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Low-Order H∞ Sub-Optimal Controllers
Copyright © IFAC 12th Triennial World Congress, S ydne y, A ustrali a, 1993
LOW-ORDER I r SUB-OPTIMAL CONTROLLERS D.-W. Gu, B.W. Choi and I. PostJethwaite Control Systems Research, Department 0/ Engineering, University 0/ Leicester, Leicester LEl 7RH. UK
Abstract. This paper presents a methodology for the problem of reducing the order of H oo sub-optimal controllers. It is shown that the order of HOO SUb-optimal controllers may be reduced to n - Pl and that for some plants the order may be less than n - Pl, where n is the size of the system matrix of the generalized plant and Pl is the number of process outputs. A characterization of a set of (n-Pl)th-order Hoo sub-optimal controllers is given. A numerical example is given to illustrate the results. Keywords_ H OO sub-optimal controllers, Low-order stabilizing controllers.
1. INTRODUCTION
D II is partitioned as DII = [Dllll Dll12] where Dll21 D lln ' D lln has m2 rows and P2 columns.
A methodology for deriving low-order stabilizing controllers was discussed (Choi et al. , 1992; Gu et al., 1993). In this paper, we extend the results to characterize loworder HOO sub-optimal controllers.
4. The physical plant is strictly proper (D22 = 0). 5. Rank[ A cjwI g122] = n
Consider a generalized plant, pes) described by
x(t) z(t) = yet) =
Ax(t) + Blw(t) + B2U(t) Clx(t) + DllW(t) + D I2 U(t) C 2x(t) + D2IW(t) + Dnu(t)
6. Rank[ A
where x(t) E Rn is the state, wet) E Rm, is the exogenous inputs, u(t) E Rm, is the controlled inputs, z(t) E RP' is the controlled outputs, and yet) E RP> is the measured outputs. The HOO sub-optimal problem of finding a stabilizing controller Koo( s) such that
Define R :=
D' R-'-D ..1.1 where Dlo
(1)
• [ RIC
_[,2
= [Dll
6 m
,
~]
and
I p 0] 0 , 0 ' DI2 ] and D.I
= [ g~:
].
A - BR-I D' .CI
-BR-I B'
-C~(I - DloR-I D~.)CI -(A - BR-I D~.CI)'
]
. [ A - BID:i'l-lc -CR-IC] Rlc -BI(I - D:IR-ID.I)B~ -(A - BID:IR-IC)' . F and H are defined, from Xoo and Yoo as above, by F := -WI(D;.CI + B'Xoo ) = [F{I F{2 F;]' H := -(BID:I + YooC')R- I = [Hll HI2 H2 ]
The stabilizing solution to algebraic lliccati equation (ARE)
E'X+XE-XWX+Q=O
where F ll , FI2 and F2 have m 1 - Pl, Pl and m2 rows, respectively, and H ll , H12 and H2 have PI - 1n2, m2 and P2 columns, respectively.
::::~].
The followin~ Theorem 1 parametrizes all solution controllers Koo( s) via a free stable 4>( s).
2. Hoo SUB-OPTIMAL CONTROLLERS
Theorem 1 (Glover and Doyle, 1988): (1) A 3tabilizing controller exi3u, 3uch that IIFI(P, I<)1100 < " if and only if (i) , > max(u[D llll , D llI2 J,u[Dilll! D~l21]) and (ii) there exi3t Jolution3 Xoo 2: 0 and Yoo 2: 0, reJpectively, 3uch that p(XooYoo) < ,2, where p(.) denoteJ the Jpectral
In this section, a brief summary of all solutions to the general Hoo sub-optimal control problem (1) is given, as stated in Glover and Doyle (1988). The following assumptions are made:
radiUJ .
1. (A,B 2 ,C2 ) is stabilizable and detectable.
2. Rank(D I2 )
D~.Dlo - [ ,2
Vw E R.
and Yoo 2: 0 to be a 3tabilizing solution to ARE
It is shown in Section 3 that the order of HOO sub-optimal controllers may be reduced to n - P2 and that for some plants the order may be less than n - P2 . A characterization of a set of (n - Pl)th-order Hoo sub-optimal controllers is also given. A numerical example is given in Section 4 and conclusions in Section 5.
[!Q
f];I] = n + Pl
Vw E R.
Also define Xoo 2: 0 to be a 3tabilizing solution to ARE
for some prespecified value of, E R has been well solved in Glover and Doyle (1988), where Fi(P, Koo) is a lower linear fractional transformation (LFT) of pes) and Koo(s), and is the closed-loop transfer function from w to z. All solutions to the Hoo sub-optimal problem are briefly summarized in Section 2.
is denoted as X := Ric
~wI
+ m2
(£) If (i) and (ii) above are JatiJfied, then all (rational) Jtabilizing controllerJ K oo , for which IIFi(P, l\)1I00 < " art given by
= m2 and rank(D 2d = Pl.
3. DI2 and D21 are transformed into D12 = [0 I]' and D21 = [0 I ] by a scaling of u and lI, together with a unitary transformation of w and z .
Koo := Fi(K., 4»
813
(2)
for any rational ~(s) E Hoo $uch that K. ha$ the realization
11~(s)"oo
< 1, where
From the realization of Koo(8) in (6), a reduced-order realization K:;" (s) can be obtained as
(3)
(7) if there exists a matrix X E Rn.xn satisfying [(.21 = 0 and Kcl = 0, i.e. , satisfying the following two matrix equations:
and
Du = D12 E
-D1l21D~1\1(-y2I - DllUD~1\1r1D1112 - Dun.
n.m,xm,
and D21 E
n."'x",
A.X -XA CqX - DqC 2
are any matrice$ $ati$fy-
mg
where
1- D1l21(-y2I - D~1\1Dullr1D~121 I - D~112(-y2I - D ll11 D;1\1r 1D1\12
It is also interesting to see that (8)-(9) are similar to those required for finding low-order stabilizmg controllers as described in Choi et al. (1992) and Gu et al. (1993). Thus, the methodology proposed in Gu et al. (1993) for solving those equations might be applicable here. However, in the present work, we have a new constraint, i.e.,
+ DnD211C2
+ HC + B2D121C1'
3. LOW-ORDER SUB-OPTIMAL Hoo CONTROLLERS
which is not required in the case of low-order stabilizing controllers, and, in addition, freedom in the choice of F is considerably limited. These two constraints indicate that the problem of finding low-order Hoo suboptimal controllers is more difficult than that of finding low-order stabilizing controllers.
The formulae cited in the previous section represent an important result in optimal control theory. It is clear that we are always able to obtain sub-optimal controllers of size equal to, or less than, n by simply choosing ~(s) as a constant matrix with largest singular value less than 1, provided the feasibility conditions (i)-(ii) in Theorem 1 are met. In this section, we use this parameterization of Koo(s) to reduce the order of HOO sub-optimal controllers.
In summary, the existence of the low-order H"" suboptimal controllers depends on the solution of (8)-(9), and an HOO_ norm constraint (10). In what follows we show how to find a solution matrix X and a suitable free parameter ~(s) to satisfy the two matrix equations, and then propose a way to tackle the H""-norm constraint.
Let ~(s) in (2) have a state-space realization
(4)
3.2 HOC Sub-Optimal Controllers of Order n - P2
where Aq E n.n,xn•. Then, using (3) and (4), [(oo(s) of (2) can be expressed as
Equation (8) can be rather easily solved for the solution matrix X having the smaller dimension of nq, whereas the solvability of (9) may be limited due to the lack of freedom on F. In this subsection, we consider the special case of nq = n-P2 where Cq and Dq can always be found regardless of the structure of F, and characterize all matrices X, E q , C q and Dq to be found in (8)-(9) in terms of an arbitrary matrix A q •
(5) making use of a state-space realization of an LFT given in Postlethwaite et al. (1988). It is observed from (5) that all controllers [(oo(s) have a state dimension of n + nq if there does not occur any cancellations between K.(s) and ~(s).
Following Konstantinov et al. (1979), the pair (A, ( 2 ) can be reduced, provided the pair (A, ( 2 ) is completely observable, via a matrix M which is a product of an orthogonal matrix and a diagonal matrix and a nonsingular matrix N, into the orthogonal canonical form (Ao, Co) :=
3.1 Derivation of Reduced-Order Controllers In order to derive reduced-order controllers, we begin b~ applying a state similarity transformation to [(oo(s) of (5) using a nonsingular matrix T1 := realization given by
[~ I~.]
(MAM- I ,NC2 M- I )
and find a new An A21
Kb1 Kb2
Allo,!
A + B2DqC 2 - B 2CqX B 2Cq XA + XB 2DqC 2 + B qC 2 - XB 2Cq X - AqX X B 2 Cq + Aq B1 + B 2DqD 21 X B1 + X B 2 D q D 21
Co =
0
[ 16
3
0
]
[ III
0 0
A Yo - I .2 A Yo A Yo ,2
I ,3
A Yo •3
[
I~~
]
Allo , lIo
... o 1
where Vo is the oburvability index of (A, ( 2), and Ai,i(i = 1,' .. ,vo ) are I. x I. matrices, and the numbers
are the conjugate Kronecker indices of the pair
+ E qD 21
KcI = Cl + D 12 DqC 2 - D I2 CqX Kc2 D12 Cq Kd
A22
0
0 A Yo -1.1
where
K.n
[ ~, 1
Ao
(6)
K.12 K.21 K.22
F := DI2I CI .
of [(00(8) of (5). By formal order we mean the order which occurs when no pole-zero cancellations are made. The realization (7) is in a convenient form for computing a set of reduced-order HOO sub-optimal controllers.
(B2 + H 12 )D12 -D 21 (C2 + F12 )(I - 1- 2Yoo X oo)-1
A
and
(9)
It is emphasized that [(:;.,( s) in (7) has an order of at most nq, which is obviously less than n + nq, the 'formal' order
and
-H2 + B2D121 Dll F 2(I - 1-2YooXoo)-1
A := A - B2 D';}Cl
(8)
(A , ( 2 ),
Using the form (A o, Co), (8)-(D) can therefore be transformed into: AqX - XAo
Du + D12DqD21'
CqX - jjqCO
814
(11) (12)
where
(25)
XM- I BqN- 1 DqN- 1 FM-I.
X Bq Dq
I'
(13) (14) (15)
with Ao22 and A o23 fixed. Consequently, the matrices X, B q , C q and Dq are computed by
(16)
To solve the problem for the case of nq = n-P2, we partition the form (Ao, Co) as shown below:
(17)
Ao
The pair ({ Ao22 A o23 ], [ [
pletely ob3ervable if the pair able.
(A, C2)
~,] 0])
[XI
Bq Cq Dq
(Aq_tl - XIA olI - Ao2dN F2 (F2XI - FI)N.
I]M
In the following, we assume for the sake of simplicity that 12 = P2, and suppose that X is partitioned by
3.3 HOO-Norm Constraint on with XI and X 2 having P2 and n - P2 columns, respectively. Note that we set nq as n - P2, which will be the order of the reduced-order Hoo sub-optimal controller. Then, from (11), we have the following two equations:
J
+ X;I XI
Bq (19) O. (20)
Lemma 4
[I 0 ])X;I = Aq
~(s)
Having found all the element matrices required for K:'( s) in (7), the Hoo-norm constraint on ~(s) as in (10) is to be met. Lemma 4, of which proof can be found ill Gu et al. (1992), shows a connection between the H oo-norm bound of a transfer function matrix and the existence of a positive definite solution to a certain ARE.
Equation (20) is equivalent to
X 2([ Ao22 A o23
(27) (28) (29)
The other consideration to solve (21) is to choose Aq as an arbitrary, but st~ble matrix with linear elementary divisors. Then, XI and X 2 can be calculated from (21) and consequently B q, Cq and Dq all depend on Aq only. Therefore, Aq would be an adequate candidate if such formed ~(s) satisfies the H OO-norm constraint of (10) . In this paper, we adopt the first parametrization approach .
i3 com-
i3 completely obuMJ-
AqXI - XIA olI - X 2A o21 AqX2 - "tl [I 0 J - X 2 [A022 A o23 J
(26)
Hence, a solution matrix X and a suitable free p-arameter ~(s) to the two simultaneous matrix equations (8)-(9) are all characterized in terms of XI only, which can be chosen arbitrarily subject to Aq in (25) being stable. In addition , such a characterization of the parameters A q , B q , Cq and Dq in terms of XI may simplify the solvability of the Hoo_ norm constraint on ~(s) which we discuss in the following subsection.
where AolI := All : P2 x P2 , Ao21 : (n - P2) x P2, Ao22 : (n - P2) X 12, and Ao23 : (n - P2) x (n - P2 - 12)' The following Fact 2 is useful in solving (11 )-(12). Fact 2
X
If the following ARE
(Aq - BqRq -I D~Cq)'Xq + Xq(Aq - BqRq -I D~Cq) -,XqBqRq- I B;Xq -,C;Sq -ICq = 0 (30)
(21)
from which we can find XI and an invertible X 2 for any 3table Aq with linear elementary divisors. Obviously, X obtained is of full rank . And, in turn, Bq can be computed from (19) .
ha3 a p03itive definite 30lution X q, then Aq i3 3table and 11~(s)ll oo 5" where Rq := D~Dq-,2IP'2 and Sq:= DqD~ I m, .
,2
Remark 3 pair (A, C2 )
The assumption of the observability of the - and hence the observability of the pair ([ Ao22 A o23 ]'1 I P'2 0]) can be relaxed in finding the solution matrix X, since X can also be found even when the pair (A, C2 ) is not completely observable. For details, refer to Gu et al. (1992).
Since A q, B q, Cq and Dq are characterized in terms of XI as in (25), (27)-(29), the existence of a positive definite solution matrix Xq to (30) depends on the choice of XI' By search over an arbitrarily stable Xl> the positive definiteness of Xq can be checked. From the above Lemma 4, the positive definiteness of Xq guarantees the stability of A q •
In order to solve (12), partition F as
To be more complete, an optimization technique may be adopted here. That is, to meet the requirements that ~(s) E nHCO and 11~(s)llco < " we define'l as := , - f with a small positive number f. We may then consider a set of constraints (31)-(34) as below. Using the fact that a symmetric matrix can be transformed to a diagonal one_ by an orthogonal transformation, we can try to find an Xq and an orthogonal matrix U such that Xq = U'XqU solves the following ARE
F = [FI 1'2
l:
,I
m2 x n
where 1'1 and 1'2 have P2 and n - P2 columns, respectively. Equation (12) can then be rewritten as
and, from (22), Cq and Dq are computed as follows:
U'(A q - BqR. -I D;Cq)'U Xq
+ _tqU'(Aq -
BqR. -I D~Cq)U
-,IXqU'BqIlq-1 B;UXq -,IU'C;Sq -ICqU =
Hence, we have Aq, X, B q, Cq and D q, and can therefore compute an Hoo sub-optimal controller of (n - P2)th-order by making use of the realization (7), provided Aq is chosen such that 11~(s)lIoo < , . Note that X, Bq and Dq can be computed from (13)-(15), respectiyely.
Xq Xq, i X q,nq
Without loss of generality, the identity matrix can be chosen as a candidate for X 2 in (21) . In this instance, (21) becomes to
3.4
Hco
~
diag(xq,I " ' " xq,n.) i = 1, · . . , nq - 1
Xq ,i+1 > O.
Sub-Optimal Controllers of Order
(32) (33) (34)
< n - P2
Recall that the order of [(:,( s) is n q , which is the number of rows of a solution matrix X . The following Theorem 5, of which proof can be found in Gu et al. (1992) , shows the possibility of lowering the order of ](:,(s) , since nq may be reduced down to 1_ (5 P2)'
(24) Thus, XI can be chosen arbitrarily subject to the stability of Aq, where Aq is computed from (24) by
0
815
Theorem 5
X
Equation (11) ha3 full row rank 30lution3
5. CONCLUSIONS
E R1voxn
This paper has considered the problem of reducing the order of Hoo sub-optimal controllers. Starting from a parametrization (5) of all solutions to the general HOC sub-optimal control problem, we first derived a reduced-order realization (7) on the assumption of the existence of a solution matrix X to (8)-(9). The aim was to eliminate any unobservable modes in the 'formal' order of controllers given by (5). We then showed how to solve the two matrix equations using an orthogonal canonical transformation. An interesting result is that the order of the HOO sub-optimal controllers may be less than or equal to n - P2.
While, for an arbitrarily given A q , Bq can be obtained to satisfy (11), the corresponding Cq and iJ q can be found from (12) . Usinl$ a similarity transformation given by a nonsingular matnx T2 such that
o 11 o 0],
[0 0 [I 0
we can rewrite equation (12) as
It should be noted that the Hoo-norm constraint (10) can be tackled by checking the positive definiteness of a solution matrix Xq to a certain ARE (30), and that the existence of a positive definite matrix Xq depends heavily on the choice of the matrix Aq which can, otherwise, be arbitrarily chosen as a stable matrix.
So, if there exist F's such that FT2 has the form (36)
It is obvious that the dual work (by eliminating the uncon-
then we can find C q and iJ q to satisfy (12), where * denotes a nonzero nonspecified block matrix. However, there is little freedom on F, since F = D'2ICIM-1 is almost completely decided by plant data. Although, for this reason, the particular F will exclude some choices of nq for which the corresponding C q and iJ q do not exist, we may still expect in some cases H OO sub-optimal controllers of order less than n - P2 .
trollable modes) can be carried out in a similar manner. 6. ACKNOWLEDGEMENT The authors are grateful to the U.K. Science and Engineering Research Council for financial support. 7. REFERENCES
4. A NUMERICAL EXAMPLE
Chiang, R.Y. and M.G . Safonov (1990) . Robust Control Toolbox: U!ler '!l Guide. The MathWorks, Natick, MA.
We consider the S1S0 digital hydraulic actuator design example demonstrated in Chiang and Safonov (1990). Our problem is to find a reduced-order stabilizing controller K~(o5) such that
Choi, B.W. , D.-W. Gu and 1. Postlethwaite (1992). Loworder stabilizing controllers and pole-assignment. Proc. Amer. Contr. Conf., Chicago, 11.
(37)
Glover, K. and J.C. Doyle (1988). State-space formulae for all stabilizing controllers that satisfy an Hoo-norm bound and relations to risk sensitivity. Sy3t. Contr. Lett. , 11:167-172.
The continuous time plant G(05) is given by Go5 _ () -
9000 05 3 + 3005 2 + 700s
Gu, D.-W. , B.W . Choi and 1. Postlethwaite (1993). Loworder stabilizing controllers . To appear in IEEE Tran3. Automat. Contr.
+ 1000
Gu, D .-W., B.W. Choi and 1. Postlethwaite (1992). Loworder H OC sub-optimal controllers. Tech . Report 92-19, Dept . of Engineering, Univ. of Leicester, U.K.
G(s) is preceded by a ZOH with a sampling period of 0.01 sec. From the corresponding discrete time (z-transfer function) model, a w-plane description of the plant can be obtained in the usual way. This model will now be used in problem (37) with the weights as in Chiang and Safonov (1990), i.e. ( .!..
+ 1)2 + 1)2
W (s) - -::-,:,,30,,-;----,::-,-;; I
-
O.Ol(s
WC) 2
s
Konstantinov, M .M., P.Hr. Petkov and N.D. Christov (1979) . Synthesis of linear multivariable systems with prescribed equivalent form. SY!liem!l Science (Poland), 5:381-394.
fo+ 1
= 3.16(3~ + 1)
Postlethwaite, 1., D.-W. Gu and S.D. O'Young (19881 Some computational results on size reduction III H design. IEEE Trans . Automat. Contr., 33(2):177-185.
and the same -y given as 1.5. For this problem, we have n = G and P2 = 1. Although a 6th-order controller is expected as a 'central' solution, we
,.
~[~ ~-·'r1~~~
derive a 5th-order controller. After finding the two equations corresponding to (11)-(12), we set nq = n -]J"J = 5, X2 = Is and choose XI . Then Aq, B q, Cq and Dq can be found as per (25), (27)-(29), for which we have 11~(s)"oo = 1.0565 « -y) . Hence, using (7) we can compute a 5thorder H OO sub-optimal controller K~( s) which is found to be stable. The resulting poles of the closed-loop are all in the 1.H.P. and the HOO-norm of the cost function is satisfied since 11F1(P,I{~)ll oo = 0.9546 « -y).
10.2
10-1
lOO
10 1
1()1
10'
1()4
Fn;queocy (~I
Using the same P( s) and ~(s), we compute the (n + nq )thorder controller Koo(s) as per (5) and then examine the normalized Hankel singular value3 of Koo(s) . The result exhibits clearly that the 5th-order controller obtained is a minimal realization of the 'formal' order controller, in the sense of Balanced Truncated model reduction.
0.81---___-:.. 0.6 0.4 0.2
With the same pes) but with ~(s) = 0 we have computed 'central' controllers using Matlab files hinf.m and hinf3yn.m. The frequency response of these controllers and the controllers given by formulae (5) and (7) are shown in Figure 1. As seen in Figure 1 and as expected theoretically, both controllers generated by formulae (5) and (7), although having different sizes, have exactly the same lrequency responses.
Figure 1: (a) Singular Values of Controllers and (b) Singular Values of Cost Functions for the Example.
816
LOW-ORDER I r SUB-OPTIMAL CONTROLLERS D.-W. Gu, B.W. Choi and I. PostJethwaite Control Systems Research, Department 0/ Engineering, University 0/ Leicester, Leicester LEl 7RH. UK
Abstract. This paper presents a methodology for the problem of reducing the order of H oo sub-optimal controllers. It is shown that the order of HOO SUb-optimal controllers may be reduced to n - Pl and that for some plants the order may be less than n - Pl, where n is the size of the system matrix of the generalized plant and Pl is the number of process outputs. A characterization of a set of (n-Pl)th-order Hoo sub-optimal controllers is given. A numerical example is given to illustrate the results. Keywords_ H OO sub-optimal controllers, Low-order stabilizing controllers.
1. INTRODUCTION
D II is partitioned as DII = [Dllll Dll12] where Dll21 D lln ' D lln has m2 rows and P2 columns.
A methodology for deriving low-order stabilizing controllers was discussed (Choi et al. , 1992; Gu et al., 1993). In this paper, we extend the results to characterize loworder HOO sub-optimal controllers.
4. The physical plant is strictly proper (D22 = 0). 5. Rank[ A cjwI g122] = n
Consider a generalized plant, pes) described by
x(t) z(t) = yet) =
Ax(t) + Blw(t) + B2U(t) Clx(t) + DllW(t) + D I2 U(t) C 2x(t) + D2IW(t) + Dnu(t)
6. Rank[ A
where x(t) E Rn is the state, wet) E Rm, is the exogenous inputs, u(t) E Rm, is the controlled inputs, z(t) E RP' is the controlled outputs, and yet) E RP> is the measured outputs. The HOO sub-optimal problem of finding a stabilizing controller Koo( s) such that
Define R :=
D' R-'-D ..1.1 where Dlo
(1)
• [ RIC
_[,2
= [Dll
6 m
,
~]
and
I p 0] 0 , 0 ' DI2 ] and D.I
= [ g~:
].
A - BR-I D' .CI
-BR-I B'
-C~(I - DloR-I D~.)CI -(A - BR-I D~.CI)'
]
. [ A - BID:i'l-lc -CR-IC] Rlc -BI(I - D:IR-ID.I)B~ -(A - BID:IR-IC)' . F and H are defined, from Xoo and Yoo as above, by F := -WI(D;.CI + B'Xoo ) = [F{I F{2 F;]' H := -(BID:I + YooC')R- I = [Hll HI2 H2 ]
The stabilizing solution to algebraic lliccati equation (ARE)
E'X+XE-XWX+Q=O
where F ll , FI2 and F2 have m 1 - Pl, Pl and m2 rows, respectively, and H ll , H12 and H2 have PI - 1n2, m2 and P2 columns, respectively.
::::~].
The followin~ Theorem 1 parametrizes all solution controllers Koo( s) via a free stable 4>( s).
2. Hoo SUB-OPTIMAL CONTROLLERS
Theorem 1 (Glover and Doyle, 1988): (1) A 3tabilizing controller exi3u, 3uch that IIFI(P, I<)1100 < " if and only if (i) , > max(u[D llll , D llI2 J,u[Dilll! D~l21]) and (ii) there exi3t Jolution3 Xoo 2: 0 and Yoo 2: 0, reJpectively, 3uch that p(XooYoo) < ,2, where p(.) denoteJ the Jpectral
In this section, a brief summary of all solutions to the general Hoo sub-optimal control problem (1) is given, as stated in Glover and Doyle (1988). The following assumptions are made:
radiUJ .
1. (A,B 2 ,C2 ) is stabilizable and detectable.
2. Rank(D I2 )
D~.Dlo - [ ,2
Vw E R.
and Yoo 2: 0 to be a 3tabilizing solution to ARE
It is shown in Section 3 that the order of HOO sub-optimal controllers may be reduced to n - P2 and that for some plants the order may be less than n - P2 . A characterization of a set of (n - Pl)th-order Hoo sub-optimal controllers is also given. A numerical example is given in Section 4 and conclusions in Section 5.
[!Q
f];I] = n + Pl
Vw E R.
Also define Xoo 2: 0 to be a 3tabilizing solution to ARE
for some prespecified value of, E R has been well solved in Glover and Doyle (1988), where Fi(P, Koo) is a lower linear fractional transformation (LFT) of pes) and Koo(s), and is the closed-loop transfer function from w to z. All solutions to the Hoo sub-optimal problem are briefly summarized in Section 2.
is denoted as X := Ric
~wI
+ m2
(£) If (i) and (ii) above are JatiJfied, then all (rational) Jtabilizing controllerJ K oo , for which IIFi(P, l\)1I00 < " art given by
= m2 and rank(D 2d = Pl.
3. DI2 and D21 are transformed into D12 = [0 I]' and D21 = [0 I ] by a scaling of u and lI, together with a unitary transformation of w and z .
Koo := Fi(K., 4»
813
(2)
for any rational ~(s) E Hoo $uch that K. ha$ the realization
11~(s)"oo
< 1, where
From the realization of Koo(8) in (6), a reduced-order realization K:;" (s) can be obtained as
(3)
(7) if there exists a matrix X E Rn.xn satisfying [(.21 = 0 and Kcl = 0, i.e. , satisfying the following two matrix equations:
and
Du = D12 E
-D1l21D~1\1(-y2I - DllUD~1\1r1D1112 - Dun.
n.m,xm,
and D21 E
n."'x",
A.X -XA CqX - DqC 2
are any matrice$ $ati$fy-
mg
where
1- D1l21(-y2I - D~1\1Dullr1D~121 I - D~112(-y2I - D ll11 D;1\1r 1D1\12
It is also interesting to see that (8)-(9) are similar to those required for finding low-order stabilizmg controllers as described in Choi et al. (1992) and Gu et al. (1993). Thus, the methodology proposed in Gu et al. (1993) for solving those equations might be applicable here. However, in the present work, we have a new constraint, i.e.,
+ DnD211C2
+ HC + B2D121C1'
3. LOW-ORDER SUB-OPTIMAL Hoo CONTROLLERS
which is not required in the case of low-order stabilizing controllers, and, in addition, freedom in the choice of F is considerably limited. These two constraints indicate that the problem of finding low-order Hoo suboptimal controllers is more difficult than that of finding low-order stabilizing controllers.
The formulae cited in the previous section represent an important result in optimal control theory. It is clear that we are always able to obtain sub-optimal controllers of size equal to, or less than, n by simply choosing ~(s) as a constant matrix with largest singular value less than 1, provided the feasibility conditions (i)-(ii) in Theorem 1 are met. In this section, we use this parameterization of Koo(s) to reduce the order of HOO sub-optimal controllers.
In summary, the existence of the low-order H"" suboptimal controllers depends on the solution of (8)-(9), and an HOO_ norm constraint (10). In what follows we show how to find a solution matrix X and a suitable free parameter ~(s) to satisfy the two matrix equations, and then propose a way to tackle the H""-norm constraint.
Let ~(s) in (2) have a state-space realization
(4)
3.2 HOC Sub-Optimal Controllers of Order n - P2
where Aq E n.n,xn•. Then, using (3) and (4), [(oo(s) of (2) can be expressed as
Equation (8) can be rather easily solved for the solution matrix X having the smaller dimension of nq, whereas the solvability of (9) may be limited due to the lack of freedom on F. In this subsection, we consider the special case of nq = n-P2 where Cq and Dq can always be found regardless of the structure of F, and characterize all matrices X, E q , C q and Dq to be found in (8)-(9) in terms of an arbitrary matrix A q •
(5) making use of a state-space realization of an LFT given in Postlethwaite et al. (1988). It is observed from (5) that all controllers [(oo(s) have a state dimension of n + nq if there does not occur any cancellations between K.(s) and ~(s).
Following Konstantinov et al. (1979), the pair (A, ( 2 ) can be reduced, provided the pair (A, ( 2 ) is completely observable, via a matrix M which is a product of an orthogonal matrix and a diagonal matrix and a nonsingular matrix N, into the orthogonal canonical form (Ao, Co) :=
3.1 Derivation of Reduced-Order Controllers In order to derive reduced-order controllers, we begin b~ applying a state similarity transformation to [(oo(s) of (5) using a nonsingular matrix T1 := realization given by
[~ I~.]
(MAM- I ,NC2 M- I )
and find a new An A21
Kb1 Kb2
Allo,!
A + B2DqC 2 - B 2CqX B 2Cq XA + XB 2DqC 2 + B qC 2 - XB 2Cq X - AqX X B 2 Cq + Aq B1 + B 2DqD 21 X B1 + X B 2 D q D 21
Co =
0
[ 16
3
0
]
[ III
0 0
A Yo - I .2 A Yo A Yo ,2
I ,3
A Yo •3
[
I~~
]
Allo , lIo
... o 1
where Vo is the oburvability index of (A, ( 2), and Ai,i(i = 1,' .. ,vo ) are I. x I. matrices, and the numbers
are the conjugate Kronecker indices of the pair
+ E qD 21
KcI = Cl + D 12 DqC 2 - D I2 CqX Kc2 D12 Cq Kd
A22
0
0 A Yo -1.1
where
K.n
[ ~, 1
Ao
(6)
K.12 K.21 K.22
F := DI2I CI .
of [(00(8) of (5). By formal order we mean the order which occurs when no pole-zero cancellations are made. The realization (7) is in a convenient form for computing a set of reduced-order HOO sub-optimal controllers.
(B2 + H 12 )D12 -D 21 (C2 + F12 )(I - 1- 2Yoo X oo)-1
A
and
(9)
It is emphasized that [(:;.,( s) in (7) has an order of at most nq, which is obviously less than n + nq, the 'formal' order
and
-H2 + B2D121 Dll F 2(I - 1-2YooXoo)-1
A := A - B2 D';}Cl
(8)
(A , ( 2 ),
Using the form (A o, Co), (8)-(D) can therefore be transformed into: AqX - XAo
Du + D12DqD21'
CqX - jjqCO
814
(11) (12)
where
(25)
XM- I BqN- 1 DqN- 1 FM-I.
X Bq Dq
I'
(13) (14) (15)
with Ao22 and A o23 fixed. Consequently, the matrices X, B q , C q and Dq are computed by
(16)
To solve the problem for the case of nq = n-P2, we partition the form (Ao, Co) as shown below:
(17)
Ao
The pair ({ Ao22 A o23 ], [ [
pletely ob3ervable if the pair able.
(A, C2)
~,] 0])
[XI
Bq Cq Dq
(Aq_tl - XIA olI - Ao2dN F2 (F2XI - FI)N.
I]M
In the following, we assume for the sake of simplicity that 12 = P2, and suppose that X is partitioned by
3.3 HOO-Norm Constraint on with XI and X 2 having P2 and n - P2 columns, respectively. Note that we set nq as n - P2, which will be the order of the reduced-order Hoo sub-optimal controller. Then, from (11), we have the following two equations:
J
+ X;I XI
Bq (19) O. (20)
Lemma 4
[I 0 ])X;I = Aq
~(s)
Having found all the element matrices required for K:'( s) in (7), the Hoo-norm constraint on ~(s) as in (10) is to be met. Lemma 4, of which proof can be found ill Gu et al. (1992), shows a connection between the H oo-norm bound of a transfer function matrix and the existence of a positive definite solution to a certain ARE.
Equation (20) is equivalent to
X 2([ Ao22 A o23
(27) (28) (29)
The other consideration to solve (21) is to choose Aq as an arbitrary, but st~ble matrix with linear elementary divisors. Then, XI and X 2 can be calculated from (21) and consequently B q, Cq and Dq all depend on Aq only. Therefore, Aq would be an adequate candidate if such formed ~(s) satisfies the H OO-norm constraint of (10) . In this paper, we adopt the first parametrization approach .
i3 com-
i3 completely obuMJ-
AqXI - XIA olI - X 2A o21 AqX2 - "tl [I 0 J - X 2 [A022 A o23 J
(26)
Hence, a solution matrix X and a suitable free p-arameter ~(s) to the two simultaneous matrix equations (8)-(9) are all characterized in terms of XI only, which can be chosen arbitrarily subject to Aq in (25) being stable. In addition , such a characterization of the parameters A q , B q , Cq and Dq in terms of XI may simplify the solvability of the Hoo_ norm constraint on ~(s) which we discuss in the following subsection.
where AolI := All : P2 x P2 , Ao21 : (n - P2) x P2, Ao22 : (n - P2) X 12, and Ao23 : (n - P2) x (n - P2 - 12)' The following Fact 2 is useful in solving (11 )-(12). Fact 2
X
If the following ARE
(Aq - BqRq -I D~Cq)'Xq + Xq(Aq - BqRq -I D~Cq) -,XqBqRq- I B;Xq -,C;Sq -ICq = 0 (30)
(21)
from which we can find XI and an invertible X 2 for any 3table Aq with linear elementary divisors. Obviously, X obtained is of full rank . And, in turn, Bq can be computed from (19) .
ha3 a p03itive definite 30lution X q, then Aq i3 3table and 11~(s)ll oo 5" where Rq := D~Dq-,2IP'2 and Sq:= DqD~ I m, .
,2
Remark 3 pair (A, C2 )
The assumption of the observability of the - and hence the observability of the pair ([ Ao22 A o23 ]'1 I P'2 0]) can be relaxed in finding the solution matrix X, since X can also be found even when the pair (A, C2 ) is not completely observable. For details, refer to Gu et al. (1992).
Since A q, B q, Cq and Dq are characterized in terms of XI as in (25), (27)-(29), the existence of a positive definite solution matrix Xq to (30) depends on the choice of XI' By search over an arbitrarily stable Xl> the positive definiteness of Xq can be checked. From the above Lemma 4, the positive definiteness of Xq guarantees the stability of A q •
In order to solve (12), partition F as
To be more complete, an optimization technique may be adopted here. That is, to meet the requirements that ~(s) E nHCO and 11~(s)llco < " we define'l as := , - f with a small positive number f. We may then consider a set of constraints (31)-(34) as below. Using the fact that a symmetric matrix can be transformed to a diagonal one_ by an orthogonal transformation, we can try to find an Xq and an orthogonal matrix U such that Xq = U'XqU solves the following ARE
F = [FI 1'2
l:
,I
m2 x n
where 1'1 and 1'2 have P2 and n - P2 columns, respectively. Equation (12) can then be rewritten as
and, from (22), Cq and Dq are computed as follows:
U'(A q - BqR. -I D;Cq)'U Xq
+ _tqU'(Aq -
BqR. -I D~Cq)U
-,IXqU'BqIlq-1 B;UXq -,IU'C;Sq -ICqU =
Hence, we have Aq, X, B q, Cq and D q, and can therefore compute an Hoo sub-optimal controller of (n - P2)th-order by making use of the realization (7), provided Aq is chosen such that 11~(s)lIoo < , . Note that X, Bq and Dq can be computed from (13)-(15), respectiyely.
Xq Xq, i X q,nq
Without loss of generality, the identity matrix can be chosen as a candidate for X 2 in (21) . In this instance, (21) becomes to
3.4
Hco
~
diag(xq,I " ' " xq,n.) i = 1, · . . , nq - 1
Xq ,i+1 > O.
Sub-Optimal Controllers of Order
(32) (33) (34)
< n - P2
Recall that the order of [(:,( s) is n q , which is the number of rows of a solution matrix X . The following Theorem 5, of which proof can be found in Gu et al. (1992) , shows the possibility of lowering the order of ](:,(s) , since nq may be reduced down to 1_ (5 P2)'
(24) Thus, XI can be chosen arbitrarily subject to the stability of Aq, where Aq is computed from (24) by
0
815
Theorem 5
X
Equation (11) ha3 full row rank 30lution3
5. CONCLUSIONS
E R1voxn
This paper has considered the problem of reducing the order of Hoo sub-optimal controllers. Starting from a parametrization (5) of all solutions to the general HOC sub-optimal control problem, we first derived a reduced-order realization (7) on the assumption of the existence of a solution matrix X to (8)-(9). The aim was to eliminate any unobservable modes in the 'formal' order of controllers given by (5). We then showed how to solve the two matrix equations using an orthogonal canonical transformation. An interesting result is that the order of the HOO sub-optimal controllers may be less than or equal to n - P2.
While, for an arbitrarily given A q , Bq can be obtained to satisfy (11), the corresponding Cq and iJ q can be found from (12) . Usinl$ a similarity transformation given by a nonsingular matnx T2 such that
o 11 o 0],
[0 0 [I 0
we can rewrite equation (12) as
It should be noted that the Hoo-norm constraint (10) can be tackled by checking the positive definiteness of a solution matrix Xq to a certain ARE (30), and that the existence of a positive definite matrix Xq depends heavily on the choice of the matrix Aq which can, otherwise, be arbitrarily chosen as a stable matrix.
So, if there exist F's such that FT2 has the form (36)
It is obvious that the dual work (by eliminating the uncon-
then we can find C q and iJ q to satisfy (12), where * denotes a nonzero nonspecified block matrix. However, there is little freedom on F, since F = D'2ICIM-1 is almost completely decided by plant data. Although, for this reason, the particular F will exclude some choices of nq for which the corresponding C q and iJ q do not exist, we may still expect in some cases H OO sub-optimal controllers of order less than n - P2 .
trollable modes) can be carried out in a similar manner. 6. ACKNOWLEDGEMENT The authors are grateful to the U.K. Science and Engineering Research Council for financial support. 7. REFERENCES
4. A NUMERICAL EXAMPLE
Chiang, R.Y. and M.G . Safonov (1990) . Robust Control Toolbox: U!ler '!l Guide. The MathWorks, Natick, MA.
We consider the S1S0 digital hydraulic actuator design example demonstrated in Chiang and Safonov (1990). Our problem is to find a reduced-order stabilizing controller K~(o5) such that
Choi, B.W. , D.-W. Gu and 1. Postlethwaite (1992). Loworder stabilizing controllers and pole-assignment. Proc. Amer. Contr. Conf., Chicago, 11.
(37)
Glover, K. and J.C. Doyle (1988). State-space formulae for all stabilizing controllers that satisfy an Hoo-norm bound and relations to risk sensitivity. Sy3t. Contr. Lett. , 11:167-172.
The continuous time plant G(05) is given by Go5 _ () -
9000 05 3 + 3005 2 + 700s
Gu, D.-W. , B.W . Choi and 1. Postlethwaite (1993). Loworder stabilizing controllers . To appear in IEEE Tran3. Automat. Contr.
+ 1000
Gu, D .-W., B.W. Choi and 1. Postlethwaite (1992). Loworder H OC sub-optimal controllers. Tech . Report 92-19, Dept . of Engineering, Univ. of Leicester, U.K.
G(s) is preceded by a ZOH with a sampling period of 0.01 sec. From the corresponding discrete time (z-transfer function) model, a w-plane description of the plant can be obtained in the usual way. This model will now be used in problem (37) with the weights as in Chiang and Safonov (1990), i.e. ( .!..
+ 1)2 + 1)2
W (s) - -::-,:,,30,,-;----,::-,-;; I
-
O.Ol(s
WC) 2
s
Konstantinov, M .M., P.Hr. Petkov and N.D. Christov (1979) . Synthesis of linear multivariable systems with prescribed equivalent form. SY!liem!l Science (Poland), 5:381-394.
fo+ 1
= 3.16(3~ + 1)
Postlethwaite, 1., D.-W. Gu and S.D. O'Young (19881 Some computational results on size reduction III H design. IEEE Trans . Automat. Contr., 33(2):177-185.
and the same -y given as 1.5. For this problem, we have n = G and P2 = 1. Although a 6th-order controller is expected as a 'central' solution, we
,.
~[~ ~-·'r1~~~
derive a 5th-order controller. After finding the two equations corresponding to (11)-(12), we set nq = n -]J"J = 5, X2 = Is and choose XI . Then Aq, B q, Cq and Dq can be found as per (25), (27)-(29), for which we have 11~(s)"oo = 1.0565 « -y) . Hence, using (7) we can compute a 5thorder H OO sub-optimal controller K~( s) which is found to be stable. The resulting poles of the closed-loop are all in the 1.H.P. and the HOO-norm of the cost function is satisfied since 11F1(P,I{~)ll oo = 0.9546 « -y).
10.2
10-1
lOO
10 1
1()1
10'
1()4
Fn;queocy (~I
Using the same P( s) and ~(s), we compute the (n + nq )thorder controller Koo(s) as per (5) and then examine the normalized Hankel singular value3 of Koo(s) . The result exhibits clearly that the 5th-order controller obtained is a minimal realization of the 'formal' order controller, in the sense of Balanced Truncated model reduction.
0.81---___-:.. 0.6 0.4 0.2
With the same pes) but with ~(s) = 0 we have computed 'central' controllers using Matlab files hinf.m and hinf3yn.m. The frequency response of these controllers and the controllers given by formulae (5) and (7) are shown in Figure 1. As seen in Figure 1 and as expected theoretically, both controllers generated by formulae (5) and (7), although having different sizes, have exactly the same lrequency responses.
Figure 1: (a) Singular Values of Controllers and (b) Singular Values of Cost Functions for the Example.
816