the extended formulationwith integrators is presented, in Section IV the example of a

three-tank benchmark is presented and some simulationresults are commented.

II. MPC CONTROL WITH MLD MODELS

A. Mixed Logical and Dynamical (MLD) SystemsThe idea in the MLD framework is to represent logical

propositions as equivalent integer expressions. MLD form is obtained by three basic steps. The first step is to associate abinary variable {0,1} with a proposition S, that may be true or false. is 1 if and only if proposition S is true. A composed proposition of elementary propositions S1,…,Sqcombined using the boolean operators like AND(^), OR ( ),NOT(~) may be expressed like integer inequalities overcorresponding binary variables i, i=1,…,q.

The second step is to replace the products of linear functions and logic variables by a new auxiliary variable z =aTx where aT is a constant vector. The z value is obtained

by mixed linear inequalities evaluation.The third step is to describe the dynamical system, binary

variables and auxiliary variables in a linear time invariant(LTI) system.

An example illustrating this procedure can be applied to a system described by two equations as follows:

0.8 ( ) ( ) if ( ) 0( 1)

0.8 ( ) ( ) if ( ) 0( ) [ 10,10], ( ) [ 1,1]

x t u t x tx t

x t u t x tx t u t

Using a binary variable [ ( ) 1] [ ( ) 0]t x t , the systemcan be re-written as,

( 1) 1.6 ( ) ( ) 0.8 ( ) ( )x t t x t x t u tUsing a variable ( ) ( ) ( )z t t x t and some relations, a valid

representation of the system is, ( 1) 1.6 ( ) 0.8 ( ) ( )x t z t x t u t

subject to (s. t.) - 10 -10

( 10) -10

- 1010(1- )

- - 10(1- )

xxzz

z xz x

José Luis Villa is with Universidad Tecnologica de Bolivar, Km 1 via Turbaco, Cartagena de Indias, Colombia. (e-mail:jvilla@unitecnologica.edu.co)

A hybrid system MLD described in general form isrepresented by (1).

(1)1 2 3

1 2 3

2 3 1 4 5

( 1) ( ) ( ) ( ) (

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x k Ax k B u k B k B z k

y k Cx k D u k D k D z k

E k E z k E u k E x k E

+ = + + +

= + + +

+ + +

)

l

Mauricio Duque and Alain Gauthier are with Universidad de los Andes,Kra65B No 17A-11, Bogota D.C., Colombia(e-mail: {maduque,agauthier}@uniandes.edu.co.)

Naly Rakoto-Ravalontsalama is with Ecole des Mines de Nantes, 4 rueAlfred Kastler, 44307 Nantes, France, (e-mail: rakoto@emn.fr).

where are the continuous and [ ] {0,1}cT T n nC lx x x= ×R

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TA3.1

0-7803-9354-6/05/$20.00 ©2005 IEEE 641

C. MPC with MLD modelsbinary states, u u are the inputs,

the outputs, and ,, represent the binary and continuous auxiliary

variables, respectively. The constraints over state, input,output, z and variables are included in the third term in(1).

[ ] {0,1}cT T m mC lu= ×R{0,1}c lp p×R

l

[ ]T TC ly y y=crz R

{0,1} lr

2

11 2

1

1

ˆ, , ) ( )( ( )

( )((1 ) (

. .

(̂ ) ( ( 1),(1 ) ( )

( )

( )

u

N

uk N

N

k

N k y t k t

k z u t

s t

y k g y k z u k

y kA bu k

=

=

= +

+

=

2

1 2

( ))

))

)

k+ +min (J N N

In [1], Bemporad and Morari show that MLD systems are well posed for optimal control and, in particular, predictivecontrol.The predictive control problem can be posed as,

1 210

3 4

5

11 2

0{ } 0

2 2

2

1 2 3

1 2 3

2 3 1

min ( , ( )) || ( ) || || ( ) ||

|| ( ) || || ( ) ||

|| ( ) ||

. .

( )

( 1) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

T

TT

e Q e Qv k

e Q e Q

e Q

e

J v x t v k u k

z k z x k x

y k y

s t

x T x

x k Ax k B v k B k B z k

y k Cx k D v k D k D z k

E k E z k E v k

=+

+ +

+

=

+ = + + +

= + + +

+ 4 5( )E x k E+ +

2

(3)B. Model Predictive Control (MPC)

In the MPC structure, the future outputs are predicted ateach instant t using the process model. The future output andcontrol values are computed by an optimization procedure.A general scheme for MPC problem is shown in Fig. 1.

The optimal solution v* to this problem is chosen as control signal of the system u(t). That means, the whole problem should be solved at each iteration, and, due to integer variables , a mixed integer programming algorithmshould be used. Fig. 1. General Scheme of MPC

The set of future control values [u(t) u(t+1) . . . u(t + N+1)] is computed such that the process output stays as close as possible to the reference trajectory. The control value u(t) is chosen and applied to the process, and the problem iscomputed in the next iteration. The strategy of the MPCframework is depicted in Fig. 2.

III. EXTENDED FORMULATION

As stated in section I, a given reference is not always a steady-state value of the system. In this section we study theexplicit introduction of integrators into the predictivescheme in order to reduce the oscillations and the erroraround the set-points.Two cases are considered, the first one is the case where the yC errors between the outputs and the references are directly integrated and introduced to the optimizer algorithm (Model1). For the second case we introduce mC integrators at theoutput of the optimizer (Model 2). The two cases are depicted in Fig. 3

Fig. 2. MPC StrategyThe common MPC formulation is as follows,

(2)1

,

k

k Fig. 3. Predictive MLD Control with Integrators Scheme

A. Model 1. Error Signal IntegrationIntroducing integrators to the error signals betweenreference and process output, the extended MLD modelincluding the dynamics of integrators becomes as follows,where,

N1 and N2 are the minimum and maximum cost horizons, N1has a particular meaning when the system has a dead time. 1 21

0

3 4

5

11 2

0{ } 0

2 2

2

min ( , ( )) || ( ) || || ( ) ||

|| ( ) || || ( ) ||

|| ( ) ||

. .

T

TT

e Q e Qv k

e Q e Q

Q

J v x t v k u k

z k z x k x

k

s t

=+

+ +

+

2

Nu is the control horizon.The coefficients (k) and (k) are sequences that consider the future behavior.

(k) is the reference trajectory.

642

1

1

2 3

2 3

2 3 1 4

( 1) ( ) 0 ( )0

( 1) ( ) ( )

( ) ( )

( )( ) ( ) 0 0

( )

m

x k x k B u kA

Tm D TmTm C Ik k

B Bk z k

Tm D T D

u kE k E z k E E

ref k

+= +

××+

+ +× ×

+ + 5

( )

( )

x kE

k+

ref k+

5+

(4)

B. Model 2. Control Signal IntegrationIntroducing integrators to the continuous control signals, theMLD model including the dynamics of integrators becomesas follows,

1 210

3 4

5

11 2

0{ } 0

2 2

2

min ( , ( )) || ( ) || || ( ) ||

|| ( ) || || ( ) ||

|| ( ) ||

. .

T

TT

e Q e Qv k

e Q e Q

Q

J v x t v k u k

z k z x k x

k

s t

=+

+ +

+

(5)

2

1 01 03

2 02 03

13 1 3 13

23 2 3 23

( )

( )

( )

( )

Z Z Z V

Z Z Z V

Z h h V

Z h h V

=

=

=

=

11

2 3

1 1

( 1) ( ) ( )

0( 1) ( ) ( )

( ) ( )0 0

( ) ( )( ) 0

( ) ( )

br brr

r r r

brr br

r

x k x k B u kA B

Tm IIu k u k u k

B Bk z k

x k u ky k C D D

u k u k

+= + +

×+

+ +

= + 2 3

2 3 1 4 1

( ) ( )

( ) ( )( ) ( ) 0

( ) ( )b

b rr

D k D z k

u k x kE k E z k E E E E

u k u k

+ +

+ +

C. Computational ComplexityThe main challenge to be approached using the MLDframework is the computational complexity introduced bythe Mixed Integer Quadratic Programming (MIQP) or Mixed Integer Linear Programming (MILP) algorithms usedby the MLD methods. In this case an increasing on thecontrol horizon induces an exponential increasing on thecomputation time.This topic has been addressed by several authors in theliterature, c.f. [3], where efficient techniques have beendeveloped in order to compute explicit solutions of the closeloop controller.

IV. SIMULATION RESULTS

The three-tank system was presented in [5] as a suitable benchmark for testing algorithms and theories related tohybrid systems.

Fig. 4. Three Tank SystemThis system has three tanks each of them interconnected

with another as depicted in Fig. 4. Each tank is characterized

by its capacity Ci and each valve by its resistance Ri . The state variables of the system are the three levels of each tank hi. Each valve Vi can be open or closed.

The model is written using binary variables ( i) and relational expressions,

01 1 11 1

2 2 02 2

3 3 03 3 3

( )1

1 (

1 ( )

vv

v v

v v

Z h hh h

h h Z h h

h h Z h h

== >

= > =

= > =2)

1

2

1 1 13 11 1

1 1 13 1 1

( ) ( ) ( ) ( )( 1) ( ) * (s

L

q k h k Z k Z kh k h k T

C R C R C RC+ = +

1)

2 23 22 2

2 23 2 2 2

( ) ( ) ( )( 1) ( ) * (

q k Z k Z kh k h k Ts

C R C RC+ = )

3 133 3

3 13 3

23 1 2

23 2 1 3 2 3

( ) ( )( 1) ( ) * (

( ) ( ) ( ))

N

h k Z kh k h k Ts

R C R CZ k Z k Z kR C RC RC

+ = + + +

+ + +

We apply the different control models to this benchmark.Using the original algorithm proposed in [1] and tuning theweighting matrices of the formulation, some simulationresults of the level behavior in the tanks are shown in Fig. 5

Fig. 5. Simulation Results with the Original MLD ModelSimulation results using the model 1, with the integral of

the error signals, are shown in Fig. 6

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[2]A. Bemporad. Efficient Conversion of Mixed Logical Dynamical Systems into an Equivalent Piecewise Affine Form.IEEE Trans. Automatic Control, vol. 49, no. 5, pp. 832-838,2004

[3]A. Bemporad, F. Borelli, and M. Morari, “Optimal controllers for hybrid systems: Stability and piecewise linear explicit form,” in Proceedings of 39th IEEE Conf. On Decision andControl, pp. 1810—1815, Dec. 2000.

[4]C.E. Garcia, D.M. Prett and M. Morari. Model Predictive Control: Theory and Practice – A Survey, Automatica, 25, pp. 335-348, 1989.

Fig. 6. Simulation Results (Error Signal Integrator) In this case, the control system drives the process near to

zero, with a high frequency oscillation trying to reduce theerror in tank-3 level using the tank-1 level.

[5]B. Heiming, J. Lunze. Definition of the Three-Tank BenchmarkProblem for Controller Reconfiguration. European Control Conference, 1999

Simulation results using the model 2, with the integral ofthe control signals, are shown in Fig. 7

[6]W.P.M.H. Heemels, B. De Schutter, and A. Bemporad.Equivalence of hybrid dynamical models. Automatica, vol. 37, no. 7, pp. 1085-1091, July 2001.

[7] A. Van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems. Springer-Verlag. Springer-Verlag. LNCIS 251. 2000.

[8]J.L. Villa. Modeling and Control of Hybrid Systems: The MLDApproach. PhD. Thesis. Université de Nantes, France. 2004

[9]J.L. Villa, M. Duque, A. Gauthier and N. Rakoto-Ravalontsalama. A New Algorithm for Translating MLD Systems into PWA Systems. Proc. of 2004 American Control Conference (ACC'04), pp. 1208-1213, Boston, USA. 2004

[10] F. Wiedijk, O. Maler and A. Pnueli . Hybrid Systems:Computational and Control. HSCC2003. Springer-Verlag.LNCS 2623. 2003

Fig. 7. Simulation Results (Control Signal Integrator)

In this case the control has a good performance with highovershoot in tank-1 level whereas tank-3 level has a highoscillation. Other results can be found in [8]

V. CONCLUSIONS

In this paper we study the introduction of integrators inthe formulation of an MLD system to be used as model for aModel Predictive Control. We study two formulations, thefirst one uses the error signal integral, and the second oneuses the control signal integral. These two formulations have been tested in a three-tank benchmark.

The two models keep the signal around the referenceproducing an oscillatory behavior. Using the first model, thecontrolled system has higher oscillatory behavior, whereasusing the second model, the responses have high overshoot.

Ongoing work is focused on formal definition of the weighting matrices, and the study of the robustness and sensitivity of different MLD formulations.

REFERENCES

[1]A. Bemporad and M. Morari. Control of Systems Integrating Logic, Dynamics, and Constraints. Automatica 35, pp. 407-427.1999.

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