Synthesis of State Feedback for Linear Systems Subject to Control Saturation by an LMI-Based Approach

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

SYNTHESIS OF STATE FEEDBACK FOR LINEAR SYSTEMS SUBJECT TO CONTROL SATURATION BY AN LMI-BASED APPROACH J.M. Gomes da Silva Jr. *,1 A. Fischman .. ,1 S. Tarbouriech* J.M. Dion" L. Dugard"

* LAAS-CNRS, 7 Avenue du Colonel Roche, ;11077 Toulouse Cedex 4, France. e-mail: (jmgomes.tarbour)@laas.fr .. LAG-INPG, ENSIEG, BP 46,38402 St.-Martin-d'Heres, France. e-mail: (arao.dion.dugard)@lag.grenet.fr

Abstract: This paper addresses the problem of synthesis of state feedback control laws subject to input saturation. By using a quadratic approach, conditions guaranteeing the local asymptotic stability of the non1inear (saturated) closed-loop system are stated. From these conditions, a framework based on Linear Matrix Inequalities (LMIs) is proposed to compute a state feedback matrix in order to ensurp. the asymptotic stability of a given admissible set of initial states. The trade-off between saturation tolerance, performance requirements and the size of the region of initial states to be stabilized is discussed. Keywords: control saturation, stabilizing feedback, stability domains, LMI.

1. INTRODUCTION

stability of the closed-loop system with respect to the control limits should be considered . This fact has motivated, in the last years, the development of design techniques that consider, a priori, the control bounds (Bernstein and Michel, 1995).

Control systems are often linearly designed. The plant is represented by a local linear model and the control law is, in general, a linear state or output feedback one. The modern theory of linear control provides efficient techniques and methods to compute such control laws that guarantee both stability and some performance and robustness requirements with respect to (w.r. t) the linear closed-loop model of the process. In general, this kind of design does not explicitly consider the bounds on the control inputs and saturation can cause instability of the system . Thus, it is important to focus another aspect of the robust stability: the control law has to be able to preserve the stability of the closed-loop system when the actuators saturate. In other words, the robustness of the 1

It is well-known (Burgat and Tarbouriech, 1995), (Sussmann et al., 1994) that the global stabiliza-

tion of linear systems subject to control saturation can be achieved if and only if the openloop system is not strictly unstable. Furthermore, when the linear system is obtained from a linearization of a nonlinear system, the global stabilization makes no sense. In this case, we should consider the semi-global stabilization (Lin and Saberi, 1995),(Alvarez-Ramirez et al., 1994) and the local stabilization (Gutman and Hagander, 1985),(Burgat and Tarbouriech, 1995). Then the objective is to guarantee the asymptotic stability with respect to a given set Xo of admissible initial states. In this sense, two approaches can be cited. The first one consists in computing the

Authors supported by CAPES and CNPQ, Brazil.

207

It is worth noticiug that inside the domain S(F, p) defined as :

control law in order to guarantee the positive invariance of a set S ::> Xo such that S is contained in the region of linear behavior of the closedloop system. Hence, the saturation is avoided in S. However, by this methodology, if the set Xo is relatively large, it may not exist a solution to the problem. The second approach consists in considering the saturation and consequently the nonlinear behavior of the closed-loop system.

t::.

S(F,p)={xE~n; -p~Fx~p}

the control inputs do not saturate and therefore the evolution of the closed-loop system IS described by the following linear model:

x(t) = (A + BF)x(t)

This paper is concerned with the second approach. The objective is to propose an LMI-based framework for the synthesis of a state feedback control law that guarantees both the local stability of the closed-loop saturated system for a given polyhedral set Xo and a certain time-domain performance when the system is not saturated.

Consider now a given polyhedral set Xo of admissible initial conditions represented by the convexhull of its vertices: Xo = Co{ Xl , ••. , x,,}, Xi E ~n, i = 1, .. . , v. The problem to be solved in the sequel is stated as follows : Problem 1. Consider system (1). Compute a saturated state feedback control law defined by (3) such that for all the initial states belonging to X o, the corresponding trajectories converge asymptotically to the origin. In addition, this law should also guarantee a given time-domain performance specification inside the domain of linearity of the system defined by (5) .

This problem can be interpreted as a problem of Local Asymptotic Stability. In fact, to solve it we should calculate a state feedback that guarantees the local stability of system (4) in a region that contains the set Xo . Note that if we are able to compute a matrix F that makes a set S ::> Xo positively invariant and contractive w.r.t the saturated system (4), then the problem is solved (Gutman and Hagander, 1985). In this paper we deal with a quadratic approach for synthesis and thus we are particularly interested in ellipsoidal domains of invariance and contractivity.

2. PROBLEM STATEMENT

Consider the following linear continuous-time system defined as :

(1)

where x(t) E ~n and u(t) E ~m are respectively the state vector and the control vector. Matrices A and B are real constant matrices of appropriate dimension . For system (1) the following hypotheses hold :

(1) The control vector is subject to linear constraints which define the polyhedral compact region n c ~m: t::.

n={uE~m;-p~u~p}

3. LOCAL STABILIZATION OF THE SATURATED SYSTEM

In this section, conditions that guarantee the local asymptotic stability of the saturated system in an ellipsoidal set are stated.

(2)

with p>- O. (2) The pair (A, B) is assumed to be stabilizable.

Note that the saturated control law defined in (3) can be also written as :

Consider the following saturated state feedback control law : u(t) = sat(Fx(t)), F E ~m.n , where each component is defined, Vi 1, . .. ,rn, by :

=

(sat(Fx(t)))i

=

(sat(Fx(t)))i = a(x(t));FiX(t)

-Pi if FiX(t) < -Pi FiX(t) if - Pi :; FiX(t) :; Pi (3) { Pi if FiX(t) > Pi

where 0 as :

the closed-loop system is given by the following nonlinear model :

x(t) = Ax(t)

+ Bsat(Fx(t))

(6)

Outside S(F, p), the control inputs saturate and the stability of the system must be analyzed by considering equation (4) .

Notations. For any vector x E ~n, X !: 0 means that all the components of x, denoted Xi, are nonnegative. For two vectors x, Y of ~n, the notation x !: y means that Xi - Yi ~ 0, Vi = 1, ... , n . For a vector x E ~n and a positive scalar s, xis denotes the vector whose components are defined by xii s . Ai denotes the ith row of matrix A. X(A) denotes the maximal eigenvalue of A. For two symmetric matrices, A and B , A > B means that A - B is positive definite. AT denotes the transpose of A. 11 . 11 denotes the Euclidean norm and 11·1100 the oo-norm.

x(t) = Ax(t) + Bu(t)

(5)

< a(x(t));

a(x(t))i

(4) 208

=

~

(7)

1, Vi = 1, . .. , rn, is defined

-Pi -(-) if FiX(t) < -Pi FiX t 1 if - Pi ~ Fix(t) :; Pi { ~() if FiX(t) > Pi FiX t

(8)

and contractive w.r.t system (12), Vc > O. Since this system represents the behavior of (4) only for the states belonging to S(F, pia), thus there exists Cmaz > 0, such that that £(cmaz ) C S(F, pia) . It follows that, £(c), with C ::; Cmax , is a contractive set w.r .t system (4). Therefore, the local asymptotic stability of this system in £(c), C ::; Cmaz , is guaranteed. 0

Thus, system (4) can be rewritten as

x(t) = (A + BD(a(x(t)))F)

(9)

where D(a(x(t))) E ~m*m is a diagonal matrix with a(x(t)); , i = 1, ... , m as diagonal elements. Let 0

< a ::; 1 be a given lower bound for

a(x(t));, Vi = 1, . .. , m . The set of states such that a(x(t))i ~ a > 0 is given by : S(F,plo:)

= {x E ~n ;

-plo:

~

Fx

~

plo:}

(10)

3.2 ?loo Approach

From these definitions, we present in the sequel, two approaches in order to formulate conditions for the local asymptotic stability of system (4) in an ellipsoid.

Note that equation (4) is equivalent to : .i;(t)

= (A + BF)x(t) + B(D(o:(x(t))) -

I)Fx(t) (14)

By considering the lower bound a of the saturation term, a(x(t));, Vi, system (4) can be represented, in S(F, pia) , by the following system :

3.1 Polytopic Approach

x(t) = (A + BF)x(t)

Define matrices A(j) E 1Rn *n as follows:

+ B~(t)Fx(t)

(15)

with 11~(t)11 ::; (1 - a). For this kind of system the following result is known. where DU) E ~m.m is a diagonal matrix whose elements arbitrarily take the value 1 or a, j = 1, ... , 2m (Molchanov and pyatnitskii, 1989). From a convex linear combination of such matrices A(j) , the motion of system (4) in the set S(F,pla) can be represented by (Burgat and Tarbouriech, 1995) :

Theorem 1. (Petersen, 1989) The system (15) is quadratically stable if and only if the following conditions hold:

(i) All the eigenvalues of (A+BF) have negative real part (ii) IIF(sI - (A + BF))-l Blloo < (1- a)-I.

p

x(t) =

L: >'(j)(x(t))A(j)x(t)

(12)

The problem of local stabilization of a saturated system in an ellipsoidal domain can be formulated as an ?loo problem as stated in the following proposition.

j=l p

with p

=2

m

,

L: >'(j)(x(t)) = 1, >'(j)(x(t)) ~ O. j=l

Proposition 2. If there exist matrices W = wr > 0, Y and a diagonal scaling matrix S = SI' > 0 such that:

Note that the matrices A(j) are the vertices of a convex poly to pe of matrices. From this poly topic representation, the following proposition can be stated. Proposition 1. If there exist a matrix W = WT o and a matrix Y such that , Vj = 1, . .. , 2m :

WA T

+ AW +BY + yTBT + y T S-ly +BSBT(1 - 0:)2

>

<0

(16)

Then, for F ~ YW-I, there exists a positive scalar Cmaz , such that the ellipsoids £(c) ~ {x E ~n ; xTW-1x::; cl, with C::; Cmaz , are domains of local asymptotic stability for system (4).

Then, for F ~ YW- 1 , there exists a positive scalar Cmax , such that the ellipsoids £(c) ~ {x E ~n ; xTW- 1X ::; c}, with c ::; Cmax , are domains oflocal asymptotic stability for system (4).

Proof: If (16) holds it follows that conditions of Theorem 1 hold and the control law u(t) = Fx(t) with F = YW- l stabilizes the system (15) and V(x) ~ xTW-Ix is a Lyapunov function for this system (Boyd et al., 1994) . From this point the

= 1, . . . , 2m , it follows that1 = Fx(t) with F = YW-

Proof: If (13) holds Vj the control law u(t)

proof mimics the one of Proposition 1. 0

stabilizes system (12) and V(x) ~ XTW-1x is a Lyapunov function for this system (Boyd et al., 1994). By definition, £(c) is positively invariant

Remark 1. Notice that all the trajectories of system (4) in S(F, pia) can be represented by those 209

p- 1 ~ {:} wl I ~ {:} wl ~ {:} wl >

of system (12) or (15) but the converse is not true. In fact, (12) and (15) are conservative representations of (4). Furthermore, the Lyapunov function V(x) = xTW-1x is a global Lyapunov function for system (12) or (15). However, since (12) or (15) represents (4) only for the states belonging to S(F, p/o:), V(x) is a local Lyapunov function for this system.

{:} w~ •

(GP-l)T LT Li(GP-l)w;2 p 1/ 2(GP- 1f LT Li(GP-l)pl/2 5..(pl/2(GP-l)T LT Li(GP-l)pl/2) max IIL;GP- 1/ 2yI12

-lIyll9 max IILiGxl12 - IIP1/2 Z 1151

>

{:} wl ~ ~llIIL;GxW {:} wl ~ (G;x)2 , 'fix E £ {:} -Wi :s; (GiX) :s; Wi , 'fix E £

Remark 2. : We have considered the case of an homogeneous saturation, in the sense that the same lower bound (0:) for the saturation term o:(x(t)) is applied to all the control inputs: 0: < O:(X(t))i:S; 1, 'Vi = 1, . . . ,m.

0

Lemma 2. (Boyd et al., 1994) Consider a polytope described by its vertices, S = Cot Vl, •. . , v$}. The ellipsoid £ contains the polytope S if and only if

VT1] ->0 [ Vi1 P4. SYNTHESIS OF THE CONTROL LAW

'Vi=I, ... ,s.

(18)

Definition 1. Let V be a subregion of the complex plane. A dynamical system x(t) Ax(t) is called V-stable if all the eigenvalues of A lie in V.

=

Now, given the control constraints (2), a domain of initial admissible states Xo , and a coefficient of saturation 0: , we propose a methodology to compute a control law of type (3) that solves Problem 1. The objective is to compute a matrix F that guarantees the stability of the closed-loop system (4) inside a positively invariant and contractive set £ . Such a set must contain Xo and be contained in S(F,p/o:). In addition, the computed feedback law should satisfy some given time-domain performance requirements when the closed-loop system is not saturated, i.e., in the region of linear behavior (5). These kinds of requirements can, in general, be achieved by placing the poles of (A+BF) in a suitable region in the complex lefthalf plane. Thus, we consider also that matrix F should place the poles in a specific region in the complex plane.

Let H, Q E ~., be symmetric matrices and z a complex number with its conjugate denoted by z. Then, for 1 :s; i, j :s; 1, define:

Consider now a region V ~ {z E C ; I-o(z) < O} This region is gener:ically called LMI region (Chiali and Gahinet, 1996). The following theorem gives a condition for the V - stability of a system x(t) = Ax(t) .

Theorem 2. (Chiali and Gahinet, 1996) The system x(t) = Ax(t) is V-stable if and only if there exists a symmetric matrix W > 0 such that, for 1 :s; i,j:S; 1:

Before giving a solution for Problem 1, some preliminary results are presented in the sequel.

[hij W

Consider a symmetric polyhedral set S(G , w) ~ {x E !Rn ; -w :5 Gx :5 w}, G E ~9·n, W E !R9 ,

+ qij(AW) + qij(AW)T] < 0

and an ellipsoidal set £ ~ {x E ~n ; xT Px :s; 1}, P = pT > O. Let Li be the ith row of a g-order identity matrix. Then we can state the following two lemmas.

From the results presented above we are able now to formulate sufficient conditions to be verified in order to solve Problem 1. Consider a set of admissible initial states Xo = CO{Xl' ... ' x tl } , an LMI region V and a coefficient of saturation 0:.

Lemma 1. £ c S(G , w) if and only if, 'Vi

Proposition 3. If there exist matrices W = WT > 0, Y, and a diagonal matrix S = SI' > 0 satisfying the following LMls :

1, ... ,g

(i) [hijW+qij(AW+BY)+%(AW+By)T] < 0 (ii) LMIs (13) or LMI (16) yT LT ] > 1 m ••• Li Y (pdo:)2 - 0 , '>-I; v. = , .. . , .

(;;;) [W Proof: By using the Schur's complement, (17) is equivalent to:

(iv) 210

[1Xi xT] ~O , 'Vi = 1, ... ,v. W

Then F = YW- 1 solves Problem 1 and £ = {x E ~n ; X T W- 1 x ::s I} is a domain of asymptotic stability for system (4) .

linear performance requirements and to guarantee the stability of X o, it is often necessary to consider the saturation.

Proof : From Theorem 2, the LMI (i) implies that the poles of (A + BF) are clustered in region V . From Lemmas 1 and 2, LMIs (iii) and (iv) guarantee that Xo C £ C S(F, pia) . Finally, from Proposition 1 or 2, the LMls (ii) implies that £ is a domain of asymptotic stability for system (4) and F = YW- 1 solves Problem 1. 0

When, for the specified V and a there is no feasible solution for LMIs (i), (ii), (iii) , (iv), we should relax the performance requirements (by choosing a new less restrictive region V) or consider a smaller a of tolerance. Nevertheless, it is important to note that for the considered domain of initial admissible states it may not exist a stabilizing feedback for the system which is the case when Xo is not included in the controllable region of the constrained input system.

~

~

The conditions stated in the above proposition are in an LMI form . These constraints can be incorporated in a convex optimization scheme and efficiently solved. Hence, LMIs (i) ,(ii),(iii) and (iv) can be viewed as a basic framework to synthesize saturated control laws.

5. NUMERICAL EXAMPLE Consider the control of two inverted pendulums in cascade where the system matrices are:

The LMI (i) represents the time-domain requirement as a pole placement of (A + BF) in the specified region V . Remark that this performance requirement concerns only the non-saturated behavior of the system. If the system is saturated our objective is only to guarantee the local asymptotic stability. The choice of V and a represents the trade-off between the performance requirement and the saturation tolerance. In general, the more stringent is the performance requirement (in terms of pole placement), the higher is the feedback gain associated to satisfy it. High gains are associated with small regions of linear behavior, in other words, the possibility of control saturation is greater. The smaller is a, the larger is the zone of nonlinear behavior of the system and consequently, the general performance of the closed-loop system is degraded. However this allows to consider larger domains of admissible initial states. Hence, we propose now an algorithm to consider this trade-off and compute a matrix F that solves Problem 1.

A _ -

[

0] 0 01 -9.80 0] 9.8 0 . B = [0 1 -2 00 01' 00 -9.8 0 2.94 0 -2 5

Consider that the control bounds are defined by (2) with P = [10 lof. Suppose that we want to stabilize the following set of initial conditions: 0.5]

Xo

=Co{ [ 0:5

[0.5] '

-~.5

'

[ -0.5] [ _0.5] 0:5 ' }

with a tolerance of saturation given by:

-~.5

atol

= 0.5

Now a matrix F that solves Problem 1 is computed by applying the algorithm described in section 4. We illustrate the solution by considering two examples of V regions: a Jl-shift (~e(z) < -Jl) and a disk centered at Jl with radius d. Jl-shift In this case LMI (i) is given by:

Algorithm

W AT

• Step 0: Fix the region V from the timedomain requirements. • Step 1: Choose the tolerance of saturation:

+ AW + BY + yT BT + 2~W < 0

Consider as requirement Jl = 3. For this value of J.L we do not find a feasible solution for a = 0.5, however, for a = 0.4 the solution exists. If we relax the time-domain requirement by considering J.L = 2.5, there exists a feasible solution for a = 0.5. In this case, one matrix F that solves the problem

atol

• Step 2: Set a = 1 • Step 3: Solve LMI's (i),(ii),(iii) and (iv) as a feasibility problem . If there is no solution then decrease a . Otherwise compute F and stop. • Step 4: If a > atol goto step 3. Otherwise it is not possible to obtain a solution for the problem from the proposed methodology with the given data.

IS :

F

= [-27.2730 -10.2347 10.3811

11.3835 -2.2541] 3.7590 -10.4068 -0.8179

and a domain of local asymptotic stability for the saturated system is given by the ellipsoid £ = {x E ~4 ; X T W- 1 x::s I} with:

Note that we begin the algorithm with a = 1, that is, we try first to solve the problem linearly, without saturation. However, to satisfy both the

W- 1

211

=

1.9621 0 .7317 -0.6974 0.1852] 0.7317 0.2801 -0.2929 0.0712 -0.6974 -0.2929 0.5503 -0.0383 [ 0.1852 0 .0712 -0.0383 0.0273

It can be verified that Xo C £ C S(F, p/0.5).

The trade-off between linear performance and saturation tolerance can be illustrated by the following table. For a given J.L, it is showed the approximated maximum a for which it is possible to obtain a solution by using the proposed framework: 1.9 1

Disk In this case LMI (i) is given by: W WAT+ILW+yTB T ] AW+ILW+BY 02W >0 [

Since there exist efficient methods to solve LMIs and dedicated software is available, this approach represents an interesting way to compute feedback control laws with saturation tolerance and thus, to consider the control constraints directly in the design step of the control system. Then, we can conclude that the computed feedback guarantees the stability of the closed-loop system with a certain degree of robustness with respect to the actuators limits. It can be pointed out that the presented method can be straightforwardly extended by considering uncertainties in the matrices of the system.

Consider as a requirement J.L = 10 and 0 = 8. For these values we do not find a feasible solution . If we relax this requirement by considering a disk with radius 0 = 8.2, there exists a feasible solution for a = 0.5. In this case, one matrix F that solves the problem is: F

= [-22.4403 7.0591

7. REFERENCES Alvarez-Ramirez, J., R. Suarez and J. Alvarez (1994) . Semiglobal stabilization of multiinput linear system with saturated linear state feedback. Syst. & Contr. Let. (23),247254. Bernstein, D.S. and A.N. Michel (1995). A chronological bibliography on saturating actuator. Int. J. of Rob. and Nonlin. Contr. 5, 375380. Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. Burgat, C. and S. Tarbouriech (1995). Some control strategies for discrete-time saturated state feedback regulators. In: Non-Linear Systems. Vol. 2. Chap. 4. Chapman & Hall. Chiali, M. and P. Gahinet (1996). Hoo design with pole placement constraints: an lmi approach. IEEE Trans. Autom .. Contr. 41(3), 358-367. Gutman, P.O. and P. Hagander (1985). A new design of constrained controllers for linear systems. IEEE Trans. Autom .. Contr. 30, 2233. Lin, Z. and A. Saberi (1995). Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedback. Syst. & Contr. Let. 24, 125-132. Molchanov, A.P. and E.S. Pyatnitskii (1989). Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst. & Contr. Let. 13,59-64. Petersen, LR. (1989). The matrix riccati equation in state feedback, H 00 control and in the stabilization of uncertain systems with norm bounded uncertainties. In: Workshop on the Riccati Eq. in Cont. Syst. and Sig.. Como, Italy. pp. 51-56. Sussmann, H.J., E.D. Sontag and Y. Yang (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Autom .. Contr. 39(12), 2411-2425.

-8.6673 15.0748 -0.8084] 2.2226 -8.0464 -1.2620

and a domain of local asymptotic stability for the saturated system is given by the ellipsoid £ = {x E 1R4 ; X T W- 1 x ~ I} with: W- 1

=

1.4896 0.5873 -0.8840 0.0987] 0.5873 0.2352 -0.3623 0.0406 -0.8840 -0.3623 0.7081 -0.0332 [ 0.0987 0 .0406 -0.0332 0.0152

It can be verified that Xo C £ C S(F, p/0.5).

The trade-off between linear performance and saturation tolerance can be illustrated by the following table. By considering J.L = 10, for a given 0, it is showed the approximated maximum a for which it is possible to obtain a solution by using the proposed framework:

I ! I 8~0 I ~:~ I ~:: I 8~6 I In this case, with 0 = 8 we cannot find a solution whatever a is.

6. CONCLUSION The problem of state feedback synthesis subject to control saturation was addressed. Local stability conditions for the saturated system were stated by considering two different representations for the nonlinear closed-loop system. From these conditions, an LMI-based framework was proposed to compute a state feedback matrix that guarantees the stability of the closed-loop system for a given set of admissible initial conditions. In addition, this feedback guarantees a time-domain performance requirement (in terms of pole placement) when the system is not saturated. 212