Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993
ROBUST PERFORMANCE DESIGN OF SYSTEMS WITH PARAMETRIC UNCERTAINTY Y.A. Jiang and DJ. elements School of Electrical Engineering, University of New South Wales, PO Box 1, Kensington 2033, Australia
Abstract. For systems with norm bounded structured uncertainties, we propose a method, based on a scaled Riccati equation, to construct a state feedback control law with a guaranteed quadratic cost. The resulting state feedback control quadratically stabilizes the uncertain system and provides an upper bound for the LQ performance of the perturbed closedloop system .
Key Words. Linear optimal regulator; parameter uncertainty; quadratic stability; robust performance
1. INTRODUCTION
without matching conditions. A scaled algebraic Riccati equation (ARE) approach is derived for constructing a control law which guarantees the quadratic stability and an LQ performance level in the presence of uncertainties. This scaled ARE approach uses optimal constant inputjouput scaling H 00 state-feedback control theory (Packard et al., 1992) to deal with the robust LQ performance. The robust control law is obtained by searching for the optimal scaling matrix which minimizes an upper bound for an LQ cost function while maintaining the quadratic stability. We show that the descent direction of the scaling matrix can be determined by the solution of a Lyapunov equation and an algorithm is proposed.
When the states are available, linear-quadratic (LQ) optimal control theory is a useful tool for feedback control of a nominal system. However, the remarkable performance of LQ optimal control is lost in the presence of plant uncertainties. Although many types of modelling errors must be considered, one class of uncertainty model that has been used extensively is the parametric uncertainty model (Barmish, 1985), in which the plant is represented by a nominal linear system containing time-varying bounded uncertain parameters. One possible robust LQ control problem for such systems is to guarantee the quadratic stability while maintaining an acceptable level of performance in the presence of the parametric uncertainty.
2. ROBUST PERFORMANCE
In recent years, various robust LQ control problems have been proposed. For example, Peres et al. (1992) formulated the problem as a convex minimization problem for systems with parametric uncertainties which belong to a convex polyhedral domain. Alternatively, a robust LQ control problem for systems with one norm bounded unstructured uncertainty on the state matrix was considered and a Riccati equation approach was used to obtain a guaranteed quadratic cost in closed-loop by Petersen and McFarlane (1992) .
We consider uncertain linear systems described by state equations of the form x(t) = (A + ~A(t))x(t) z(t) = Cx(t) + Eu(t) y(t) = x(t),
+ (B + ~B(t))u(t) (1)
x(O) = xo
where x(t) E Rn is the state, u(t) E R m is the control vector, z(t) E RP is the performance vector, y(t) is the observation vector and ~A(t) and ~B(t) are the norm bounded time-varying structured uncertainties which are described by
This paper considers a robust LQ control problem for systems with norm bounded uncertainties in both the state and input matrices
~A(t) ~B(t)
45
= FA~A(t)EA'
= FB~B(t)EB,
aA E aB
~A(t) E
~B(t)
(2)
3. A RICCATI EQUATION APPROACH
where AA := {diag(Ll. 1(t), ... , Ll.'. (t» : Ll.iLl.T ::; p2 11•XI., Vi} AB:= {diag(Ll.IC+l(t), ... ,Ll.,(t»: Ll.iLl.T ::; p2I , • x,., Vi}
(3)
In this section, we present a scaled Riccati equation approach for the robust performance design. Define
To simplify the statement, we suppose that
(A, B) is stabilizable, (C, A) is observable, ET E > 0, and C T E = O.
F=[O,FA,FB],
=
100
(u T ET Eu
+ X T C T Cx)dt
(4)
D :={diag(Iq,DA,DB):DAEDA,DBEDB} DA := {diag(d1IIl"'" d'.!'I.) : di > o} D B := {diag(d,c+lIII.+" ... ,ddl,): di > o} (9)
=-
together with the Riccati operator R(P,D,,) := PA + AT P - PB(N T DN)-lB T P +,-2 PFD-1F T P+ HT DH (10)
r
The minimal value of the cost function is = xl Pxo. Further, with Ac = A - BK, and Cc = C - E K, we see that P > 0 satisfies the Lyapunov equation
Theorem 1 Suppose that there exist a P > 0 and a D E D such that R(P, D, I) = 0 for some
1 < {3-1 . Let the state feedback law with u - [{ x with gain
=0
(11)
On the other hand, for any feedback gain I( for which Ac is stable, the associated cost is xl Loxo where La > 0 is the solution of the Lyapunov equation LoAc
1
N= [iB
and a set of input/ouput scaling matrices which are commutative with all uncertainties in ~--t and ~B as
Without uncertainties, it is known that the optimal controller which minimizes this cost function is a state feedback law u K x with gain J( = (ET E)-l BT P where P> 0 is the stabilizing solution of the Riccati equation
PA c + A~ P+ C;Cc
1,
(8)
In this paper, we seek quadratic stability (Barmish, 1985) in conjunction with an associated cost function J
H= [ {A
be stabilizing. Then, the closed-loop system of (1) is quadratically stable and an upper bound for the cost (4) is J r xl Pxo .
=
Proof: With K given by (11), the nominal closed-loop system is given by
+ A~ La + C; Cc = 0
In the presence of the uncertainty ~ = (~A' ~B)' Ac and Cc deviate from their nominal values and so therefore does the solution L(~) of the Lyapunov equation deviate from P.
= Acx(t) = Ccx(t) BI( and Cc = C £(t) z(t)
where Ac = A EK, and the perturbed closed-loop state matrix is
The robust performance problem is to find a state feedback gain I( which quadratically stabilizes the system (1) and minimizes the cost function: sup xJ L(~)xo
For any X, Y, D, and ~ with D > 0, and ~~ T {32 I, we have
D~
Using this result on the uncertainties separately, it easily follows that
~A
:s
A
=
~D,
In this paper, we show that there exists an upper bound for this minimum of the form
(6) where P
> 0 satisfies P Ac + A~ P + C; Cc ~ 0
and
~B
pA c + AJ P + C.; Cc < R(P, D,p-l)
(7)
for all uncertainties. With a stabilizing control gain K, the existence of such a P quarantees the quadratic stability of the closed-loop system and provide~ an upper bound for L(~) since P ~ L(~). Thus, our objective for the robust performance design is to select a state feedback gain K to obtain a suitably small upper bound (6) .
:s
R(P, D, '1)
=
0
ODD Thus, to achieve the quadratically guaranteed performance, we want to select a scaling matrix such that 1 < {3-1 and xJ Pxo is small in the ARE R(P, D, I) = o.
4. OPTIMAL INPUT jOUTPUT SCALING 46
not attained. If 'ip is attained, a Riccati equation approach for searching for a descent D can be found in Jiang and elements (1992). Now, suppose that 'ip is not attained. In this case, as shown in Lemma 1, the ARE solution Q is singular at the optimal point. So, a descent D will be one that causes an increase in the smallest eigenvalue of Q. Let D(t) = e bt E D and Q(t) be the solution to R(Q(t),D(t),'ip ) = o. It is known that Q(t) is an analytic function of t and Q, its derivative at t = 0, can be computed from the Lyapunov equation
In this section, we use input/output scaling to obtain a suitably small upper bound (6). We first consider the optimal constant input/output scaling which achieves the minimum robustness level -y; := infb : R(P, D, -y)
~
0, -y
> 0, P > 0, DE
D} (12) It is known (Packard et al., 1992) that this minimization problem is convex and thus can be solved by searching for D and P separately. For a fixed D, absorbed into F, H, and N,
'ip
:= inf{-y : R(P, I, r)
::; 0, r > 0, P > O} (13)
.
T .
QAQ +AQQ+ Wb
is an H 00 optimization problem and a fast algorithm is available in Sherer (1990). However, to deal with the optimal value 'ip , it is convenient to consider Q = p-l and the Riccati operator
Q(t)
Lemma 1 Consider system (1) and the Riccati operator (14). The following statements hold. a) If the optimal value 'ip is attained with P, there exists Q > 0 such that R(Q,I,'ip ) = o.
Ai(t)
(15)
inf{-y : R(Q,D,r):S O,r 000
Proof: Under the assumptions, from Lemma 1, AQ is stable and there exists a unique solution Q to the Lyapunov equation (19). So, for any iJ E D, Q(t) can be expanded as in (20). Thus, the condition that Amin(X T QX) ::; 0 and Amax(XT QX) ~ 0 for every iJ E D is a necessary condition for 'ip = This completes the proof.
ER} (16)
so, the search for a suitable D E D can be replaced by a search for a suitable iJ E D. Let ZD
= QHT DHQ+i- 1 FD- 1 FT _B(NT DN)-l BT
r;.
Then, we have R(Q,D,r) = AQ+QA T +ZD. With Zb(t) := ZeD" we have Zb(O) = Zr. further, ZD(t) is an analytic function of t and its derivative at t = 0 is
000
Noting the structure of D, an algorithm for the search for D can be derived as follows. Partition U and V as
(17) where
~
Uo Wij = U··DU - V· DV U = FT ftp V = HQ + N(N T N)-l BT
> OQ ~ O} < 'ip
if there exists iJ E D such that XT Q X is .definite where X spans the null space of Q and Q is the solution of the Lyapunov equation (19).
For a fixed 'ip , P, and D = I, we want to find a new D with a smaller 'ip . Any D E D can be written as D = eb where
D:= {diag(Oq,ddt" ... ,dlltr ): di
= tAi(XT QX)
Lemma 2 Suppose that 'ip is not attained, and that the solution Q of the ARE R(Q, I,'ip ) = 0 is singular. Then there exists D E D such that
is stable . Moreover, Amin(Q) = O.
DE
(20)
Thus, if XT QX > 0, then, all Ai(t) are positive for t near 0 and D = e bt is a descent direction .
b) If the optimal value 'ip is not attained with P, then there exists Q ~ 0 such that R(Q, I, 'ip ) = 0 and HT HQ
= Q + tQ + o(t)
where limt-+o o(t)/t = O. Let X span the qdimensional null space of Q. Since Q(t) is analytic in t, and symmetric for each t, there exist analytic functions Ai(t) describing the eigenvalues of Q(t). Let the Ai(t) be ordered so that Ai(O) = 0 for i = 1, ... , q. To first order, these q smallest eigenvalues are
The characterization of 'ip is given by Scherer (1990) .
= _AT -
(19)
where AQ is given by (15) and Wb is given by (18). So, a Taylor series for Q(t) in terms of t is
R(Q, D, -y) := AQ + QAT + QHT DHQ +y- 2 FD- 1 F T _B(NTDN)-IBT (14)
AQ
=0
r
(18)
1 ,
HQ+N(NTN)-'n
.~ r11
T
Vo
(21)
and consider the Lyapunov equations
From Lemma 1, two cases need to be considered: the optimal value 'ip is attained and 'ip is
.
0= QiAQ
47
.
TT·
+ AQQi + Ui
Ui - Vi Vi, VI
(22)
For any
B .
I '
.
>0
XT(Ld;Q;)X
*
XTQX
>0
1
=
rp
f>
E
E
rp
rp)
b
FA
rp
Theorem 2 Suppose that there exists a singular 0 such that R(Q, I, = 0 and the matrix AQ is stable. Then, there exists D E D such that R(Q,D,,) = 0 for some, < and Q > 0 if there exists
:t: 1 ~ [! U ~
=
i=1
Q~
[
with block structured uncertainty .6. A diag(8 1 ,82 ) , 18;1-1 ~ 4.5. Without scaling, the minimum = 30.92. With Theorem 2, a suboptimal scaling matrix D = diag(19.7, 1.4) is obtained and the resulting minimum is = 3.5. For this D, the cost function with respect to , = 4.5 is J r = 15.8. To reduce the cost,. we use Theorem 3 to adjust this scaling matr~and obtain another scaling matrix D = diag(9.9, 1.4) which reduces the cost function to J r 8.8 with respect to the same uncertainty norm bound {3-1 4.5.
let Q := Li;::1 diQi. It is clear that Q satisfies the Lyapunov equation (19) and thus I
~
=
such that
6. CONCLUSION
I
XT (L diQi)X > 0
We have presented an approach to deal with robust LQ control. The resulting full-state controller guarantees quadratically stability and provides an upper bound for the LQ performance of the perturbed closed-loop system. The design procedure is to select an optimal input/output scaling matrix to minimize the robustness level, and then to adjust this scaling matrix to minimize the H2 cost function while maintaining quadratic stability. The trade-off between the robustness and the performance level can be handled easily by choosing the robustness level,.
i=l
where X spans the null space of Q and the Qi are the solutions of the Lyapunov equations (22). Moreover, there exists t > 0 such that D e Dt .
=
Now, let 'Y* be the minimum achieved with the optimal scaling matrix D*. If 'Y* < /3-1, then, the robust LQ problem is solvable. For a 'Y with 'Y* < 'Y < (3-1 , we want to search for D such that R(P, D, 'Y) = 0 and xri' PXo is smaller. Let D be absorbed by F, Hand N. Similar to Theorem 2, we have the following result.
7. REFERENCES
Theorem 3 Suppose that there exist P > 0 and 'Y > 0 such that R(P, I, 'Y) = o. If there exists
b
E
f>
Barmish, B. R . (1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain system . J. Optimiz. Theory Appl. Vol. 46, No.4, pp .399-408 . Jiang, Y. A., and D. J . Clements (1992). A Riccati equation approach to approximate Il-norm computation. Control 92, Perth, pp. 233-238. Packard, A., K. Zhou, P. Pandey, J. Leonhardson and G . Balas (1992) . Optimal, constant I/O similarity scaling for full-information and state-feedback control problems. Syst. Contr. Leti., vol. 19, pp. 271-280 . Peres, P . L. D ., S. R . Souza and J. C. Geromel (1992). Optimal H2 control for uncertain linear systems. Proc. of A CC, pp. 2916-2920. Petersen, I. R. and D. C. McFarlane (1992). Optimal guaranteed cost control of uncertain linear systems. Proc. of A CC, pp. 2929-2930. Sherer C. (1990). H oo-control by state-feedback and fast algorithms for the computation of optimal Hoo-norms. IEEE Trans. Auto. Contr. Vol. 35, No .lO, pp. 1090-1099.
such that I T~
.
Xo (~diPi)XO < 0 1=1
and
where the Pi are the solutions of Lyapunov equations similar to (22), then, there exists t > 0 such that P 1 Xo < PXo
xri'
xri'
where P 1 > 0 is the solution of the Riccati equation R(Pl , D, 'Y) 0 and D e Dt .
=
=
5. EXAMPLE Consider the system:
A~
[l
C=[10
1
0 0 0
0
0 0 0
-2 .3685
-oLJ o .5
48