Systems & Control Letters 23 (1994) 205 215 205 North-Holland

Robust performance of linear parametrically varying systems using parametrically-dependent linear feedbac, k

G. Becker and A. Packard Department of Mechanical Engineering, University of California, Berkeley CA 94720, USA,

Received 13 May 1993 Revised 13 September 1993

Abstract." In this paper a parameter-dependent control problem for linear parametrically varying (LPV) systems is presented. Sufficient conditions are given that guarantee an LPV system is exponentially stable and achieves an induced L2-norm performance objective from the disturbance to error signals. The usefulness of parameter-dependent controllers is motivated from a gain-scheduling viewpoint. The resulting synthesis problem is reformulated into a convex optimization problem, which can be solved using efficient new algorithms.

Keywords: Gain scheduling; parameter dependent systems; o~g'; robust control; linear matrix inequalities; affine matrix inequalities; convex optimization.

I. Introduction

In this paper we consider a finite-dimensional linear system whose state-space entries depend continuously on a time-varying parameter vector p(t)~ R s. We assume that the trajectory of p(t) is not known in advance, although its value is known in real-time. This measurement of p(t) gives real-time information on variations in the plant's characteristics, so it is desirable to design a controller which uses this information. The usefulness of such a parameter-dependent controller can be motivated from a gain-scheduling viewpoint. The synthesis problem is to find a linear finite-dimensional controller, whose state-space entries can also depend continuously on p, such that the closed-loop system is exponentially stable and achieves good performance (as defined in Section 2.2) with respect to variations in p. The main analysis tool for assessment of stability and performance makes use of a single quadratic Lyapunov function. Note that this problem differs from standard linear time-varying system stabilization, since the controller dynamics are restricted to depend causally on the variations in the plant dynamics.

Relevant work includes the following: the existence of a stabilizing, parameter-dependent controller for a class of parameter-dependent systems was studied in [17, 14]; linear parameter varying (LPV) systems, 'quasi-LPV' systems, and their importance in gain-scheduling design is discussed in [30-33], and an analytical approach to gain-scheduled design is discussed in [26, 35]. In this paper, we simply begin with LPV systems as the starting point from which analysis and synthesis then proceeds.

The notation is standard. •" ×= is the set of real n x m matrices. The transpose of a matrix M is denoted M T. If M = M r, then the eigenvalues of M are real, and the notation 2max(M) is clear. Also for M --- M T, the notation M > 0 (M > 0) indicates that M is positive definite (positive semi-definite), and M < 0 indicates that M is negative definite. For M > 0, M 1/2 is the Hermitian square root. A vector u ~ " has Euclidean norm denoted by n u fl- L~'(~ +) is the set of m-dimensional vector valued, measurable, square integrable functions

Correspondence to." Dr. A. Packard, University of California, Department of Mechanical Engineering, Berkeley, CA 94720, USA.

0167-6911/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-6911(93)E0129-5

206 G. Becker, A. Packard / Robust performance of linear parametrically varyino systems

with norm I1"11 z. The extended space L~' ~ (N +) consists of functions, f such that P J e L z , for all t > 0, where Pt is the truncation operator defined as Ptf(r) = f ( r ) for z < t, and 0 for r > t.

2. Analysis of LPV systems using a single quadratic Lyapunov function

In this section we give the definition of an LPV system, and define an induced L2-norm performance objective. Related, motivating work on quadratic stability and on the control of linear parametrically dependent systems (including gain-scheduling) is found in [-1, 18, 17, 14, 12, 26, 30-33, 35].

Definition 2.1. Given a compact subset ~ ~ E~, the parameter variation set ~ denotes the set of all piecewise continuous functions mapping R (time) into ~ with a finite number of discontinuities in any interval.

Definition 2.2. Assume that the following are given: a compact set ~ c R S, and continuous functions A : ~ s--, R n×n, B:Rs--*E "×"", C : E ~--, R "e×" and D : ~ ~--, E'~×". These represent an nth order linear parametrically varying (LP V) system, whose dynamics evolve as

e(t)_] = C(p(t)) D(p(t)) d ( t ) ] ' where pe~-~,. (2.1)

The following definition summarizes some notation that will be used throughout this paper.

Definition 2.3. Denote the LPV system in Definition 2.2 as 2;(~, A, B, C, D), or Z~ for short, then for any p ~ - ~

• the linear time-varying system described in equation (2.1), will be denoted Xp, • q~o(t, to) is the state-transition matrix of Z'p, • for x(O) = O, the causal, linear operator Gp'L"S, e --* L"2~.e is given by

e(t) = Go(d)(t ) := [ ' C(p(t)) ¢o(t, r) B(p(z)) d(z) dr + D(p(t)) d(t). 3o

Finally, define G~-:= {G o :pe~-~}.

2.1. Quadratic stability of LPV systems

We now state the definition of a quadratically stable LPV system. To begin with, suppose that we are given and the continuous matrix function A : ~s ~ R" × ~. Consider the undriven LPV system

2(0 = A(p(t))x(t), (2.2)

where p e ~ . Define the scalar valued function V: [~" ~ R as V(x):= xTPx, where PER n× n, p = pT > 0. For any p ~ , along trajectories of (2.2), the time derivative of V(x) is given by

d dt V(x(t)) = xT(t) [AV(p(t))P + PA(p(t))] x(t).

Definition 2.4. The function A is quadratically stable over ~ (or QS over ~ ) if there exist a P e R "×", p = pT > 0, such that for all p ~

AT(p)P + PA(p) < 0. (2.3)

Since A depends continuously on the vector parameter p ~ and ~ is compact, it is clear that equation (2.3) implies that the left-hand side is uniformly negative definite.

G. Becket, A. Packard / Robust performance of linear parametrically varyin 9 systems 207

Remark 2.5. If equation (2.3) is satisfied, then there exist constants 7~, 72 > 0 such that for any p~#r~, the state-transition matrix ~p(t, to), which characterizes all solutions to equation (2.2), satisfies II qip(t, to) I[ < 7~ exp [ - Y2 (t - to)]. Therefore, quadratic stability gives a strong form of robust stability with respect to time-varying parameters.

For an LPV system 2:~,, if A is quadratically stable, we say that 2:a, is quadratically stable.

2.2. Induced L2-norm performance objective for LP V systems

We now introduce an induced L2-norm of a quadratically stable LPV system _r~. We also state a sufficient condition that guarantees 2:~, is quadratically stable and achieves an induced L2-norm performance objective. The following preliminary lemma establishes the existence of an induced Lz-norm for a quadrati- cally stable LPV system. Using Remark 2.5, the proof follows from classical results in LTV systems theory [7].

Lemma 2.6. Given a quadratically stable L P V system 2:(~, A, B, C, 0), there exists a finite scalar M > 0, such that for zero initial conditions,

PleJI2 sup sup < M < ~ .

d~L2

(2.4)

Using Lemma 2.6 we can now define the induced L2-norm of a quadratically stable LPV system.

Definition 2.7. Given a quadratically stable LPV system ~ , , for zero initial conditions, define

IfeJl2 IIa~,ll:= sup sup

d~L2

(2.5)

The following fact will be used in the proof of Lemma 2.9. Again, from Remark 2.5 the proof follows from standard LTV systems analysis.

Lemma 2.8. Given a quadratically stable L P V system S,a,,for deL2 and any x(O)e~",for all Pe~a' ,

lim x(t) = O. t ~ o O

Before starting the performance lemma for the general LPV system 2:(~, A, B, C, D), we state the following preliminary lemma for the case when D is identically zero.

Lemma 2.9. Given the LP V system X(~', A, B, C, 0). I f there exists an X e ~n× ,, X = X r > 0 such that for all p ~ ,

AX(p)X + XA(p ) + CT(p)C(p) + XB(p )BT(p )X < O, (2.6)

then (1) the function A is quadratically stable over ~', (2) there exists a fl < 1 such that II G~, II < ft.

Proof. (1) Suppose that (2.6) holds. Since A depends continuously on the vector parameter p ~ ' and ~ is compact, it is clear that the condition is uniformly negative definite. Since CT(p)C(p) + XB(p)B~(p)X is uniformly positive semidefinite, AT(p)X + XA(p) is uniformly negative definite. Hence from Definition 2.4, A is quadratically stable over ~. Therefore, S~, is quadratically stable.

208 G. Becker, A. Packard / Robust performance of linear parametrically varying systems

(2) Since equation (2.6) is uniformly negative definite, there exists a fl < 1 such that

1 1 x AT(p)X + XA(p) + ~ CT(p)C(p) + -fl XB(p)B (p)X < 0 (2.7)

! e(t) to be the disturbance and error to the scaled LPV system, for all p e ~ . Define d(t):= x/fld(t) and ~(t):= ,/9

2~(~, A, 1-- B ~ C, 0). Next, define the scalar valued function V: 0~" ~ ~ as V(x):= xTXx. Using equation ¢~, ,

(2.7), a standard 'completion of the square' argument [36, 5] and the fact that l im,~o x(t) = 0, it can he shown that for any pa~-~, along trajectories of

1 Yc(t) = A(p(t))x(t) + - - ~ B(p(t))d(t),

V(x(t)) can be integrated between 0 and ~ . The resulting relationship shows that for any d¢Lz,

1 BTXx 2 + IJ dJ122 - JI ~ IJ ~, o < _ -

which implies that

sup sup < 1. p~.~. I1,., ~,0 il a7112

dEL;

Since Ilai12/lldl12 =~ Ilell2/l ldll2, it is clear that liG~,ll <- fl < 1. [ ]

The main analysis lemma used in solving the quadratic LPV y-performance problem follows.

L e m m a 2.10. Given the L P V system, Z(~, A, B, C, D), and scalar 7 > O. I f there exists an X e ~ n×n, X = X r > 0 such that for all p 6 ~ ,

-AT(p)X + XA(p) XB(p) y - ' CT(p)]

BT(p)X - 1 7-XD;(p) J < 0, 7 - ' C ( p ) 7-1D(p)

then (1) the function A is quadratically stable over ~ , (2) there exists a fl < 7 such that tl G~II <- ft.

(2.8)

Remark 2.11. The condition stated in (2.8) can be transformed into the more familiar Riccati inequality using Schur complements

AX(p )X -b XA(p) + 7- 2CX(p)C(p)

+ (XB(p) + 7 2 C~(p)D(p))(i _ 7-2 DT(p)D(p))-X(BT(p)X + 7-2DT(p)C(P)) < 0. (2.9)

This transformation is commonly found in the earlier state-space o'¢g~ research, [15, 16, 24, 37]. Note that if the state-space data is linear time-invariant (LTI), then Lemma 2.10 can be written as a necessary and sufficient condition and is equivalent to A being stable and the ~ norm from d ~ e being less than 7 [23].

P r o o f . (1) Suppose that equation (2.8) holds. Then it is clear that the (1, 1) block is uniformly negative definite, therefore A is quadratically stable.

(2) Equation (2.8) implies that

- I

G. Becket, A. Packard / Robust performance of linear parametrically varying systems 209

therefore (I - 7-2DX(p)D(p)) is uniformly positive definite, so we can always take its inverse and matrix square root. Define d(t):=d(t), and ~(t):~ ?-te(t) , then for any p ~ ' ~ , we can write the state-space representation of the scaled LPV system, 2;~,, as

j = L C(p(t)) D(p(t)) l Ld(t) J'

where (~(p):= 7-1C(p) and /)(p):= ?- lD(p) . Define [25, 27]

O(t) J = ~ [ I -- D ( p ( t ) ) D T ( p ( t ) ) ] '/2 - D (p ( t ) ) J Ld(t)J Note that oT(t)O(t) + t~r)(t)t~(t) = ~T(t)~(t) + dT(t)d(t). It is straightforward to verify that applying this

transformation to the LPV system in equation (2.10) results in

O(t) J [ A(p(t)) = d ( p ( t ) ) ]'

(2.10)

the unitary parameter-dependent transformation

(2.11)

where

,4(p) := A(p) + 7 - 2 B ( p ) ( ! -- 7 - 2 D T ( p ) D ( P ) ) - I D T ( p ) C ( P ) ,

/}(p) := B(p)( l -- 7-2 DT(p)D(p)) - 1/~,

C(P) := 7- ~ [I + T - 2 D (p)(I -- 7- 2 DT(p)D(p)) - 1DT(p)] 1/2 C(p).

From Lemma 2.9, there exists a/~ < 1 such that for ,~(~, A,/~, C,0), IJ G~,, </~, if there exists an X ~ R n×~, X = X T > 0, such that for all p e ~ ,

.4T(p)X + X,4(p) + CT(p)t~(p) + X B ( p ) B T ( p ) X < 0. (2.12)

It is straightforward to verify that the condition in equation (2.12) is identical to equation (2.9),where X is the solution to equation (2.8). Since [Id II - If e [[ = II a~ll - fl ell, Lemma 2.9 implies that fl G~, ][ < ft. Recalling the fact that ~(t) = 7-1e(t), we get IIa~,[I < Vfl. If we take fl = Vfl, it is clear that fl < 7, and [Ia~,,/I </~. []

3. Control of LPV systems for quadratic L2 performance

In this section, we use the LPV system analysis in Section 2.2 to formulate a performance oriented parametrically dependent output feedback synthesis problem, which we call the quadratic LP V 7-performance problem.

Given a compact set ~ R s, consider the open-loop LPV system,

e(t) = ICl (p( t ) ) ]Dxl(p(t)) Dl l (p( t ) ) d(t) , (3.1)

y(t) LC2(p(t)) IDEl(p(t)) Oz2(p(t)) u(t)

where p ~ , . If, for all p ~ , we restrict the matrix functions D12 to be full-column rank and D21 to be full-row rank, then it is possible to derive a control synthesis solution to the quadratic LPV 7-performance problem for the LPV system in (3.1) [2]. However, to simplify the derivation of the control synthesis result in Section 4, we make the following restrictive assumption on the state-space data [8]: DiE and Dzl are constant matrices, D l l = 0(n, xn,), D22 = 0(n~xnu), DIT2D12 -- In,, D21DT1 = I,~, DT2CI(p) = 0(n.×n) and B1 (p)D2rl = 0(n × ,~) for all p ~ . The solution to the synthesis problem is conceptually the same when these assumptions are relaxed, however, the algebra is considerably more complicated [2]. These restrictions have

210 G. Becker, A. Packard / Robust performance of linear parametrically varying systems

been motivated for linear time-invariant (LTI) ~o~ suitable norm preserving transformations on d and LPV system can be written as

:t(t)

el(t) e2(t) y(t)

"A(p(t)) B~(p(t)) c,(p(t)) o

0 0 C 2 ( p ( t ) ) 0

000i Bz(p(t))o0l ]

where pe~,, x(t)e~", [dl (t) v, d2(t)T]X~ ~ "a, Eel(t) T,

control problem in [8]. Under these assumptions, after e, and invertible transformations on u and y, the causal

x ( t )

dl(t) d2(t) u(t)

(3.2)

e 2 ( t ) T ] T 6 R n ' , u(t)~R "", and y(t)eR "~ for all t > O. For the rest of this paper it is assumed that the open-loop LPV system is described by equation (3.2). To define the synthesis problem, describe the m-dimensional linear feedback controller as follows: Given a nonnegative integer m, and continuous functions AK : R s ~ •" × m, Br : R s ~ R m × ~', Cr : R ~ ~ •" × m, and Dr : ~ --* ~"" × ~, the dynamical p-dependent linear feedback can be written as

u(t) ~ DK(p(t)) y(t) q Cx(p(t))

~(t) J = x~(t) " Ar(p(t)) T Let X~p(t):= [xT(t) xJIt)], eT(t):= [e~[(t) e~(t)], and dT(t):; [dI(t) d~(t)]. Then using the above controller

the closed-loop system becomes

[#clp(t) l=[Ac|p(p(t)) Bclp(p(t))][xclp(t)l, e(t) _] Cdp(p(t)) O¢,p(p(t)) d(t) ] (3.4)

where

A¢lp(P): = [ A(p) + BE(p)Dr(p)C2(p) Bz(p)CK(p) ] Br(p)C2(p) At(p) '

B°'p(P):=[B'~ p) B2(p)Dx(P)IBK(p) / ' C,,p(p):=~LDr(p)C2(p)C'(P) CK0(p)], (3.5)

I: o ] and Dclp(p):= Dr(p) "

Using the above definitions, the synthesis problem can be started as follows.

Definition 3.1. Given the LPV system, satisfying the assumptions in equation (3.2), and y > 0. The quadratic LPV y-performance problem is solvable if there exist an m > 0, a finite-dimensional, m-state controller (3.3), and an XeR ("+")×l"+m), X = X r > 0 such that for all p e g ,

I A~p(p)X + XAclp(p) XBc,p(p) y-~C~p(p) 1 B~p(p)X - I y-~D~p(p) < 0. (3.6)

y - I C c l p ( P ) y - 1 D c l p ( P ) - - I

The advantage of writing equation (3.6) as an affine-matrix inequality (AMI), rather than the Riccati inequality in equation (2.9) is apparent when solving the quadratic LPV y-performance problem in Theorem 4.2.

Note that this problem is a generalization of standard .,~¢goo optimal control, and as such, conceptually expands the applicability and usefulness of the Ae~o methodology. Also, the solution can be put inside a larger design iteration, such as a D - K iteration, to achieve robustness to other perturbations, such as unmodeled dynamics, and errors in sensing of the scheduled parameter, p. Finally, note that the synthesis problem exploits the realness of the parameter p, and should be less conservative than methods based on structured

G. Becker, A. Packard / Robust performance of linear parametrically varying systems 211

small-gain arguments [22]. The solution to the quadratic LPV ),-performance problem is given in the following section.

4. LPV control for quadratic L 2 performance: controller synthesis

In this section, we formulate necessary and sufficient conditions for the solution to the quadratic LPV ?-performance problem. The following preliminary lemma will be used in the proof of the main result. The proof involves Schur complements and the matrix inversion lemma.

Lemma 4.1. Suppose that X~j~ff~ n×n, Y I I ~ n×n, with Xla = X~a > O, and YII = Y~I > O. Let m be a posit- ive integer. There exist matrices Xxzeg~ n×m, X22eR m×m such that X22 = Xr22,

X,l Xx2] IX,1 X12]-' [Y11 *1' ('*'means'don'tcare') XIz X221 >0' and Xrz X221 = * *

if and only if

>0, and rank < ( n + m ) . I_ I . r11 - k l_ I . r l ~ -

Hence, given any two positive-definite matrices Xt t , Ytt e R n× n, there exists an integer m such that the dilation can be completed if and only ifXxx - Yll 1 > O. If this condition holds, then the rank of X11 - Yi-11 determines the dimension necessary for the dilation. Since n x n matrices have rank of at most n, the maximum dimension needed for the dilation is mmax = n, and this occurs when XI t - Y{11 > O.

We now state the main synthesis result of this paper.

Theorem 4.2. Given ~, the open-loop system in equation (3.2), and y > 0, the quadratic LPV ),-performance problem is solvable if and only if there exist matrices X l l e R n×n, X l l =X~I >0 , and Yll~R n×n, Yll = Y~I > O, such that for all p ~ :

A(p) Y,, + YlxA(p) T - B2(p)B~(p) YlxC~(p) ) , - ' B , ( p ) ]

CI(p) Yll - I 0 < O, ),- ' B~(p ) 0 - I

I AT(p)X,, + X , a A ( p ) - CXe(p) C2(p)XI,B,(p) 7-'C'[(p)']

B~(p)Xlx - I 0 J < O, ),- 1 C1 (p) 0 - I

(4.1)

Xl l 7 - 1 1 . ] > 0 .

),-11, Yli

We explicitly state the formulas for one n-dimensional, strictly proper controller that solves the quadratic LP V

),-performance problem. Let Z := (X11 - 7- x Y× 1 )- 1 and define

H(p):= - EY{xX A(p) + AT(p) Y?I 1 -- Yixl B2(p)BT(p) Y{I'

+ Cry(p) C, (p) + ),- z y? , B, (p)BT(p) Yi-~ 1 ]. (4.2)

212 G. Becker, A. Packard / Robust performance of linear parametrically varying systems

Then we define state-space data for the parametrically-dependent controller (3.3) as

AK(p) :----- A(p) + 7- 2 BI(p)BT(p) Y;I 1 - B2(p)BT(p) - Z[CT(p)C2(p) + v-ZH(p)], (4.3)

Br(p):= ZCTE(p), Cr(p):= - B~(p) Y;I l, and Dr(p):= 0.

Proof. Although the third AMI in equation (4.1) guarantees that ( X l t - 7 - 2 y l t 1) > 0, we can always perturb X ~ such that the three AMIs in equation (4.1) still holds and (X11 - 7 -2 Yi-~ ~) > 0, and hence is invertible.

Given an m > 0 and an X e R In+"}×!n+mt, X = X T >0 , partition X as

[ X~ ~ X1 z and define := X = L X T 2 X22 ' Vl2 Y22

Then, using Lemma 4.1, we can rewrite this constraint in terms of Xt~, Y ~ e ~ × n , X ~ = xT~ > 0, YI~= yT > 0 s u c h t h a t

Xl l 7-1I . 1 7-1I , Ytl j > o ,

which is the third AMI in equation (4.1). Using the closed-loop data in equation (3.5), define the left-hand side of equation (3.6) as G(p):= R(p) + U(p)K(p)VV(p) + V(p)KT(p)UT(p), where

AT(p)X11+ XllA(p) AT(p)X12 XxlBI(p) 0 7-1CX(p) 0 1 XTEA(p) 0 X~2BI(p) 0 0 0

J

BT(p)X,I BT(p)X,2 - td, 0 0 0 R ( p ) = 0 0 0 - Id~ 0 0 '

y- lC l (p ) 0 0 0 - Ie, 0 0 0 0 0 0 - I~

U(p) =

XllB2(p) X121 X[2B2fp) X3 0 0 0 0 ' 0 0

7 - 1 I~ 0

Now, define matrices Ul(p) and [U(p), U±(p)], [V(p), V~(p)] are full

[Dr(p) Cr(P) l V(p) = K(p) = [_Br(P) Ar(p) ] '

C~(p) 0 0 Is 0 0

le~ 0

0 0 0 0

Vl(p) such that for all p ~ , UXL(p) U(p)= 0, V~_(p) V(p)= 0, and rank. From the above definitions it is straightforward to verify that

YAp)=

vVll 0 0 0

Y~2 0 0 0 0 Id, 0 0

0 0 ld~ 0

0 0 0 le, - BT2(p) 0 0 0

and V±(p) =

I. 0 0 0 0 0 0 0 0 Id, 0 0

-- C 2 ( p ) 0 0 0

0 0 I~ 0 0 0 0 I~

Since both U~_(p) and V.(p) are full column rank for all p e ~ , it is clear that if G(p) < 0 for all p e ~ , then

U~(p)G(p)U.(p) < 0 and VT(p)G(p) V±(p) < 0

for all p e ~ , which is equivalent to

UTi(p)R(p)U±(p) < 0 and VT(p)R(p) V L(p) < 0 (4.4)

G. Becker, A. Packard / Robust performance of linear parametrically varyin# systems 213

for all pe#~. Carrying out the algebraic manipulations, it can be readily seen by taking Schur complements of the resulting expressions in (4.4) that the first two conditions in equation (4.1) are satisfied.

~= For sufficiency, we simply need to verify that, using the strictly-proper controller in (4.3), we can find an X ~ ( n + m)x(n + m), X = X T > 0, that solves the quadratic LPV ?-performance problem. Indeed, define X as

[ X l l -- ( X l l -- ~'-2 y l l l ) 1 X ' = _ ( X l t -- ? - 2 Ylll ) X l l - ? - 2 Ylll •

By taking Schur complement with respect to the (1, 1) block, it is clear that X > 0. Recall that we can rewrite the AMI that solves the quadratic LPV ?-performance problem in equation (3.6) as the Riccati inequality in equation (2.9). Since D¢lv = 0, this becomes

A~p(p)X + XA~p(p) + XBc,p(p)BT, v(p)X + 7 -2 CTp(p)Cclp(p) < 0. (4.5)

Using algebraic manipulations similar to [28], where the LTI case is studied, it can be shown that equation (4.5) holds if equation (4.1) holds. []

Note that the three AMIs in (4.1) represent convex constraints on the matrices Xl l and Y11, and that X11 = xTi > 0, and Yll = Y~I > 0 trivially represent convex constraints. This is true even when the simplifying assumptions are relaxed, although the dependence of the first two AMIs on the state-space data is more complicated [2]. Since the first two AMIs in (4.1) depend continuously on p e # , the solution to the quadratic LPV ?-performance problem can be formulated as a convex feasibility problem with an infinite number of constraints. Typically, one must resort to gridding the set ~ , and solving approximations to the actual constraints (4.1). If no solution can be found for a finite subset of ~ , then clearly no solution exists for ~. On the other hand, ira solution is found satisfying (4.1) on a finite subset o f ~ , then, in general, the best one can do to ensure that (4.1) is satisfied for all p~#~ is to check that (4.1) holds on a very dense grid o f # . As the number of parameters increase, the number of grid points will generally increase in an exponential fashion. Hence we expect reasonable success using the single quadratic Lyapunov approach in cases where there are a few (3 or 4) parameters.

The quadratic LPV ?-performance problem can be reduced to a convex problem with a finite number of constraints by imposing additional restrictions to # , the open-loop state-space data, and its dependence on pe~. For the general open-loop LPV system in (3.1), it is also required that (A1) the allowable parameter set

is a convex polytope whose finite set of extreme points is denoted 6 ~, (A2) B2 and C2 are constant matrices (i.e. parameter independent sensors and actuators), (A3) D12 and D2t are constant matrices (which is already assumed for the system (3.2)), (A4) the dependence of the remaining continuous matrix valued functions, A(p), B1 (p), Ct (p), DI 1 (P), and D22(P) on pe#~ is multi-affine. Under these additional assumptions, to solve the quadratic LPV ?-performance problem, it is sufficient that the AMI constraints hold on the finite set 6 ~ of extreme points of ~. For the open-loop system considered in this paper (3.2) this is stated as in the following theorem.

Theorem 4.3. ? > O, the quadratic LPV v-performance problem is solvable if and Xll = X~I > O, and YIIE~ n×n, Y11 = Y~I > O, such that for all ee8,

Given ~, an open-loop system (3.2) satisfyino the additional restrictions (A1)-(A4) above, and only if there exist X l 1 ~ n×n,

A(e)Y11 + YllA(e) T -- BzBr2 YI1C~(e) ?-lBl(e) 1 Cl(e) YII -- I 0 < O, ? - t B~(e) 0 - I

[AT(e)Xll + XIIA(e)-CTC2 XllBI(e) 7-1C~(e) 1 BT(e)Xt, - I 0 < O, 7- X Cl(e) 0 -- I

y - t l . Y11 j > O .

(4.6)

214 G. Becket, A. Packard / Robust performance of linear parametrically varying systems

Proof. (Outline). The important fact to note in the theorem statement is that if equation (4.6) holds at the extreme points o f ~ , then it holds for all p~#. Indeed it can readily be verified that the maximum eigenvalue of the first two AMIs is convex in (A, B1, Ct). Since the dependence of A, Bt and Ct on p e ~ is affine, it is clear that equation (4.6) can be written as a convex function on the convex polytope, ~. By properties of convex functions, the maximum eigenvalue of equation (4.6) occurs at one of the extreme points, e~8. []

This set of AMIs constitutes a finite-dimensional convex feasibility problem. The special structure of AMIs is being studied extensively by many researchers, and remarkably efficient algorithms for their solution have been developed and continue to be refined. Relevant references are [4, 13, 34, 19-21, 9].

If the open-loop state-space data is constant (i.e. no p dependence), then the quadratic LPV v-performance problem formulated in Definition 3.1 is equivalent to the suboptimal goo control problem. The existence of the suboptimal controller is characterized by the feasibility of the three conditions in equation (4.6) as stated in Theorem 4.3. This type of characterization is now well-known, with versions in 1-29, 10, 11, 28].

Acknowledgements

We would like to thank P. Apkarian, G. Balas, P. Gahinet, P. Khargonekar, W. Lu, K. Nagpal, D. Philbrick, B. Reichart, M. Rotea, C. Scherer, S. Shahruz and J. Shamma for helpful discussions. The authors gratefully acknowledge financial support from the National Science Foundation, CTS-9057420.

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