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Directional interpolation in the state space
Systems & Control Letters 10 (1988) 317-324 North-Holland
317
Directional interpolation in the state space Hidenori K I M U R A Department of Mechanical Engineeringfor Computer-ControlledMachinery, Faculty of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 560. Japan Received 7 September 1987 Revised 25 November 1987 and 13 February 1988
Abstract: This paper presents a new state-space characterization of the directional interpolation based on the observation that the generalized Pick matrix is regarded as a solution of a Lyapunov-type equation. A simple representation of the :~, of all solutions is derived in terms of linear fractional transformations. The computation of the associated J-unitary matrix is simpler than the well-known procedure based on the Glover's scheme or the J-spectral factorization of Ball and Ran.
Keywords: Model matching, H**-control, Nevanlinna-Pick interpolation.
1. Introduction In H~-optimization and robust stabilization, it is important to characterize the class of all • ~ R H ~ m for a given I"1 ~ RHea, and T2 ~ RH~×m such that • is strictly contractive and T2-1(T1 - O) ~ R / / , ~ , , where RH~×,, denotes the set of all m × r rational matrices whose entries are all in H ~. We denote this class by ~9°, i.e., 00
•
< 1 , h ffi Y l - T 2 Q , Q
} .
The problems are to find the condition on/'1 and T2 under which the set is non-empty, to parametrize all • ~ if ~' is non-empty, and to compute the corresponding Q. The relevance of the set ~" to H®-optimization and robust stabilization is found, e.g., in [6,7,15]. There are several approaches to treat the class ~ . In the frequency domain, Francis et al. [6] gave a complete parametrization of ~ ' based on the BaH-Helton theory. A simpler version of this result is found in [10]. In these contributions, a more general class is treated in which the strict inequality II• II ® < 1 is replaced by the inequality II• II ® < 1. Another frequency-domain approach is to reduce the problem to the matrix Nevanlinna-Pick problem [3], or the directional interpolation problem [11,13], which was originally formulated by Fedeina [5] and discussed in [1]. In particular, the directional interpolation ap-proach [9,11] gives a simple and highly structured representation of ~' when the RHP zeros of T2 are all simple. There, a classical Schur-Nevanlinna algoritlun was extended to the directional interpolation problem, the state-space version of which was developed in [13]. A disadvantage of the interpolation approach is that the solution is represented sometimes in the complex field [4]. In the state space, no explicit characterization of ~ has been obtained so far. Instead, a related problem of model reduction has been solved in the state space initially by Glover [8] and simplified later by B',dl and Ran [2], which can be applied to characterize the set ~'. This was fully discussed in Francis' book [7]. In this paper, art explicit state-space parametrization of ~ is obtained based on the observation that the generalized Pick matrix for the directional interpolation problem introduced in [1,11,13] is a solution to a Lyapunov-type equation. The parametrization obtained in this paper is regarded as a state-space interpretation of the directional interpolation theory developed in [11]. The parametrization is simply obtained by solving two Lyapunov-type equations. 0167-6911/88/$3.50 ~ 1988, Elsevier Science Publishers B.V. (North-Holland)
H. Kimura / Directional interpolation in the state space
318
2. State-space characterization of the Pick matrix Let T~(s) = Di + C~(sl- A,)-~B~, i = 1, 2, be canonical state-space realizations of Ta(s) and T2(s), respectively. The McMillan degrees of T~(s) and T2(s) are n~ and n2, respectively. Throughout the paper, we use the Doyle convention
r~(s)=
c~
o~'
r~(~)=
02"
In order to simplify the subsequent development, we make two assumptions on T2(s ) as follows: (A1) D 2 ~ Rmxm is non-singular. (A2) All the transmission zeros of Tz(s) have positive real parts. These assumptions imply that the A-matrix of T~(s)-~ given by
Au'= Az - LCz,
L'= BzD2 ~,
(1)
is antistable, i.e., all the eigenvalues of A~ are in the closed right half plane. Since Aa is stable, there exists a unique solution R of the equation
A u R - RA1 - LC1
(2)
Consider a Lyapunov-type equation
Aoe + ea~ = LL~ - MM ~,
(3) (4)
M = R B 1 + L D I.
Since A,, is antistable, the equation (3) has a unique solution P. To see the meaning of the equation (3), consider a special (but generic) case where the Jordan form of A u is diagonal. In this case, we write XAuX - ~ - A = diag(Xl, X2,..., Xn).
(5)
The i-th row x~ of X satisfies xr~A,- h~x~, where 2~ is a transmission zero of T2(s ). Let ~ "-- LT£i . Since
~,*(z>~+ c~(x,/- A~) -'B~) = x,'B~D~-'(D~ + C~(X,I- A~) -'n~) = xr, ( l + n 2 0 ; ' C 2 ( X , l -
a2)-')n2
= ~,(X,1 - a , ) ( x , 1 - a , ) - ' n ~ = o, we see that ~* satisfies
~?re(x,) - 0 ,
(6)
i.e., ~* is the 'zero vector' of T2(s) at the transmission zero h~. Also, from (4), ~ : - M r 2 ~ satisfies n* = xr~M = x~RBI + I~*Dl. Due to (2), x~RA~ = ;k,xr~R - $,*C~. Hence, x~R - 12*C~(XJ- A~) -~. Therefore, we conclude that
~* • = ~?rl(x,).
(7)
The relations (6) and (7) imply that any • ~,9° must satisfy ~*O(X,) = ~1", i = 1, 2,..., n.
(8) Thus, the problem of finding a • ~ 5,' is equivalent to the directional interpolation problem defined in [11], that is, the problem of finding a contractive q~ satisfying the directional interpolation conditions (8). Premultiplication by x r and postmultiplication by ~j of (3) yield (h~+~,j)xrp2j=~*~j-~l*vb. Therefore, we have
( XPX* )0 =
x, + ~j
(9)
H. Kimura / Directionalinterpolationin the state space
319
which implies that XPX* is the Pick matrix for the directional interpolation problem [9]. It was shown in [1,11,13] that the directional interpolation problem is solvable if and only if the corresponding Pick matrix is positive definite. Actually, Limebeer and Anderson [13] derived a solvability condition for a more general type of interpolation problem. In the case where A u is diagonalizable, the class b ° is identical to the class of the solutions of the directional interpolation problem. Therefore, we conclude that Y' is non-empty if and only if P is positive definite when A,, is diagonalizable. Actually, the same assertion is valid even when A~ is not diagonalizable. For space limitation, its proof is omitted [12]. Theorem 1. The set
Y' is non-empty, if and only if the solution P of the equation (3) is positive definite.
3. Construction of a J-unitary matrix
We assume that the Pick matrix P is positive definite. Define a O(s)
[e,,(;) e(,) = [e.,(;)
AT[ - P - 1 L
e,.(s) e,~(,)
=
~
RH~r+m)x(r+m) by
p-I M"
LT
Im
0
MT
0
I,
(lo)
The important properties of Û(s) are exploited in the following lemmas. Lemma
1. 1]"P > O, O( s ) satisfies
O(s)*JO(s)-JffiO,
s=j~,
(lla)
O(s)*JO(s)-J
Re s > O ,
(llb)
o
O1
-L"
Proof. From (4), it follows that
(~I + Au)P + P(sl + A~) - (s + ~)P + LL T - MM T. Hence,
p - 1 ( ~ I + A u ) - , ( L L T _ M M T ) ( s I + A u )T
p_,(~l+Au)-, + ( s I + A r ) - ' p - l f
-1
p-1
+ (s + ~)P-I(Yd + A.)-1P(sl + AT)-IP -1. Direct manipulations using above identity yield
O ( s ) * J O ( s ) - - - ( [ ImO I~ +
.(['o
o]
-Z
,
M T P-1(sI+A")- [L M] +
- M T ( s l + A ~ ) - ' , ~-
= J - (s + ~)H(s)* P H ( s ) ,
H(s)=(sI+ATu)-~P-~[-L M]. This establishes (11).
',
-L
)
320
H. Kimura / Directional interpolation in the state space
Let us defh~e
[nl,(~)
r/~(~)
"=T(s)-'O(s)
/'/22($)
(12)
where 0
L
(13)
"
Lemma 2. A state-space realization of II(s) is given by n(s)--
(14)
c~ ' D~
with A,,- [ A1 0
B1MT"
0 B,, = - P-1L
-A T '
B1 ] P- IM
G= [D;I(G + GR) D21( DIMT- C2P- LT) ]
[
MT
0
D~ ~
/)2-I,lDz 1"
0
Proof. From the definition of H(s),
/'/($) ~_ T2 I(TlO21 - ~11)
T2"-1(TI~2~ - ~12 ) ]
O,~1
O.
"
0
"LT
(15)
It is not difficult to see that
Tl@21 - @11 -
D
M1
lm
-
0
-Aru
-~1 Ds M-''-~-L Since
r2- l(s) -- D~_~c,
02 -1
1G
the product rule yields
A~,O T21(TI02~ - Oil) -
The non-singular matrix U-
0 0
I., 0
0 I~
0
-LCIA1 0
-L(DIMT-Aru
0 ]
t
-p-1L "
-P-IL "---~
H. Kimura / Directionalinterpolationin the state space yields the congruence
[A~-LC1-L(DIMT-LT) U A1 B1M T 0 --AT.
]
0]
U-1 ffi
A 1 BIM T , o
D21[C2 C1 D1MT-LT]U-1---D21[C2
321
-AT
C1+ C2R D,M T - C2P- IT].
Here, we have used the relations (2), (3), (4). Therefore, the first part of the state space turns out to be uncontrollable. Hence,
r;l(r~021 - 011) =
[
A1
B1M T
I[
o
-aZ
I
D;I(c, +C2R )
0
]
-P-1L]"
D~'I(D, M T - C 2 P - L T) [ - D ; '
Analogously, we have
T21(T1022- 012 ) -~
0
P'IM •
-A T
~,(c~ + c~R) ~ , ~ i ~ , ~ = c~p- L~)
~;'~,-----~
In view of (10) and (15), the assertion has been established. Since both A1 and --AT are stable, so is A,,. Therefore,/i(s) ¢ RH~m+,)×tm+,). Furthermore,//(s) -1 exists and is in RH~m+,y×(m+,),as is shown in the following lemma. Lemma 3.
II(s) has stable inverse, i.e.,
n(~)-'
~
RH~m+r)×(m+O .o
Proof. Obviously, D~-1 exists and is given by D~'I~
0
L
"
From the inversion formula, it is sufficient to show that A . using (1) and (4) yield
A~t- B~rD~IC~r-~- _p-1L(C1 + C2S )
B,,D~I¢~ is
p-1A2P •
Since both A1 and A 2 are stable, the assertion follows immediately.
4. Parametrizafion of all so|ufions
We are now in a position to state the main result of the paper.
stable. Direct manipulations
322
H. Kimura / Directional interpolation in the state space
Theorem 2. Under the assumptions P > 0 and (A1), (A2), • ESP if and only if there exists a contractive
S ~ H~mx~ such that (~-- (OllS "{- O,2)(O2,S -[" 022) -1
(16)
The matrix Q which yields the representation • = 7"1 - T2Q is given by Q- (II,,S ¢ .r/,:)(H:,S + HE:)-'.
(17)
Proof. Assume that • is represented as (15) for some S ~ H ~ × , with [] $ [[ ~ < 1. Let
(1,, Due to (llb),
X*X-Y*Y=[S*
[s]
l,]@*JO i,
<[S*
/r]J
[s]
/r = S ' S - / , < 0 ,
for each Re s >_0. This implies that • = X Y - 1 ~ R H ~ x , and II O II oo < 1. Let
(M
s
From the d~finition (12) of H and (18) we have
r( M
,.)--
Hence,
O= X Y - ' = ( T , N - T 2 M ) N - 1 = T l - T2MN -1. This implies that • ES~' with Q = M N -~. Since Y ffi N, N -~ ~ R H ~ , . Thus, Q ~ RH~x.. This establishes the sufficiency part of the theorem. Let a • ~,~q' be repres~,.ed by • ffi I'1- T2Q, and let Q ffi M N - l be a coprime factorization of Q. Write V .ffi 1 I - ' V
(20)
N "
Due to Lemma 3, U ~ H~x,, V ~ H,~,. Note that
(21) Since • is contractive, it follows, from (20), that
O
V*)O*Jo[U]=-(U*
V*)J[UvI=V*V-U*U,
s=j(o.
This implies that S = U V - ' satisfies .I, - S*S > 0 for each s = jto. It remains to show that S ¢ H~×,. Since U ~ H~,, oo×,, it suffices to show that V-1 ~ H ~ × , . The proof of this assertion is rather involved and is found in [7], p. 124. The linear fractional transformation (15) can also be represented as
• = q n + xi'12S(I- g'22S)-1~/'21,
(22)
323
H. Kimura / Directionalinterpolationin the state space where
~ ~. [ ~/Zll ~tt12] .._ [ 012~ 1
011 -- e12~1~21
From the realization (10) of O, one obtains by straightforward but tedious computation
-'o
(23)
Note that
p(-A T- p-IMMT ) + (-A T- p-1nnT)Tp--
- L L T _ n n T,
which implies that '/'(s) is stable. Analogously, we can obtain another representation for Q in (16) by
Q "- ~11 -i- ~ 1 2 s ( I - -Y22S)-1~21,
(24)
where
Zll
Z12 ]
[ 1-I12111
/'/11 -- H12II2~1-I21
~v- L~21 ' 22 -- L
- H~II21
From the definition of H in (12), we have
T2-1(T1 - ~P11) ~Tr21
- T;1
1,1.
~Tt22 J
-
Aa
0
B1
0
0
--ATe -- P - 1 M M T
p-1M
p-I L
D21(C1 "t" C2R )
- D 2 1 ( LT + C2P )
D21D1
D~ 1
0
-M T
~
o
The representation (15) or (21) is totally different from the one obtained in [7] based on the method of [2]. The main distinction is that the J-unitary matrix obtained in [7] is not stable and is of McMiHan degree 2n2, while the one given in (10) is of McMillan degree n2. The major computations required for obtaining (10) are to solve the two linear equations (2) and (3), which is comparable to the computation of the two Gramians required in the formula of [7]. However, the method described in [7] requires the additional task of extracting the antistable part of T2(s)-lTl(S), which amounts to solving a linear equation [14]. An expression of the J-unitary matrix similar to (10) is found in [9] associated with a scalar Neva~nna-Pick problem. The results obtained here are much more general and represented in the new framework of state space.
5. Conclusion
A new state-space characterization of the class of interpolation matrices is presented. The characterization is based on the state-space interpretation of the Pick matrix for the directional interpolation problem. The computation is essentially reduced to solving two linear equations which is simpler than the method described in [7].
324
H. Kimura / Directional interpolation in the state space
The extension of the results to the so-called two-sided problem, in which S° is replaced by So--- { ¢ ~ n.~%.. II * II oo < 1. • = T1 - T2 Q T3 }. is straightforward and is reported in the forthcoming paper [12], in which the whole algorithm of H°%optimization is derived from the view point of directional interpolation. The author is grateful to one of the reviewers for his enlightening comments, in which he derived the main results of this paper based on the approach of [7].
References [1] J.A. Ball and J.W. Helton, Lie group over the field of rational functions, signed spectral factorization, signed interpolation, and amplifier design, J. Operatory Theory 8 (1982) 19-64. [2] J.A. Ball and A.C.M. Ran, Optimal Hankel norm model reductions and Wiener-Hopf factorizations I: The canonical case, SlAM J. Control Optim. 25 (1967) 362-382. [31 B.C. Chang and J.B. Pearson, Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control 29 (1984) 880-887. [41 P. Dorato and Y. Li, A modification of the classical Nevanlinna-Pick interpolation algorithm with applications to robust stabilization, IEEE Trans. Automat. Control 31 (1986) 645-648. [51 I.P. Fedcina, A criterion for the stability of the Nevanlinna-Pick tangent problem, Mat. Issledoaniya 7 (1972) 213-227 (Russian). [61 B.A. Francis, J.W. Helton and G. Zarnes, H=-optimal feedback controllers for linear multivariable systems, IEEE Trans. Automat. Control 29 (1984) 888-900. [71 B.A. Francis, A course in H= control theory (Springer, Berlin-New York, 1987). [8] K. Glover, All optimal Hankel-norm approximations of linear multivanable systems and their L=-error bounds, Internat. d. Control 39 (1984) 1115-1193. [91 V.E. Katsnerson, Methods old-Theory in Continuous Imerpolation Problems of Analysis, Pt. I, Monograph (1982); translated by T. Ando. [IO1 H. Kimura, On interpolation-minimization problem in H ~, Control: Theory Advat~,ced Technology 2 (1986) 1-25. [111 H. Kimura, Directional interpolation approach to H~-optimization and robust stabilization, IEEE Trans. Automat. Control 32 (1987) 1085-1093. [121 H. Kimura, Conjugation and model-matching in H ®, under preparation. [131 D.J.N. Limebeer and B.D.O. Anderson, An interpolation theory approach to H ~ controller degree bou,'ds, Linear Algebra Appl., to appear. [14] M.O, Safonov, E.A. Jonckheere, M. Verma and D.J.N. Limebeer, Synthesis of positive real multivariable feedback systems, lnternat. J. Control. 45 (1987) 817-842. [151 M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 1985).
317
Directional interpolation in the state space Hidenori K I M U R A Department of Mechanical Engineeringfor Computer-ControlledMachinery, Faculty of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 560. Japan Received 7 September 1987 Revised 25 November 1987 and 13 February 1988
Abstract: This paper presents a new state-space characterization of the directional interpolation based on the observation that the generalized Pick matrix is regarded as a solution of a Lyapunov-type equation. A simple representation of the :~, of all solutions is derived in terms of linear fractional transformations. The computation of the associated J-unitary matrix is simpler than the well-known procedure based on the Glover's scheme or the J-spectral factorization of Ball and Ran.
Keywords: Model matching, H**-control, Nevanlinna-Pick interpolation.
1. Introduction In H~-optimization and robust stabilization, it is important to characterize the class of all • ~ R H ~ m for a given I"1 ~ RHea, and T2 ~ RH~×m such that • is strictly contractive and T2-1(T1 - O) ~ R / / , ~ , , where RH~×,, denotes the set of all m × r rational matrices whose entries are all in H ~. We denote this class by ~9°, i.e., 00
•
< 1 , h ffi Y l - T 2 Q , Q
} .
The problems are to find the condition on/'1 and T2 under which the set is non-empty, to parametrize all • ~ if ~' is non-empty, and to compute the corresponding Q. The relevance of the set ~" to H®-optimization and robust stabilization is found, e.g., in [6,7,15]. There are several approaches to treat the class ~ . In the frequency domain, Francis et al. [6] gave a complete parametrization of ~ ' based on the BaH-Helton theory. A simpler version of this result is found in [10]. In these contributions, a more general class is treated in which the strict inequality II• II ® < 1 is replaced by the inequality II• II ® < 1. Another frequency-domain approach is to reduce the problem to the matrix Nevanlinna-Pick problem [3], or the directional interpolation problem [11,13], which was originally formulated by Fedeina [5] and discussed in [1]. In particular, the directional interpolation ap-proach [9,11] gives a simple and highly structured representation of ~' when the RHP zeros of T2 are all simple. There, a classical Schur-Nevanlinna algoritlun was extended to the directional interpolation problem, the state-space version of which was developed in [13]. A disadvantage of the interpolation approach is that the solution is represented sometimes in the complex field [4]. In the state space, no explicit characterization of ~ has been obtained so far. Instead, a related problem of model reduction has been solved in the state space initially by Glover [8] and simplified later by B',dl and Ran [2], which can be applied to characterize the set ~'. This was fully discussed in Francis' book [7]. In this paper, art explicit state-space parametrization of ~ is obtained based on the observation that the generalized Pick matrix for the directional interpolation problem introduced in [1,11,13] is a solution to a Lyapunov-type equation. The parametrization obtained in this paper is regarded as a state-space interpretation of the directional interpolation theory developed in [11]. The parametrization is simply obtained by solving two Lyapunov-type equations. 0167-6911/88/$3.50 ~ 1988, Elsevier Science Publishers B.V. (North-Holland)
H. Kimura / Directional interpolation in the state space
318
2. State-space characterization of the Pick matrix Let T~(s) = Di + C~(sl- A,)-~B~, i = 1, 2, be canonical state-space realizations of Ta(s) and T2(s), respectively. The McMillan degrees of T~(s) and T2(s) are n~ and n2, respectively. Throughout the paper, we use the Doyle convention
r~(s)=
c~
o~'
r~(~)=
02"
In order to simplify the subsequent development, we make two assumptions on T2(s ) as follows: (A1) D 2 ~ Rmxm is non-singular. (A2) All the transmission zeros of Tz(s) have positive real parts. These assumptions imply that the A-matrix of T~(s)-~ given by
Au'= Az - LCz,
L'= BzD2 ~,
(1)
is antistable, i.e., all the eigenvalues of A~ are in the closed right half plane. Since Aa is stable, there exists a unique solution R of the equation
A u R - RA1 - LC1
(2)
Consider a Lyapunov-type equation
Aoe + ea~ = LL~ - MM ~,
(3) (4)
M = R B 1 + L D I.
Since A,, is antistable, the equation (3) has a unique solution P. To see the meaning of the equation (3), consider a special (but generic) case where the Jordan form of A u is diagonal. In this case, we write XAuX - ~ - A = diag(Xl, X2,..., Xn).
(5)
The i-th row x~ of X satisfies xr~A,- h~x~, where 2~ is a transmission zero of T2(s ). Let ~ "-- LT£i . Since
~,*(z>~+ c~(x,/- A~) -'B~) = x,'B~D~-'(D~ + C~(X,I- A~) -'n~) = xr, ( l + n 2 0 ; ' C 2 ( X , l -
a2)-')n2
= ~,(X,1 - a , ) ( x , 1 - a , ) - ' n ~ = o, we see that ~* satisfies
~?re(x,) - 0 ,
(6)
i.e., ~* is the 'zero vector' of T2(s) at the transmission zero h~. Also, from (4), ~ : - M r 2 ~ satisfies n* = xr~M = x~RBI + I~*Dl. Due to (2), x~RA~ = ;k,xr~R - $,*C~. Hence, x~R - 12*C~(XJ- A~) -~. Therefore, we conclude that
~* • = ~?rl(x,).
(7)
The relations (6) and (7) imply that any • ~,9° must satisfy ~*O(X,) = ~1", i = 1, 2,..., n.
(8) Thus, the problem of finding a • ~ 5,' is equivalent to the directional interpolation problem defined in [11], that is, the problem of finding a contractive q~ satisfying the directional interpolation conditions (8). Premultiplication by x r and postmultiplication by ~j of (3) yield (h~+~,j)xrp2j=~*~j-~l*vb. Therefore, we have
( XPX* )0 =
x, + ~j
(9)
H. Kimura / Directionalinterpolationin the state space
319
which implies that XPX* is the Pick matrix for the directional interpolation problem [9]. It was shown in [1,11,13] that the directional interpolation problem is solvable if and only if the corresponding Pick matrix is positive definite. Actually, Limebeer and Anderson [13] derived a solvability condition for a more general type of interpolation problem. In the case where A u is diagonalizable, the class b ° is identical to the class of the solutions of the directional interpolation problem. Therefore, we conclude that Y' is non-empty if and only if P is positive definite when A,, is diagonalizable. Actually, the same assertion is valid even when A~ is not diagonalizable. For space limitation, its proof is omitted [12]. Theorem 1. The set
Y' is non-empty, if and only if the solution P of the equation (3) is positive definite.
3. Construction of a J-unitary matrix
We assume that the Pick matrix P is positive definite. Define a O(s)
[e,,(;) e(,) = [e.,(;)
AT[ - P - 1 L
e,.(s) e,~(,)
=
~
RH~r+m)x(r+m) by
p-I M"
LT
Im
0
MT
0
I,
(lo)
The important properties of Û(s) are exploited in the following lemmas. Lemma
1. 1]"P > O, O( s ) satisfies
O(s)*JO(s)-JffiO,
s=j~,
(lla)
O(s)*JO(s)-J
Re s > O ,
(llb)
o
O1
-L"
Proof. From (4), it follows that
(~I + Au)P + P(sl + A~) - (s + ~)P + LL T - MM T. Hence,
p - 1 ( ~ I + A u ) - , ( L L T _ M M T ) ( s I + A u )T
p_,(~l+Au)-, + ( s I + A r ) - ' p - l f
-1
p-1
+ (s + ~)P-I(Yd + A.)-1P(sl + AT)-IP -1. Direct manipulations using above identity yield
O ( s ) * J O ( s ) - - - ( [ ImO I~ +
.(['o
o]
-Z
,
M T P-1(sI+A")- [L M] +
- M T ( s l + A ~ ) - ' , ~-
= J - (s + ~)H(s)* P H ( s ) ,
H(s)=(sI+ATu)-~P-~[-L M]. This establishes (11).
',
-L
)
320
H. Kimura / Directional interpolation in the state space
Let us defh~e
[nl,(~)
r/~(~)
"=T(s)-'O(s)
/'/22($)
(12)
where 0
L
(13)
"
Lemma 2. A state-space realization of II(s) is given by n(s)--
(14)
c~ ' D~
with A,,- [ A1 0
B1MT"
0 B,, = - P-1L
-A T '
B1 ] P- IM
G= [D;I(G + GR) D21( DIMT- C2P- LT) ]
[
MT
0
D~ ~
/)2-I,lDz 1"
0
Proof. From the definition of H(s),
/'/($) ~_ T2 I(TlO21 - ~11)
T2"-1(TI~2~ - ~12 ) ]
O,~1
O.
"
0
"LT
(15)
It is not difficult to see that
Tl@21 - @11 -
D
M1
lm
-
0
-Aru
-~1 Ds M-''-~-L Since
r2- l(s) -- D~_~c,
02 -1
1G
the product rule yields
A~,O T21(TI02~ - Oil) -
The non-singular matrix U-
0 0
I., 0
0 I~
0
-LCIA1 0
-L(DIMT-Aru
0 ]
t
-p-1L "
-P-IL "---~
H. Kimura / Directionalinterpolationin the state space yields the congruence
[A~-LC1-L(DIMT-LT) U A1 B1M T 0 --AT.
]
0]
U-1 ffi
A 1 BIM T , o
D21[C2 C1 D1MT-LT]U-1---D21[C2
321
-AT
C1+ C2R D,M T - C2P- IT].
Here, we have used the relations (2), (3), (4). Therefore, the first part of the state space turns out to be uncontrollable. Hence,
r;l(r~021 - 011) =
[
A1
B1M T
I[
o
-aZ
I
D;I(c, +C2R )
0
]
-P-1L]"
D~'I(D, M T - C 2 P - L T) [ - D ; '
Analogously, we have
T21(T1022- 012 ) -~
0
P'IM •
-A T
~,(c~ + c~R) ~ , ~ i ~ , ~ = c~p- L~)
~;'~,-----~
In view of (10) and (15), the assertion has been established. Since both A1 and --AT are stable, so is A,,. Therefore,/i(s) ¢ RH~m+,)×tm+,). Furthermore,//(s) -1 exists and is in RH~m+,y×(m+,),as is shown in the following lemma. Lemma 3.
II(s) has stable inverse, i.e.,
n(~)-'
~
RH~m+r)×(m+O .o
Proof. Obviously, D~-1 exists and is given by D~'I~
0
L
"
From the inversion formula, it is sufficient to show that A . using (1) and (4) yield
A~t- B~rD~IC~r-~- _p-1L(C1 + C2S )
B,,D~I¢~ is
p-1A2P •
Since both A1 and A 2 are stable, the assertion follows immediately.
4. Parametrizafion of all so|ufions
We are now in a position to state the main result of the paper.
stable. Direct manipulations
322
H. Kimura / Directional interpolation in the state space
Theorem 2. Under the assumptions P > 0 and (A1), (A2), • ESP if and only if there exists a contractive
S ~ H~mx~ such that (~-- (OllS "{- O,2)(O2,S -[" 022) -1
(16)
The matrix Q which yields the representation • = 7"1 - T2Q is given by Q- (II,,S ¢ .r/,:)(H:,S + HE:)-'.
(17)
Proof. Assume that • is represented as (15) for some S ~ H ~ × , with [] $ [[ ~ < 1. Let
(1,, Due to (llb),
X*X-Y*Y=[S*
[s]
l,]@*JO i,
<[S*
/r]J
[s]
/r = S ' S - / , < 0 ,
for each Re s >_0. This implies that • = X Y - 1 ~ R H ~ x , and II O II oo < 1. Let
(M
s
From the d~finition (12) of H and (18) we have
r( M
,.)--
Hence,
O= X Y - ' = ( T , N - T 2 M ) N - 1 = T l - T2MN -1. This implies that • ES~' with Q = M N -~. Since Y ffi N, N -~ ~ R H ~ , . Thus, Q ~ RH~x.. This establishes the sufficiency part of the theorem. Let a • ~,~q' be repres~,.ed by • ffi I'1- T2Q, and let Q ffi M N - l be a coprime factorization of Q. Write V .ffi 1 I - ' V
(20)
N "
Due to Lemma 3, U ~ H~x,, V ~ H,~,. Note that
(21) Since • is contractive, it follows, from (20), that
O
V*)O*Jo[U]=-(U*
V*)J[UvI=V*V-U*U,
s=j(o.
This implies that S = U V - ' satisfies .I, - S*S > 0 for each s = jto. It remains to show that S ¢ H~×,. Since U ~ H~,, oo×,, it suffices to show that V-1 ~ H ~ × , . The proof of this assertion is rather involved and is found in [7], p. 124. The linear fractional transformation (15) can also be represented as
• = q n + xi'12S(I- g'22S)-1~/'21,
(22)
323
H. Kimura / Directionalinterpolationin the state space where
~ ~. [ ~/Zll ~tt12] .._ [ 012~ 1
011 -- e12~1~21
From the realization (10) of O, one obtains by straightforward but tedious computation
-'o
(23)
Note that
p(-A T- p-IMMT ) + (-A T- p-1nnT)Tp--
- L L T _ n n T,
which implies that '/'(s) is stable. Analogously, we can obtain another representation for Q in (16) by
Q "- ~11 -i- ~ 1 2 s ( I - -Y22S)-1~21,
(24)
where
Zll
Z12 ]
[ 1-I12111
/'/11 -- H12II2~1-I21
~v- L~21 ' 22 -- L
- H~II21
From the definition of H in (12), we have
T2-1(T1 - ~P11) ~Tr21
- T;1
1,1.
~Tt22 J
-
Aa
0
B1
0
0
--ATe -- P - 1 M M T
p-1M
p-I L
D21(C1 "t" C2R )
- D 2 1 ( LT + C2P )
D21D1
D~ 1
0
-M T
~
o
The representation (15) or (21) is totally different from the one obtained in [7] based on the method of [2]. The main distinction is that the J-unitary matrix obtained in [7] is not stable and is of McMiHan degree 2n2, while the one given in (10) is of McMillan degree n2. The major computations required for obtaining (10) are to solve the two linear equations (2) and (3), which is comparable to the computation of the two Gramians required in the formula of [7]. However, the method described in [7] requires the additional task of extracting the antistable part of T2(s)-lTl(S), which amounts to solving a linear equation [14]. An expression of the J-unitary matrix similar to (10) is found in [9] associated with a scalar Neva~nna-Pick problem. The results obtained here are much more general and represented in the new framework of state space.
5. Conclusion
A new state-space characterization of the class of interpolation matrices is presented. The characterization is based on the state-space interpretation of the Pick matrix for the directional interpolation problem. The computation is essentially reduced to solving two linear equations which is simpler than the method described in [7].
324
H. Kimura / Directional interpolation in the state space
The extension of the results to the so-called two-sided problem, in which S° is replaced by So--- { ¢ ~ n.~%.. II * II oo < 1. • = T1 - T2 Q T3 }. is straightforward and is reported in the forthcoming paper [12], in which the whole algorithm of H°%optimization is derived from the view point of directional interpolation. The author is grateful to one of the reviewers for his enlightening comments, in which he derived the main results of this paper based on the approach of [7].
References [1] J.A. Ball and J.W. Helton, Lie group over the field of rational functions, signed spectral factorization, signed interpolation, and amplifier design, J. Operatory Theory 8 (1982) 19-64. [2] J.A. Ball and A.C.M. Ran, Optimal Hankel norm model reductions and Wiener-Hopf factorizations I: The canonical case, SlAM J. Control Optim. 25 (1967) 362-382. [31 B.C. Chang and J.B. Pearson, Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control 29 (1984) 880-887. [41 P. Dorato and Y. Li, A modification of the classical Nevanlinna-Pick interpolation algorithm with applications to robust stabilization, IEEE Trans. Automat. Control 31 (1986) 645-648. [51 I.P. Fedcina, A criterion for the stability of the Nevanlinna-Pick tangent problem, Mat. Issledoaniya 7 (1972) 213-227 (Russian). [61 B.A. Francis, J.W. Helton and G. Zarnes, H=-optimal feedback controllers for linear multivariable systems, IEEE Trans. Automat. Control 29 (1984) 888-900. [71 B.A. Francis, A course in H= control theory (Springer, Berlin-New York, 1987). [8] K. Glover, All optimal Hankel-norm approximations of linear multivanable systems and their L=-error bounds, Internat. d. Control 39 (1984) 1115-1193. [91 V.E. Katsnerson, Methods old-Theory in Continuous Imerpolation Problems of Analysis, Pt. I, Monograph (1982); translated by T. Ando. [IO1 H. Kimura, On interpolation-minimization problem in H ~, Control: Theory Advat~,ced Technology 2 (1986) 1-25. [111 H. Kimura, Directional interpolation approach to H~-optimization and robust stabilization, IEEE Trans. Automat. Control 32 (1987) 1085-1093. [121 H. Kimura, Conjugation and model-matching in H ®, under preparation. [131 D.J.N. Limebeer and B.D.O. Anderson, An interpolation theory approach to H ~ controller degree bou,'ds, Linear Algebra Appl., to appear. [14] M.O, Safonov, E.A. Jonckheere, M. Verma and D.J.N. Limebeer, Synthesis of positive real multivariable feedback systems, lnternat. J. Control. 45 (1987) 817-842. [151 M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 1985).