Feedback control of linear discrete-time systems under state andcontrol constraints

In this paper the problem of stabilizing linear discrete-time systems under state andcontrol linear constraints is studied. Based on the concept of positive invariance,existence conditions of linear state feedback control laws respecting both theconstraints are established. These conditions are then translated into an algorithmof linear programming.

1. IntroductionMost industrial systems must operate within fixed bounds and are subject to strict

control limitations. The determination of closed-loop controls for such systems bystate or output feedback often reduces to solving an associated unconstrainedproblem and then modifying the solution by superimposition of state and controllimitations. The global stability of these control schemes is usually not guaranteed.Another approach that is more rigorous, consists of explicitly introducing theconstraints in the lagrangian formulation of an optimal control problem (Mouradi1979, Franckena and Sivan 1979, Gauthier and Bornard 1983). However, itsimplementation is not simple, because as an open-loop scheme it implies considerableoff-line computation and as a closed-loop scheme it is represented by a non-linearcontroller.

The concept of invariance (or positive invariance), which is related to the notion ofLyapunov functions, is a convenient tool both for guaranteeing stability andrespecting the constraints. In the general case of constrained controllers for linearsystems, Gutman and Hagander (1985) used quadratic Lyapunov functions todetermine non-linear feedback controllers. However, for linear systems with linearconstraints on state and control variables, non-quadratic Lyapunov functions must beused in order to generate the biggest positively invariant set included in the domain ofconstraints. Such Lyapunov functions have already been applied for improving linearconstrained controllers of linear systems characterized by a stable non-negativedynamic matrix (Chegan<;as and Burgat 1985).

In this paper the problem of design of linear state feedback controllers for lineardiscrete-time systems under linear constraints on the state and control variables isstudied. In § 2 the linear constrained regulation problem is formulated. Then, usingrecent results on the existence of polyhedral positively invariant sets (Bitsoris 1986,1988 a), existence conditions of a solution to this problem are obtained (§ 3). Finally, in§ 4, an approach for the determination of such a control law is established.

Received 14 April 1987.t Laboratoire d'Automatique et d'Analyse des Systemes du C.N.R.S., 7 Avenue du Colonel

Roche, 31077, Toulouse, France.t Control Systems Laboratory, Electrical Engineering Department, University of Patras,

26110 Patras, Greece.

2. Problem statementThroughout the paper, capital letters generally denote real matrices, lower case

letters denote column vectors or scalars, Rn denotes the euclidean n-space and Rn x mthe set of real n x m matrices. For a real matrix A = (aij), IAI denotes the matrix IAI=(Iaij!)' For vectors x = [XI Xz ... Xn]T and Ixi = [ixil IXzl Ixnl]T. Finally:lLdenotes a unity matrix.

We consider discrete-time linear system described by the difference equation

where X ERn, U E Rm, A ERn XI', BERn x m and k belongs to the set of non-negativeintegers T= {a, 1,2, ... }.

The control vector u(k) is subject to constraints

where P = [PI pz ... Pm]T with Pi> 0, i = 1,2, ... , m.

There is also given a bounded set of initial states Xo defined by the inequalities

whereGERqXnwithq~n,rankG=nandw=[wl Wz ... wq]Twithwi>O,i=1, 2, ... , q. These inequalities can also be considered as state constraints.

The problem to be studied is the determination of a linear state feedback controllaw

that satisfy constraints (2) are transferred asymptotically to the origin while thecontrol vector u(k) does not violate the constraints (1). We call this problem the linearconstrained regulation problem (LCRP).

If the equilibrium x = ° of the open-loop system

is stable in the sense of Lyapunov or asymptotically stable, then the above problemadmits the trivial solution u(k) = 0. If, on the contrary, the open-loop system isunstable, then the LCRP may not possess any solution. Therefore, we shall say thatconstraints (1) and (2) are compatible with respect to system (S) if the LCRP has atleast one solution.

3. Existence conditions of linear constrained controllersLet us associate to each linear state feedback control law u(k) = Fx(k) with

F E Rm x n the set

R(F, p) = {x E R": - P ~ Fx ~ p}

It is clear that the polyhedral set R(F, p) is the region of initial states of the closed-loopsystem (4) at which the linear state feedback control u(k) = Fx(k) does not initiallyviolate the constraints (1).

According to the above notation the set of initial states defined in (2) is expressed

It is obvious that the control law u = Fx is a solution of the LCRP ifand only if theresulting closed-loop system (4) is asymptotically stable and every trajectory x(k; xo)emanating from the region R(G, w) does not leave the region R(F, p) for any instantk E T. This condition can also be expressed as follows (Bitsoris 1988 b).

Proposition 1The control law u = Fx with FE Rm x n is a solution to the LCRP if and only if

(a) the eigenvalues Ai>i = 1,2, ... , n, of the matrix A + BF are in the open disklAd < 1;

(b) there exists a positively invariant sett Q £; W of the resulting closed-loopsystem (4) such that

There are many ways to use this result for obtaining a solution on to the LCRP(Bitsoris 1986). An approach can be based upon the following corollary of thepreceding proposition.

Corollary 1If F is a real m x n matrix such that

(a) the eigenvaluesA;, i = 1, 2, ... , n of the matrix A + BF are in the open disk IAI< 1,(b) R( G, w) is a positively invariant set of the closed-loop system (4); and

(c) R(G, w) £; R(F, p)

then the control law u = Fx is a solution to the LCRP and the state vectors satisfyconstraints (2) for all initial states Xo E R( G, w) and k E T.

It is clear that for applying this result to the derivation of a solution to the LCRP,one must first establish conditions guaranteeing that R( G, w) is a positively invariantset of the resulting closed-loop system (4). It is well known that an asymptoticallystable system possesses positively invariant sets having a quadratic boundary. Suchpositively invariant sets are generated by quadratic Lyapunov functions of the typev(x) = xT Px where P E Rn xn is a symmetric positive-definite matrix. By usingLyapunov-like functions we can also generate positively invariant polyhedral setsR(G, w).

It can be easily seen that the polyhedral set R(G, w) can be equivalently defined bythe expression

*( ) t::. {I(GXLI}v x = max --1 ~i~q Wi

(Gx); denoting the ith component of the vector Gx.The following proposition provides a necessary and sufficient condition for v*(x)

defined by (5) to be a Lyapunov function, and, accordingly, for R(G, w) to be apositively invariant set of system (S).

t A non-empty subset Q of R" is said to be a positively invariant set of (4) if and only ifXo E Q implies x(k; xo) E Q for all k E T.

Proposition 2 (Bitsoris 1988 a)The polyhedral set R( G, w) is a positively invariant set of system (S) if and only if

there exists a matrix H E Rq x q such that

(IHI- :ll.)w:::; 0

GA-HG=O

By a direct application of this result to the closed-loop system described by (4) weestablish the following.

Proposition 3If FER'" x n and there exists a matrix HE Rq x q such that

(IHI-:ll.)w:::;O

GA + GBF= HG

(iii) the eigenvalues Ai = 1,2, ... , n of the matrix A + BF are in the open disk IAil<1

then the control law u = Fx is a solution to the LCRP.Observe that condition (iii) of the preceding proposition can be replaced by an

asymptotic stability condition on the matrix H because G(A + BF) = HG and, byhypothesis, G E Rq x 11 with rank G = n. Note also that, since matrix IHI -:ll. has non-negative off-diagonal elements and w is a vector with positive components, inequality(6) implies stability in the sense of Lyapunov of matrix H and accordingly of theclosed-loop system (4) also (Bitsoris and Burgat 1977). Moreover, in the case whereinequality (6) is strictly satisfied, that is (IHI - :ll.)w< 0, or if matrix H is irreduciblethen matrix H and accordingly system (4) are asymptotically stable (Gantmacher1960). Consequently, condition (iii) of the preceding proposition can be replaced bythe condition that H is irreducible or by imposing that condition (6) is strictlysatisfied.

In order to facilitate the application of the result presented in the precedingproposition to the design of constrained regulators, condition (8), which expresses thecompatibility of the constraints on the control vector and the initial states, must bereplaced by an equivalent algebraic condition. To this end, we can use the followingresult.

Proposition 4If

ProofFrom (9) it follows that if

IF( GT G) -1 GTyl :::; IF( GT G) -1 GTIlyl

:::; IF( GT G) -1 GT Iw :::; p

Now, setting y= Gx in (10) and (11) we conclude that if

IGxl~w

or, equivalently if x E R( G, w) then

IF(GTG) -I GTGxl = IFxl ~ p

It must be noted that in the case where G E R" x n, inequality (9) becomes IFG-11w ~ p and is, in addition, a necessary condition for R(G, w) c:;: R(F, p) (Bitsoris 1988 b).

Now, by combining the results stated in Propositions 3 and 4 we conclude that ifFE Rm x n and there exists an asymptotically stable matrix HE Rq x q satisfying (6), (7)and (9), then with the control law u = Fx all the states Xo E R( G, w) are transferredasymptotically to the origin while the control and state vectors satisfy inequalities (I)and (2) respectively.

4. Design by means of linear programmingA straightforward application of the preceding result to the design of constrained

linear controllers seems to be a very difficult problem. However, by an appropriatetransformation of conditions (6), (7) and (9), the determination of a solution to theLCRP can be reduced to a linear programming problem.

Observe that (7) is satisfied with

Now, by introducing matrix DE Rm xq such that

D = F( GT G) - 1GT

relation (12) can be written as

H=GA(GTG)-IGT +GBD

Insertion of (13) and (14) into (9) and (6), respectively, yields

IDlw~p

IGA(GTG)-IGT +GBDlw~w

(15)

( 16)

These conditions on matrix D guarantee the existence of a linear control lawu = Fx = DGx such that R( G, w) is a positively invariant set of the resulting closed-loop system

and R(G, w) c:;: R(F, p). However, conditions (15) and (16) do not imply the asympto-tic stability of system (17). The asymptotic stability of (17) is guaranteed if inequality(16) is strictly satisfied. For this reason, inequality (16) is replaced by the inequality

IGA(GT G) -I GT + GBDlw ~ GW (18)

where G is a real number such that 0 ~ G < 1.Therefore, the existence condition of a solution to the LCRP reduces to the

existence of a matrix DE Rm x q and a real number G, 0 ~ G < 1, satisfying inequalities(15) and (18).

Non-linear inequalities (15) and (18) do not seem easy to solve. However, oneshould take into account the fact that the vector w has positive components. Thus, wecan use the following result.

Proposition 5The matrix inequality

where Y = (YiJ, Y E RP x r, r:J.E R' and fJ E RP with r:J.j;;::'0 and fJi ;;::,0, is equivalent to theset of equations

where es = [e1S ezs ersJT denotes one of the distinct vectors es E R' withcomponents equal to + 1or -1.

ProofAssume that Y=(Yij), YERPxr satisfies inequalities (19). Then

r r r

I Yijejsr:J.j ~ I !Yijllejslr:J.j = I IYij!r:J.j ~ fJij~l j~l j~l

for all i= 1,2, ... , p and s = 1,2, ... , 2r, because r:J.j;;::' O.Therefore, (19) implies (20).Conversely, if Y = (Yij), Y E RP x r satisfies (20) then for ejS = sign (Yij) we obtain

r r

I !Yij!r:J.j = I Yijejsr:J.j ~ fJi' i= 1,2, ... , pj~ 1 j~ 1

The application of this result to the determination of a solution to the LCRP isstraightforward. The system of inequalities (15) and (18) can be equivalently replacedby a system of linear inequalities with unknown variables, the elements dij of matrix Dand the positive variable B. If the set of solutions to these inequalities is non-empty,then such a solution can be obtained by minimizing any linear function of theunknown variables dij and B.

Since it is very important not only to stabilize the system but also to increase therate of convergence to the equilibrium, we can choose as the objective function of thelinear programming problem, the function

J(D, B) = B

Indeed, if B satisfy inequality (18), then by virtue of (14)

!Hlw~BW

v(x) = max {!(GX);l}t Wi

is positive-definite. Therefore v(x) can be considered as the distance of x from theorigin. (It can easily be proved that v(x) represents the distance of x from the origin, inthe space R" with the distance

{V(X) + v(y)

d(x, y) = oifx#y

if x = Y

Now, taking into account (7), we get

v(x(k + 1))= m~x f(GX(~~ 1))il}

= m~x {I(G(A + ~:)X(k));i}

= m~x fHG:;k))il}

~ m~x {(IHII~~(k)l)i}~ w(x(k))

because from (22) it follows that

Therefore, minimization of 8 increases the rate of convergence of the state variablex(k) to the origin.

If the optimal solution D*, 8* of the linear programming problem is such that8* < 1, then the asymptotic stability of the resulting closed-loop system, is guaranteedbecause inequality (6) is strictly satisfied. This is also true if 8* := 1 and the resultingmatrix H is irreducible. Finally if 8* = 1 and matrix H is not irreducible one mustapply the Kotelyansky-Sevastianov conditions (Gantmacher 1960) to examinewhether matrix IHI is asymptotically stable.

This approach is illustrated in the following example.

Example

Let us consider the second-order system described by the difference equation

[ 0·8 0'5J [OJx(k + 1) = x(k) + u(k)-0,4 1·2 1

IX1+2x21~5

l-l'5xl+2x21~10I

(25 a)

(25 b)

The problem to be solved is the determination of a linear state feedback controlu = [fl f2]X such that the resulting closed-loop system is asymptotically stable andall initial states satisfying (25) are transferred to the origin while the control vectordoes not violate constraints (24).

A~[_~: ~~Jare Al = 1 + jO-4 and .1.2 = 1 - jO·4.

Now, setting

B~ [~lG~[-:5 ~lw~ [la p~7

and taking into account that det G #- 0, conditions (15) and (18) become

IDlw":;p

IGA(GTG)-lGT + GBDlw= IGAG-I + GBDlw,,:;ew

51d11+ 1OId21 ,,:; 7

510'87 + 2dll + 1010'58 + 2d21 ,,:; 5e

51-0'305 + 2dll + 1011'13 + 2d21,,:; lOe

(26 a)

(26 b)

(26 c)

Thus, the LCRP for system (23) under constraints (24) and (25) is reduced to thedetermination of dl, d2 and e which minimize the objective function

under constraints (26).Transformation of inequalities (26) to a system of linear inequalities and

application of a standard algorithm of linear programming gave the optimal values

u ~ [-0435 -04825{ -:5 ~J[::JWith this control, the resulting closed-loop system becomes x(k + 1) = (A

+ BD*G)x(k) where

[0·8

A+BD*G=-0,11125

0·5 ]-0,635

The eigenvalues of matrix A + BD*G are Al = 0'76, .1.2 = 0·59 and it is worth noticingthat the optimal value of parameter e is a good upper bound of max (1.1.11, 1.1.21). Due tothe optimality of control law (26), the intersection of the boundary of set R( G, w) andthe boundary of set R(F, p) is not empty. This is shown in the following Figure.

5. ConclusionThe Linear Constrained Regulation Problem has been analysed by determining

positively invariant sets associated to non-quadratic Lyapunov functions. Existenceconditions oflinear state feedback control laws have been obtained and translated intoan efficient algorithm based on linear programming. The control laws obtained by thisapproach not only transfer to the origin all the initial states belonging to a polyhedralsubset of the state space but also optimize the convergence rate, while respectingcontrol constraints.

Although in the formulation of the LCRP no constraints on the state vector havebeen imposed, the proposed algorithm also provides a solution to the case where thestate vector must satisfy linear inequalities.

REFERENCES

BITSORIS,G., 1986, Sur I'existence des ensembles invariants polyhedraux des systemes lineaires,Technical Report 86015 (L.A.A.S.-CN.R.S, Toulouse, France); 1988 a, Int. J. Control,47, 1713; 1988 b, J. Large-scale Systems, to be published.

BITSORIS,G., and BURGAT,C, 1977, Int. J. Control, 25, 413.CHEGANc;AS,J., and BURGAT, C, 1985, Actes du Congres Automatique d'AFCET, Toulouse,

France, 193.FRANKENA,J. F., and SIVAN,R., 1979, Int. J. Control, 30, 159.GANTMACHER,F. R., 1960, The Theory of Matrices (New York: Chelsea).GAUTHIER, 1. P., and BORNARD,G., 1983, Rev. Autom. Inf. Res. Oper., 17, 205.GUTMAN, P.O., and HAGANDER,P., 1985, IEEE Trans. autom. Control, 30, 22.MOURADI, M., 1979, Rev. Autom. Inf. Res. Oper., 13, 127.