H∞ control with state feedback

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ELSEVIER

S iTULS & ¢ONTIIOL LITTWIS Systems & Control Letters 25 (1995) 289-293

A generalization in mixed

~2/~

control with state feedback

E i t a k u N o b u y a m a *,a, P r a m o d P. K h a r g o n e k a r a Department of Control Enoineerin9 and Science, Kyushu Institute of Technolooy, lizuka, Fukuoka 820, Japan b Department of Electrical Enoineerin9 and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122, USA

Received 24 November 1993; revised 27 June 1994

Abstract In this note, we consider a mixed o,ug2/~'~oocontrol problem with a performance cost which combines the o~'2 norm cost and the so-called auxiliary cost. The main result is to give a formula for the optimal cost and a nearly optimal controller. It is assumed that the state of the plant is available for feedback. Keywords." ~ o o control; Mixed oct~2/~go~ control problem; State feedback; Dynamic state-feedback controller

1. Introduction Consider the finite-dimensional linear time-invariant system shown in Fig. 1, where G and K denote the transfer matrices o f the plant and the controller, respectively. Let a state-space description o f G be given by = Ax + BlWl +B2w2 + B3u, zl = Clx + Dlu, Z2 -~- C2x -~ D2 u, y~x.

(1)

Thus, the plant state is assumed to be available for feedback. A controller K is said to be admissible i f K internally stabilizes the closed-loop system. The (pure) mixed 3¢gz/oct~ control problem is defined as the problem to find an admissible controller K that achieves inf{llTz, w, H2 : K admissible and IITz2w211o~< ~}

(2)

for a given 7 > 0, where Tz,wj denote the transfer matrix form wj to zi. (Dependence of Tz,wj on K will be suppressed unless it is needed for the sake o f clarity.) This problem can be motivated as a problem o f optimal nominal performance subject to a robust stability constraint (see [ 1, 3, 5-7] and the references therein). Despite significant advances in o~¢g2and ~ control theories, this mixed o~2/,JCt'~ problem has turned out to be surprisingly difficult. No analytical solution has been found in the general case, although in the special case o f state feedback with imBj N imB2 = 0, a complete solution was given in [6]. "~"This research was supported in part by the Airforce Office of Scientific Research under grant no. F49620-93-1-0246 and the Army Research Office under grant no. DAAL03-02-G-0163. * Corresponding author. Tel.: (948) 29-7728. Fax: (948) 29-7709. E-mail: [email protected]. 0167-6911/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 01 67-6911 (94)00080-8

290

E. Nobuyama. P.P. Khargonekarl Systems & Control Letters 25 (1995) 289-293

tO1

w2

~

,~

G

k

[

: Zl

, z2

Fig. 1. Feedback system A somewhat related mixed Ae2/~o~ control problem was formulated in [1] in the case where B1 = B2. In [1], instead of the (pure) 3¢f2 norm cost IITz, w, 112 an upper bound on IlTz,w, II2 is adopted as a cost function to be minimized. The upper bound is sometimes referred to as the "auxiliary cost". It is shown in [3] that (nearly) optimal value of the auxiliary cost is obtained with a constant-gain state feedback, and the gain matrix can be computed as the solution of a convex program. These results address the mixed af'~2/~oo control problem in the special cases where imB1 A imB2 = 0 or B1 = B2. However, in the case of general B1,B2, the only analytical result is found in [6] in which a sufficient condition is given for the pure mixed acg2/Jfoo control problem to have a solution. Also, some numerical methods have been explored in [5] which apply to the general output feedback case. For discrete-time SISO systems, an iterative method has been presented for a mixed 3¢f2/afoo suboptimal problem in [7]. In this note, motivated by this fact and the results in [1, 6, 3] we will introduce a performance measure in the general case, which represents an upper bound on the (pure) o~¢g2norm cost 11Tz,w, II2. This performance measure is defined as a natural extension of the fir2 norm cost and the auxiliary cost in the following sense: it coincides with the deg2 norm cost if im B l fq im B2 = 0, and with the auxiliary cost if B1 = B2. The main result of this note is to characterize the optimal value of the performance measure and give an admissible controller which achieves (nearly) optimal value of the cost. It turns out that the optimal value and a suitable controller can be obtained by separately solving an fir2 optimal control problem and an auxiliary cost optimization problem.

2. Problem formulation Consider the feedback system shown in Fig. 1 where the plant P is given by (1), and assume that (A, B3) is stabilizable. Partition B1 as Bl = [Bll Bl2] where imBll N imB2 = 0 and imB,2 C_imB2, and wt as wl = [W'll w'12]' conformably. (This decomposition can always be done by using an orthogonal transformation on the input vector wl .) Let an admissible controller K be given, and suppose its state-space description is ~ = A{ +/~x.

u=d~+13x,

(3)

With this controller the closed-loop system is given by (k = F ~ + G l l w l l + GI2Wl12 + G2w2,

zl=HlqJ, F:=

z2 = H2qJ,

c,,:=["a,l o12:E 21 c2:

/~

H1 := [CI +OIL}

Did],

1-/2 := [(72 q- D2/)

D2C],

where ~ := [x' {']'. Standard properties of the ~f°2 norm lead to

IIT~,w, II2 = IIT=,w,, [I2 + IITz, w,2112.

(4)

We will now introduce an upper bound on IITz,w, 112 which will allow us to simultaneously generalize the results in [ 1, 6] The basic idea is to introduce an upper bound of the form

E. Nobuyama, P.P. Khargonekar/Systems & Control Letters 25 (1995) 289 293 J(Tzw) : = IITzlw,, I1~ + Ja(Tz~),

291 (5)

where Ja(Tz~) will be defined next. We begin by noting that UTz,w,21122is given by [[Tz, w,2112-of (F, G12), which satisfies

tr(H, LcH() where Lc is the controllability Gramian

FLc + LcF I + GI2G;12 = 0.

(6)

Now let 7 > 0 be given and suppose HTz2w2U~ < ~. Then there exists the unique real symmetric matrix Y ~ 0 such that

. H~H2 FY + YF' + r - 7 - Y + GzG 2 = 0

(7)

and F + YH~H2/7 2 is asymptotically stable (see, for example, [2]). Since imB12 C_imBz it follows that im GI2 C im G2. Let fl ~>0 be the smallest number such that G12 Gtl2 ~
(8)

Note that fl is uniquely determined only by the plant data. From (7) and (8) we have , = _R,. 2 Y H ~72 H2Y F(fl2Y) + (fl2Y)F' + G12G12

(fl2G2G, 2 _

G,2G,2)~<0"

Now it follows from (6) (see [8]) that Lc ~<(fl2y), and

[iTs,w,2ll2 = tr(H,L~H~)<~tr(H,(fl2y)H().

(9)

This motivates the following upper bound on 11Tz,w,, 1122: Ja(Tzff) : = tr(Hl(fl2y)H(),

(10)

where z := [z~ z~]' and ff := [w{2 w~]'. In the case where B1 --= B2, fl = 1 and Ja(Tz~) coincides with the auxiliary cost introduced in [1]. Using J~(Tz#) the total performance cost J(T~w) to be minimized is defined by (5) where z := [z'1 z~]', w := [w'1 w~]'. It is easy to see that J ( T ~ ) has the following properties: (i) J(T~w) represents an upper bound on IIT~w, II2, i.e., IIT~,w,112<<.J(Tzw). (ii) J(T~w)=

II ~,w,n2 i f i m B 1 N i m B 2 = 0 , J~(Tzw) if BI = B2. (iii) Iffl = 1, J(T~w) --+ [ITz,w,H2 as y ---+cx).

3. Main result

We are now ready to present the main result of this paper. This result gives a formula for the optimal cost. The proof is constructive and presents a nearly optimal controller. The proof relies heavily on a certain controller construction first introduced in [6, Lemma 3] which was later generalized in [4]. As a matter of fact, this controller construction leads to solutions to a class of multiobjective controller design problems. Theorem 3.1. Consider the system given in (1) and let 7 > yopt be given where 7°pt :=

inf{ltTz2w211o~: K

admissible }. Then the following holds: _opt , inf{J(T+):g admissible, IITz2w211~ < ~ } - - s 0_opt + °ta

(11)

E. Nobuyama, P.P. Kharoonekar/Systems & Control Letters 25 (1995) 289-293

292

where %opt := inf{ IIT~,w,, 112:g admissible },

(12)

a : ~ inf{da(T~):K admissible and IIT~=w=ll~ < ~}. ~opt

(13)

ProoL It is obvious t h a t CXqpt -~- ~a°pt represents a lower bound of J(Tz~). Hence, it suffices to show that given any e > 0 there exists an admissible controller K such that IlTz2w211~ < 7 and J(Tzw) < ,qpt + opt + e. First, we use some results on the o'~ 2 control and the auxiliary o v ¢ ' 2 / ~ control. The following is a standard result from the ovg2 control theory: for any e > 0 there exists a constant matrix Fq such that

IITz,w,,(Fq)ll~ = II(G +D1Fq)(SI - A - B3Fq)-'B~, 1122 < ~q-opt + e/2"

(14)

In [3], it is shown that for any e > 0 there exists a constant matrix F~ such that IIT~w2(F~)II~ < ~ and ..opt Ja(Tz~(Fa)) < =a + e/2. Since IITz2w~(Fa)ll~ < ~, there exists Ya~>0 such that , C2F, Ya/]2 2 + B2Bt2 = 0, AFaYa + YaAtFa + YaC2F

(15)

and A& + YaC~FC2FJ72 is asymptotically stable where AF, := A + B3Fa and CZFo := C2 + D2F~. Moreover, since Ja(Tz~(F~)) < opt + e/2 it follows that tr((C1 +DIFa)(fl2Y~)(G +DIF~)') < c~aopt + e / 2 .

(16)

Next we will use a key controller construction first introduced in [6]. The basic idea is to combine controllers Fq and Fa. Define the controller K,: as follows: K~ := (Fa - Fq)(sl - A1 )-~(A1A - AAF,) + Fq(l - A) + FaA,

A1 : = A + ( I -

A)B3Fa + AB3Fq, AFq :=A+B3Fq,

A := B2V+H1,

Hi := I - BllB~-I,

(17)

V2 := HIB2.

Here M + denotes the Moore-Penrose generalized inverse for a matrix M. In the sequel we will show that K~ satisfies IITz~w~(K~)ll~ < ~, and J(Tzw(K~)) < ~pt + ~aOpt+ e. Note that since imB11 N imB2 = 0 and imBj2 C_imB2, from the proof of Lemma 3 in [6], we get

ABI1 =0,

AB12 = BI2, AB2 = B2.

Let x and ~ denote the states of G and Ke, respectively, and transform the states as £ := (I - A)x - ~ , ~, := Ax + ~ . Then the closed-loop equation in Fig. 1 is described by

~, = PC' + 011Wll + Oj2wl2 + 02w2, Z I = IYIl~J,

z 2 = B2@ ,

0 AF~ '

/if" : =

all

:=

/]~ := [C1 + D1Fq C1 + D1Fa],

1

GI2 : =

0

O12

'

G2 :=

B

,

/]z := [C2 + D2Fq Cz + DzFa],

where ~ := [£' ~'~]'. Hence, from (14)

IIT~,w,,(K,)II~ = II(C, + D , F q ) ( s l - . 4 - BsFq)B,11{22 <

_ opt + e/2. ~q

Let Ya be the real symmetric matrix which satisfies (15). Define

?:=

[0 0

0 1 Ya "

(18)

E. Nobuyama, P.P. Kharyonekar/ Systems & Control Letters 25 (1995) 289 293

293

Then it is easy to see that P~" + YF' + Y(/~t2/~2Y)/y 2 ~- G2Gt2= 0 and that P + Y/-tt2/~2/72 is asymptotically stable. This implies that IlTz2w2(K,)]l~< 7. Moreover, from (16) Ja(Tzff~(Ke )) =

tr(H1 (f12]~)/~rtl )

=tr((C1

+DIFa)(fl2ya)(CI + D I F a ) ' )

< ~aopt +/3/2.

(19)

_

Consequently, from (18) and (19) we obtain

J(Z=.w(g~)) = IITz, w,,(g~)ll~

opt _~_~qpt ..1_g. + Ja(Tzff,(ge)) < OCq _

[]

Remark. It is a standard result in the ~ 2 control theory that Fq satisfying (14) can be computed via solving an algebraic Riccati equation. It is shown in [3] that Fa satisfying (16) can be computed via solving a convex program. Once Fq and Fa are obtained, K~ defined by (17) gives a solution to our problem. Remark. In the case of imBl N imB2 --- 0 a general result is given in [4], in which it is shown that there exists a single controller K for any given Ci (i = 1, 2) such that Tz, w~( K ) = Tz, w~( C l ) and Tz2w2(K) -- Tz2w2(Cz). Thus, norms other than the ~ 2 and ~ norms can also be approached using this result.

4. Concluding remarks

In this note, we considered a mixed ~,~2/~eoo state-feedback control problem with a performance cost which combines the ~ 2 norm cost and the so-called auxiliary cost. By using the controller structure introduced in [6] we have given an admissible controller which achieves the (arbitrary nearly) optimal value of the performance cost. Corresponding results for the minimum-phase output-feedback case can be easily obtained by using the results in [4] to recover the state-feedback closed-loop properties.

References [1] D.S. Bemstein and W.M. Haddad, LQG control with an Hoo performance bound: a Riccati equation approach, IEEE Trans. Automat. Control 34 (1989) 293-305. [2] J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis, State-space solutions to standard H2 and Ho¢ control problems, IEEE Trans. Automat. Control 34 (1989) 831-847. [3] P.P. Khargonekar and M.A. Rotea, Mixed H2/Hoo control: a convex optimization approach, IEEE Trans. Automat. Control 36 ( 1991 ) 824--837. [4] P.P. Khargonekar, M.A. Rotea and N. Sivashankar, Exact and approximate solutions to a class of multiobjective control problem, Proc. ACC (1993) 1602-1606. [5] D.B. Ridgely, C.P. Mracek and L. Valavani, Numerical solution of the general mixed H2/Hoo optimization problem, Proc. ACC (1992) 1353-1359. [6] M.A. Rotea and P.P. Khargonekar, H2 optimal control with an Hoo-constraint: the state feedback case, Automatica 27 (1991 ) 307-316. [7] M. Sznaizer, An (almost) exact solution to general SISO mixed )f2/~t%o problems, Proc. ACC (1993) 250-254. [8] J.C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control 16 (1971) 621~534.