Constrained regulation of linear continuous-time dynamical systems

Systems & Control Letters 13 (1989) 247-252 North-Holland

247

Constrained regulation of linear continuous-time dynamical systems Marina VASSILAKI AMBER S.A. Computer Systems, P.O.B. 3500, Athens, Greece

Georges BITSORIS Control Systems Laboratory, Electrical Engineering Department, University of Patras, 26500 Patras, Greece Received 7 January 1989 Revised 12 June 1989

Abstract: In this paper the Linear Constrained Regulation Problem (LCRP) for continuous-time dynamical systems is studied. The first part of the paper deals with the existence of linear state-feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space, while respecting linear constraints on both state and control vectors. Then, the LCRP is formulated as an optimization problem which can be solved by applying linear programming techniques.

Keywords: Linear systems; constrained regulation; state feedback; asymptotic stability.

I. Introduction

Although most industrial systems are subject to state and control limitations, little work has been done to develop efficient design algorithms of constrained controllers [8,10]. In practice, the determination of such a controller is cartier out by solving un associated unconstrained control problem and then modifying the solution for both the state and control constraints to be satisfied. Usually, this modification consists in saturating the control vector, but in this case the stability of the resulting closed-loop system is not always guaranteed. More efficient approaches to this problem are based on the properties of positively invariant sets. In [5] necessary and sufficient conditions for the state of the system to belong to a given set under bounded control are obtained. A similar problem concerning sets of initial states that can

be transferred to a given target set without violating the control constraints has been studied in [1,4,7,8]. Algorithms for computing admissible sets of initial states and open-loop or feedback control laws are presented in [1,4,7,8,11,12]. In this paper, positive invariance conditions of polyhedral sets are applied to the problem of derivation of state feedback control laws for linear continuous-time systems under linear state and control constraints. This problem, known as the Linear Constrained Regulation Problem [1], is formulated in the next section of the paper. In Section 3 conditions guaranteeing the existence of a solution to this problem are obtained. Finally, in Section 3, an algorithm based upon linear programming is established and an illustrative example is presented.

2. Problem statement Throughout the paper, capital letters denote real matrices, lower case letters denote column vectors or scalars, R n denotes the Euclidean nspace and R n×m the set of all n x m matrices. For two vectors x = [x I

x2

'''

xn] T

Y2

"'"

Yn] T,

and Y = [Yl

x < y (x ~
3;(t) = A x ( t ) + Bu(t)

(S)

with A ~ R ~×~, B ~ R ~×m, x ~ R ~, u ~ R m, and t ~ T, where T is the time set T = [0, oo). The control vector u(t) is subject to linear constraints of the form

--pL <~u<~pU

0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

(1)

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M. Vassilaki, G. Bitsoris / Linear constrained regulation

where pL and pu are real vectors with positive components. The state vector is also constrained to belong to a closed convex polyhedral domain defined by the inequality

which is the region of states where the control vector does not violate constraints (1). Consider also the set P(D,c)

A = { x ~ R " : Dx<~c}

where D ~ R px", with rank D = n , p>~n, and c>0. Finally there is given a closed set of initial states defined by the inequality

of states satisfying constraints (2). According, to this notation, the set of initial states defined by inequality (3), is denoted by P(G, w). It is obvious that a control law u = F x is a solution to the L C R P if and only if the resulting closed-loop system

Gx <~w

Yc = ( A + B F ) x

Dx <~c

(2)

(3)

where G E R q×n with rank G = n, q >~ n, and w > 0. The problem to be studied is the determination of a linear state feedback control law u(t) =Fx(t) such that all initial states x 0 satisfying inequalities (3) are transferred to the origin asymptotically, while both the state and the control vectors do not violate constraints (1) and (2) respectively. Due to the linear form of the constraints and the polyhedral form of the set of initial states we call this problem the Linear Constrained Regulation Problem (LCRP). It is obvious that the L C R P is well posed if all the initial states belonging to the set defined by (3) satisfy the state constraints (2). The well-posedness however, does not guarantee the existence of a solution to the LCRP, even if the open-loop system is asymptotically stable.

3. Existence conditions for linear constrained controllers

The development of the conditions established in this paper is based on the concept of positive invariance. We recall that a non-empty set I2 is said to be a positively invariant set of system (S) if and only if x 0 ~ $2 implies that x ( t ; x o, t o ) ~ 2

for all t 0 ~ T a n d

t>~t o ,

x(t; t o, Xo) being the trajectory of S with initial conditions (t 0, x0). To each linear state feedback control law u = Fx we associate the polyhedral set Q ( F , p', ou) g ( x ~ R " :

--oL ~ Fx <~ou )

(4)

is asymptotically stable and every trajectory x(t; t o, Xo) emanating from the region P(G, w) does not leave the regions Q(F, pL, ou) and P ( D , c) for any t >/t 0. This condition can also be expressed as follows [1]: Proposition 1. The control law u = Fx with F G R m×n is a solution to the L C R P if and only if (i) The eigenvalues hi, i = 1, 2 . . . . . n, of the matrix A + B F have negative real parts. (ii) There exists a positively invariant set $2 c R n of the resulting closed-loop system (4) such that P(G, w)C~2cP(D,

c)

and 12c_Q(F, 0 L, ou). Many different algorithms can be established by using this result. The approach presented in this paper is based on a corollary of this proposition: Corollary 1. Suppose F is a real m X n matrix such that (i) The eigenvalues ~i, i = 1, 2 . . . . . n, of the matrix A + B F have negative real parts. (ii) P(G, w) is a positively invariant set of the closed-loop system (4), and (iii) P(G, w ) c Q(F, pL, pU). Then the control law u = Fx is a solution ot the LCRP. For the application of this result to the derivation of a solution to the LCRP, one must establish conditions guaranteeing that P(G, w) is a positively invariant set of (4) and P(G, w)c_ Q(F, pL, pU).

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We first establish algebraic conditions for P(G, w) to be positively invariant with respect to the closed-loop system (4). It is well known that an asymptotically stable system possesses hyperellipsoidal positively invariant sets. These sets are generated by quadratic Lyapunov functions xTpx, where P ~ R n×" is a symmetric positive definite function satisfying the matrix equation PA + ATp - Q = O, Q ~ R "x~ being a symmetric positive definite matrix. By using non-quadratic Lyapunov functions we can also generate positively invariant sets of the polyhedral type as is the case of the set P(G, w). Real n x n matrices of class M, defined below are of great importance for the definition of such Lyapunov-like functions.

Definition. The n X n real matrix B = (bij) is said to belong to the class M, if and only if bij >I 0, for all i ~ j. Since P(G, w) is a closed set, it can be defined by the expression P(G, w)=fx~R":v(x)~
where

(iv) the eigenvalues hi, i = 1, 2 . . . . . n, of the matrix A + B F have negative real parts, and (v)

P(G, w)c

Q ( F , pL, pC).

Then the control law u = F x is a solution to the LCRP.

Condition (iv) can be replaced by a condition on matrix H. Indeed, since H has non-negative off-diagonal elements and w is a vector with positive components, if H w <~O, H w ¢ 0 and H is irreducible or if inequality (6) is satisfied strictly, then the matrix H has eigenvalues with negative real parts [6]. This implies that the closed-loop system (4) is asymptotically stable, because from (7) and the hypothesis that rank G = n it follows that the eigenvalues of matrix A + B F are also eigenvalues of matrix H. The application of the result stated in Proposition 3 to the design of constrained controllers is considerably simplified if condition (v) is replaced by a set of linear inequalities. To this end, let x ti), i = 1, 2 . . . . . r, be the vertices of the dosed polyhedron P(G, w). The vertices x (i) are solutions of the linear equations G(')x (i) = w (/),

(Gx)i denoting the i-th 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 P(G, w) to be a positively invariant set of system (S); see [21.

Proposition 2. The polyhedral set P(G, w) is a positively invariant set of system ~ = A y if and only if there exists a matrix H ~ R q×q such that H ~ Mq,

GA - H G = 0

H w ~
A straightforward consequence of the above result is the following proposition: Proposition 3. Suppose F ~ R " x " and there exists a matrix H ~ R qxq such that (ii) (iii)

P ( G , w)c_ Q ( F , pL, pV) if and only if --oL<,Vx(i)<~O U,

i = 1 , 2 . . . . . r,

(8)

Proposition 5. I f there exist a stable matrix H ~ Mq and a matrix F ~ R "×" satisfying relations (6), (7) and (8), then the control law u = Fx is a solution to the LCRP.

4. A design algorithm

HEMq, H w <~0, GA + G B F = HG,

Proposition 4. The polyhedral sets P(G, w) and Q( F, OL, 0u) satisfy the relation

A direct consequence of the results stated in Propositions 3 and 4 is the following.

and

(i)

i = l , 2 , . . . , r, where G f~) ~ R "×" are nonsingular submatrices of G and w (i) the corresponding subvectors of w. With this notation we can state the following proposition which is a basic fact of linear programming [9]:

(6) (7)

The application of the preceding result to the design of linear constrained controllers is straight-

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M. Vassilaki, G. Bitsoris / Linear constrained regulation

forward. One has to resolve the set of linear equations or inequations (6), (7), (8) and hi: >~ 0 for i ~ j and then, by applying one of the tests stated in the preceding section, to check the stability of matrix H. Such a solution can be obtained by defining a linear programming problem with conditions (6), (7), (8) and hij>~O for i ~ j as constraints. Since, however, it is very important not only to stabilize the system but also to increase the rate of convergence to the equilibrium one should choose an appropriate objective function. This can be done by defining a linear programming problem with objective function

J(F,

n, e)= e

hi j>10,

(10a) (10b) i = 1 , 2 . . . . . r,

= ( axlk. q

= E hk,,(Gx), +

i~j,

(10d) (10e)

where e is a real number. Then maximization of the objective function increases the rate of convergence of the state variable to the equilibrium. Indeed, consider the positive definite function v(x) defined by relation (5) and define its total derivative with respect to system (4) as

q

<~ E hGiwiv(x) + h/,A:wGv(x) i~1 i4~k:

,(x) <~ - e w k v ( x ) because h i >/0 and Therefore

(Gx)i <~wiv(x ) for

i ¢ kj.

which implies that maximization of e increases the rate of convergence of the state to the equilibrium. This approach is illustrated in the following example. Example. Let us consider the second order system

22 =

--1.6

3.4][x 2 +

u

and a set of initial states defined by the inequalities

=limsup{~-~[v(x(t+At; t, x))--v(x)]). At--,O +

If for x~P(G, w), kg, j = l , denote the indices for which

hkA~(Gx)kj

i~1 i~kj

(10C)

e >/0,

2..... s, j<~m,

(Gx ) ~, .(x) =

(G(A + BF)x)G

(9)

and constraints

GA + GBF = HG, Hw <~- ew, --pL<~Fx(')<~pu,

Furthermore

0.5x 1 - x 2 >/ - 1 . 5 ,

(lla)

x 1 - 8x 2 ~< 4,

(llb)

X 2 ~ 0.5.

(11c)

- - ,

W k:

The state vector is subject to constraints

then taking into account that v(x) and :~(t; t, x) are continuous we conclude that [1]

O(x)=max ( Go,-xat, (lim t++ A (t ;1t[' x ) ) G k j

wk:

- 6 ~< x a ~< 8,

(12a)

-2

(12b)

~< x 2 ~< 1,

and the control vector must satisfy the inequality - 8 ~< u ~< 10. W kj

- max ( G j ( x ) ) k:

= max k:

(G(A +

BF)x)kj }

W k:

k:

}

(13)

The problem to be studied is the determination of a linear state feedback control law u = [fl f2] x such that the resulting closed-loop system is asymptotically stable and all initial states satisfying inequalities (11) are tranferred to the origin asymptotically while the state and control vectors satisfy inequalities (12) and (13) respectively.

251

M. Vassilaki, G. Bitsoris / Linear constrained regulation

gives

First observe that, since the eigenvalues of the matrix

F=[1.5 A=[

-1"751.6 -1.213.4 H =

and ~k2-~-3.7491, the unforced system is unstable. Furthermore, the problem is well posed because the set of initial states defined by inequalities (11) is a subset of the set of states satisfying the constraints (12). The vertices of the closed polyhedron defined by inequalities (11) are

-4],

-0.8 0 0.2

0 -0.2 0

1.8] 0 -0.8

a r e ~k1 = - 2 . 0 9 9

3116

and e = 0.2. Since e > 0 the resulting control law u = 1.5x~ 4x2 is a solution to the LCRP. With this control law the eigenvalues of matrix A + B F are hi = - 0 . 2 and X2 = - 1 . 4 , that is the dominant pole of the resulting closed-loop system is equal to the optimal value of - e . It should also be noted that due to the optimality of the control law, a vertex of the polyhedral set P ( G , w ) lies on the boundary of the set Q ( F , pL, pu). This is shown in Figure 1.

],

Maximization of the objective function (9) under constraints (10) with

B=

W

~

[0.5) [_0.s 1] 1

,

[i-5]

G=

oL=8,

1 0

-8, 1

5. Conclusion

In this paper the Linear Contrained Regulation Problem for continuous-time systems has been studied. By applying recent results on the positive

pU = 10.

.5

~X2

Q(Fj,pLp U)

Fig. 1. The sets P(G, w) and Q(F, pL, 01-1)for the closed-loopsystem.

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M. Vassilaki, G. Bitsoris / Linear constrained regulation

invariance of polyhedral sets, c o n d i t i o n s for the existence of a solution to this p r o b l e m were obtained. Then, a design algorithm was developed, b y which a n o p t i m a l solution to the L C R P can be o b t a i n e d b y solving a linear p r o g r a m m i n g p r o b lem. The control law derived b y this a p p r o a c h n o t only transfers to the origin all the initial states belonging to a polyhedral subset of the state space b u t in a d d i t i o n optimizes the convergence rate, while respecting both state a n d control constraints. It should be emphasized that the design algorithm presented here was based o n Corollary 1 that was derived b y setting 12 = P ( G , w ) i n hypothesis (ii) of Proposition 1. It is evident that one could o b t a i n a n o t h e r solution to the L C R P b y setting 12 = P ( D , c). T h e design algorithm would be similar to that presented i n Section 4. However, the solution o b t a i n e d b y the latter a p p r o a c h generally would give a worst rate convergence coefficient e because P ( G , w ) c_ P ( D , c). It is also evid e n t that one could o b t a i n other design algorithms b y a different choice of the positively i n v a r i a n t set 12 [11.

References [1] G. Bitsoris, The linear constrained regulation problem, Technical Report UP CSL 87002, Dept. of Electr. Engineering, Uniyersity of Patras (1987). [2] G. Bitsoris, On the existence of positively invariant polyhedral sets for continuous-time linear systems, Technical

Report UP CSL 88003, Dept. of Electr. Engineering, University of Patras (1988); also submitted to IEEE Trans. Automat. Control (1988). [3] J. Chegancas and C. Burgat, Regulateur P-invariant avec contraintes sur les commandes, Actes du Congr& Automatique 1985 d'AFCET (1986) 193-203. [4] M. Cwikel and P.O. Gutman, Convergence of an algorithm to find maximal state constraint sets for discretetime linear dynamical systems with bounded control and states, IEEE Trans. Automat. Control 31 (5) (1986) 457-459. [5] A, Feuer and M. Heymann, $Llnvariance in control systems with bounded controls, J. Math. AnaL Appl. 53 (1976) 266-276. [6] F.R. Gantmacher, The Theory of Matrices, Vol. II, Chelsea, New York, 1960). [7] P.O. Gutman and M. Cwikel, An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states, IEEE Trans. Automat. Control 32 (3) (1987) 251-254. [8] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems, IEEE Trans. Automat. Control 30 (1) (1985) 22-33. [9] G. Hadley, Linear Programming (Addison-Wesley,Reading, MA, 1974). [10] F.H. Moss and A. Segall, An optimal control approach to dynamic routing in networks, IEEE Trans. Automat. Control 27 (2) (1982) 329-339. [11] M. Vassilaki and G. Bitsoris, Optimum algebraic design of continuous-time regulators with polyhedral constraints, to be presented at IFAC Conference on Advanced Information Processing in Automatic Control, Nancy, France (1989). [12] M. Vassilaki, J.C. Hennet and G. Bitsoris, Feedback control of linear discrete-time systems under state and control constraints, Internat. J. Control 47 (6) (1988) 1727-1735.