fAutomatica, Vol. 27, No. 3,(pp. 549-554, 1991Printed in Great Britain.

0005-1098/91 $3.00 + 0.00Pergamon Press pic

© 1991 International Federation of Automatic Control

A Class of Invariant Regulators for theDiscrete-time Linear Constrained Regulation

Problem*

Key Words-Discrete-time linear systems; Linear Constrained Regulation Problem; stability; positiveinvariance; variable regulator; linear programming.

Abstract-Stable dynamic systems admit positively invariantdomains associated to their Lyapunov functions. Conversely,some domains can be made positively invariant for systemswith state feedback controllers designed in such a way thatsome associated non-negative definite functions are bound todecrease. In particular, this approach can be used toestablish conditions on the gain matrix for LinearConstrained Regulation Problems (LCRP). We constructfixed and variable regulators easy to compute through linearprogramming, for a class of constrained linear systems.

IntroductionTECHNICALCONTROLlimitations have long been considered amajor problem in control engineering. Actually, most controlschemes do not integrate constraints in their design. Inpractice, control laws often have to be complemented byadequate control limiting devices. However, saturated linearcontrollers may fail to stabilize unstable linear systems.

For some sets of initial conditions, stabilizing saturatedlinear controllers can be designed by the method of Gutmanand Hagander (1985). Their approach rests on theconstruction of an elliptic positively invariant domainincluded in the polyhedral domain of constraints andincluding the set of initial states. The shape of the invariantdomains is directly induced by the selected Lyapunovfunctions. So, the choice of classical quadratic Lyapunovfunctions is not the most efficient for the Linear ConstrainedRegulation Problem (LCRP); it does not maximize the sizeof the domain of initial states for which a stabilizingconstrained regulator can be computed. This limitation canbe overcome by the use of non-quadratic Lyapunov functionsof the type introduced by Rosenbrock (1963). Along thisline, some authors (Vassilaki et al. 1988; Benzaouia andBurgat, 1988) have proposed methods for constructingpolyhedral positively invariant sets better fitted or evenperfectly matching the domain of linear constraints.

In particular, for a linear state feedback, the set of controlconstraints generates a convex polyhedron in the state space.This polyhedron can be made invariant by specific matrixconditions. A simplified version of the invariance conditions

* Received 19 July 1989; revised 3 September 1990;received in final form 28 September 1990. The originalversion of this paper was presented at the 11th IFAC WorldCongress Symposium on Automatic Control at the Service ofMankind which was held in Tallinn, Estonia, during August1990. The published proceedings of this IFAC Meeting maybe ordered from: Pergamon Press pic, Headington Hill Hall,Oxford OX3 OBW, U.K. This paper was recommended forpublication in revised form by Associate Editor R. V. Patelunder the direction of Editor H. Kwakernaak.

t Laboratoire d'Automatique et d'Analyse des Systemes,7, avenue du Colonel Roche, 31077 Toulouse, France.* Author to whom all correspondence should beaddressed.

is proposed in this paper. It allows for an easy on-lineimplementation of the control scheme with possibleextensions to adaptive cases (Beziat and Hennet, 1988). Thismethod is based on linear programming. In the feasiblecases, it generates a rapidly converging control law;computation of the gain matrix can also be frequentlyupdated to accelerate the convergence speed by taking intoaccount the current state of the system. Global stability ofthe variable regulating scheme is proven under a localstability condition.

Positive invariance of polyhedral setsConsider the discrete-time linear dynamical system

described by the equation:

with Xk ErJ/" for any k E}{, Ao ErJ/"*".Let R(G, w) be a not-empty convex polyhedral set defined

by:

G E rJ/g*" and w is a vector in rJ/g. The inequalities betweenvectors are componentwise. For instance, in definition (2),GX:5 w stands for (GX);:5 (w); for i= 1, ... , g.

According to the selected pair (G, w), R (G, w) can be anytype of polyhedral set (bounded or unbounded, including ornot the origin point). The case of proper cones is alsoincluded in this representation, for w = O.

By definition, R(G, w) is said to be positively invariant forsystem (1) if and only if:

X,ER(G, w)::}X'+k=A~X,ER(G, w)

VrE}{, VkE}{,

This definition of posllive invariance can be found, inparticular, in Lasalle (1976).

The property of Q-invariance defined in Gutman andCwikel (1986)-is closely related to this definition. But itcombines the positive invariance of a domain of the statespace with linear constraints on the control vector and withan asymptotic stability requirement. In contrast, positiveinvariance of an unbounded polyhedral set does notgenerally require or imply the asymptotic stability of thestate trajectories emanating from this set.

Existence of positively invariant polyhedral sets for system(1) is a generic property which covers different types ofdynamical behaviours. However, if rank G = nand (W)i >0for i= 1, ... , g, then, positive invariance of R(G, w)implies the stability of system (1). This last property caneasily be shown as in Bitsoris (1988a) by selecting as aLyapunov function of the system:

V(X) = max {I(GX);I}.; (W)i

polyhedral set R(G, w). It provides a necessary and sufficientcondition for R(G, w) to be a positively invariant set ofsystem (1). This basic result on polyhedral invariant sets wasinitially established by Bitsoris (1988b) under some morerestrictive conditions (in particular, all the components ofvector w were supposed strictly positive). Here, theseconditions are relaxed and a more direct proof is presented.

Proposition 1. The convex polyhedral set R(G, w) ispositively invariant for system (1) if and only if there exists amatrix HE '?IIK*gsuch that:

Hii ~ 0, for i = 1, ... ,g; j = 1, ... , g (4)

~=~o ~

~s~ WProof. A necessary and sufficient condition for R(G, w) tobe positively invariant for system (1) is:

GAoX S w, V X E '?II" : GX S w. (7)

Condition (7) is a special case of inclusion of a polyhedralconvex set in an other polyhedral convex set. The extendedFarkas' lemma (Hennet, 1989) provides an algebraiccharacterization of such an inclusion.

Any point of R(G, w) also satisfies the set of linearinequalities P . X S 1/J with P E '?IIP*"and 1/J E '?liPif and onlyif there exists a (dual) matrix H of '?IIp*gwith non-negativecoefficients satisfying conditions HG = P and Hw S 1/J.

This result can easily be proven by concatenation ofnecessary and sufficient conditions related to each row P ofmatrix P. By the standard Farkas' lemma (see e.g. Schrij~er,1986), a necessary and sufficient condition for:

P;XS1/Ji' VX:GXsw

3HiE'?III*g:HiG=P;, Hi,wS1/Ji' (HJi~O, Vj=1, . , . ,g.

The simultaneous satisfaction of all these elementaryconditions for any point of R(G, w) is equivalent to theexistence of p row-vectors Hi satisfying the same types ofcondition as above. The extended Farkas' lemma is simplyobtained by constructing matrix H from these prow-vectors.In particular, by setting P = GA and 1/J = w, the necessaryand sufficient condition for R(G, w) to be positivelyinvariant can be written:

Hii ~ 0 for i = 1, ... ,g; j = 1, ... ,g. 0

In variance and stability under input constraintsNow, the considered multivariable linear systems can be

represented by state-space equations of the following type:

Xk E '?II" being the state vector, Uk E '?11mthe control vector,A E '?II"*", BE '?II"*m. The control vector is subject toconstraints:

with (Urn)i~O, (UM)i~O, for i=I, ... ,rn and k=0, 1,2, . , ., Assuming that the state of the system isobservable, we want to control its closed-loop dynamics.Then, the selected control law is a linear state feedback, withFE '?IIm*":

Uk = FXk (10)

The closed-loop evolution of the system is described by

Xk+I=AoXk with Ao=A+BF. (11)

The general problem called LCRP (Linear ConstrainedRegulation Problem) consists of determining a matrix F suchthat the state vector of system (11) converges to 0 whilerespecting constraints (9) and relations (8), (10).

From Kalman and Bertram (1960), it is well-known that ifsystem (11) has all its eigenvalues in the unit circle of thecomplex plane, it admits elliptic positively invariant domainsassociated to its quadratic Lyapunov functions. The squareroot of a quadratic Lyapunov function is a contracting normfor the state vector.

From the properties of equivalence between norms (seee,g. Glazman and Liubitch, 1974), existence of a L2contracting norm is equivalent to the existence of apolyhedral norm for which the state is contracting. This normcan be written as in relation (3). Existence of such a norm isequivalent to the positive invariance of R(G, w). Hennet andLasserre (1990) have presented a scheme for constructing apolyhedral positively invariant domain for any asymptoticallystable system. The converse problem, analyzed in this paper,is to shape the dynamical behaviour of the controlled systemso that a particular domain is made positively invariant. Theconvex polyhedron generated in the state space byconstraints (9) is:

R[F, Urn, UM) = {X E '?II" : - Urn S FX SUM}.

A specialized version of invariance conditions (4-6) isprovided by the following proposition:

Proposition 2. A sufficient condition for R[F, Urn, UM) tobe positively invariant is the existence of a pair ofnon-negative matrices (H+ E '?IIm*m,H- E '?IIm*m)such that:

F(A + BF)=HFfIp sp

(12)

(13)

(UM) 2mP = Urn ' p E '?II .

This condition is also necessary when rank F = rn.

ProofSufficiencyInvariance of R[F, Urn, UM) is readily derived from relations(12) and (13) by a direct application of Proposition 1 tosystem (11) with:

G=[~~] and w=p.

Necessity when rank F = rnAssume now that R(G, w) is a positively invariant set ofsystem (11), with G and w defined as above.

Then, from Proposition 1, there exists a matrix withnon-negative elements,

fill E '?II';' *m for I, J E (1, 2) such that:

G(A + BF) = fiG; fIw S w.

Then, (fill - fI12)F = (fI22 - fI21)!' = FSA + ElF). If rankF = rn, this relation implies: Htt - H12 = H22 - H2t. SetH = fill - H12. Let H+ be the matrix of the non-negativecomponents of Hand H- the matrix of the non-negativecomponents of (- H). Then, matrix H = H+ - H- satisfiescondition (12). And

{(H_+)ii s mt,'n {(~lI)ij' ~22)ii} (1 )_ .. for i, j E " .. , rn .(H )ii S mll1 {(H12)ii' H21)iJ

Thus, matrices H+ and H- satisfy the necessary condition(13):

~t2)W S W.H22

Proposition 2 can also be derived from Proposition 4 inBitsoris (1988b), which treats the case of given non-symmetrical polyhedral domains of the state-space.

The algorithm of the next section solves relations (12),(13) by an indirect technique. A solution matrix H computedby this technique also guarantees the invariance of a givenpolyhedral domain of the state-space, R[G, Um, UMJ with Gfull-rank. Under this scheme, the class of matrices H can bereplaced by the class of matrices fI with no loss of generality.

In the special case of symmetrical constraints (Um =UM "" Om)' relation (13) can be equivalently replaced by(1 -IHI)Um ~ 0). A necessary condition for this inequalityto be satisfied is (1 -IHI) to be an M-matrix (Benzaouia andBurgat, 1988).

The LCRP can be solved whenever it is possible to find(H+, H-, F) such that:• Matrix (A + BF) has all its eigenvalues located in the unit

circle of the complex plane.• Positive invariance conditions (12), (13) are verified.• The initial state vector, Xo e R[F, Um, UM].The algebric formulation of this last condition is:

-Um~FXo~UM.

In this paper, we propose a method for finding an easilycomputable pair (H, F) belonging to a subclass of solutionsof equation (12). This design technique is based on thefollowing proposition:

Proposition 3. If R[G, Um, UM] is a positively invariantdomain of system (11), and H an associated solution of thesystem:

G(A+BF)=HG; fIp~p,

then any polyhedral domain R[Q, Um, UM] is also apositively invariant domain of (11) if

{Q=!1G; !1erJlmom

!1H = H!1. .

Proof G(A + BF) = HG and !1H = H!1 imply Q(A +BF) = HQ. Therefore, under the assumed condition Hp ~ p,R[Q, Um, UM] is also positively invariant. 0

Now, a controller Uk = FXk letting R[F, Um, UM]positively invariant can be constructed as follows:• Select a fixed matrix G e rJlmom for which R[G, Um, UM]

can be made positively invariant by state feedjJack. Then,3(H e rJlmom, Fe rJlmon); HG = G(A + BF), Hp ~ p.

• Solve the linear system:

{

HG = G(A + BDG)HD = DH; De rJlmomfIp ~ p.

• Set F= DG.The subclass of controls obtained by this scheme ischaracterized by the additional property of lettingR[G, Um, UM] positively invariant. It is therefore importantto handily select matrix G so as to obtain a nonemptysubclass of controls for most feasible problems.

A basic condition for the existence of non-negative vectors(Um, UM) such that R[G, Um, UM] can be made positivelyinvariant by state feedback is that Ker( G) should be an(A, B)-invariant subspace in the sense of Wonham (Hennetand De Bona Castelan Neto, 1990).

For any system in its minimal representation, theassumption rank B = m with m ~ n is ..$uite general. Underthis assumption, there exist matrices B e rJlmon such that

In particular, we can select the left pseudo-inverse ofB; BX = (BTB)-IBT. The kernel of BX is the quotient spacerJln/Im(B), which is an (A, B)-invariant subspace, sincerJln/Im(B) + Im(B) = rJln. (Wonham, 1985).

If we select G = BX and apply Proposition 3, we canobtain a subclass of positively invariant controllers by

imposing the following set of conditions:

BXABH= HBxAB

F=HBx -BxA

fIp <po

(16)(17)(18)

Equation (16) is obtained from condition DH = HD withD = H - BX AB, since H commutes with itself. And clearly,the choice of F from relation (17) satisfies conditionBX (A + BF) = HBx. In the case m < n, a positivelyinvariant controller satisfying conditions (16), (17) and (18)does not generally guarantee the overall stability of theclosed-loop system. Some additional stability conditions haveto be introduced to get a stabilizing positively invariantregulator.

An algorithm for solving the LCRPA possible way to simplify the stability analysis is to

impose as a positively invariant set a polytope R( G, w) withR(G, w)cR[F, Um, UM], w>O and rank G=n (Vassilakiet aI., 1988). Under this assumption, the function VeX)defined by relation (3) is positive definite and can be chosenas a candidate Lyapunov function for system (11). In thispaper, the polytope to be maintained invariant will bedirectly constructed by completing R[F, Um, UM] under theassumption: (Um)i >0, (UM)i > 0, for i= 1, ... , m.

In the case m <n, it is always possible to add dummycontrol variables (Uk)m+t"",(Uk)n, to add n-mindependent columns in B so that rank B = n, and to imposethe following constraints:

-(Um)m+i~(Uk)m+i~(UM)m+i (19)

with, for instance, (Um)m+i=(UM)m+i=8, for i=1, ... , n - m, and 8> 0 as small as desired.

Under these extra conditions, it can be assumed that Xk

and Uk have the same dimension, n, and that rank B = n. Inthis case, the resolution of the LCRP can be directlyobtained from the following Proposition:

Proposition 4. If m = n and rank B = n, conditions (20), (21)and (22) guarantee the positive invariance of R[F, Um, UM]and the asymptotic stability of the controlled system (11).

B-1ABH = HB-tAB (20)

F = HB-1 - B-1A (21)

fIp < p. (22)

Proof From relations (20) and (21), we can derive condition(12):

F(A + BF) = (HB-t - B-1A)BHB-1

= H(HB-1 - B-1A)

=HF.

Thus, under inequality (22), R[F, Um, UM] is a positively setof the closed-loop system.

Under condition (21), the dynamic matrix of theclosed-loop system is:

Matrices Ao and H being similar, asymptotic stability of thecontrolled system (11) is equivalent to the asymptoticconvergence to 0 of the control sequence, which satisfies, fork = 0, 1, ... , the recurrent relation (24):

Uk+1 = F(A + BF)Xk

=HFXkUk+1 = HUk·

Constraint (22) consists of the two inequalities:

H+UM+H-Um<UM

H-UM + H+Um < Um

The summing up of these two inequalities yields:

(H+ + H-)(UM + Um) < UM + Um

Therefore, matrixsatisfy:

IHI = «/Hijl)) and vector W = UM + Urn

IHIW S (H+ + H-)W

<W.

W is a positive vector of '!JI.n. Then (1 -\HI) is an M-Matrix,and from a classical result presented in Lasalle (1976), it is anecessary and sufficient condition for system (24) to beasymptotically stable. 0

The coefficients of matrices H+ and H- can be taken asthe unknown variables of a linear programming problem,denoted problem (II) and formulated as follows:

Hp S Ep (26)

B-1AB(H+ - H-) = (H+ - H-)B-1AB (27)

B-1AXo - Urn S (H+ - H-)B-1XO SB-1XO + UM.

(28)

Note that inequalities (28) simply express that the initial stateshould belong to R[F, Urn, UM] (relation 14) for a gainmatrix F satisfying relation (21).

An efficient way of solving the LCRP can be derived fromthe following proposition.

Proposition 5. If problem (II) has a solution (H+, H-, E)with E< 1, then the LCRP is solved by using the control:

Uk = F· Xk with F = (H+ - H-)B-l - B-IA. (29)

Proof. From Proposition 4, pOSItIve invariance ofR[F, Urn, UM] and asymptotic stability of the closed-loopsystem derive from the respect of conditions (27) and (26)when E < 1; and condition (28) in problem (II) guarantees:

If the optimal value of E, EO, is strictly smaller than 1,compute F by (29) and set Uk = FXk.

Note that the solution of problem (II) explicitly dependson the initial state of the system, Xo, through inequalities(28). However, it is clear that the computed gain matrix Fcan also stabilize the system from any other initial pointbelonging to R[F, Urn, UM], and that the control trajectoryalways remains feasible. If the initial state of the system isnot perfectly known, a design technique imposing theinvariance of a domain containing the domain of possibleinitial states (Gutman and Hagander, 1985; Vassilaki et aI.,1988) is probably more appropriate.

If E* 2: 1, stability of the closed-loop system is notachieved by this algorithm.

The fact of imposing the invariance of R[B-1, Urn, UM]and relations (20) and (21) constrains the closed-loopeigenvectors to belong to some subspaces. But under theassumptions of this paragraph, this is not a severe restriction.Any set of n independent directions can generate aninvariant domain of the closed-loop system.

It is only the size of the domain of stabilizable initial stateswhich may be reduced by using a control belonging to theinvestigated subclass. This possibility constitutes the onlycase of "conservativeness" of the proposed algorithm, whena suitable feedback exists but cannot be found by solvingproblem (II).

The efficiency of this algorithm can be improved when thecurrent state of the system can be observed. A variableregulating scheme can then he implemented. The gain matrixis periodically updated by solving a linear problem denoted(Ilk)' similar to problem (II) except for relations (28), inwhich the initial state vector, X 0' is replaced by the currentstate vector, Xk:

Hk • P S Ek • P (31)

B-1AB(H; - HI:) = (H; - HI:)B-1AB (32)

B-1AXk - Urn S (H; - HI:))B-1Xk S B-1AXk + UM.

(33)

The variable regulating law is:

Uk = FkXk

and Fk is computed from the optimal solution (H;, HI:, Ek)of problem (Ilk) by relation:

Fk = (H; - Hi:)S-1 - B-IA. (35)

The closed-loop evolution of the system is described by:

Xk+l = (A + BFk)Xk.

Replace Fk by its expression (35). It yields

Xk+1 = BHkB-1Xk·

The eigenvalues of Hk are also the eigenvalues of BHkB-1•

Proposition 6. If there exists an integer r such that problem(II,) has the optimal solution (H" E,) with E, < 1, then theoptimal solution of problem (Ilk) exists and verifies E, < 1for any integer k 2: r, and the controlled system isasymptotically stable.

Proof. From Proposition 5, relations (20), (21) and (22) aresatisfied whenever the optimal solution of problem (Ilk) issuch that Ek< 1. Then, from Proposition 4, Xk+l ER[Fk> Urn, UM], and (Hk> Ek) is also a feasible solution ofproblem (Ilk+I)' Consequently, the set of feasible solutionsof problem (Ilk+l) is not empty and by the choice of theobjective function, we must have: Ek+1S Ek> and, byinduction, E, < 1 implies Ek< 1 for any integer k 2: r.

Note that in the case of time-varying linear systems, localstability conditions Ek< 1 for k 2: r do not automaticallyimply the global stability of the system. Asymptotic stabilityof the system under the variable feedback law can be provenby showing the existence of a Lyapunov function for theclosed-loop system. Since vector W has positive components,the function L(X), defined as follows, is positive definite:

The difference between two successive values of functionLU is (jLk = L~fk+l) - L(Xk) .. From relation (36), deriveB Xk+1 = HkB Kk. Then, maJorate L(Xk+l) usmg (38):

L(X ) = {1(B-IXk+l)il}k+1 m~x W;

I(B-1Xk+I).1 I(HkB-1Xk).1W; W;

=~ Ii(Hk)iiW;(B-1Xk);1W; j~l "'i(IHkl W)i I(B-IXk),1

< max--------W; , W;

I(B-IX ) IS Ek' max k I .

I W.Then, L(Xk+1) S EkL(Xk) and since Ek< 1 for k 2: r,(jLk < O. L(X) is a Lyapunov function of the sequenceof state vectors, which asymptotically converges to 0:limk~ooXk = O. 0

X2

R(FUm UM)'v----

// // // / X10

/ /

/ /_1 / /

/ /

/ /_2 I/ I

/ _./

_3 //

_4_5 _4 _3 _2 _1 0 4

FIG. 1. Trajectory of the state-vector Xk with the constantregulator.

ExampleConsider the second order system described by the

state-space equation:

x _ (1.7 -3.3) ( 3.0 2.0)k+! - 1.3 0.3' Xk + -2.0 2.0 . Uk

(1.5) (1.0)- 1.5 :sUk:s 1.4'

x = (-5.0)o -3.5'

The unforced system is unstable. The eigenvalues of matrixA are: A1•2 = 1 ±j1.95.

The constant regulator gives the optimal values:

H = (0.380.93

-0.35)0.0

U = (-0.076 0.536).k -0.544 0.384 Xk•

Simulation results are presented in Fig. 1.The same problem can be solved using the variable

regulator. The simulation results presented in Fig. 2 showthat the variable regulator considerably increases the speedof convergence.

The same system now has to be controlled from a differentinitial state vector:

(3.5)Xo= 3.0 .

The successive values of the rate of convergence Ek obtainedby the linear programming algorithm are: Eo = 1.53,E1=1.68, E2 = 0.96, E) = 0.92, E4=0.82, Es=0.6.

We can see in Fig. 3 that although Eo> 1, the systemconverges since we get Ek < 1 after two steps. The origin can

o_1_2

_3

_4_ 5 _4 _3 _2 _1

FIG. 2. Trajectory of the state-vector Xk with the variableregulator.

FIG. 3. Trajectory of the state-vector Xk with the variableregulator in a case Eo> 1.

be reached in 3 time units. So, in particular, Xo belongs tothe maximal Q-invariant set defined and constructed inCwikel and Gutman (1986) and Gutman and Cwikel (1987).

Since EO> 1, the constant regulator is unable to stabilizethe system. In contrast, the variable regulating scheme canbe satisfactorily applied.

ConclusionsMany linear constrained regulation problems can be solved

by constructing positively invariant domains associated withLyapunov functions. The invariance conditions obtained bythis approach can be used as constraints on the gain matrix ofthe control law. A linear simplified version of theseconditions is presented in this paper. The LCRP can then besolved by a standard linear program. The selected objectivefunction to be maximized is denoted EO' It measures the rateof convergence of the system to the origin.

Two types of regulators are proposed. In the fixed scheme,the same gain matrix is applied at each period, while in thevariable scheme, a new gain matrix is computed at eachperiod, from the knowledge of the current state of thesystem.

If Eo < 1, stability of the closed-loop system is guaranteedwith the fixed and with the variable controller, but thevariable regulator may considerably increase the speed ofconvergence.

If Eo 2: 1, the fixed regulator is unable to stabilize thesystem. The variable regulator generates a feasible control aslong as the solution of the linear program remainsnumerically finite. Several iterations of the algorithm canthen be computed off-line. And stability of the closed-loopsystem can often be obtained by getting Er < 1 after someperiods of time. But if the constraints are too severe, theprocess should rather be stopped to avoid divergence. Thisvariable control law can also be very efficient in an adaptivecontext, with a-priori unknown parameters. Then, at eachupdating time k, the best current estimates of matrices A andd B are updated by a recursive identification algorithm andthe computation of the gain matrix also directly uses theinformation on the current state of the system throughresolution of problem (fIk).

Acknowledgements-The authors want to thank Professor G.Bitsoris and the anonymous referees for their helpfulcomments on a previous version of the paper, which was alsopresented at the XIth IFAC World Congress in Tallinn.

ReferencesBenzaouia, A. and Ch. Burgat (1988). The regulator

problem for a class of linear systems with constrainedcontrol. Syst. Control Leu., 10,357-363.

Beziat, l.-P. and l.-C. Hennet (1988). Stability andinvariance conditions in generalized predictive control.IMACS Int. Symp. on System Modelling and Simulation,Cetraro, Italy, pp. 163-167.

Bitsoris, G. (1988a). Positively invariant polyhedral sets ofdiscrete-time linear systems. Int. J Control, 47, 1713-1726.

Bitsoris, G. (1988b). On the positive invariance ofpolyhedral sets for discrete-time systems. Syst. ControlLeu., 11,243-248.

Cwikel, M. and P. O. Gutman (1986). Convergence of analgorithm to find maximal state constraint sets for-discrete-time linear dynamical systems with boundedcontrol and states. IEEE Trans. Aut. Control, AC·31,457-459.

Glazman, I. and Y. Liubitch (1974). Analyse Lineaire dansles Espaces de Dimension Finie. Editions MIR, Moscow.

Gutman, P. O. and P. Hagander (1985). A new design ofconstrained controllers for linear systems. IEEE Trans.Aut. Control, AC·30, 22-23.

Gutman, P. O. and M. Cwikel (1986). Admissible sets andfeedback control for discrete-time linear systems withbounded control and states. IEEE Trans. Aut. Control,AC·31, 373-376.

Gutman, P. O. and M. Cwikel (1987). An algorithm to findmaximal state constraint sets for discrete-time lineardynamical systems with bounded control and states. IEEETrans. Aut. Control, AC·32, 251-254.

Hennet, J. C. (1989). Une extension du lemme de Farkas etson application au probleme de regulation lineaire souscontraintes. C. R. Acad. Sciences, t. 308, serie 1, pp.415-419.

Hennet, J. C. and E. De Bona Castelan Neto (1990).

Invariance and stability by state feedback for constrainedlinear systems. Note LAAS-CNRS (submitted to theEuropean Control Conference, Grenoble).

Hennet, J. C. and J. B. Lasserre (1990). Spectralcharacterization of linear systems admitting positivelyinvariant polytopes. Note LAAS 90010 (submitted toMath. Control Signals and Systems).

Kalman, R. E. and J. E. Bertram (1960). Control systemsanalysis and design via the second method of Lyapunov.Trans. A.S.M.E., D82, 394-400.

Lasalle, J. P. (1976). The stability of Dynamical Systems.SIAM Regional Conference Series in Applied Mathemat-ics, SIAM, Philadelphia.

Rosenbrock, H. N. (1963). A method of investigatingstability. Proc. IFAC, 590-594.

Schrijver, A. (1986). Theory of Linear and IntegerProgramming. Wiley, Chichester, U.K.

Vassilaki, M., J. C. Hennet and G. Bitsoris (1988).Feedback control of linear discrete-time systems understate and control constraints. Int. J. Control, 47,1727-1735.

Wonham, W. M. (1985). Linear Multivariable Control-AGeometric Approach. Springer, Berlin.