New Algorithms for the Quadratic Stabilization of Uncertain Linear Systems

Copyright © IFAC System Structure and Control, Nantes, France, 1995

NEW ALGORITHMS FOR THE QUADRATIC STABILIZATION OF UNCERTAIN LINEAR SYSTEMS F. AMATO·, A. PIRONTI* and S. SCALA·· • Universitd di Napoli Federico II, Dipartimento di Informatica e Sistemistica, via Claudio 21 , 801 25 Napoli , Italy •• Centro Italiano Ricerche Aereospaziali, via Maiorise , 81043 Capua , Italy Abstract. In this paper we consider a linear system depending in non-linear fashion on a vector of uncertain parameters. Asswning that the parameters range in a hyperrectangle, we provide a necessary and sufficient condition for the quadratic stabilizability of the system in tenns of convex optimization procedures . Resume. Dans ce texte nous traitons un systeme lineaire qui depend dans une maniere pas lineaire d'un vecteur de parametres incertains. Suppose que les parametres se trouvent dans un hyperectangle. no us donnons une condition necessaire et suffisante pour la stabilisabilite quadrati que du systeme en termes des procedures d 'optimisation convexe.

Key Words. Linear systems; uncertain parameters ; quadratic stabilization ; linear matrix inequalities : convex programming

1. II\TRODUCTION

A more general result can be found in Bernussou et a/ (1989) , where a system depending affinely on parameters is proved to be quadratically stabilizable if and only if a certain convex optimization problem admits a solution.

The concept of quadratic stability dates back to the late Seventies with the germinal papers by Leitmann (1979) and Barmish (1983). An uncertain linear system is said to be quadratically stable if there exists a fixed (that is time-invariant) Lyapunov function whose derivative along the trajectories of the system is negative definite for all possible values of the uncertain parameters. In this case uniform asymptotic stability of the origin of the state space is guaranteed with respect to all possible temporal behaviour of the parameters within their bounding set.

In this paper we consider the case in which the system and input matrices depend non-affinely on parameters. First we show that if the dependence is multi-affine it is still possible to solve the quadratic stabilization problem via convex optimization algorithms. Then we deal with the case of non-affine dependence. In Section 3 we introduce a procedure presented in Amato et a/ (1994) to cover the image of a non-affine matrix valued function by the image of a function depending multi-affinely on parameters: via this procedure we can substitute the original non-affine system matrices with multi-affine ones and apply the previous algorithms. Since the covering introduces conservatism , the solvability of the original non-affine problem does not guarantee that of the problem with the new matrices; in Section 4 we show how to reduce this conservatism. Finally in Section 5 we provide an example of applicaton of our technique.

In the same way, considered a forced linear plant with uncertain system and input matrices, it makes sense to consider the problem of finding a linear state feedback controller which quadratically stabilizes the overall closed loop system. A sufficient condition guaranteeing quadratic stab ilizability via linear control is the so-called Matching Assumption (see Barmish , 1983) . However , since this structural requirement is too restrictive , the effort of the researchers has been devoted to understand when it is possible to state conditions which are both nece.ssary and sufficient . In Khargonekar et a/ (1990) it is shown that when the dependence on parameters is in the so-called norm bounded , one block form , the stabilizing controller exists if and only if a suitable algebraic Ri ccati equation has a positive definite solution .

2. PRELIMINARIES Consider an uncertain system in the form

x(t)

= A(p)x(t) ,

t E IR+ ,

pER C IR n •

,

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205

n where z(t) E rn. , p (PI pz . . . Pnp)T is the vector of uncertain parameters, R is a hyperrectangle, i. e.

=

As shown in Bernussou et al (1989), letting

s=p- 1 ,

(7)

it is readily seen that system (5) is quadratically stabilizable iff the following problem admits a solution. and A( . ) is continuous . We denote by 1, ... , 2np , the vertices of R.

P(i) ,

i =

Problem 1: Find a symmetric matrix S E rn.nxn and a matrix L E rn.mxn .such that

Definition 1: Quadratic stability.

i) S

The uncertain system (J) is said to be quadratically stable if there exists a positive definite matrix P E rn.n xn such that for all pER

L(p)~-(AT(p)P+PA(p»)>O .

ii) Ls(p) ~ - (SAT(p) LT BT(p)

(3)

= A(p(t»z(t) ,

t E rn.+

(4)

Theorem 1: Assume that the dependence of A and

(5)

B on parameters is multi-affine, i. e.

with u(t) E rn.m , and A( . ) and B( . ) continuous, we can state the follov"ing definition .

np A(p) -- '"'A' . pilpi2 .. . pin, ~ 11,12,· · ·,ln." 1 2

(8a)

i1 ,i:l I... ,i ftp

Definition 2: Quadratic Stabilizability.

np B(p) -- '"'B· · . pilpi2 ~ ll , Z2 , ... ,Znp 1 2 ... pinp

The uncertain system (5) is said to be quadrat-

i l ,i 2 , ... ,i n

ically stabilizable via linear state feedback control if there exists a matrix KErn. m X n such

,

(8b)

,

where ij E {O, I}, j = 1, .. . , n p , Ai l .i 2 •...• i np E Bil.i2 ..... inp E rn.nxm . Then system (5) is quadratically stabilizable via linear control iff there exist S > 0 and a matrix L E rn.mxn such that for all i = 1, . .. , 2np

that the unforced system , obtained from (5) letting u Kz ,

rn.nxn and

=

x(t) = (A(p)

Vp En .

convex hull is positive definite. When the convex hull is a polytope, the set of matrices is positive definite iff the vertices are.

In the same way, given a forced uncertain linear system in the form

+ B(p)u(t) ,

+ B(p)L) > 0

Fact 1: A set of matrices is positive definite (i. e. each matrix in the set is positive definite) iff its

for all Lebesgue measurable functions p : rn.+ ,....... R , t ,....... pet) .

x(t) = A(p)z(t)

+ A(p)S+

When A and B depend multi-affinely on parameters we can state the next fundamental theorem. To this end we need the following well known result .

Quadratic stability guarantees uniform asymptotic stability of the linear time-varying system x(t)

> 0;

+ B(p)K)z(t)

is quadratically stable .

We recall that quadratic stabilizability via dynamic, time-varying state feedback linear control implies quadratic stabilizability via memoryless, time-invariant, linear state feedback control (see Petersen , 1988) . Hence we do not lose any generality in Definition 2. On the contrary. as shown in Petersen (1985) , quadratic stabilizability without any other specification does not imply quadratic stabilizability via linear control , hence this specification in Definition 2 is mandatory.

P{i )

Proof. Under the hypothesis of the theorem , the matrix function Ls(p) defined in ii) of Problem 1 depends multi-affinely on parameters . The well known Mapping Theorem (Zadeh and Oesoer , 1963) states that the convex hull of a multiaffine function , defined on a hyperrectangle, coincides with the convex hull of the images of the vertices of the hyperrectangle . The proof follows from Fact 1. •

From Definition 2 follows that the uncertain system (5) is quadratically stabilizable via linear control if and only if there exist a positive definite PE rn.n xn and a matrix K E rn.mxn such that LK(p) ~ -(A(p) +P(A(p)

Remark 1: Note that Theorem 1 recovers as particular case Theorem 1 in Bernussou et al (1989) , where the affine dependence on parameters was considered.

+ B(p)Kfp

+ B(p)K» > o.

being the i-th vertex ofn .

By virtue of Theorem 1, when A and B depend multi-affinely on parameters, Problem 1 is equivalent to the following.

Vp E R .

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206

Problem 2: Find a symmetric matrix S E IR and a matrix LE IR mxn such that

nxn

practical applications because in many situations the parameters have a physical significance and hence are inherently positive. In any case we can always split, prior to the application of the algorithm , the original hyperrectangle in smaller hyperrectangles, each contained only in one orthant of IRnP .

i) S> 0 ; ii) LS(P(i)

> 0,

Since the set of positive definite matrices is convex , and the constraints in ii) of Problem 2 contain linear matrix inequalities which are of a convex nature , Problem 2 can be solved via iterative algorithms based on cutting plane techniques. Recently Bernussou et al (1989) have proposed an algorithm which , in a finite number of iterations, establishes whether Problem 2 has a solution or not.

3.1. An Algorithm to Cover the Image of a NonAffine Function by That of a Multiaffine Function

Consider a matrix valued function

i l ,i 2

When the dependence on parameters of the system matrices in equation (5) is more complex than multi-affine , constraint ii) in Problem 1 is no longer reducible to a finite number of convex constraints as in ii) of Problem 2, with the obvious consequence that the previous algorithm cannot be utilized .

, ··

.,iv

satisfying Assumption 1. The next algorithm constructs a multi-affine function whose image covers that of F( . ) . Algorithm 1: Step 1 For j 1, ... , v if /j(p) in (10) is not multi-affine , apply the procedure detailed below to construct a couple of affine functions f . and 7)· such that

=

In the next section , making use of a procedure recently proposed in Amato et al (1994) (see Algorithm 1 below), we will show how to cover the image of a non-affine function by that of a multiaffine one; via this procedure we can substitute the original non-affine system matrices with multiaffine ones and apply the above-mentioned algorithm.

-)

"ipEn

We show how to find these bounding functions in case of convex functions (the procedure is similar for the concave case) . Let p' E nand

3. A PROCEDURE FOR NON-AFFINE FUNCTIONS In this section we treat the case in which the system matrices in (5) depend non-affinely on parameters. The class of functions considered is specified in the next assumption .

Then set

f .(p)

-)

= /j(P·)+7rj(P*)(p-p*)·

Now consider the images of the vertices of

(13)

n under

t:;.

/j, namely /ji = /j(P(l» , . .. , /j2np = fj(P(2 np ), and order them with decreasing magnitude , say ... > f)· h 2 n l' • Simple considerations from f)·h 1 > linear algebra show that there exists an integer r E {np + 1, ... , 2"p} such that the hyperplane hj(p) in the space IR"p+i , connecting the points (p[h,) , fjh , ),.. ., (p[hr) , fjhJ, is univocally determined . By virtue of the convexity of fj we have that , for all pEn , /j(p) S hj(p) ; hence set

il,i2 ,···, iv

E {O ' I} A·t 1 · ., tv . E IRnx " } nxm B i l,i2 , ... ,i~ E IR and the aj '5 and bj '5 are continuous scalar functions . For j 1, . . . , v, if aj J ~2 , · ·

the gradient of /j (p) in

lar, 1970 , Theorem 25.1)

t:;.

iJ'

7rj (p')

p' . The following inequality holds (see Rockafel-

Assumption 1: The matrix valued functions A( . ) and B( . ) in (5) can be written in the following way

where

(11)

J

=

is not multiaffine it is assumed to be convex or concave; the same holds for the bj '5.

(14)

Note that Assumption 1 recovers as a particular case the multi-affine dependence .

Step 2 Construct the matrix function ip(p,8) obtained from (10) substituting for /j(p) , j = 1, . . . , v , the multi-affine function

Remark 2: It is interesting to note that many functions are convex or concave only if the parameters on which they depend do not change sign . This fact does not cause any loss of generality in

207

(1- OJ)[)p) { /j (p)

+ OJ/j(p)

if /j is not multi-affine if /j is multi-affine, ( 15)

where the -) f. 's and the /)' 's are affine functions constructed according to Step 1 and

O. I. ~ { [0, 1] ) E ) {O}

6.

6.

6T

gT)T .

Step b) Define

~Ls(W) ~ S~~(W)+~A(W)S

F(R).

+LT ~~(W)

Observe that !Ji(p , 6) is an affine function of the Oi 'S and a polynomial function of the Pi'S; moreover since in (10) there are at most v products, the maximum exponent of each Pi cannot exceed

Then solve Problem 2 with

~ Ls

+ ~B(w)L. (19) replacing Ls·

In this case, the existence of a solution to Problem 2 is no longer equivalent to the quadratic stabilizability of the system under consideration. This is obviously due to the fact that the immersion of the images of A and B introduces conservatism. In the next section we show how to reduce this conservatism.

v.

Step .) Replace each pf;, Cli > 1, with a multi-affine function as in (15); obviously in this case the bounding functions (which can be easily computed applying Step 1) will depend only on Pi . The number of replacements , say I, may vary between zero (when !Ji is already multi-affine, that is !Ji does not contain powers of PI, . . . , Pn p greater than 1) and np(v - 1) .

4. A REFINEMENT PROCEDURE Let us denote with "co ( . )" the operation of taking the convex hull of the argument ; for a given S and L, recalling that ~Ls(St) 2 Ls(R) and that ~ Ls is multi-affine, application of Fact 1 enables to conclude that

In this way we introduce a new vector of fictitious variables g ranging in the hyperrectangle £ defined in the same way as V and define the new function ~(p , 6, g) ; in Theorem 3 of Amato et al (1994) it is shown that ~ is multi-affine, moreover

x V x £)

E IRI'A+I'B and

=

Let V ~ 11 X 12 X ... x Iv ; hence !Ji(p , 6) is defined over R x V. It is simple to recognize that

~(R

f

define V = V A X VB and £ = £ A X £B; note that V and £ have 2I'A+I'B and 21'A+1'B vertices respectively. Finally let !l ~ R x 1> x £ and 2"n 2 np +I'A+I'B+1'A+1'B the number of its vertices. In order to simplify the notation. we will consider ~ A and ~ B functions of the whole vector w (pT

(16)

2

o~

g~)T E IR1'A+1'B; in the same way we

=

if /j is not multi-affine if /j is multi-affine.

!Ji(R x V)

Let us define 6 ~ (6~

g ~ (g~

co (Ls(R)) ~ co ({~Ls(W(i)) , i = 1, .. . , 2nn}) . (20) where , as usual, we denote the vertices of n by

::>

!Ji(R x V)

W(i ) '

::>

F(R) .

In the next theorem we show that splitting the hyperrectangle R in smaller hyperrectangles and considering the union of the images of these subhyperrectangles, the right hand side in (20) converges to the left hand side when the splitting is made finer and finer.

Remark 3: Note that RxV x£ is a hyperrectangle with 2 n p+I'+1' vertices, where jJ. is the number of non-multi-affine functions in the expression (10) of F and, as said , 1 is the number of replacements in Step 3 of Algorithm 1.

The solution of Problem- 1 when A( . ) and B( . ) satisfy Assumption 1 is composed of two steps:

We say that T = {RI , R 2 , ... , Rd is a rectangular covering of the hyperrectangle R if U~=I Rr = R . We define T as the set of all rectangular coverings of R.

Step a) Apply Algorithm 1 to construct two multi-affine functions

Now let C(IRn) the set composed of all closed sets in IRn . We equip this set with the Hausdorff metne

:3.2 . Solution of Problem 1 in the Non-Affine Case

~ A (p , 0 A , g A) :

£A

.......

~B(p, OB, gB) : R x VB x £B

.........

R x VA

X

IR n x n IR nxm

dH(SI,

(21) with 5 1 , 5 2 E C(IRn) and where d(x , S) denotes the usual Euclidean distance of the point x from the set 5 .

sllch that ~A(R , VA ' £A)

::>

A(R)

( 17)

~B(R , VB'£B)

::>

B(R).

(18)

52)~max { X,ES, max d(Xl' 5'2), max d(SI ' X2J} X 2 ES,

Let {ThhEIN , Th E T, a sequence of rectangular coverings, set up as follows . TI {R}; T2 is

=

208

constructed by splitting R into 2np hyperrectangles by lines parallel to the coordinate axes; T:3 is obtained applying the same procedure to each hyperrectangle in T2 and so on; note that Th is composed of 2(h-l)n p hyperrectangles. We denote by R hr the r-th hyperrectangle in Th .

Step 3 Split each element belonging to S in 2np hyperrectangles, put them into S and go to Step 2.

The next theorem studies the convergence properties of Algorithm 2. Theorem 3: Algorithm 2 converge.s in a finite number of iterations.

The rigorous proof of the next theorem is long and technical and is omitted for brevity (the interested reader is referred to Amato, 1994). Theorem 2: Con.sider the .sequence {Th hEIN, then ..,(h-l)n p

hl!..ri U co ({~t(w(;)), i = 1, ... , 2nn}) = Ls(R) r=l in the metric .space (C(IRn) , d H ), where ~t i.s the function con.structed on Rhr according to Algorithm 1 and Rhr X D x [ .

2(·-1)np

U co ({PLs(W([»), i = 1, ... , 2nD}) C Ls(fJ).

w(;) i.s the i-th vertex of rhr ~

r=1

At iteration s Problem 3 admits a solution and therefore Algorithm 2 stops.

Based on Theorem 2 we propose an algorithm for the solution of Problem 1 when the system matrices are non-affine and satisfy Assumption 1.

Conversely suppose that system (5) is not quadratically stabilizable. Since L s ( . ) is continuous , two situations can happen

Given a set S, Sj denotes the j-th element of S and card(S) the number of elements of S. Given two sets A and B , with B ~ A, A - B denotes the set of the elements of A which do not belong to B.

i) the set R equals the union of a finite number of subdomains r;, i = 1, ... , r , such that for each of these domains there exists a class of positive definite matrices Si and a class of matrices £i satisfying , for i = 1, ... , r,

Algorithm 2: Step 1 (Initialization) Let S {R} , h O.

=

=

Ls(p»O,

Step 2

=

=

Let h h + 1; for j 1, ... , card( S) ~ mh repeat the following procedure: apply Algorithm 1 to A( . ) and BC' ) restricted to Sj and solve the following problem.

and

i.s the i-th vertex of Dj ~

Sj

x

1)

x [.

r

If Problem 3 admits a solution then stop, the system is quadratically stabllizable and K LS- 1 is a compensator which quadratically stabilizes the system: otherwise test the necessary condition described below.

=

.5 . AN EXAMPLE We present the following example to clarify how the algorithm works when the system depends non-affinely on parameters.

Problem 4: Necessary condition Find S > 0 and L E IR mxn .such that LS(Pj(i»)

> 0, i

n;=1 Si = 0 or n;=l £i = 0;

VLE£i , (22)

For what concerns case i) , suppose, for istance. that r = 2. It is obvious that in this case Problem 3 does not have solution. On the other hand , there certainly exists a finite iteration index s such that the points LS(Pj(i») ' i = 1, . .. , 2np, j = 1, ... , rn" belong in part to r 1 and in part to 2 . Therefore at iteration s Problem 4 does not have solution and the algorithm stops. The discussion of case ii) is similar. •

Find S > 0 and L E IR mxn .such that

Wj(i)

VSESi

VpEri

ii) there is a subset Q of R such that the pair (A(p), B (p)) is not stabilizable for p E Q .

Problem 3: Sufficient condition

where

Proof. If system (5) is quadratically stabilizable there exist S > 0 and L E IR mxn such that Ls(p) > 0 for all pER. Following continuity arguments there exists an open set fJ ~ R such that system (5) is quadratically stabilizable in fJ. Obviously Ls(fJ) ~ Ls(R); therefore by virtue of Theorem 2 there will exist an index s such that

= 1, ... , 2np,j = 1, . .. , Tnh ,

Example 1: Let a ll~(P )

where Pj(i ) is the i-th vertex of Sj. A(p)

If this problem does not admit solution then stop, the system is not quadratically stabilizable; otherwise goto Step 3.

B(p)

209

(

(1

2{

0 1 a32(p)

0 ) -11

(23a) (23b)

where

a11(p)

=

a32(p)

-1 + exp(pdP2

(24a)

2 + P2sin(P2)

(24b)

6 . CONCLUSIONS In this paper we have considered the problem of quadratically stabilizing a linear system whose system matrices depend non-affinely on parameters . Using an algorithm which covers the image of a non-affine function by that of a multi-affine function, we have reconducted our original problem to one which can be solved via standard convex optimization procedures. Moreover we have provided an algorithm which reduce at will the conservatism due to the covering procedure .

with P E [0, .8]2. Note that B does not depend on parameters, so that ~B(W) = B . Now we have to compute ~ A(W) , The function A(p) can be written in the standard form (9) by letting al(p)

= exp(pt};

a2(p)

= P2 ;

a3(p)

= sin(p2) ' (25)

ACKNOWLEDGEMENTS

Step 1 In this case the non-mul ti-affine functions are a 1 , and a3 . Using Step 1 of Algorithm 1, a choice of the bounding functions is

The work of the first author was supported by Consiglio N azionale delle Ricerche under Grant n. 201.15.5. 7. REFERENCES

(ll(P) = 1 + 1.72p1

0.82+ 1.64p1 ;

{!l(P) !!3(p)

0.84p2 ;

(l3(P)

Amato, F. (1994) . Stability analysis for linear systems depending on uncertain time-varying parameters. PhD Thesis, Dipartimento di Informatica e Sistemistica, Napoli. Amato, F., F . Garofalo , L. Glielmo , and A. Pironti (1994) . Robust and quadratic stability via Poly topic set covering. 1nl. l . Robust and Nonlinear Control, to appear. Barmish , B. R. (1983). Stabilization of uncertain systems via linear control. IEEE Trans. Aut. Contr., AC-28 , 848-850. Bernussou, J., P. L. D. Peres, and J. C. Geromel (1989). A linear programming oriented procedure for quadratic stabilization of uncertain systems. Syst . Contr. Lett., 13 , 65-72 . Khargonekar , P. P. , I. R. Petersen , and K. Zhou (1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and Hoo control theory. IEEE Trans. Aut. Contr., AC-35 , 356-361. Leitmann , G . (1979). Guaranteed asymptotic stability for some linear systems with 1. Dyn . M eas. bounded uncertainties. Contr. " 101 , 212-216 . Petersen , I.R. (1985) . Quadratic stabilizability of uncertain linear systems: existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control. IEEE Trans. Aut. Contr., AC-30, 291-293 . Petersen, I. R. (1988) . Quadratic stabilizability of uncertain linear systems containing both constant and time-varying uncertain parameters. lOTA , 57 , 439-461. Convex Analysis. Rockafellar, R. T . (1970). Princeton University Press , Princeton . Zadeh, L. A., C. A. Desoer (1963) . Linear System Theory. McGraw-Hill, New York .

= 0.05 + 0.88P2.

Step 2 Substituting aj to aj . j = 1, 3, we obtain

1/;11 (p . 8) !li A (p, 8)

=

o1

~

(

1/;32(P , 8)

0 )

-1 1

with

'1/; 11 (p , 8)

= 0.82 +

'1/;32(p , 8)

=2+

1.64p1 + 0.186 1 + 0.08p161 (26a)

0.05p263 + 0 . 84p~ + 0 . 04p~63 (26b)

Step 3 The only non-multi-affine function in (26) is g(p) = p~ ; as bounding functions we choose g(p) -0.25 + P2 and g(p) P2. Substituting 9 and 9 in (26b) we obtain that -

=

=

o 1
where 1.79 + 0.84P2 - 0.0163 +

+0.09p263 +

0.0163~ .

0 . 21~

(27)

Note that ip A is multi-affine, moreover 3X3 where : [l -> rn.

~A [l

~R xD x[

= [0 , 0.8]2

X

[0 ,

IF x [0 , 1] . (28)

The algorithm solving Problem 2 converges after 55 iterations, and the resulting K is:

K

= (-24 .8003

29.3263

-7.2079) .

(29)

210