Parameterized polyhedra approach for robust constrained generalizedpredictive control

Sorin Olaru, Didier Dumur and Sihem Tebbani

Abstract— The paper considers the discrete time-invariantlinear systems affected by input disturbances and construct theexplicit description of the constrained Generalized PredictiveControl (GPC) law taking in account the constraints existencefrom the design stage.

The explicit formulation of the predictive controllers givesa useful insight on the closed loop capabilities. The GPCis a special case of Model Predictive Control (MPC) whoseexplicit formulation is known to be a piecewise affine func-tion of state. As novelty the present work shows that thispiecewise linear dependence on the context parameters canbe found by exploiting the fact that the optimum of a min-max multiparametric program is placed on the vertices ofa parameterized polyhedron. As these parameterized verticeshave specific validity domains, the control law is expressed asa piecewise linear function of the current system parameters.

The resulting GPC law is formulated in terms of a look-uptable with two-degree of freedom polynomials in the backwardshift operator (also known as the RST formulation).

I. INTRODUCTION

Predictive Control is an optimization based control de-sign technique which enjoys a remarkable reputation forits simplicity and versatility. Its unconstrained formulationis known to result in a low complexity linear control lawwhich can further be completed with a mechanism [6] to dealwith constraints violation a posteriori. However, due to thetime domain formulation of the predictive laws, the inclusionof constraints from the design stage was investigated withexcellent results towards stability, feasibility or robustness[9]. The drawback is the relatively high complexity of theoptimization problem to be solved at each sampling time.

Improvements in the on-line computations can be achievedif the explicit solution of the multiparametric optimizationproblem from the background of the predictive law iselaborated. Thus at each sampling time, a piecewise linearfunction has to be evaluated instead of running an iterativeoptimization routine. In the nominal case corresponding witha quadratic optimization problem and linear constraints, theexplicit solution was investigated with success using an al-gebraic approach in [1], geometrical arguments [4], [11] anddynamic programming [4]. These alternatives with differentmaturity degrees converged towards similar formulations andrepresent valuable results.

In the case of robust MPC, the explicit solution is some-how more difficult to achieve as the optimization problem

This work was supported by the European Commission, Directorate forResearch.

All authors are with the Automatic Control Department,Supelec, Plateau de Moulon, 91192 Gif-sur-Yvette, France.sorin.olaru@ieee.org; didier.dumur@supelec.fr

is based on a min-max cost function. It was successfullytackled in the studies of [2], but the alternative methods hada small delay in presenting similar solutions.

The current work is trying to compensate this setbackby presenting an explicit solution for the robust predictivecontrol through geometrical arguments in the continuity of[11]. The approach is based on the concept of parameterizedpolyhedra [8] and related parameterized vertices, where theoptimal solution can be found. Another particularity will bethe specific focus on Generalized Predictive Control (GPC)which has a great success in industry applications [3].

In the following, Section 2 formulates the robust GPCproblem and the related optimization. The main result is pre-sented in Section 3 in terms of the explicit multiparametriclinear optimization problem defining the robust GPC law. InSection 4 the design procedure is applied for an applicationto a positioning benchmark including an induction machine.

II. ROBUST GENERALIZED PREDICTIVE CONTROL

A. GPC formulation

Model-based Predictive Control implies the idea of mini-mizing a cost function based on the predicted plant evolution.This strategy is also called the ”receding horizon principle”and differs from one algorithm to another by the plantmodel chosen or by the cost function considered. The RobustGeneralized Predictive Control (RGPC) is characterized bytwo major features:

• It uses a CARIMA plant model:

A(q−1) yt = B(q−1)ut−1 + ξt

/∆(q−1)︸ ︷︷ ︸

vt

(1)

with u, y the system input and output, ξt a centeredGaussian white noise, A and B polynomials in thebackward shift operator q−1 of respective degree na

and nb, and ∆ = 1 − q−1 the difference operator.• The cost function is penalizing the tracking error and

control effort over a receding horizon:

U =N2∑

j=N1

|wt+j − yt+j | +Nu∑j=1

λj |∆ut+j−1| (2)

with y(t + j) the output prediction, N1, N2 the predic-tion horizons, Nu the control horizon, λj the controlweighting factor and w the set-point. Note that the costfunction U corresponds to the sum of ∞ -norm termsin other robust MPC formulations.

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

MC2.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 428

Using this model, the j-step ahead predictor will be:

yt+j = Fj(q−1)yt + Hj(q−1)∆ut−1︸ ︷︷ ︸freeresponse

+

+ Gj(q−1)∆ut+j−1 + Jj(q−1)vt+j︸ ︷︷ ︸forced response

(3)

where the Fj , Gj ,Hj , Jj polynomials are solution of theDiophantine equations:

∆(q−1)A(q−1)Jj(q−1) + q−jFj(q−1) = 1Gj(q−1) + q−jHj(q−1) = B(q−1)Jj(q−1) (4)

and Jj(q−1) = Jj(q−1)∆(q−1) is multiplying the distur-bance vt+j . The disturbances are included in a polyhedraldomain containing the origin:

V = {v |Mv � l; l � 0} (5)

Remark: The coefficients of Gj represent in fact the stepresponse coefficients for the system described by the polyno-mials (A(q−1), B(q−1)). Similarly, Jj are formed with theimpulse response coefficients of the system described by thepolynomials (A(q−1), 1).

B. Robust GPC

Robust GPC has to consider the worst combination ofdisturbances and thus to solve at each sampling time a min-max optimization problem, then only the first component ofthe optimal sequence ku is effectively applied, as at the nextsampling time ”a receding horizon strategy” is followed:

minku(t)

maxkv(t)

N2∑j=N1

|yt+j − wt+j | +Nu∑j=1

λj |∆ut+j−1| (6)

with the vectors ku ={∆ut, . . . ,∆ut+Nu−1

}and kυ =

{vt+1, . . . , vt+N2}. The optimization (6) has to be solvedsubject to diverse types of constraints rising from operationalor performance considerations. These are represented gener-ally as a set of linear inequalities:

⎧⎨⎩

Cku + Dkv � d + Ep,Mvt+k � l, k = 0, . . . N2 − 1yt+N2 ∈ Yf

(7)

where p = [ut−1 · · ·ut−nbyt · · · yt−na wt+N1 · · ·wt+N2 ]

T ,Yf is a terminal set and the terminal constraint in (7) isimposed from the stability considerations.

Remark: The terminal constraints in the classical GPC for-mulations are represented by equality constraints at the endof the prediction horizon. In the robust GPC, such constraintsare very difficult to fulfill as they cancel the control degreesof freedom implying severe feasibility problems.

The constrained optimization (6-7) provides a robust con-trol sequence but is quite conservative as it is consideredfor all disturbance realizations ignoring that measurementsare available as time progresses. The control potential is

improved if a feedback approach is adopted resulting in anested min-max formulation:

min∆ut

{λ0 |∆ut| + maxvt

{|yt+1 − wt+1| +

+ min∆ut+1

{. . . + min∆ut+Nu−1

{λNu−1 |∆ut+Nu−1|+

+ maxvt+Nu−1,...,vt+N2−1

N2∑k=Nu

|yt+k − wt+k|}}

· · ·}}(8)

subject to (7). This represents a ”closed loop” formulation[12], avoiding the feasibility problems in comparison withthe ”open-loop” formulations.

The robust GPC formulation is based on the on-linesolving of the associated min-max optimization problem (6-7) or (7-8). Both the cost function and the set of constraintsdepend on the parameter’s vector p. The alternative solutionto the on-line iterative routines is to explicitly formulate off-line the optimal solution k∗

u(p) further evaluate it on-line.

C. The linear multiparametric optimization

The disturbances in the form (6) can be considered usingthe extremal possible combinations of vertices in V for eachprediction stage completing the sequence kυ:

vt ∈ V ⊂ R ⇒ kυ ∈ V N2 ⊂ RN2 (9)

The optimum for the inner optimization in (6) will be onthe border of the feasible domain, more precisely on one ofthe vertices of V N2 as long as it is defined as a polytope andthe objective function is convex. Thus (6) becomes:

minku

maxvkυl

U(p,ku,kυl)

subj.to : Ain1ku + Ain2kυl� bin + Binp

l ∈ L,kυl∈ V N2

(10)

with L = {1, 2, . . . , Nv} and Nv = 2N2 is the number ofvertices in V N2 . This means that the inner optimization in (6)will act only on the set of vertices of V N2 . The inequalitiesin (10) represent a compact rewritten of those in (7).

The main impediment in finding the explicit formulation ofthe solution for (10) is the expression of the cost function,given as a collection of ∞-norm terms. In order to avoidthe inherent difficulty of handling it, an equivalent linearprogram (LP) [5] formulation must be achieved. The op-timization problem is equivalent with the minimization ofthe sum of bounds for these terms. This is resumed by thefollowing result where the cost function is considered asa sum of linear terms in the vector of unknowns ku andparameters p (the disturbance vector is ignored at this stage).

Proposition 1: Relations (a) and (b) are equivalent:

(a)K(p) = minku

U(ku,p) = minku

n∑i=1

‖Siku + Pip + si‖∞subject to Ainku � bin + Binp

(b) K(p) = minρ,{σi},ku

ρ

subject to

⎧⎪⎪⎨⎪⎪⎩

−1σi � Siku + Pip + si � 1σi, 1 � i � nn∑

i=1

σi � ρ

Ainku � bin + Binp

429

with σi, ρ ∈ R, 1 a unit entries vector of appropriate lengthand n the number of ∞-norm terms in U .

With the previous transformations the optimization (6) canbe rewritten as a compact multiparametric linear program:

ku ∗ (p) = arg minρ,ku,{σj

i}ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−1σji � Siku + Pip + Wikυl

+ si � 1σji ,

1 � i � n, 1 � l � Nv⎡⎢⎢⎢⎢⎢⎣

n∑i=1

σ1i

...n∑

i=1

σNvi

⎤⎥⎥⎥⎥⎥⎦

� 1ρ

Ain1ku + Ain2kυl� bin + Binp,

1 � l � Nv

(11)

III. THE EXPLICIT SOLUTION

In the following, the closed form of the RGPC law isthe main objective which implies the construction of theexplicit solution of multi-parametric linear programs (MPLP)(11). A procedure will be proposed focusing on the set ofconstraints and its geometrical correspondence [11] usingthe concept of parametrized polyhedron. Thus the resultingalgorithm differs from the solutions based on branch andbound methods or other mixed integer linear solvers, beingmainly focused on the enumeration of the edges of anaugmented dimension polyhedron. In order to get familiarwith these concepts, some brief definitions are introduced(details can be found in [7], [8]). It is noted in this part x =ku for the simplicity of development (x ∈ R

n,p ∈ Rm).

A. Parameterized Polyhedra

A system of linear constraints as in (11) defines a poly-hedron [8]:

P = {x ∈ Rn |Aeq x = beq;Ainx � bin} (12)

or in a dual Minkowski representation of generators:

P = conv.hull {x1, ..,xv}+ cone {y1, ..,yr}+ lin.spaceZ(13)

where conv.hullX denotes the set of convex combinationsof points in X , coneY denotes nonnegative combinationsof unidirectional rays and lin.spaceZ represents a linearcombination of bidirectional rays. It can be also rewritten:

P ={

x ∈ Rn|x =

v∑i=1

λixi +r∑

i=1

γiyi +l∑

i=1

µizi

}

0 � λi � 1,v∑

i=1

λi = 1 , γi � 0 , ∀µi

(14)A parameterized polyhedron is defined in the implicit form

by a finite number of inequalities and equalities with the note

that the affine part depends linearly on a parameter vector p:

P ′(p) ={x ∈ R

n |Aeq x = Beqp+ beq;Ainx � Binp + bin}

={

x(p)| x(p) =v∑

i=1

λi(p)xi(p)+

+r∑

i=1

γiyi +l∑

i=1

µizi

0 � λi(p) � 1,v∑

i=1

λi(p) = 1 , γi � 0 , ∀µi

(15)

with zi the lines, yi the rays, xi the vertices and µi, γi, λthe corresponding coefficients.

Remark: Only the vertices are concerned by theparametrization of the polyhedron (parameterized vertices- xi(p), the rays and the lines do not change with theparameters’ variation.

The parameterized polyhedron P ′(p) has a non-parameterized correspondent in an augmented space:

P ′ ={(

xp

)∈ R

n+m|[Aeq/in

∣∣− Beq/in

] ( xp

)= beq/in

} (16)

The form (15) faces an important difficulty, the de-scription of the parameterized vertices xi(p) and theirvalidity domains. This can be achieved using the factthat they correspond to m-faces in the augmented data(Rn)+parameter(Rm) space for the polyhedron (16):

xi(p) = Projn(Fm

i (P ′) ∩ S(p))

(17)

where Projx (.) projects the combined space Rn+m onto the

original space Rn, S(p) is the affine subspace:

S(p) ={(

xp

)∈ R

n+m∣∣p = p

}(18)

and Fmi (P ′) is a m-face of P ′ found as the intersection

between P ′ and the supporting hyperplanes [8].For each face of the polyhedron P ′, a set of active

constraints is well defined, resulting in the fact that eachpoint

(xi(p)T pT

)T ∈ Fmi (P ′) lies in a subspace of

dimension m and thus x and p are related by:[

Aeq

Aini

]x =

[A′

eq

A′ini

]p +

[beq

bini

](19)

where Aini , A′ini

, bini are the subset of the inequalitiesdefined previously, satisfied by saturation. If the matrix[

ATeq AT

ini

]Tis not invertible, it corresponds to faces

Fmi (P ′) where for one given p more than one point x ∈ n

is feasible and such combinations do not match a vertex ofP ′(p) and are the zones where P ′(p) changes its shape. Inthe invertible case, the dependencies are:

xi(p) =[

Aeq

Aini

]−1 [ A′eq

A′ini

]p +

[Aeq

Aini

]−1 [ beq

bini

]

(20)For the implementation of these theoretical results, an enu-

meration of the m-faces must be available. If the projection

430

of this face on the parameters space is a polyhedron ofdimension less than m, then the already mentioned conditionof an m-face that does not define a vertex of P ′(p) is active.Otherwise the projection corresponds to the validity domainV D of the parameterized vertex given by (20).

B. Explicit solution of multiparametric LP (MPLP)

Recalling the problem to be solved similarly to (11):

x∗(p) = minx

fT x

subject to Ainx � Binp + bin

(21)

one can use the geometrical description of the feasibledomain in terms of a parameterized polyhedron in orderto construct the explicit solution. The result (which revealsalso the piecewise affine dependence on the parameters) isresumed by the following proposition, (a generalization forparameterized polyhedra of the geometrical description of anLP given in relation with the Chernikova algorithm [7]):

Proposition 2: The solution of a MPLP optimizationproblem is characterized by the followings:

a) If there exists a bidirectional ray z such that fT z = 0 ora unidirectional ray y such that fT y � 0, then the minimumis unbounded;

b) For the subdomains of the parameter spaceDifez ∈ R

n where the associated polyhedronP = {x|Ainx � Binp + bin,p ∈ Difez} is empty,the problem is infeasible;

c) All bidirectional rays z are such that fT z = 0 and allunidirectional rays y are such that fT y � 0. For the subdomains Dk where the minimum:

min{fT xi(p)|xi(p) vertex of P(p)

}

is attained by a constant subset of vertices of P (p), thecomplete solution is:

Sk(p) = conv.hull {x∗1k(p), . . . ,x∗

sk(p)}++ cone {y∗

1, . . . ,y∗r} + lin.spaceP (p)

where x∗i are the vertices corresponding to the minimum and

y∗i are such that fT y∗

i = 0�Adapting this result to our goal of finding the explicit

solution for the LP in robust GPC yields to:Proposition 3: The solution of a MPLP within robust GPC

satisfies:a) The problem is infeasible for the subdomains Difez ∈

Rn where no parameterized vertex is available;b) Subdomains Dk are defined for the zones where the

solution Sk(p) = conv.hull {x∗1k, . . . ,x∗

sk} is given by thesame set of parameterized vertices satisfying:

fT x∗1k = . . . = fT x∗

sk == min

{fT xi(p)|xi(p) vertex of P(p)

}

Remark: As the parameters in (21) vary in the parameterspace, the vertices of the optimization domain may split,shift or merge. The optimum will follow this evolution in theparameter space as it is a continuous function of parameters.

From a practical point of view the implementation of thisresult is direct and follows the steps:

1) Find the expression of the parameterized feasible do-main in the augmented data+parameter space:

Ainx � Binp + bin ⇔ [Ain −Bin

] [ xp

]� bin

2) Find the m-faces where m is the dimension of theparameter space.

3) Retain only those corresponding to parameterized ver-tices by ignoring those with non-invertible projection on theparameter space

4) Compute validity domain Dk for each parameterizedvertex

5) Compare each pair of vertices. In the case of a non-empty intersection of their validity domains split them usingthe linear cost function. The final expression will be aunion of regions corresponding to the parameterized verticescharacterizing the optimum.

Remark: The difference of two convex domains is not aclose operation and the output of the procedure is a union ofconvex sub-domains of the parameters’ space, not necessarycovering entire R

m (step 4).From the robust GPC point of view, the difference:

ℵ = m\ {∪Dk; k = 1..nD} (22)

describes the regions of infeasible parameters.A special attention must be given to the step 5 with

the iterative comparison of the vertices and their validitydomains. A possible routine may be based on the followingprocedure:

procedure CutDomains (V D: the set of validity domains)n=cardinal (V D); i=1; j=2;while i < n + 1

while j < n + 1if V Dj ∩ V Di = ∅

if fT xi � fT xj then V Dj = V Dj − V Di

else V Di = V Di − V Dj

endifj = j + 1;

endifendi = i + 1

end

C. Explicit robust GPC law in RST form

The result of the presented algorithm will be a set of rvalidity domains and for each such domain a linear functioncan be designed as corresponding to the analytical solution ofthe optimization (11). The overall function will be piecewiselinear and continuous and it can be represented by a look-uptable (Table I).

Remark: In the case of the RGPC law, the parameters arethe past inputs, past outputs and future setpoints and thusthe control law for each validity domain can be rewritten asa polynomial law in the delay operator q−1.

Indeed, one can recall from section II, the fact that theparameters vector was defined as:

p = [ut−1 · · ·ut−nbyt · · · yt−na wt+N1 · · ·wt+N2 ]

T

431

TABLE I

LOOK-UP TABLE OF EXPLICIT MULTIPARAMETRIC LP SOLUTION

Validity domain Linear lawV D1 : (M1p < m1) x = F1p + f1

. . . . . .V Dr : (Mrp < mr) x = Frp + fr

and thus the optimal linear laws in table I can be rewrittenas polynomial control laws due to the fact that the GPC lawapply effectively only the first control action:

u(t) = T (q) w(t) − S(q−1) u(t) − R(q−1) y(t) + f

The complete explicit formulation of the constrained GPClaw is stated as a collection of such affine RST laws:

TABLE II

LOOK-UP TABLE OF EXPLICIT RST LAWS

Validity domain GPC lawV D1 : (M1p < m1) ut = T1wt − S1ut − R1yt + f1

. . . . . .V Dr : (Mrp < mr) ut = Trwt − Srut − Rryt + fn

Once this look-up table is available, an efficient position-ing mechanism can be constructed such that the followingon-line routine can find the control action according to therobust GPC philosophy.

Algorithm (on-line solver)1. Find the corresponding set Dk, k = 1..r for the current

p. Return infeasible if no Dk found.2. Compute uGPC using the RST law for Dk.3. Update the set of context parameters p.4. Restart from 1 with the new p.

IV. EXAMPLE

Consider a benchmark system for the positioning controlof asynchronous machines [10]. The predictive control lawwill be designed starting from a model of the mechanicaland electrical part together with the zero order hold and thesampler - Fig. 1. However, the influence of the electricalpart can be neglected, compared with the mechanical timeconstant. The mechanical part is characterized by the motorinertia J , the friction coefficient f and Γ the load torque.The discrete time transfer function between the electro-mechanical torque and the angular displacement for a sam-pling time Te = 14 × 76.6µs = 1, 0724ms is given by:

θ(q−1)u(q−1)

=q−1B(q−1)

A(q−1)=

10−4(0.821q−1 + 0.8206q−2)(1 − q−1)(1 − 0.998q−1)

.

Fig. 1. Model for the MPC design.

In order to validate different controllers and their capabil-ities, a position benchmark cycle is used, where the speed

reference increases to the nominal speed and descends to zerowith different profiles. This cycle enables to test the positionloop behavior at very slow and zero speeds at nominal speedand during parabolic position tracking.

Using classical GPC procedures (quadratic cost functions),in the unconstrained case, a two degree of freedom RSTcontroller can be designed as in Fig. 2.

Fig. 2. RST controller scheme.

The time-domain simulation of such a control law basedon a prediction horizon of N2 = 5, Nu = 2 and a end-pointconstraint can be visualized in Fig. 3.

−8 −7 −6 −5 −4 −3 −2 −10

20

40

60

temps

Pos

ition

−8 −7 −6 −5 −4 −3 −2 −1

−20

0

20

temps

Con

trol

−8 −7 −6 −5 −4 −3 −2 −1 0−0.01

0

0.01

0.02

temps

Err

or

Fig. 3. Classical GPC controller.

The same control idea can be used for the constrained (onthe overshoot) case:

yt+k � wt+k,∀1 � k � N2

but it doesn’t offer a robust constraint satisfaction for distur-bances affecting the model:

A(q−1) yt = B(q−1)ut−1 + vt

−0.001 � vt � 0.001

The closed loop system with the classical GPC is farfrom being optimal (Fig. 4a). The tracking error is importantand more than that, it is not always positive (Fig. 4b) asdemanded by the constraints stated previously. The solutionis to implement a robust GPC scheme as presented in thetheoretical part of this article. By applying the parameterizedpolyhedra approach presented earlier, one can obtained forN2 = 4; Nu = 1 an explicit GPC law expressed by a

432

0 1 2 3 4 5 6 70

20

40

60

temps

Pos

ition

0 1 2 3 4 5 6 7

−60

−40

−20

0

20

temps

Con

trol

0 1 2 3 4 5 6 7 8−0.01

0

0.01

0.02

temps

Err

or

(a) Position,control and error

1 2 3 4 5 6 7−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

temps

Err

or

(b) Zoom on the error

Fig. 4. Classical GPC with constraints

piecewise linear law of 4 RST polynomials in the backwardshift operator.

The validity domains of these control laws are unionsof convex sets in the space of parameters. An interestingaspect, in this case, is that the RST laws are not containingaffine terms. This is due to the fact that the inequalityconstraints are formulated exclusively on previous inputs,previous outputs and future references and no constant termsinterfere. The time simulation confirms the improvement ofthe closed loop behavior as it can be seen in Fig. 5 (there isno overshoot, the constraints are robustly fulfilled).

0 1 2 3 4 5 6 70

20

40

60

temps

Pos

ition

0 1 2 3 4 5 6 7

−60

−40

−20

0

20

temps

Con

trol

0 1 2 3 4 5 6 7 8−0.01

0

0.01

0.02

temps

Err

or

Fig. 5. Robust GPC law with constraints. Prediction horizon N2 = 4.

If the prediction horizon is augmented to N2 = 14 one canobserve that the control signal is less sensitive to the presenceof disturbances. This is obtained with a deterioration of thetracking error (Fig. 6) which represents a classic compromisein the robust control theory.

V. CONCLUSIONS

The paper presented a geometrical approach for the robustGeneralized Predictive Control under constraints, confirmingthe formulation of the optimal sequence as a solution of amulti-parametric linear problem.

The explicit solution of this problem was synthesized bymeans of parameterized polyhedra. This approach proposesan alternative to the recent methods presented in the literature

0 1 2 3 4 5 6 70

20

40

60

temps

Pos

ition

0 1 2 3 4 5 6 7

−60

−40

−20

0

20

temps

Con

trol

0 1 2 3 4 5 6 7 80

0.02

0.04

0.06

temps

Err

or

Fig. 6. Robust GPC law with constraints. Prediction horizon N2 = 14.

with an insight on the geometry of the feasible set. Itsadvantages might be the fact that optimum lies on the pa-rameterized vertices providing a direct constant linear affinedependence in the context parameters. The disadvantagesare linked to the fact that the procedure of identifying theparameterized vertices might be computationally demandingfor long prediction horizons even if on-line procedures.

REFERENCES

[1] A. Bemporad, M. Morari, V. Dua, E. Pistikopoulos, ”The explicit linearquadratic regulator for constrained systems”, Automatica vol.38, 2002,pp. 3-20.

[2] A. Bemporad, F. Borelli and M. Morari. ”Robust Model PredictiveControl: Piecewise Linear Explicit Solution”. Proc. European ControlConference, pp. 939-944, Karlsruhe, 2001.

[3] E.F. Camacho and C. Bordons, Model Predictive Control, Springer,2003.

[4] G.C. Goodwin, M.M. Seron and J.A. De Dona. Constrained Controland Estimation. Springer-Verlag, London, 2004.

[5] E.C. Kerrigan, and J.M. Maciejowski. ”Feedback min-max modelpredictive control using a single linear program: robust stability andthe explicit solution ”. International Journal of Robust and NonlinearControl , vol. 14-4, 395-413, 2004.

[6] M. V. Kothare, P.J. Campo, M. Morari, and C. N. Nett. ”A unifiedframework for the study of anti-windup designs”, Automatica, vol.30, pp. 1869-1883, 1994.

[7] H. Leverge. ”A Note on Chernikova’s Algorithm”. Technical Report635, IRISA, France, 1994.

[8] V. Loechner, and D.K. Wilde. ”Parameterized Polyhedra and theirVertices”. Int. Journal of Parallel Programming, 25(6), December,1997.

[9] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert, ”Con-strained model predictive control: Stability and optimality”, Automat-ica, Vol.36, pp. 789-814, 2000.

[10] E. Mendes, and J.P. Barbot, ”Benchmark transitique rapide”, APIIJournal Europen des Systmes Automatiss, Vol. 36-5, pp. 701-708,2002.

[11] S. Olaru and D. Dumur, ”A Parameterized Polyhedra Approach forExplicit Constrained Predictive Control”, 43rd IEEE Conference onDecision and Control, Bahamas, 2004.

[12] P.O.M. Scokaert, and D.Q. Mayne. ”Min-max Feedback Model Predic-tive Control for Constrained Linear Systems”. IEEE Trans. AutomaticControl, 43-8, pp. 1136-1142, 1998.

433