H∞- suboptimal control problem with boundary constraints

Systems & Control Letters 13 (1989) 93-99 North-Holland

93

H -suboptimal control problem with boundary constraints * Toshiharu SUGIE Automation Research Laboratory, Kyoto University, Uji, Kyoto 611, Japan

Shinji H A R A Department of Control Engineering, Tokyo lnsitute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan Received 11 February 1989 Revised 26 May 1989

Abstract: A necessary and sufficient condition is derived for the existence of a stabilizing controller such that (i) a specified closed loop transfer function Ge.(s ) has H~o-norm less than one and (ii) Ger(S) satisfies generalized boundary constraints on the jto-axis.

Keywords: H~-control; boundary interpolation; linear multivariable system, mixed sensitivity; J-lossless function.

1. Introduction

The objective of this note is (i) to formulate an H ~ - p r o b l e m with generalized boundary constraints, which covers m a n y important problems including the mixed sensitivity problem with an integral type controller shown in Section 5 and (ii) to give a complete solvability condition for the problem. This problem is closely related to the so-called boundary interpolation problem. However the results obtained so far are restricted to the SISO case [7,8] or a specific M I M O case [12], which does not cover important problems such as the mixed sensitivity problem. Notation. The sets of real and complex valued m x n matrices are denoted by R mxn and C T M , respectively. The sets of all proper and stable proper real rational matrices of size m X n are mXn denoted by R(s) T M and RI-I~ , respectively. For G(s) ~ R H '~xn , we define --oo

II G II ~ '= sup II a ( j w ) II, 60

In the past several years, the Hoo-control theory initiated by Zames [13] has been developed remarkably (see [4]). In particular, the procedure for solving the H~-control problem in the state space setting has been considerably simplified [1,3,6,9111. However, in order to make the Hoo-control proble0a more practical, it is necessary to take the design requirements on the boundary (i.e., on the j~o-axis) into consideration. For example, the controller must have poles at the origin in order to achieve robust tracking for step type c o m m a n d signals. It is therefore important to know when we can obtain such an integral type controller that achieves the given Hoo-norm design specification.

where I1" II denotes the largest singular value. The subset of R H mxn consisting of all S(s) satisfying II S II < 1 is denoted by B H ~mXn. If no confusion arises we drop the size 'm × n'. Jm.n denotes the matrix block diag{ I m, - I n }, where I,, is an m-dimensional identity matrix. G ( s ) ~ R ( s ) which has no poles on the j~-axis is said to be (arm.,- J k j ) lossless [5,9], if it satisfies the following two conditions:

G*(joJ)Jm,.G(joJ) = Jk,,, G*(s)Jm,nG(s )
V~, VRe[s] > 0,

where * denotes conjugate transpose.

2. Problem formulation * This research was carried out while the authors were visiting the Dept. of Electrical Engineering, University of Waterloo, and was supported by the Manufacturing Research Corporation of Ontario.

In this note, we consider a finite dimensional linear time invariant control system ~ ( G , K ) de-

0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

T. Sugie, S. Hara /H~-suboptimal controlproblem

94 scribed by

(2)

u=K(s)y,

where G(s) := {G 0 ~ R ( s ) pi×m, } and K ( s ) R ( s ) m~xp~ are transfer matrices of the plant and the controller, respectively, and r, u, e and y are m v d i m e n s i o n a l external input, m2-dimensional control input, p~-dimensional controlled output, and pR-dimensional measurement output, respectively. Let G~,(s) be the transfer matrix from r to e, that is,

G~r = G u + GazK(K - G22K)-lG2a

(3)

and consider the following design specifications: (sl) K ( s ) stabilizes G(s) internally, (52) 11Ger1[ ~ < 1, (s3) LiGe~(J~i)R i = ~i (i = 1. . . . . n), where L, ~ C ~'xp', R i E C m,x~,, ~oi ~ C ~,×~, and 0~, > 0 are given in advance. Without loss of generality, we assume rank L i = # i ,

rank R i=pi,

For simplicity, in other words, we treat the 2-block problem. Also, in order to avoid complex notation, we give the proofs only for the single b o u n d a r y constraint case, i.e., n = 1, L~ = L, R~ = R, ~1 = ~, ~g = °~1, and drop the suffix i in the proofs of all lemmas and theorems.

3. Boundary constraints

3.1. Admissible boundary constraints We first consider the admissibility of the b o u n d a r y constraints. We say that the constraints { L,, R~, ~ } are admissible, if there exists a controller K ( s ) such that (sl) and (s3) hold. Then, we have the following lemma. Lemma 3.1. Suppose X( G, K ) satisfies the assumptions (A1)-(A4). The constraints { Li, Ri, ~ } are

admissible, if and only if there exist constant matrices Fi ~ C m2×ml such that L,[GH(jcos)+G12(jw,)F,]R,= ~

( i = 1 . . . . . n).

(6)

(4)

Proof. (Necessity) Let G22 = ND -a = / ~ - 1 ~

for i = 1 . . . . . n, and

0
(5)

The first two specifications (sl) and (s2) are required in the original H~-problem. The additional requirement (s3) gives b o u n d a r y constraints on Get(s). If there exists a K ( s ) such that (sl) and (s2) hold, we say that the H~-problem is solvable. If there exists a K ( s ) such that (sl)-(s3) hold, we say that the H~-problem with boundary constraints

{ L,, Ri, ~ } is solvable. It is known that we can easily check wether the H ~ - p r o b l e m is solvable or not by solving only two Riccati equations [6,9,11] under the following assumptions: (A1) G(s) is stabilizable by u and detectable from y. (A2) G(s) has no poles on the imaginary axis. (A3) rank Ga2 = m 2, rank G21 =P2, Vs =jo~ U In this note, we further assume: (A4) P2 = m p

be coprime factorizations of G22 over R H ~ , and (X, Y) be a solution pair of the Bezout identity N X - D Y = I e 2 . Then it is well k n o w n that all stabilizing controllers K ( s ) are given by

K=(X+DW)(Y+NW)

-1

VW~RHoo.

(7)

Substituting (7) into (3), we have

L G e r R = L [ G a 1 + G ~ 2 ( X + DW)[)G21]R.

(8)

Therefore, if (s3) holds, we get (6) by letting F be F = [( X + DW)DGza ] (jo~a).

(9)

(Sufficiency) Suppose (6) holds. Since [ D(j~1) [ ~ 0 , [D(j0~l) [ ~ 0 , and [Gzl(Jr01)] 4=0, from the assumptions (A2)-(A4), there exists a W ( s ) RI-I~ satisfying (9). (Note that we can easily construct a W(s) satisfying (9) in the case of multipoint constraints, i.e., in the case of n >_>_2, by e.g. Lagrange interpolation, because W(s) has no restriction on its H ~ - n o r m . ) F r o m (7) and (8), this implies that there exists a K ( s ) such that (sl) and (s3) hold. []

T. Sugie, S. Hara / Ho~-suboptimalcontrolproblem 3.2. Basic constraints Besides the given constraints (s3), the system ,~(G, K ) has its inherent constraints. We will clarify the relation between this inherent constraints and the admissibility of the given constraints• Let G(2(s ) G R(s) O',-'~)×m be a left annihilator of G~2(s) such that

G~(s)Gl2(S) =0, rankG~(joa)=p,-m2=:#o,

Vto~R.

95

(L, g'). Since LB is a left annihilator of G12(J~l) with maximum rank, (16) implies that LcGa2(j%) is right invertible. Also, R is left invertible because it has full column rank from the assumption. These invertibilities assure that for any '/'c there exist a matrix F ~ C " = x " ' such that

t c ( Gn(J6%) + Ga2(Jt°I)F)R = ~c

(17)

holds• From (13) we have

(10)

LB(GI,(jaq) + Ga2(jwa)F)R = ~BR

(11)

for any F. The above two equations and (16) yield

(18)

Then, from (3), we always have

G,~(s)Ge,(S ) = G ~ ( s ) G , , ( s )

(12)

for any choice of the controller K(s). This is an inherent constraint on G~,(s). If we define •

l

LBi.= G,2(joa,),

x/"m := G~2(jo:,)G,,(jooi)

(13)

for each boundary point s =j~0~, then

LmG~(J%) = g'm

( i = 1 ..... n)

(14)

must be satisfied. Hence we say (Lm, ~si ) are

( L2 ) ( G , , ( j ~ o l ) + G 1 2 ( j ~ l ) F ) R = ( f f ' ; R ) .

(19)

With the aid of Lemma 3.1, the lower block of the above equation implies that { L, R, q } is admissible. [] Remark 3.1. Proposition 3.1 gives us not only the relation between the basic constraints and the admissibility but also an explicit (rank) condition by which we can check the admissibility directly.

basic constraints with respect to the boundary points jo:i (i = 1. . . . . n). Now we can state the admissibility condition in terms of the basic constraints.

Proposition 3.1. Under the same hypothesis as in Lemma 3.1, the constraints ( Li, Ri, vPi} are ad-

missible if and only if rank( L [ i

'tI'tBiRi~i ]l = rank( Lm]zi]

(15)

for i = l ..... n.

Remark 3.2. If G12 is square (i.e., the so called 'l-block problem' case), no basic constraint such as (14) is required, and hence all triples ( Li, Ri, ~tti} are admissible. Taking the basic constraints into consideration, the actual constraints on Ge,(j~ ) are given by (14) and (s3). If the constraints given by (s3) are admissible, (14) and (s3) are equivalent to (14) and (s3)' L, Ger(j~i)R, = ~ R , with

LBi I,

~BiRi

Proof. (Necessity) Eq. (14) leads to LBG~r(j~I)R = ~/'BR. This together with (s3) implies (15) in the case of n = 1. (Sufficiency) Suppose that (15) holds for i = 1. . . . . n. Then there exists a nonsingular matrix T ~ C (~°+m)×(~o+m) such that

for i = 1. . . . . n, where Lci and ~c~ are defined by (16)• The above constraints (s3)', which are an extended form of (s3), play an important role in solving the boundary constraint problem in the next section.

(16) L

=

%

0

'

4. Solvability of H~-problem with boundary constraints

( LLcB ) is r°w full r a n k The right hand side of (16) is obtained by eliminating the linearly dependent rows from

In order to derive the main result, we need three lemmas. The first lemma is a result on the parametrization for the Hoo-suboptimal problem.

T. Sugie, S. Hara /H~-suboptimal control problem

96

Lemma 4.1 [1,9,10]. Suppose X(G, K ) satisfies the assumptions (A1)-(A4). I f the H~-problem is solvable, then the class of all solutions Ge~(S) satisfying [1Ger II ~ < 1 is given by Ger = (011S q- 012)(021 3 q- 022) 1

n

012)

}P'

021

022

}ml

m2

(22)

The next two lemmas give the properties of O(s) defined in the above lemma. Lemma 4.2 [9,10]. Suppose O(s) given by (22) is (Jp .... -J,,>p~)-lossless. Then there exists a OE(s) R(s) (p'+m')×(p'-':) such that 0 : = ( 0 E O) -

-

Theorem 4.1. Suppose the system X( G, K ) satisfies the assumptions (A1)-(A4). Then the Ha-problem with boundary constraints (Li, Ri, vt i ) is solvable, if and only if the following three conditions hold: (a) The H~-problem is solvable. (b) ~,L~, R i, ~ } are admissible. (c) LiL* > g'i(R*Ri)-'fZ,* (i = 1 . . . . . n).

P2

is ( Je,,,m - Jm>e=)-l°ssless" Furthermore, the stabilizing controllers K ( s ) satisfying [I G~r II ~ < 1 and the parameters S(s) in (21) are in one-to-one correspondence.

is ( Jp,, ml

Now we are in the position to state the main result.

(21)

where S(s) is an arbitrary element in BH m2×p2 and

0 ( S ) :=

other words, it holds for any choice of S(s) in (21), we obtain (25). []

(23)

Proof. (Necessity) Suppose (sl), (s2) and (s3) hold. Then, it is obvious that (a) and (b) are satisfied. Since the extended constraint (s3)' holds now, we have LGer(Jo~I)R(R.R)

(28)

while it is easily checked that IiGerOo31)R(R*R)-|/2][~_~

[] GerO~.ol) l[ < ]

(29)

holds from (s2). Lemma A in the Appendix with (28) and (29) leads to (c). (Sufficiency) Without loss of generality we assume

Jp ..... )-lossless. Moreover, LL* = I,

(24) holds for any ¢o.

[,emma 4.3. Suppose Get is described by (21). Then O(s) given by (22) satisfies

(LB,

,/2 = q , ( R . R ) - I / 2 ,

--~Bi)OOgaJi)=O

( i = 1 . . . . . n).

(25)

R * R = I.

(30)

This implies that (c) is equivalent to (c)' IIq'll < 1 . Now, suppose (a), (b) and (c)' hold. From Lemma 4.1 and the definition of the extended constraint ( L , ~') given by (20), it is enough to show that there exists an S(s) ~ BI-I~ satisfying L ( OllS + Oi2)( O~lS + O = ) - I R = q,

at s =jc0 I.

(31)

Proof. Let Y:-- 021S + 022; then we have

The condition (a) implies that there exists a controller K ( s ) such that IfGer(J~°l)II < 1, and the basic constraint requires LBG~,(j~Ol)= ~B-Hence, from Lemma A, we get

from (21). On the other hand, (14) means

•I*Bg"; < LBL ~ .

(LB

--~B)(Ger(JCOl)) = 0 " I

(27)

Since the above equation holds for any K(s), in

(32)

Since (30) implies that L B L ~ = I from the definition of L, we obtain [I q'B II < 1.

(33)

97

T. Sugie, S. Hara /H~-suboptimal control problem

On the other hand, from (30), there exists a matrix such that k := (R, RE) is unitary. Now, (33) and (c)' yield

is row full rank at s =jto 1, because t~:= (0 E, 0) is nonsingular and L B has full row rank. This and (25) lead to

II('PBR

(34)

det{(L B

(35)

This implies that S A = 0 from (42). Consequently, it is seen from (39) and (41) that there exists an S , such that

R E E C m'×(m'-v')

'/5,RE)II = II'PolI < 1 ,

{ Ill xoc ] = l [ { ' l l < 1 -

The above two inequalities imply that there exists a matrix Q such that tlAII
A.'=

( vt'sR

g'sRE )

'Pc

- g's)0e(j~o,)) 4= 0.

(43)

II SB II < 1,

(44)

(36)

Q

holds (see e.g. [6]). Let {'c := ('/'c

Q)~-I

r . '

.

.

.

Lc

(37) ~t"c

Then from (30) and Lemma 4.2, we have

O
-

~c

Remark 4.1. In the case of the 1-block problem, the solvability condition can be reduced to (a) and

FJpl,mlr*

=F(a E

(C) :~: L i L ? ~" ~ i ( R ? R i ) - l ~ i

O)J? . . . . (0 E o ) * r *

at s=j~01,

(38)

where 0E is defined in Lemma 4.2. From Lemma A, this implies that there exist SA ~ C (p~-"~)×m~ and SB E C m~×m, satisfying

I~SB )

The above equation implies that (31) holds for S = SB from the definition of ~'cFrom Lemma B in the Appendix, there exists an S(s) ~ BHoo such that S(jtol) = S B holds, and hence it satisfies (31). This completes the proof (note that Lemma B assures this argument is valid for multi-point constraints).

(39)

<1,

00,0,))

(::/--o.

(40)

The last equation is equivalent to

* (i = 1 . . . . . n),

because no extension is required for the boundary constraints. The latter condition (c) n is an extension of the previous results for the scalar case [7,8], and for the MIMO case with m I = mz, Pl = P2, R i = I and L i= I [12]. Remark 4.2. It is easy to check the conditions (a)-(c) in the state space setting. In fact, we can check the condition (a) by solving two algebraic Riccati equations, while the conditions (b) and (c) contain nothing but constant matrices. Note that it is straightforward to obtain the basic constraint ( Lsi , XPsi ) for each given point s =jto i by (13) via the state-space form of G(s). As a corollary of Theorem 4.1, we have the following result for the optimal Ho:norm.

Now we claim that SA --0. Since (25) holds from Lemma 4.3, (25), (37) and (41) yield ( L.

-

=0.

On the other hand, we can see that

(L B

--~I'B)(OE(s )

O(s))

(42)

Corollary 4.1. Under the same hypothesis as in Theorem 4.1 and admissibility of the given con-.

straints { Li, Ri, vt"i}, we have that y := inf II ae, II ~

K(s)

subject to (sl) and (s3)

= max{ Y0, 71 . . . . . ~,~ }

(46)

T. Sugie, S. Hara /H~o-suboptimalcontrolproblern

98

where

After some straightforward calculation, we have

subject to (sl)

70 := inf [1Get II ~ K(s)

(47)

(P(0)W,-I(0))(P(0) wt-l(0)) * > I

(48)

from the condition (c) of Theorem 4.1. Therefore, the above condition must be satisfied in order to achieve Ilae, ll~ < 1 with (51). Note that (56) comes from the implicit requirements

^ -1/2 V,'.=II(L,L*) - 1 / 2 ,/,,(R'R,) II .

.

.

.

(i = 1 . . . . . n).

II WtK(I+

5. Examples Let the plant

G(s) be

0

O21=

PK) -1 l[ ~

(56)

< 1

and

given by

PK(I+PK) -1 =I '

G21 = I,

G22 =

Wt

],

P,

(49a) (49b)

where ~ and Wt are unimodular weighting matrices and P is the plant to be stabilized. In this case, Ger is expressed as

( Ws(I+PK) -1 ) • Get= W,K(I+PK)-I

(50)

So, this is a so-called mixed sensitivity problem. Suppose we impose a boundary constraint: ( I + P ( 0 ) K ( 0 ) ) -1 = 0,

(51)

as well as (sl) and (s2). The above constraint implies that zero steady-state error for step-type external input is required as an additional design specification. The corresponding {L, R, ~o} in (s3) is given by L=(W~-I(0)

0),

R=I,

~=0.

(52)

One may think that this boundary constraint can be always achieved whenever the Hoo-problem is solvable, because the interpolation value if" is zero. However, according to Theorem 4.1, this is not the case. The basic constraint (LB, ~B } in (14) is described by

=

= (w/l(o)

@B = O~(O)GI,(O) = I.

v(o)w:'(o)),

(53) (54)

It is easily checked that {L, R, ~ } is admissible from Proposition 3.1, and the extended boundary constraint { £, ~' } is given by

at s = 0. It should be stressed that we need no such knowledge of implicit requirements by using the extended boundary constraints. Next, we consider another problem with a boundary constraint which corresponds to the zero steady-state error requirement in the case where some elements of r are step type signals and others become zero as the time t ---, oo. The constraint can be expressed as

(I + P(O)K(O))-'R =

0

(57)

where R is, for example, given by R = (1, 0) T. In this case, corresponding to (c) of Theorem 4.1, we have

(e(o)w,-'(o))(e(o)w,-l(o))

*>

R(R*R)-II¢*. (58)

As we can guess, this condition is weaker than (56) because R(R*R)-IR * < I.

6. Conclusion A necessary and sufficient condition for solvability has been derived for the 2-block Hoo-suboptimal control problem with generalized boundary (jw-axis) constraints by considering the basic constraints included in the given plant. Our result can be extended to the so-called 4-block problem case by taking account of the annihilator of G21 as well as that of G12. However it requires much more complicated calculations to derive the solvability condition; the result therefore will be reported in a forthcoming paper•

T. Sugie, S. Hara /H~-suboptimalcontrolproblem

99

These observations imply that the above S(s) satisfies all conditions in L e m m a B. []

Appendix L e m m a A. S u p p o s e that A ~ C/×m a n d B ~ C t×n are given, a n d r a n k A = l. Then there exists an X ~ C "~×n such that

AS=B,

IlSll < 1 ,

(59)

i f a n d only i f

Acknowledgement

( o r equivalently ( A , B ) J m , n ( A , B ) * > 0).

T h e a u t h o r s w i s h to e x t e n d t h e i r g r a t i t u d e to Prof. M. V i d y a s a g a r for v a l u a b l e d i s c u s s i o n s a n d h e l p f u l c o m m e n t s , e s p e c i a l l y o n t h e p r o o f of L e m m a B.

P r o o f . S i n c e n e c e s s i t y is o b v i o u s , we will s h o w s u f f i c i e n c y . First, w e a s s u m e A A * = 1 w i t h o u t loss o f g e n e r a l i t y . T h e n , t h e r e exists a n A E s u c h t h a t

References

AA* > BB*

(60)

is u n i t a r y . N o w , let

t h e n it is e a s y to c h e c k t h a t X = X o satisfies (59). T h i s c o m p l e t e s t h e p r o o f . [] L e m m a B. S u p p o s e t h a t {S,~CP×m,a~i>O,i=l

..... n}

are given, a n d [1 S i II < 1 holds f o r each i. Then there exists an S ( s ) ~ B H ~ xm such that S(jtoi) = S i holds f o r each i. P r o o f (Sketch). C o n s i d e r a P i c k m a t r i x d e f i n e d b y

= {

:=

{ i-s,*sj } 2e-~,--o~j)

'

e>0;

(61)

then, clearly, P / , ( e ) > 0 a n d II Pii(e) II --' oo as ---, 0. O n t h e o t h e r h a n d , if i ~=j, II P~j(e) II -< a as e---, 0 f o r s o m e f i n i t e n u m b e r a. T h e r e f o r e , t h e r e exists a n e 0 > 0 s u c h t h a t P ( e o ) > 0 holds. T h i s i m p l i e s t h a t t h e r e exists a n X ( s ) ~ B H pxm satisfying X(e o +j~)= S i (see [2,12]). N o w , let S ( s ) := X ( s + %). T h e n o b v i o u s l y S(jt~i) = S i for e a c h i, a n d X ( s ) ~ R I - I ~ i m p l i e s t h a t S ( s ) ~ R H ~ . A l s o , we h a v e t h e f o l l o w i n g r e l a t i o n s :

II S II = = sup II S ( j . ) II 6d

= sup II S(eo +jo~) II < I l g l l ~ < 1. ¢d

(62)

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