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A Two-Riccati, Feasible Algorithm for Guaranteeing Output L∞ Constraints
A Two-Riccati, Feasible Algorithm for Guaranteeing Output L Constraints Guoming G. Zhu Cummins Engine Company, Inc. Columbus, IN 47202-3005
Robert E. Skelton Space Systems Control Lab., Purdue University West Lafayette, IN 47907
I. I N T R O D U C T I O N In the presence of all L2 disturbances whose outer products are bounded by a known matrix N, this chapter develops controllers to minimize the worst case control L norm bound subject to the worst case L constraints on each output group. The resulting controller is an LQG controller with a special choice of the output weighting matrix and input noise intensity matrix. Both weighting matrices may be determined by an iterative algorithm. The design algorithm turns out to be feasible if the algorithm converges. It has been shown in [ 1-3,10] the standard Output Covariance Constraint
(OCC) problem has two different interpretations; stochastic and deterministic. One deterministic interpretation of the OCC problem is the output L constrained control problem ( O L ) , described as follows. Consider the system CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
97
98
G U O M I N G G. Z H U A N D R O B E R T E. S K E L T O N
A Px P + B Pu + D
P
where
xp e 9~"~,
yp
=
Cpx p
Z
=
M xp
y p ~
9~"~,
z~
P
wP
(1.1) +v,
p
9~~ ,
ve
9~": ,
u~
9~"" , and
wp ei 9~"
are,
respectively, vectors of the plant state, plant output, measurement signal, measurement noise, control input and deterministic exogenous signal. We assume that the closed loop system with some static or dynamic controller u(s) =[C (sl - A)-' B c + D]z(s)
is strictly proper, then it can be expressed in
the following state space form .~
=
Ax+Dw
; w e g l "~"~
y
=
Cx ; ye91""',
(1.2)
w h e r e C = [ C r , C [ ] r , w = [ w p, r v r] r and y = [yp,u r r ] r , and matrices Cy a n d C . depend on the coordinate of the closed loop system state. Here we assume that the plant performance outputs are divided into m output groups y, = C x ~ 91" (i = 1,2, .. . , m ) , that is yp = [ y , ~ ,Y2~ ..... Y.~]~ . Let the square of the L 2 norm of w(.) be def'med by
IIw<
(1.3)
(t)W-'w(t)d,,
=
where W is a symmetric positive definite matrix, and define the L
norm of
each output group y, (-) to be
Ily,<>11
yr (t)y, (,), i : 1,2 ..... m,
(1.4)
and note that
II.,<.>ll = up.
, (t),
i=
1,2 .....
nu.
(1.5)
Define the L 2 disturbance set (1.6)
GUARANTEEING OUTPUT L CONSTRAINTS
99
Then the O L problem can be stated as follows. THE OL PROBLEM .o
For the given plant (1.1)find a (static or dynamic) controller to minimize
s ug.llu,r
JoL = E r
n:}
; r > 0 , i = 1,2,. .. ,nu
(1.7)
i=1
subject to
Ily, )ll <
i=1,2
,m, V w es
w
(1.8)
where r > 0 (i = 1,2 ..... nu) and e, > 0 (i = 1,2 ..... m) are given constants.
E! The significance of this problem is obvious: under the given L2 disturbance environment we seek a controller which satisfies the output L constraints with minimum control effort J o,.. This is a practical controller design problem. In physical hardware there are always hard limits due to sensor and actuator signal saturation, and such systems (e.g., the Hubble Space Telescope) must be evaluated based upon its ability to keep pointing errors less than values specified by optical resolutions. Such missions might not be feasible if such L inequality constraints cannot be guaranteed. The O L problem has been solved in [3,10]. Also, the O L design algorithm has been applied to the "NASA Minimast' flexible structure lab facility at Langley Research Center and the results have been reported in [ 1]. Often, in practical problems, much more information about the disturbance vector w(.) is known than simply the L: norm in (1.6). For example, we may know bounds for the L2 norm of each w, (-). To improve the O L method, we presume to have more information about the disturbance than the L2 norm bound in (1.6). Suppose we know that w(.) belongs to the following disturbance set s
= w(-): t ~ 9~''+'z ,
w~ (t)dt < Ix, 9
One can still use the OL results over the disturbance set .a
(1.9)
100
GUOMING G. ZHU AND ROBERT E. SKELTON
= {w(.):,
IIw()ll:
=
w" ( , ) W - ' w ( t ) d t
< ~},
(1.10)
nw+nz
by choosing 13= ~ Ix~ and W = I, that is, to enlarge the disturbance set K2 so i=l
that ff~ _c ~ . It can be easily shown that the set (1.10) is always equal to or large than the set (1.9). Therefore, the old O L approach may yield more conservative designs than the present paper. The outer product of all the input disturbances, which is a positive semidefinite matrix, will include this additional information (1.9). Hence, we presume to know a bound N>O on the outer product of w characterized by the following set f2N= {w(-): t --->9~--',
(
foW(t)w r (t)dt< N}.
(1.11)
The precise statement of the new problem, called the Extended Output L EOL ) constraint problem, will be given in the next section. The chapter is organized as follows. Section II provides the definition of and some preliminary knowledge which will be useful for
the EOI_, problem
solving the problem. Section III provides the continuous time solutions (necessary conditions and an algorithm) for the static controller (measurement feedback) and dynamic controller. Two special cases, state feedback and strictly proper full order dynamic solutions are discussed in Section III. Section IV is the discrete time version of Section III. Section V gives an example for a discrete system to demonstrate the algorithm and to compare with the O L designs. Section 6 adds some conclusions.
II.
STATEMENT OF THE
Before precisely stating the Lemmas. Lemma 2.1
EOL EOL
PROBLEM problem, we introduce two very useful
GUARANTEEING OUTPUT L, CONSTRAINTS
101
For the asymptotically stable and controllable system (1.2) define Y = CXC r, where X is the positive definite solution of the following Lyapunov
Equation 0 = XA r +AX +DWD r, W > 0 .
(2.1)
Let w(.) in (1.2) be any L 2 disturbance defined in (1.3) and let
Ily(.)ll = s u p y t~>O
T
(t)y(t)
(2.2)
be the square of the L norm of y(.). Then
Ily(-)ll ---
IIw(-)ll:,
(2.3)
and for all disturbances belonging to the set ~w in (1.6)
13. trj = suplly(-)ll . wr163w
(2.4)
Lemma 2.1 was proven in [4,9].
Remark 2.2 The bound given in (2.3) is for the asymptotically stable and strictly proper system (1.2). Note that for the proper (not strictly proper) system, i.e., there is a direct feed forward term in y(t), hence, there exists no L bound on the output for any disturbance which belongs to the set f2 w in (1.6) with ~ > O. Therefore, the closed loop system with finite L norm on the output for any w(.) ~f2 w should be in the form (1.2). Lemma 2.3 Let (1.3) and (2.2) respectively denote the square of the input L 2 norm and the square of output L norm of system (2.1). Under the same assumptions as in Lemma 2.1, consider the disturbance w(.) in the set (1.11), where N>O is ^ given. Suppose there exists a W such that
102
GUOMING G. ZHU AND ROBERT E. SKELTON
Y
YYo
=
L(L~L)-V2 L; N = LL;
=
CXC ~; 0 = X A r + A X + D W D r,.
=
Dr ~oea~,C r yoYoCe r ~,d t D > O ;
(2.5)
[Ylyo; Ilyoll=1.
:
Using W for W in (1.3), (2.3) leads to "~[Y]tracel~-' = sTIly(.)ll'.
(2.6)
A proof is provided in [6]. Remark 2.4 Equation (2.6) shows that the bound (2.3) will be tight over the disturbance set O N by the choice o f W as the L 2 norm weight in (1.3) to
replace W. In the E O L problem we consider disturbances in the set (1.11). It is easy to see that for any L 2 disturbance w(.) which belongs to ~N defined in (1.11), we have
=
~oo wr (t)W-~w(t)dt trace{W-"2!w(t)wr(t)dtW-"2} (2.7)
<
trace (W -''2NW-''2)
=
traceNW-t,
where W > 0 is a given weight matrix. Therefore,
Ilu,<)ll ]y,
<
[ C XC,r ],, traceNW-I
<
"o[C XC r ]. traceNW -~ , i = 1,2,...,m,
,
i = 1,2 ..... nu, (2.8)
where X satisfies (2.1) and C = [ C r , C r ..... c rT] . and C, are defined in (1.2). Then the E O L
EOL Problem:
problem can be stated as follows:
GUARANTEEING OUTPUT L CONSTRAINTS
103
Find a static or dynamic controller and the input L 2 norm weighting matrix W > 0 to minimize the EOL cost J eo~. = traceNW-' . { traceRC XC r }
(2.9)
subject to (2.1) and "o[C~XCr].traceNW -' < e 2 i = 1 , 2 , i
'
~
,m,
(2.10)
where e, (i = 1,2 ..... m) are given and R =diag[rz,r 2..... r ] > 0 .
(2.11) E!
The physical interpretation for the E O L problem is clear if one considers the inequalities in (2.8). Note that in this case
Er,
(2.12)
Ilu,
i=l
Hence, the cost function is an upper bound of the weighted summation of supremums of control signals over all L2 disturbances which belong to ~N" The EOL| problem is similar to the O L problem, but the input disturbances belong to different sets, f2 N and fir, respectively. The next section provides the solutions of the E O L problem in the continuous time case.
III. S O L U T I O N S
OF CONTINUOUS
TIME EOL
PROBLEM
In this section, we shall discuss the first order necessary conditions for static and dynamic controllers, and also present an iterative algorithm. A.
STATIC MEASUREMENT FEEDBACK CASE
For the static measurement feedback case, we seek a constant feedback gain matrix G which is the solution of the E O L problem such that
104
GUOMING G. ZHU AND ROBERTE. SKELTON (3.1)
u =Gz.
Therefore the EOL| problem becomes (3.2)
I min traceNW -~. trace R G M X M r G r P
G.W>O
P
subject to (2.1) and
[o[C,XC,r ].traceNW-I - -
<
2
9
_ e,,
t = 1,2 ..... m.
To keep the closed loop system strictly proper, we only consider the case where v = 0 for system (1.1). In this case, the closed loop system (1.2) matrices are
A = A +BpGMp, D = D Theorem
/.
, W =Wp, C=[Cp,Mr].p
3.1
Suppose that G is an optimal solution o f the EOL EOL
(3.3)
problem
is
regular.
Then
there
exists
problem, and that the a
matrix
Q = block
diag[ Q~ , Q2 ..... Q. ] > 0 such that i)
G = - ' f f - ' B r K X M r ( M , X M pr)-,.,
ii)
0 = KA + A r K + M r G r ' R - t G M p + CrQCp;
iii)
DrKD
iv)
Q{block diag[ C, x c r , C2 x c j ..... C x c r ]. t r a c e N W -' - F} = 0;
v)
A X + XA r + D pWD pr = 0 ,
P
P
= W - ' N W - ' ( t r a c e R G M X M rGr + traceQCpXC r)''2" P
P
(3.4a)
where "R = R. traceNW-' , Q = Q. traceNW-' , F = diag[ 2, I,,~.212 2 ..... 1~I 2 ], and l j is an identity matrix with dimension mj.
The proof of the above theorem is obvious. By using the following augmented cost function
H
trace NW -' { trace RGM pXM r G r } + +trace K[ X ( Ap
i=! + BpGMp ) r + ( Ap +
[ C x c ,rtrace NW -' - E I , (3.5)
BpGM, ) X
+ OpWD p r ],
GUARANTEEING OUTPUT L CONSTRAINTS
105
setting the partial derivatives with respect to X, G, K, W to zero, and adding one equation for inequality constraints in (3.2) (Kuhn-Tucker conditions), one can get the necessary conditions. By setting M = I , one obtains the results for state feedback case which is a special case of measurement feedback control.
Corollary 2.2 Suppose that G is an optimal solution of the EOL problem for the state feedback case, and that the EOL| problem is regular, that is, the EOL problem is not independent of the EOL cost function in (3.2). Then there exists a block diagonal positive semidefinite matrix Q such that i)
G = -'R-'BrKX;
ii)
0 = KA,, + A r K - KB,-R-'B rK + Cr'QCp ;
iii)
D ~, rKD
iv)
Q{block diag[ C, x c r , C~XC[ ..... C x c r ]. traceNW -' - F} = 0;
v)
(Ap + BpG) X + X (Ap + BpG) r + DpWDp r
P
m
p = W-I NW-I (traceRGXG r + traceQC X C ; ) |'~"
(3.6)
"-0,
g
where R , Q and F are the same as those in Theorem 3.1. The solution of the third equation in (3.3) and (3.6) is given in the following Lemma.
Lemma 3.3 Suppose that N and K are positive definite and D has full column rank. Then equation D r KD = W-INW-1 has one symmetric positive definite solution W~ = K~'[K~NK~] ~2K~-',
(3.7)
and one negative definite solution
w, where
r; ',
(3.8)
106
G U O M I N G G. Z H U A N D ROBERT E. S K E L T O N
(3.9)
K,~ = ( D r K D ) ''2 .
The proof was presented in [6]. An algorithm used to solve those necessary conditions is presented in Subsection C. B.
DYNAMIC FEEDBACK CASE Consider the following dynamic controller with order nc xc
=
AcXc + Bcz
u
=
C x +Hz.
(3.10)
It is well-known that by setting
=[APo ;x=
(3.1 la) =
;~=
;G= o
s
Ix:] [':] [5] ;y=
~
;D=
;D
=
x
"
;I)= B,,
; A
[01
(3.1 lb)
'
with compatible dimensions, the closed loop system can be written as (3.12) y
----
[d' , ,~'c" l" X-F H v
In order to obtain a bounded output L
9
norm for an L2 input, system
(3.12) should be strictly proper, i.e.,/4 = 0 or v - 0. Here we consider the case that v = 0. Then the closed loop system is in the form (1.2) with A=A+BGtr
C
[~r
21~/rGr]r
(3.13)
Let (3.14)
GUARANTEEING OUTPUT L CONSTRAINTS
107
where R = diag[r~,r2..... r ] > 0. By solving the following static measurement feedback EOL problem
trace NW -l . trace [CGl~I X~Ir G r
min
I
G.W>O
(3.15)
P
subject to (2.1) and
---
"
^T
[o[CXC ].traceNW
-1
~
2
_ e,, i = 1,2 ..... m.
one can obtain first order necessary conditions for the dynamic controller case. Hence, when v = 0 in (1.1), the EOL problem with a fixed order dynamic controller produces the static measurement feedback EOL controller as a special case. Special Case: Full Order Dynamic Controller For this special case, we consider the following strictly proper full order dynamic controller L
=
ax+
u
=
Cx.
z
(3.16)
Let
X =
IXPx-~Xc]; Y = [ ~ I ; w=IWvl'
(3.17)
then the closed loop system can be put into the form (1.2) with A -BM
I
BM c
D=
The EOL follows:
A +BC
A +BM
p
P 0
-BM
c
C;C= n
c
P 0
-A
(3.18a)
p
P.
(3.18b)
c
problem with full order dynamic controller can be stated as
108
GUOMING G. ZHU AND ROBERT E. SKELTON
min
traceNW-'.traceRC XC r, C = [ 0 , C ]
Ac ~c .Cr .W>O
(3.19)
subject to (2.1) and
"0[ C XC r ]. traceNW-1 <_ e 2,, i = 1,2,... ,m. where, in this case, C = [ C, C p ], C, = [C, r , C r ..... C r ]r.
Theorem 3.4 Define the following matrices
w
=
r
;K=
[ ll
;X=
[xl x2] I
(3.20)
,
where Wj, K o and X o (i, j=l.2) are nx by nx matrices. Suppose that the triple ( C , A c, B ) is the optimal solution to the full order dynamic EOL problem defined in (3.19), and that the EOL| problem is regular. Then there exists a matrix Q = block diag[Q~ , Q2 ..... Q. ] > 0 such that Q{block diag[ C, x c r , C2 XC r ..... C XC r ]. traceNW-' - F} = 0,
(3.21)
and A =A,+B
C-B
M,;
(3.22)
B = ( X t , M pr + D W~2)W2-2" "~-i
C=-R
T
B K22,
where X , , and K22 satisfy X,, A ,r + A , X ,, _ ( X I M,
,r +D Wn)W~-l (X,,
M ,r
+D W~2)r +D W~,D ,r =~,
(3.23)
K=Ap + ArK= - K=B "R-'Br K= + Cr'QC = 0 , and "R = R . traceNW-' and Q = Q . traceNW-l . In addition DrKD = W-'NW-' (traceRC XC r + t r a c e Q C XCry )~2, where K is defined in (3.20) with K~2 = K22 and K~2 satisfying
(3.24)
GUARANTEEING OUTPUT L CONSTRAINTS
109
(3.25)
r ( K , , - K2:) + C~rR'C~ =0,
(K,,-K22)(A,,-BM)+(A,-BM)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying
(3.26)
X22(A +B C ) r + ( A p + B p C ) X 2 2 +BW~2Br =0.
The proof is similar to the static measurement feedback case. Consider the augmented cost function m n
_~.
(3.27)
traceNW-' . traceRC XC r,, + E ' ~ [ C XC r 9traceNW-' - E,] I, : i=l
+traceK .[ AX + XA r + DWD r ].
Then (3.27) can be rewritten into H = traceXCrO.C + traceK .[ AX + XA rDWD r ] _ traceQ. F,
where 0 = block diag['Q,'ff ] and "R = R . traceNW-' , "Q = Q . traceNW-' . By setting the partial derivatives with respect to K, X, W to zero and adding the Kuhn-Tucker condition (3.21),one can complete the proof. _...
.__
Note that (3.22) and (3.23) describe an LQG controller if Q, R, and W are given. Hence, the full order E O L controller is an LQG controller with some special choice of the weighting matrices Q, R and input noise intensity matrix W. (The forthcoming algorithm determines Q, R and W). Remark 3.5 The fact that there exists a W > 0 satisfying the first order necessary conditions requires that D rKD > O. Remark 3.6 The first order necessary conditions for the OL problem only need to satisfy (3.22) and (3.23) for fixed W, which are included in the EOL| problem. Hence the OL| problem is a special case of the EOL problem.
C.
NUMERICAL ALC~RITHM
110
GUOMINGG. ZHU AND ROBERTE. SKELTON
Subsections A and B of this section provide all the necessary conditions for the design of static and dynamic controllers. The following algorithm provides a way to design the controller by iterating on these necessary conditions. This is an extension of the O L algorithm presented in [3].
THE EOL .o ALGORITHM Step I
Given system matrices Ap,Bp,Cp,Dp. Mp, initial weighting matrices Wo , Qo , R , outer product bound N, output L bounds e,, (i = 1,2 ..... m), error tolerance e , and free parameters 0 < [~< 1 and a >0. Define ~ = R. traceNW, -! ~ = Q. trace NW -~ Let i = 0
Step 2
Controller design step: Case 1- static state feedback Compute m-!
G, = - R
7"
(3.28a)
BpK,
by solving (3.28b)
K i A p + A ; K - K B p R---' i Bp"K -t- C;'QiC p = O, and then solve for X, by (Ap+BpG,)X, + X,(Ap+BpG,) r +DpWD r =0.
(3.28c)
Let D, = Op, q = Cp, and C~ = G,. Go to Step 3. Case 2: Static measurement feedback Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G i for given Qi and W
6,
- R -, ' B prK , Xi M pr ( M P X, M rP ) -''' O= K(Ap+BpG, Mp) r + K ( A p + B p G , Mp)
(3.29a)
=
+ M;<
C,M
(3.29b) ,
0 = (Ap + BpG,Mp) X, + X, (Ap + BpG,Mp )r +DW~D r
(3.29c)
GUARANTEEING OUTPUT L CONSTRAINTS
111
Let D i = Dp, Cy = Cp, and C~ = G i Mp. Go to Step 3. Case 3: full order dynamic controller
Solve for controller matrices A~, B~, and C~ by A= i Ap+
BpC~ - B M i p, " M ,r + DpWs
8:=(x:,
(3.30a) '
;
C' = - g -i' Bp"K~2, '
and X,; At+, ApX[, + DpW/~Drp _ _ ( X I li T
i
(3.30b)
M rp+ DpW/~)(W2~2)-'(X[,M rp+ D W~;)r = 0; i
i
---1
T
i
K=A + A K= - K~:B R, B K= + C ~ , C
= O.
Then solve for K:, - K~2 and X~: by (K:,-K~2)(A P - B ; M P ) + ( A P - B ~ M P ) r ( K : , - r ~ )
(3'30c)
+<~,c =o, i
x=(a p
+BC;/~+(A+Bp C'IX= +Ew~(8;/T =o, c i
i
i
where
Di=
[o P
;K,=
)r
;X=
ix: o] l
(3.30d)
i "
Let C = [Cp, C ], and C~ = [0, C'c ]- Go to Step 3. Step 3
Compute K; = (D, TK,D, )1/2 ;
I.2, = traceRC: X, (C: )r + traceQ~C X,C r ,
(3.31)
112
GUOMING G. ZHU AND ROBERTE. SKELTON
Then solve for W§
(3.32)
)-' (K~' NK~ )"' (K~)-' ~+ (1-~)W,
Q , ( i + l ) = e , -~y,,, (i)Q,(i) ya. ,,Y=C,
XC, r , j = 1 , 2 ..... m,
a,+, = block aiagIQ, (i), Q: ( i) ..... Q. (i)]. Step 4
/:
and
IIw,.,-w,ll< ,
,,op,
where
e:l~ F = block diag[e,l,, " ~ : ..... e : l ] . E l s e , i = i + l a n d g o t o S t e p 2 . D During each iteration of the special cases, state feedback or full order dynamic controller, the resulting controller is an LQ or LQG controller. Hence, in those two special cases, under the usual stabilizability, detectability assumptions the stability of the closed loop system is guaranteed during each iteration. Remark 3.7 The algorithm is written for 3 different cases. One makes a choice in Step 2 for different controller complexity. Lemma 3.8 Suppose that the EOL
algorithm converges to some weight matrix
W > O. Then the resulting controller is feasible, i.e., the closed loop system with the resulting controller will satisfy the inequality constraints (2.10) The proof of Lemma 3.8 is similar to that in [3].
IV. DISCRETE TIME VERSION Here, we consider the following discrete system
xp(k+l)
=
A x(k)+Bu(k)+Dpwp(k)
y,(k)
=
C x,(k)
z(k)
=
M x (k)+ v(k),
(4.1)
GUARANTEEING
OUTPUT L
CONSTRAINTS
113
where the dimensions of the matrices and vectors are analogous to the continuous system (1.1). Suppose that the closed loop system with a static or dynamic controller is strictly proper, then the closed loop system can be expressed as follows
x(k+l)
=
Ax(k)+Dw(k);we91
y(k)
=
Cx(k) ; y ~ R ~+~',
n"*n~
(4.2)
where C=[C~,Cr] r , w = [ w r , v r l r and y = [y~r,u~lr . Define the square of the input e: norm of w(.) as
IIw<-)ll: = ~ w, o
(4.3)
k=O
and let the square of the output e| norm for each individual channel be
Ily,<.)ll~.= s u p y r ( k ) y , ( k ) ; Ilu,<)11: = suo k - ~ u:t
i : 1,2 ..... m;
i = 1,2,
" ' "
(4.4)
,nu.
For the strictly proper closed loop system we have the following Lemma.
Lemma 4.1 [5] For the asymptotically stable system (1.2) with (A, D) controllable define Y = CXC r, where X is the positive definite solution of the following Lyapunov equation X = AXA r + DWD r, W > 0.
(4.5)
Let w(.) in (4.2) be any e: disturbance which belongs to
a~ -
w<.): k -~ ~t -+~,llw<.)ll: = E w~
and
Uy<-)II: - s~py~
(4.6)
114
GUOMING G. ZHU AND ROBERT E. SKELTON
be the square of the ~| normx~f y(.). Then
(4.7)
Ily(.)ll <_ trlllw(.)ll . and for all disturbances belonging to f~w in (4.6)
(4.8)
otY] = s~. ~y(.)~2| .~_,. Define the input disturbance set f2 N ={w(.): k-'~91""~'~w(k)wr,~o ( k ) < N, N > 0 } .
(4.9)
The discrete time EOL| problem can be stated as follows. DISCRETE TIME EOL PROBLEM .a
Find a static or dynamic controller and an input matrix W >0 to minimize the EOL cost J EoL. = trace NW -~ "{ trace R C, x c r }
2
norm weighting
(4.10)
subject to (4.5) and "o[QXCr].traceNW -~ <_e 2,, i = 1,2,...,m,
(4.11)
where e, (i = 1,2 ..... m) are given and R =diag[r~,r 2..... r ] > 0 .
(4.12) t3
The same physical interpretation for the E O L problem as that for continuous time case can be found by using Lemma 4.1, i.e.,
(
i=1
and
wlit~ N
u:t
(4.13)
GUARANTEEING OUTPUT L CONSTRAINTS
]ly,<>11<-6[CXC,r].traceNW -~, i = 1,2 ..... m.
115
(4.14)
Hence, J EOL., is an upper bound of the weighted summation of supremums of the control signals over all the f 2 disturbances which belong to f~,, in (4.9), and inequality (4.14) guarantees that the output g norm for each individual channel will be less than or equal to the given bound. A.
STATIC MEASUREMENT FEEDBACK CASE
We seek a constant feedback gain matrix G such that the solution of the E O L problem is (4.15)
u =Gz.
Here we only consider the case that v = 0 to have a strictly proper closed loop system. After combining equation (4.1) and (4.15), the closed loop system (4.2) matrices are A = Ap + BpGMp, D = Dp , 9
and the discrete time E O L
r J'~IT
W = Wp , C = .t ~_ p , M r
(4.16)
problem for measurement feedback controller
becomes (4.17)
f min traceNW -~ 9traceRGM P X M PTG r G.W>O subject to (2.1) and o [ C XC, ]. traceNW -~ < _ E2, . . i. = . . 1. 2,
--
r
m.
Theorem 4.2 Suppose that G is an optimal solution o f the EOL problem defined in (4.17), and that the EOL problem is regular. Then, there exists a matrix Q = block diag[Q~, Q~ ..... Q= ] > 0 such that
116
GUOMING G. ZHU AND ROBERT E. SKELTON
i)
G - - ( R-- + BrpKBp) -' BrpKAFXMFr ( M pX M r ,) -l",
ii)
K = ArKA + MrG r ('R+ B;KBp)GMp +CrQCp;
iii)
D ; K D p = W - I N W -I ( t r a c e R G M p X M p r G r + trace __ QCp X C r )~,2.,
iv)
Q{ block diag[ C, X C r , C: X C r ..... C X C r ] . trace N W -' - F} : 0;
v)
A X A r + DpWD r = X ,
(4.18)
where - - = R . traceNW-', Q = Q. traceNW-', F = diag[e . 2I I ! ,e.212 2 ..... R
e21
],
(4.19)
a n d l j denotes an m j x mj identity matrix.
The proof of Theorem 4.2 is similar to that of Theorem 3.1. By setting M = I, one obtains the necessary conditions for the state feedback case, which is a special case of measurement feedback control. Corollary 4.3 Suppose that G is an optimal solution o f the E O L case, a n d that the E O L
f o r state f e e d b a c k
p r o b l e m is regular. Then there exists a matrix
Q = block diag[Q t , Q2 ..... Q. ] > 0 such that i)
G = -('R + BrKBF) -'BrKAp ;
ii)
0 = ArKAp - ArKBF ('ff+ BrKBp) -'BrKAF +Cr'QCp;
iii)
D pr K D p = W-' N W - ' ( t r a c e R G X G r + t r a c e Q C X C ; )
iv)
e { b l o c k d i a g [ C t X C r , C 2X C r ..... C X C r ]. t r a c e N W - ' - F} = 0;
v)
( Ap + B p G ) X ( A p + B pG ) r + D pW D rp= X ,
(4.20)
u2 .
where R , Q a n d I" are the s a m e as in (4.19).
The solution of the third equations in (4.18) and (4.20) can be solved by Lemma 3.3. The algorithm used to solve those necessary conditions will be given at the end of this section. B.
DYNAMIC FEEDBACK CASE We consider the fixed order dynamic controller
GUARANTEEING OUTPUT L CONSTRAINTS
x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x ( k ) + H z(k).
117
(4.21)
Let v = 0. Then the closed loop system has the following form x(k+l)
=
y(k)
=
[~r l~IrGr]rx(k)+[_iv(k),
(4.22)
where x(.), y(.), ,4, /~, /), C and M are defined in (3.11). The fixed order dynamic controller design yields the standard static measurement feedback problem, as a special case. Special Case: Full Order Dynamic Con~oller Consider the following strictly proper full order dynamic controller x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x (k).
(4.23)
By defining the closed loop system state vector x, output vector y and input vector w as (3.17), and defining the closed loop system matrix A, D and C as (3.18). The discrete time E O L problem with full order dynamic controller can be stated as follows f A~.B,.C~.W:,O min trace N W - ' . trace RC u XC ur; C,,= [0,C c] subject to (2.1) and
(4.24)
"o[C, XC 7 ]. trace NW -t < e.~, i = 1, 2 .... m. Theorem 4.4 Partition W, K and X as in (3.20). Suppose that the triple ( C , A c, B ) is an optimal solution o f the EOL problem defined in (4.24), and that the EOL problem is regular. Then there exists a matrix Q=blockdiag[Q~, Q2 ..... Q, ] > 0 such that Q{block diag[C~ XC r, C 2XC r ..... C XCr., ]. traceNW-' - F} = 0,
(4.25)
118
GUOMING G. ZHU AND ROBERT E. SKELTON
and A = Ap + B,,C - B Mp,
(4.26a)
) -'. B c = (ApX,,M pr +D,W~:)(W22 + MpX,, M T ,, C = -(R + BrK22Bp
-l
T
n K22A ,
where X,, and K~2 satisfy Xl, = ApX,,A r + DpWi,D r P
(4.26b)
P
- ( A p X, , M rp +OpW~ 2)(W22 + g p x , ] Mr)-l(ap X , Mrt, +DpW~2) r r r K22 = ApK22A p - ApK22B,, (-~ + B r K22Bp )_, BpK22A r + C Pr QC, P P m
and R and Q are defined in (4.19). In addition, DTKD =W-'NW-' (traceRC XC[ +traceQC XC T ) II2 ,
(4.27)
where K is defined in (3.20) with K,2 = K22 and K n satisfying (K,, - K 2 2 ) = ( A p - B Mp) ~(K,, -K2:)(A p - B M p ) + C [ ' R C ,
(4.28)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying X22 = ( A p + B p C ) X : : ( A p + B p C ) r +B (W:2 + MpX,,M~)B[.
(4.29)
The proof is similar to that of Theorem 3.4.
C. NUMERICAL ALGORITHM Step I
Given system matrices A, Bp ~ Cp ~ Dp.M p , initial weighting matrices Wo,Qo,R,
outer
product
bound
N,
output
L
bounds
e,, (i = 1,2 ..... ny), error tolerance e , and free parameters 0<13 < 1
and 1I >0, define R~ = R . trace lVW~-' ~ = Q trace NW. -~ Let i Step 2
Controller design step: Case 1: static state feedback
O.
GUARANTEEING OUTPUT L CONSTRAINTS
119
Compute a i -" --(R-~
"~
Brr, Bp) -' Brr, Ap
(4.30a)
by solving K, = ArK, A - A r K , B , ( ~ +BrK, Bp)-'BrK, A +Cr~Cp P
P
"
P
P
Pi
(4.30b)
P
and then solve for X, by (4.30c)
X, =(Ap+BpG,)X,(Ap+BpG,) r +DpWD r. Let D~ = Dp, Cy = Cp, and C'u = G,. Go to Step 3. Case 2: Static measurement feedback
Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G, for given Q~ and W G, =-(R~ + Brr, Bp) -' BrK, ApX, Mr (MpX, Mr)-';
(4.31a)
K, = (Ap + BpG, Mp) r K (Ap + BpG, Mp)
(4.31b)
+ MrG , ( ~ + BrKiBp)G, Mp +Cr~Cpp r X,=(Ap+BpG, Mp)X,(Ap +B pG,M p)r +D pWD , p.
; (4.31c)
Let D i = Dp, Cy = Cp, and C~u= G~M r. Go to Step 3. Case 3: Full order dynamic controller
Solve for controller matrices A i B ~ and C ~ by c ~
c p
c
(4.32a)
A: = Ap + BpC: - B: Mp ; , pr + DpW~;)(W:': + MpX,, M~) p -' 9 B: = ( ApXllM T
i
-1
T
i
C: = - ( R + BpK~2Bp) BpK22Ap, and
120
GUOMING G. ZHU AND ROBERT E. SKELTON
'
XI!
-- A p X 11 i A pr
-~-
D p W l ,' D pr
(4.32b)
X,,' M Pr + DpW/t)(W:'2 + M P X,,, M Pr)-I (ApX,,, M Pr + DpWl2, ) r . , s2 , B p ( ~ + B r K22B , r , p K n' A p - A p K p) -, BpK~sA +C ;~ C
- (A P
Kn'
-A"
P
i
P
i
"
)
i
Then solve for K,, - Ks2 and X22 by
(K[,- K~2 ) = (Ap - B~ Mp) r(K[,- K~s)(Ap - B: M) + C r ( ~ + B p KT s 2, B p ) C c
Xss' = (Ap + BpC;)Xs:(A i
(4.32c) )
p + BpC~ ) r + V~ (W~'2 + M p X , ,' M rp) ( B 'c) r ,
where
[o ,:]
D,= o
B~ ; r , = (K;,) ~ K;, ; x =
ix: o] o
(4.32d)
x~,
Let C = [Cp , Cp ] and C~ = [0, C~ ]. Go to Step 3. Step 3
Compute ,
D r
)us;
(4.33)
k', = traceRC~ X, (C~ )r + traceQ, C X,C r , Then solve for
w,., = x, (K;)-' (K; NK;)"' (K',)-'l~ + (~- 13)W,;
(4.34)
Qj(i + 1) = E-~Y j j~ (i)Qj(i) Y~" j , Y = Cj XC r, j = 1,2 , .. . ,m, Q,§ = block diag[Q, (i), Qs (i) ..... Q, (i)]. Step 4
,f
and 2
F =blockdiag[e,l,,
,op,
where
2
e212 ..... e S l ] . Else, i =i + 1 and go to Step 2. !"1
GUARANTEEING OUTPUT L CONSTRAINTS
121
Lemma 4.5
Suppose that the discrete time EOL algorithm converges to some weight matrix W>O. Then the resulting controller is feasible, i.e., the closed loop system (4.2) with the resulting controller will satisfy the inequality constraints (4.14). V.
EXAMPLE
In this section we will present an example to illustrate the EOI_, algorithm and to make comparisons with the O/., design. The example is a simply-supported Euler-Bernoulli beam of length L with a force and torque applied at the 0.45L point from left end. Using the beam model in chapter 3 of [8] and taking the first 5 modes of the model one can put the plant into the following state space model .;c
=
A x
yp
=
CpXp
z
=
M xp
P
P
+B
P
p
P
u+D
P
w
P
(5.1) +v,
where the output yp is the vertical displacement and rotation angle at the point of force and torque application which is at the 0.45L point from left end. The measurements are the vertical velocity and angular velocity at the same point (i.e., the sensors and actuators are collocated). Also, we assume that wp and v are L2 disturbances and wp is applied at the same position as that of actuators (hence, Dp = B )p . The design results for the continuous and discrete time controllers are similar, so we only present the discrete design results. For the discrete time controller designs, we choose the sampling time T = 0. 05 second and discretize the continuous model (5.1) to obtain the following discrete time model. x (k+l)
=
A xp(k)+Bu(k)+Dpw(k)
y,(k)
=
Cpxp(k)
z(k)
=
M , x (k)+v(k),
(5.2)
where system matrices Ap, Bp, Dp, C and Mp for discrete system (5.2) are presented in Appendix A. Note that the system matrices Ap, Bp, Dp, C and
122
GUOMING G. ZHU AND ROBERT E. SKELTON
Mp (5.2) are different from the system matrices in (5.1) .Under the assumption that the closed loop system input w(.) is constant during the sample period we have
IoW(t)w r (,)dt = s fj~i*"rw(t)w r (,)dt = T E w ( k ) w r (k). k~O
(5.3)
k=O
Equation (5.3) provides the connection between continuous and discrete outer products. A.
DISCRETE TIME STATE FEEDBACK CONTROLLER DESIGN
For the state feedback controller design, we assume that v = 0. Note that the dimension of wp is 2. Suppose that the components of wp satisfy
[[wp,(')~i = ~oWp' (t)wr~ (t)dt < 1~ [[wp2(')~i = Io wp2 (t)w;~ (t)dt < 1 ,
(5.4)
and
IoW ~(t)w;2 (t)dt = 0 . Using (5.4) we can have
ol
wp (k)w r (k) <
ol
=
1/T
= N.
(5.5)
20
Note that N is the upper bound matrix for the closed loop input disturbance w, and in the state feedback case w = wP . We choose, (5.6)
R =diag[1,1]; e = 10 -~', a = 1.5; [3= 1.0; F =diag[O.l,O.1]
as design parameters. The outer product bound on the input L2 disturbance is defined in (5.6). See Table 5.1 for results, Table 5.1 State Feedback Controller Design Controllers
"OL - 1 .a
0.10
[ 0.10
OL - 2
0.10
0.10
EOL
0.10
[ 0.10
GUARANTEEING OUTPUT L CONSTRAINTS
"-2
-'2
""2
"-2
U,
0.0999
0.2380
123
0.10
0.10
0.10
0.10
i.21x 103 1.72x 10~ 1.57 x 102 1.16x 102il.54x 102 1.17 xlO 2
U2 W
:4.55•
Q
I
~ 1
0
Jot,. or J cot.
%
N(see (5.5)) 82.9 0 1.27x109
0
1.73x10 s
"81.2
0
0
186
184
2.72x102
2.72x102
|
.
,,
17 Iteration # 10 16 The algorithm has not yet converged. With the choice W = I the best output g bounds that can be reached by state feedback are [0.0028, 0.2375]. Hence the algorithm would not converge since the solution does not exist. .
.
.
.
.
where
~',
m
2
(5.7)
= c, xc~, ., u, =[cxc~],,., i= 1,2,
Pole Plactm~at of EOL..inf and OL-inf 1 0.8 ...................... 0.6 .......... 0.4
~.2
~.6
..........
~.8 -I -I
......................
~.5
0
O~
l
Figure 1 Closed Loop Pole Positions for State Feedback
124
GUOMING G. ZHU AND ROBERT E. SKELTON
and the optimal weighting matrix
2.4436 -0.03721 Wsr = -0.0372
(5.8)
0.2513"
The resulting E O L controller guarantees that for any s 2 disturbance satisfying (5.5), the square of the output t ' norm of each output will be bounded by 0.1, and Table 5.1 shows that the E O L designs keep the outputs within their bounds using smaller control efforts (lower design cost E O L ) than the O L designs. If one chooses W = N in the O L design, the optimal cost of the O L design is close to that of the E O L design, and Figure 1 shows that the closed loop poles of the two design cases are very close. On the basis of our examples, W = N is a good choice for W in O L designs. To appreciate the disadvantage (conservativeness) in the O/., design, a choice W = 1 for the O L design yields unachievable design goals, (in this case when MF = 1 the lower bounds
on
the
e,,2 (i = 1,2)
achievable
in
(4.17)
is
[CDWDrCr],, p p p p
9traceNW-', but if one changes W to N for the O L design, the same design goals are achievable.
4000
C o n v e r g e n t Process of q I & q2
Solid line: q l D a s h e d line: q2 3000 ................ i.................. .................................... I
9 I
9
. 2 0 0 0 ....... i ......',~. 9
.
......................
.,.,,,,w
I
1
~
'
~
.....................................
0
:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Fig. 2
10
15
20
Iteration No.
Figure 2 Convergence Process of Q for State Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
125
Hence, in the OL design the choice of W is very important and the EOL| design provides the best choice of W. Also, the cost of the OL design is very high, which implies the bound of IIu,
+11",
is large. Figures 2 and 3
provide convergence information for the discrete time
EOL| problem. Even
though this is a 10th order system, we can see that matrices Q and W converge in only 16 iterations.
Convergent Process of W
3 2.5
.... 1.5
-..~
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii . . . .
.
.................
., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
,,
0
5
1. . . .
10
15
20
Fig. 3 Iteration No.
Figure 3 Convergence Process of W for State Feedback B.
DISCRETE TIME FULL ORDER DYNAMIC CONTROLLER DESIGN
For the full order dynamic design, we added disturbances in the measurement channels. We assume that the L: norm of the continuous time measurement disturbance for each channel is 0.1, and all the disturbances are independent, i.e., E
w, (t)wj7 (t)dt = 0; ~' i ~: j, i, j = 1,2,3,4.
Then using the same method as in the state feedback case, we obtain the matrix bound on the outer product of closed loop input g 2 disturbance as
126
GUOMING G. ZHU AND ROBERT E. SKELTON
N =diag[200,20,O.2,0.2].
(5.9)
r =[2,2].
(5.10)
The design goals are
We use parameters defined in (5.6) (except F) to design the EOL| full order dynamic controller, as in the state feedback case, we designed two OL| controllers with W = I and W = N for comparisons, see Table 5.2 for results.
Controllers
Table 5.2 Dynamic Feedback Controller Design "OL - 1 OIL - 2 2.00 10.9
F --"2
"--2
--'2
"--2
U2
U,
.o
2.00 2.00
2.00 4.57
2.00 2.00
2.00 2.00
2.00 2.00
1.68 x 10 S 1.25 x 105 1.60x 102 5.25x 10 4.14x10 1.98x 10
!
W
Q
EOL
.0
r2.49x 10"
L
o
0
!
7.14X 10'.
N(see (5.9)) "141 0" 0
43.2
(5.11)) 28.1
0
0
19.0
Jo,~.or J eo,_~
*
2.12x102 2.90x 105 6.12x10 Iteration # 10 35 25 See the notion for Table 5.1 but the bounds becomes [ 10.9192 3.6787].
The Wor in Table 5.2 is
I 2.3039 -0. 0298 W~ = / 0 . 0 3 6 4
L-0.0032
-0.0298
0.0364
-0.0032]
0.4297
-0. 0007
0.0162 | .
-0.0007
0.0146
-0.0044 /
0.0162
-0.0044
0.0694.]
(5.11)
As in the state feedback case, the full order dynamic E O L design provides the lower design cost, see Tables 5.2. Unlike the state feedback case, the results of the O/., design with W = N are no longer close to the EOI., design, around twice the difference in the design costs J o,.. and J~.o,... Also Figure 4 shows that the pole placements for the different designs are quite different.
GUARANTEEING
OUTPUT
L
CONSTRAINTS
127
Pole Placement of EOL-inf and OL-inf 1 0.8 0.6 0.4
0"20
'
~
I e
:
-0.2 -0.4 ............................ i.............................. !.............................. i................... .t ....... -0.6 -0.8
9
-1 -1
-0.5
t
0
.
x
0.5
1
Figure 4 Closed Loop Pole Positions for Full Order Feedback Hence, the example shows that in the full order dynamic controller design, the choice of weight matrix W by the E O L algorithm is much more important than in state feedback case. In the full order dynamic OL| design the maximal accuracy bounds are available. In the discrete case, X~ satisfies
X,x = ApX,, Ar -
pX, , Mr
+ M pX, , M r- )' , M X,,A r + DpWDr9
(5.12)
and the achievable e: (i = 1 2) in (4.24) are i
"8[C X,x Cr, ]. traceNW -'" i - 1,2 9
9
(5.13)
where W = block diag[W,V]. Hence the choice of W in the OL| design will directly effect the feasibility of the O L design. When we chose W = I in the O L design in Table 5.2, it turns out that the design goal is not achievable (there are no feasible solutions). While the EOI., design does not have this problem because the algorithm will choose the weight matrix W to make constraints feasible. Also from numerical experience, the controller "gain" (cost of the E O L ) will increase as the g bounds decrease, which is consistent with the OL case. .a
128
G U O M I N G G. ZHU AND ROBERT E. SKELTON
Table 5.2 shows that the convergence rate of the OL and EOL design is very close (even though EOL. is iterating on two Riccati, while OL is iterating on one). From Figures 5 and 6 we can see that, even though this is a 20th order system, the algorithm converges in only 26 iterations.
Convergent Process of q I & q 2 _
80
.~.... ]---]
60 e,q o"
Solid line: .(:11
TD he
.....
nei q2 ...........
40
o"
.....::ii:!iiiil
20 0
ii
10
0
20
30
Fig. 5 Iteration No. Figure 5 Convergence Process of Q for Full Order Feedback
Convergent Process of W
2.6 2.4 2.2
2
1.8 ................. ,
1.6 0
i ..................................... |
10
20
30
Fig. 6 Iteration No.
Figure 6 Convergence Process of W for Full Order Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
129
VI. C O N C L U S I O N S The paper presents a methodology to design measurement or dynamic controllers that guarantee output L bounds while the input L2 disturbances have an outer product which is bounded by a given constant matrix N, while minimizing a weighted summation of the L upper bound for each control channel. The feasibility of the E O L algorithm has been shown under the assumption of convergence. Finally, an example is presented for 2 special cases: state feedback and full order dynamic controller design, those example show that the EOL| design uses much less control than O L designs, given similar L2 bounds on inputs. Also the algorithms converge fast and smoothly (converges in 16 iterations for state feedback design and 26 for full order dynamic design for a 20th order system). In fact, the new E O L algorithm converges as fast as the old OL| algorithm even though the E O L algorithm iterates on two Riccati equations instead of one. Finally, inequality constraints which are not achievable using an O L algorithm might be achievable with the new E O L algorithm. The basic reason for this phenomenon is that more information about the disturbance environment is used in the E O L , where an upper bound on the L2 norm of each disturbance w~(.) is known. Only the upper bound of the L~ norm of the D
a
me ~
o
vector w(.)
utilized in the O L design, making it too qv~
-
-
i=o
conservative.
VII. A P P E N D I X A The system matrices for discrete system (5.2) in the example are Ap = block diag[ A~ , A 2 .... , A~ ],
(A.1)
where
=[0.99880.0500], [0.9801
-0.7944];,
A~ L-0-0500
0.9987j" A~ = -0.7944
0.9794 J
(A.2a)
130
A3 =
GUOMING G. ZHU AND ROBERT E. SKELTON
[0.9905
-3.9101 =[ 0.3186 A, 1-0.0377
Mr= P
-3.9101]; A, = [0.6973 0.8983 -23.5535] 0.3076 jr
[.-11.4410
0
0.9877
0.1564
0
0
O.3090
- 1.9021
0
0
=.08910
-1.3620
0
0
-.5878
3.2361
0
0
0.7071
3.5355
P
=D
P
=
(A.2b)
0.6916 J" (A.2c)
0
n
-11.4410],,
"C
7p
0.9877
0.1564
0.3090
-1.9021
-0.8910
-1.3620
-0.5878
3.2361
0.7071
3.5355
=
0.0012
0.0002
0.0494
0.0078
0.0004
-0.0024
0.0153
-0.0944
-0.0011
-0.0017
-0.0430
--0.0567
-0.0007
0.0038
-0.0263
0.1446
0.0008
0.0039
0.0266
0.1332
(A.3)
(A.4)
GUARANTEEING OUTPUT L CONSTRAINTS
131
VIII. R E F E R E N C E S 1.
C. Hsieh, J. H. Kim, G. Zhu, K. Liu, and R. E. Skelton, "An Iterative Algorithm Combining Model Reduction and Control Design," Proceeding of American Control Conference, 1990. 2. C. Hsieh, "Control of Second Order Information for Linear Systems," Ph.D. Dissertation, Purdue University, W. Lafayatte, IN, 1990. 3. G. Zhu, and R. E. Skelton, "Mixed L2 and L Problems by Weight Selection in Quadratic Optimal Control," Int. J. Contr., Vol. 33, No. 5. J. Cont. 1991. 4. D . A . Wilson, "Convolution and Hankel Operator Norms for Linear System," IEEE Trans. Auto. Contr., Vol. AC-34, No. 1, 1989. 5. G. Zhu, M. Corless, and R. E. Skelton, "Robusmess of Covariance Controllers," Allerton Conf., Monticello, IL, 1989. 6. R . E . Skelton, and G. Zhu, "Optimal L Bounds for Disturbance Robustness," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 7. G. Zhu, and R. E. Skelton, '~Robust Discrete Controllers Guaranteeing g 2 and g| performances," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 8. R.E. Skelton, Dynamics System Control, Wiley, New York, 1988. 9. M. Corless, G. Zhu, and R. E. Skelton, "Robust Properties of Covariance Controllers," In Proceedings of the IEEE Control and Decision Conference, Tampa, 1989. 10. G. Zhu, M. A. Rotea, and R. E. Skelton, "A Feasible Convergent Algorithm for Output Covariance Constraint Problem," In the Proceedings of 1993 American Control Conference, San Francisco, CA., 1993.
Robert E. Skelton Space Systems Control Lab., Purdue University West Lafayette, IN 47907
I. I N T R O D U C T I O N In the presence of all L2 disturbances whose outer products are bounded by a known matrix N, this chapter develops controllers to minimize the worst case control L norm bound subject to the worst case L constraints on each output group. The resulting controller is an LQG controller with a special choice of the output weighting matrix and input noise intensity matrix. Both weighting matrices may be determined by an iterative algorithm. The design algorithm turns out to be feasible if the algorithm converges. It has been shown in [ 1-3,10] the standard Output Covariance Constraint
(OCC) problem has two different interpretations; stochastic and deterministic. One deterministic interpretation of the OCC problem is the output L constrained control problem ( O L ) , described as follows. Consider the system CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
97
98
G U O M I N G G. Z H U A N D R O B E R T E. S K E L T O N
A Px P + B Pu + D
P
where
xp e 9~"~,
yp
=
Cpx p
Z
=
M xp
y p ~
9~"~,
z~
P
wP
(1.1) +v,
p
9~~ ,
ve
9~": ,
u~
9~"" , and
wp ei 9~"
are,
respectively, vectors of the plant state, plant output, measurement signal, measurement noise, control input and deterministic exogenous signal. We assume that the closed loop system with some static or dynamic controller u(s) =[C (sl - A)-' B c + D]z(s)
is strictly proper, then it can be expressed in
the following state space form .~
=
Ax+Dw
; w e g l "~"~
y
=
Cx ; ye91""',
(1.2)
w h e r e C = [ C r , C [ ] r , w = [ w p, r v r] r and y = [yp,u r r ] r , and matrices Cy a n d C . depend on the coordinate of the closed loop system state. Here we assume that the plant performance outputs are divided into m output groups y, = C x ~ 91" (i = 1,2, .. . , m ) , that is yp = [ y , ~ ,Y2~ ..... Y.~]~ . Let the square of the L 2 norm of w(.) be def'med by
IIw<
(1.3)
(t)W-'w(t)d,,
=
where W is a symmetric positive definite matrix, and define the L
norm of
each output group y, (-) to be
Ily,<>11
yr (t)y, (,), i : 1,2 ..... m,
(1.4)
and note that
II.,<.>ll = up.
, (t),
i=
1,2 .....
nu.
(1.5)
Define the L 2 disturbance set (1.6)
GUARANTEEING OUTPUT L CONSTRAINTS
99
Then the O L problem can be stated as follows. THE OL PROBLEM .o
For the given plant (1.1)find a (static or dynamic) controller to minimize
s ug.llu,r
JoL = E r
n:}
; r > 0 , i = 1,2,. .. ,nu
(1.7)
i=1
subject to
Ily, )ll <
i=1,2
,m, V w es
w
(1.8)
where r > 0 (i = 1,2 ..... nu) and e, > 0 (i = 1,2 ..... m) are given constants.
E! The significance of this problem is obvious: under the given L2 disturbance environment we seek a controller which satisfies the output L constraints with minimum control effort J o,.. This is a practical controller design problem. In physical hardware there are always hard limits due to sensor and actuator signal saturation, and such systems (e.g., the Hubble Space Telescope) must be evaluated based upon its ability to keep pointing errors less than values specified by optical resolutions. Such missions might not be feasible if such L inequality constraints cannot be guaranteed. The O L problem has been solved in [3,10]. Also, the O L design algorithm has been applied to the "NASA Minimast' flexible structure lab facility at Langley Research Center and the results have been reported in [ 1]. Often, in practical problems, much more information about the disturbance vector w(.) is known than simply the L: norm in (1.6). For example, we may know bounds for the L2 norm of each w, (-). To improve the O L method, we presume to have more information about the disturbance than the L2 norm bound in (1.6). Suppose we know that w(.) belongs to the following disturbance set s
= w(-): t ~ 9~''+'z ,
w~ (t)dt < Ix, 9
One can still use the OL results over the disturbance set .a
(1.9)
100
GUOMING G. ZHU AND ROBERT E. SKELTON
= {w(.):,
IIw()ll:
=
w" ( , ) W - ' w ( t ) d t
< ~},
(1.10)
nw+nz
by choosing 13= ~ Ix~ and W = I, that is, to enlarge the disturbance set K2 so i=l
that ff~ _c ~ . It can be easily shown that the set (1.10) is always equal to or large than the set (1.9). Therefore, the old O L approach may yield more conservative designs than the present paper. The outer product of all the input disturbances, which is a positive semidefinite matrix, will include this additional information (1.9). Hence, we presume to know a bound N>O on the outer product of w characterized by the following set f2N= {w(-): t --->9~--',
(
foW(t)w r (t)dt< N}.
(1.11)
The precise statement of the new problem, called the Extended Output L EOL ) constraint problem, will be given in the next section. The chapter is organized as follows. Section II provides the definition of and some preliminary knowledge which will be useful for
the EOI_, problem
solving the problem. Section III provides the continuous time solutions (necessary conditions and an algorithm) for the static controller (measurement feedback) and dynamic controller. Two special cases, state feedback and strictly proper full order dynamic solutions are discussed in Section III. Section IV is the discrete time version of Section III. Section V gives an example for a discrete system to demonstrate the algorithm and to compare with the O L designs. Section 6 adds some conclusions.
II.
STATEMENT OF THE
Before precisely stating the Lemmas. Lemma 2.1
EOL EOL
PROBLEM problem, we introduce two very useful
GUARANTEEING OUTPUT L, CONSTRAINTS
101
For the asymptotically stable and controllable system (1.2) define Y = CXC r, where X is the positive definite solution of the following Lyapunov
Equation 0 = XA r +AX +DWD r, W > 0 .
(2.1)
Let w(.) in (1.2) be any L 2 disturbance defined in (1.3) and let
Ily(.)ll = s u p y t~>O
T
(t)y(t)
(2.2)
be the square of the L norm of y(.). Then
Ily(-)ll ---
IIw(-)ll:,
(2.3)
and for all disturbances belonging to the set ~w in (1.6)
13. trj = suplly(-)ll . wr163w
(2.4)
Lemma 2.1 was proven in [4,9].
Remark 2.2 The bound given in (2.3) is for the asymptotically stable and strictly proper system (1.2). Note that for the proper (not strictly proper) system, i.e., there is a direct feed forward term in y(t), hence, there exists no L bound on the output for any disturbance which belongs to the set f2 w in (1.6) with ~ > O. Therefore, the closed loop system with finite L norm on the output for any w(.) ~f2 w should be in the form (1.2). Lemma 2.3 Let (1.3) and (2.2) respectively denote the square of the input L 2 norm and the square of output L norm of system (2.1). Under the same assumptions as in Lemma 2.1, consider the disturbance w(.) in the set (1.11), where N>O is ^ given. Suppose there exists a W such that
102
GUOMING G. ZHU AND ROBERT E. SKELTON
Y
YYo
=
L(L~L)-V2 L; N = LL;
=
CXC ~; 0 = X A r + A X + D W D r,.
=
Dr ~oea~,C r yoYoCe r ~,d t D > O ;
(2.5)
[Ylyo; Ilyoll=1.
:
Using W for W in (1.3), (2.3) leads to "~[Y]tracel~-' = sTIly(.)ll'.
(2.6)
A proof is provided in [6]. Remark 2.4 Equation (2.6) shows that the bound (2.3) will be tight over the disturbance set O N by the choice o f W as the L 2 norm weight in (1.3) to
replace W. In the E O L problem we consider disturbances in the set (1.11). It is easy to see that for any L 2 disturbance w(.) which belongs to ~N defined in (1.11), we have
=
~oo wr (t)W-~w(t)dt trace{W-"2!w(t)wr(t)dtW-"2} (2.7)
<
trace (W -''2NW-''2)
=
traceNW-t,
where W > 0 is a given weight matrix. Therefore,
Ilu,<)ll ]y,
<
[ C XC,r ],, traceNW-I
<
"o[C XC r ]. traceNW -~ , i = 1,2,...,m,
,
i = 1,2 ..... nu, (2.8)
where X satisfies (2.1) and C = [ C r , C r ..... c rT] . and C, are defined in (1.2). Then the E O L
EOL Problem:
problem can be stated as follows:
GUARANTEEING OUTPUT L CONSTRAINTS
103
Find a static or dynamic controller and the input L 2 norm weighting matrix W > 0 to minimize the EOL cost J eo~. = traceNW-' . { traceRC XC r }
(2.9)
subject to (2.1) and "o[C~XCr].traceNW -' < e 2 i = 1 , 2 , i
'
~
,m,
(2.10)
where e, (i = 1,2 ..... m) are given and R =diag[rz,r 2..... r ] > 0 .
(2.11) E!
The physical interpretation for the E O L problem is clear if one considers the inequalities in (2.8). Note that in this case
Er,
(2.12)
Ilu,
i=l
Hence, the cost function is an upper bound of the weighted summation of supremums of control signals over all L2 disturbances which belong to ~N" The EOL| problem is similar to the O L problem, but the input disturbances belong to different sets, f2 N and fir, respectively. The next section provides the solutions of the E O L problem in the continuous time case.
III. S O L U T I O N S
OF CONTINUOUS
TIME EOL
PROBLEM
In this section, we shall discuss the first order necessary conditions for static and dynamic controllers, and also present an iterative algorithm. A.
STATIC MEASUREMENT FEEDBACK CASE
For the static measurement feedback case, we seek a constant feedback gain matrix G which is the solution of the E O L problem such that
104
GUOMING G. ZHU AND ROBERTE. SKELTON (3.1)
u =Gz.
Therefore the EOL| problem becomes (3.2)
I min traceNW -~. trace R G M X M r G r P
G.W>O
P
subject to (2.1) and
[o[C,XC,r ].traceNW-I - -
<
2
9
_ e,,
t = 1,2 ..... m.
To keep the closed loop system strictly proper, we only consider the case where v = 0 for system (1.1). In this case, the closed loop system (1.2) matrices are
A = A +BpGMp, D = D Theorem
/.
, W =Wp, C=[Cp,Mr].p
3.1
Suppose that G is an optimal solution o f the EOL EOL
(3.3)
problem
is
regular.
Then
there
exists
problem, and that the a
matrix
Q = block
diag[ Q~ , Q2 ..... Q. ] > 0 such that i)
G = - ' f f - ' B r K X M r ( M , X M pr)-,.,
ii)
0 = KA + A r K + M r G r ' R - t G M p + CrQCp;
iii)
DrKD
iv)
Q{block diag[ C, x c r , C2 x c j ..... C x c r ]. t r a c e N W -' - F} = 0;
v)
A X + XA r + D pWD pr = 0 ,
P
P
= W - ' N W - ' ( t r a c e R G M X M rGr + traceQCpXC r)''2" P
P
(3.4a)
where "R = R. traceNW-' , Q = Q. traceNW-' , F = diag[ 2, I,,~.212 2 ..... 1~I 2 ], and l j is an identity matrix with dimension mj.
The proof of the above theorem is obvious. By using the following augmented cost function
H
trace NW -' { trace RGM pXM r G r } + +trace K[ X ( Ap
i=! + BpGMp ) r + ( Ap +
[ C x c ,rtrace NW -' - E I , (3.5)
BpGM, ) X
+ OpWD p r ],
GUARANTEEING OUTPUT L CONSTRAINTS
105
setting the partial derivatives with respect to X, G, K, W to zero, and adding one equation for inequality constraints in (3.2) (Kuhn-Tucker conditions), one can get the necessary conditions. By setting M = I , one obtains the results for state feedback case which is a special case of measurement feedback control.
Corollary 2.2 Suppose that G is an optimal solution of the EOL problem for the state feedback case, and that the EOL| problem is regular, that is, the EOL problem is not independent of the EOL cost function in (3.2). Then there exists a block diagonal positive semidefinite matrix Q such that i)
G = -'R-'BrKX;
ii)
0 = KA,, + A r K - KB,-R-'B rK + Cr'QCp ;
iii)
D ~, rKD
iv)
Q{block diag[ C, x c r , C~XC[ ..... C x c r ]. traceNW -' - F} = 0;
v)
(Ap + BpG) X + X (Ap + BpG) r + DpWDp r
P
m
p = W-I NW-I (traceRGXG r + traceQC X C ; ) |'~"
(3.6)
"-0,
g
where R , Q and F are the same as those in Theorem 3.1. The solution of the third equation in (3.3) and (3.6) is given in the following Lemma.
Lemma 3.3 Suppose that N and K are positive definite and D has full column rank. Then equation D r KD = W-INW-1 has one symmetric positive definite solution W~ = K~'[K~NK~] ~2K~-',
(3.7)
and one negative definite solution
w, where
r; ',
(3.8)
106
G U O M I N G G. Z H U A N D ROBERT E. S K E L T O N
(3.9)
K,~ = ( D r K D ) ''2 .
The proof was presented in [6]. An algorithm used to solve those necessary conditions is presented in Subsection C. B.
DYNAMIC FEEDBACK CASE Consider the following dynamic controller with order nc xc
=
AcXc + Bcz
u
=
C x +Hz.
(3.10)
It is well-known that by setting
=[APo ;x=
(3.1 la) =
;~=
;G= o
s
Ix:] [':] [5] ;y=
~
;D=
;D
=
x
"
;I)= B,,
; A
[01
(3.1 lb)
'
with compatible dimensions, the closed loop system can be written as (3.12) y
----
[d' , ,~'c" l" X-F H v
In order to obtain a bounded output L
9
norm for an L2 input, system
(3.12) should be strictly proper, i.e.,/4 = 0 or v - 0. Here we consider the case that v = 0. Then the closed loop system is in the form (1.2) with A=A+BGtr
C
[~r
21~/rGr]r
(3.13)
Let (3.14)
GUARANTEEING OUTPUT L CONSTRAINTS
107
where R = diag[r~,r2..... r ] > 0. By solving the following static measurement feedback EOL problem
trace NW -l . trace [CGl~I X~Ir G r
min
I
G.W>O
(3.15)
P
subject to (2.1) and
---
"
^T
[o[CXC ].traceNW
-1
~
2
_ e,, i = 1,2 ..... m.
one can obtain first order necessary conditions for the dynamic controller case. Hence, when v = 0 in (1.1), the EOL problem with a fixed order dynamic controller produces the static measurement feedback EOL controller as a special case. Special Case: Full Order Dynamic Controller For this special case, we consider the following strictly proper full order dynamic controller L
=
ax+
u
=
Cx.
z
(3.16)
Let
X =
IXPx-~Xc]; Y = [ ~ I ; w=IWvl'
(3.17)
then the closed loop system can be put into the form (1.2) with A -BM
I
BM c
D=
The EOL follows:
A +BC
A +BM
p
P 0
-BM
c
C;C= n
c
P 0
-A
(3.18a)
p
P.
(3.18b)
c
problem with full order dynamic controller can be stated as
108
GUOMING G. ZHU AND ROBERT E. SKELTON
min
traceNW-'.traceRC XC r, C = [ 0 , C ]
Ac ~c .Cr .W>O
(3.19)
subject to (2.1) and
"0[ C XC r ]. traceNW-1 <_ e 2,, i = 1,2,... ,m. where, in this case, C = [ C, C p ], C, = [C, r , C r ..... C r ]r.
Theorem 3.4 Define the following matrices
w
=
r
;K=
[ ll
;X=
[xl x2] I
(3.20)
,
where Wj, K o and X o (i, j=l.2) are nx by nx matrices. Suppose that the triple ( C , A c, B ) is the optimal solution to the full order dynamic EOL problem defined in (3.19), and that the EOL| problem is regular. Then there exists a matrix Q = block diag[Q~ , Q2 ..... Q. ] > 0 such that Q{block diag[ C, x c r , C2 XC r ..... C XC r ]. traceNW-' - F} = 0,
(3.21)
and A =A,+B
C-B
M,;
(3.22)
B = ( X t , M pr + D W~2)W2-2" "~-i
C=-R
T
B K22,
where X , , and K22 satisfy X,, A ,r + A , X ,, _ ( X I M,
,r +D Wn)W~-l (X,,
M ,r
+D W~2)r +D W~,D ,r =~,
(3.23)
K=Ap + ArK= - K=B "R-'Br K= + Cr'QC = 0 , and "R = R . traceNW-' and Q = Q . traceNW-l . In addition DrKD = W-'NW-' (traceRC XC r + t r a c e Q C XCry )~2, where K is defined in (3.20) with K~2 = K22 and K~2 satisfying
(3.24)
GUARANTEEING OUTPUT L CONSTRAINTS
109
(3.25)
r ( K , , - K2:) + C~rR'C~ =0,
(K,,-K22)(A,,-BM)+(A,-BM)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying
(3.26)
X22(A +B C ) r + ( A p + B p C ) X 2 2 +BW~2Br =0.
The proof is similar to the static measurement feedback case. Consider the augmented cost function m n
_~.
(3.27)
traceNW-' . traceRC XC r,, + E ' ~ [ C XC r 9traceNW-' - E,] I, : i=l
+traceK .[ AX + XA r + DWD r ].
Then (3.27) can be rewritten into H = traceXCrO.C + traceK .[ AX + XA rDWD r ] _ traceQ. F,
where 0 = block diag['Q,'ff ] and "R = R . traceNW-' , "Q = Q . traceNW-' . By setting the partial derivatives with respect to K, X, W to zero and adding the Kuhn-Tucker condition (3.21),one can complete the proof. _...
.__
Note that (3.22) and (3.23) describe an LQG controller if Q, R, and W are given. Hence, the full order E O L controller is an LQG controller with some special choice of the weighting matrices Q, R and input noise intensity matrix W. (The forthcoming algorithm determines Q, R and W). Remark 3.5 The fact that there exists a W > 0 satisfying the first order necessary conditions requires that D rKD > O. Remark 3.6 The first order necessary conditions for the OL problem only need to satisfy (3.22) and (3.23) for fixed W, which are included in the EOL| problem. Hence the OL| problem is a special case of the EOL problem.
C.
NUMERICAL ALC~RITHM
110
GUOMINGG. ZHU AND ROBERTE. SKELTON
Subsections A and B of this section provide all the necessary conditions for the design of static and dynamic controllers. The following algorithm provides a way to design the controller by iterating on these necessary conditions. This is an extension of the O L algorithm presented in [3].
THE EOL .o ALGORITHM Step I
Given system matrices Ap,Bp,Cp,Dp. Mp, initial weighting matrices Wo , Qo , R , outer product bound N, output L bounds e,, (i = 1,2 ..... m), error tolerance e , and free parameters 0 < [~< 1 and a >0. Define ~ = R. traceNW, -! ~ = Q. trace NW -~ Let i = 0
Step 2
Controller design step: Case 1- static state feedback Compute m-!
G, = - R
7"
(3.28a)
BpK,
by solving (3.28b)
K i A p + A ; K - K B p R---' i Bp"K -t- C;'QiC p = O, and then solve for X, by (Ap+BpG,)X, + X,(Ap+BpG,) r +DpWD r =0.
(3.28c)
Let D, = Op, q = Cp, and C~ = G,. Go to Step 3. Case 2: Static measurement feedback Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G i for given Qi and W
6,
- R -, ' B prK , Xi M pr ( M P X, M rP ) -''' O= K(Ap+BpG, Mp) r + K ( A p + B p G , Mp)
(3.29a)
=
+ M;<
C,M
(3.29b) ,
0 = (Ap + BpG,Mp) X, + X, (Ap + BpG,Mp )r +DW~D r
(3.29c)
GUARANTEEING OUTPUT L CONSTRAINTS
111
Let D i = Dp, Cy = Cp, and C~ = G i Mp. Go to Step 3. Case 3: full order dynamic controller
Solve for controller matrices A~, B~, and C~ by A= i Ap+
BpC~ - B M i p, " M ,r + DpWs
8:=(x:,
(3.30a) '
;
C' = - g -i' Bp"K~2, '
and X,; At+, ApX[, + DpW/~Drp _ _ ( X I li T
i
(3.30b)
M rp+ DpW/~)(W2~2)-'(X[,M rp+ D W~;)r = 0; i
i
---1
T
i
K=A + A K= - K~:B R, B K= + C ~ , C
= O.
Then solve for K:, - K~2 and X~: by (K:,-K~2)(A P - B ; M P ) + ( A P - B ~ M P ) r ( K : , - r ~ )
(3'30c)
+<~,c =o, i
x=(a p
+BC;/~+(A+Bp C'IX= +Ew~(8;/T =o, c i
i
i
where
Di=
[o P
;K,=
)r
;X=
ix: o] l
(3.30d)
i "
Let C = [Cp, C ], and C~ = [0, C'c ]- Go to Step 3. Step 3
Compute K; = (D, TK,D, )1/2 ;
I.2, = traceRC: X, (C: )r + traceQ~C X,C r ,
(3.31)
112
GUOMING G. ZHU AND ROBERTE. SKELTON
Then solve for W§
(3.32)
)-' (K~' NK~ )"' (K~)-' ~+ (1-~)W,
Q , ( i + l ) = e , -~y,,, (i)Q,(i) ya. ,,Y=C,
XC, r , j = 1 , 2 ..... m,
a,+, = block aiagIQ, (i), Q: ( i) ..... Q. (i)]. Step 4
/:
and
IIw,.,-w,ll< ,
,,op,
where
e:l~ F = block diag[e,l,, " ~ : ..... e : l ] . E l s e , i = i + l a n d g o t o S t e p 2 . D During each iteration of the special cases, state feedback or full order dynamic controller, the resulting controller is an LQ or LQG controller. Hence, in those two special cases, under the usual stabilizability, detectability assumptions the stability of the closed loop system is guaranteed during each iteration. Remark 3.7 The algorithm is written for 3 different cases. One makes a choice in Step 2 for different controller complexity. Lemma 3.8 Suppose that the EOL
algorithm converges to some weight matrix
W > O. Then the resulting controller is feasible, i.e., the closed loop system with the resulting controller will satisfy the inequality constraints (2.10) The proof of Lemma 3.8 is similar to that in [3].
IV. DISCRETE TIME VERSION Here, we consider the following discrete system
xp(k+l)
=
A x(k)+Bu(k)+Dpwp(k)
y,(k)
=
C x,(k)
z(k)
=
M x (k)+ v(k),
(4.1)
GUARANTEEING
OUTPUT L
CONSTRAINTS
113
where the dimensions of the matrices and vectors are analogous to the continuous system (1.1). Suppose that the closed loop system with a static or dynamic controller is strictly proper, then the closed loop system can be expressed as follows
x(k+l)
=
Ax(k)+Dw(k);we91
y(k)
=
Cx(k) ; y ~ R ~+~',
n"*n~
(4.2)
where C=[C~,Cr] r , w = [ w r , v r l r and y = [y~r,u~lr . Define the square of the input e: norm of w(.) as
IIw<-)ll: = ~ w,
(4.3)
k=O
and let the square of the output e| norm for each individual channel be
Ily,<.)ll~.= s u p y r ( k ) y , ( k ) ; Ilu,<)11: = suo k - ~ u:t
i : 1,2 ..... m;
i = 1,2,
" ' "
(4.4)
,nu.
For the strictly proper closed loop system we have the following Lemma.
Lemma 4.1 [5] For the asymptotically stable system (1.2) with (A, D) controllable define Y = CXC r, where X is the positive definite solution of the following Lyapunov equation X = AXA r + DWD r, W > 0.
(4.5)
Let w(.) in (4.2) be any e: disturbance which belongs to
a~ -
w<.): k -~ ~t -+~,llw<.)ll: = E w~
and
Uy<-)II: - s~py~
(4.6)
114
GUOMING G. ZHU AND ROBERT E. SKELTON
be the square of the ~| normx~f y(.). Then
(4.7)
Ily(.)ll <_ trlllw(.)ll . and for all disturbances belonging to f~w in (4.6)
(4.8)
otY] = s~. ~y(.)~2| .~_,. Define the input disturbance set f2 N ={w(.): k-'~91""~'~w(k)wr,~o ( k ) < N, N > 0 } .
(4.9)
The discrete time EOL| problem can be stated as follows. DISCRETE TIME EOL PROBLEM .a
Find a static or dynamic controller and an input matrix W >0 to minimize the EOL cost J EoL. = trace NW -~ "{ trace R C, x c r }
2
norm weighting
(4.10)
subject to (4.5) and "o[QXCr].traceNW -~ <_e 2,, i = 1,2,...,m,
(4.11)
where e, (i = 1,2 ..... m) are given and R =diag[r~,r 2..... r ] > 0 .
(4.12) t3
The same physical interpretation for the E O L problem as that for continuous time case can be found by using Lemma 4.1, i.e.,
(
i=1
and
wlit~ N
u:t
(4.13)
GUARANTEEING OUTPUT L CONSTRAINTS
]ly,<>11<-6[CXC,r].traceNW -~, i = 1,2 ..... m.
115
(4.14)
Hence, J EOL., is an upper bound of the weighted summation of supremums of the control signals over all the f 2 disturbances which belong to f~,, in (4.9), and inequality (4.14) guarantees that the output g norm for each individual channel will be less than or equal to the given bound. A.
STATIC MEASUREMENT FEEDBACK CASE
We seek a constant feedback gain matrix G such that the solution of the E O L problem is (4.15)
u =Gz.
Here we only consider the case that v = 0 to have a strictly proper closed loop system. After combining equation (4.1) and (4.15), the closed loop system (4.2) matrices are A = Ap + BpGMp, D = Dp , 9
and the discrete time E O L
r J'~IT
W = Wp , C = .t ~_ p , M r
(4.16)
problem for measurement feedback controller
becomes (4.17)
f min traceNW -~ 9traceRGM P X M PTG r G.W>O subject to (2.1) and o [ C XC, ]. traceNW -~ < _ E2, . . i. = . . 1. 2,
--
r
m.
Theorem 4.2 Suppose that G is an optimal solution o f the EOL problem defined in (4.17), and that the EOL problem is regular. Then, there exists a matrix Q = block diag[Q~, Q~ ..... Q= ] > 0 such that
116
GUOMING G. ZHU AND ROBERT E. SKELTON
i)
G - - ( R-- + BrpKBp) -' BrpKAFXMFr ( M pX M r ,) -l",
ii)
K = ArKA + MrG r ('R+ B;KBp)GMp +CrQCp;
iii)
D ; K D p = W - I N W -I ( t r a c e R G M p X M p r G r + trace __ QCp X C r )~,2.,
iv)
Q{ block diag[ C, X C r , C: X C r ..... C X C r ] . trace N W -' - F} : 0;
v)
A X A r + DpWD r = X ,
(4.18)
where - - = R . traceNW-', Q = Q. traceNW-', F = diag[e . 2I I ! ,e.212 2 ..... R
e21
],
(4.19)
a n d l j denotes an m j x mj identity matrix.
The proof of Theorem 4.2 is similar to that of Theorem 3.1. By setting M = I, one obtains the necessary conditions for the state feedback case, which is a special case of measurement feedback control. Corollary 4.3 Suppose that G is an optimal solution o f the E O L case, a n d that the E O L
f o r state f e e d b a c k
p r o b l e m is regular. Then there exists a matrix
Q = block diag[Q t , Q2 ..... Q. ] > 0 such that i)
G = -('R + BrKBF) -'BrKAp ;
ii)
0 = ArKAp - ArKBF ('ff+ BrKBp) -'BrKAF +Cr'QCp;
iii)
D pr K D p = W-' N W - ' ( t r a c e R G X G r + t r a c e Q C X C ; )
iv)
e { b l o c k d i a g [ C t X C r , C 2X C r ..... C X C r ]. t r a c e N W - ' - F} = 0;
v)
( Ap + B p G ) X ( A p + B pG ) r + D pW D rp= X ,
(4.20)
u2 .
where R , Q a n d I" are the s a m e as in (4.19).
The solution of the third equations in (4.18) and (4.20) can be solved by Lemma 3.3. The algorithm used to solve those necessary conditions will be given at the end of this section. B.
DYNAMIC FEEDBACK CASE We consider the fixed order dynamic controller
GUARANTEEING OUTPUT L CONSTRAINTS
x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x ( k ) + H z(k).
117
(4.21)
Let v = 0. Then the closed loop system has the following form x(k+l)
=
y(k)
=
[~r l~IrGr]rx(k)+[_iv(k),
(4.22)
where x(.), y(.), ,4, /~, /), C and M are defined in (3.11). The fixed order dynamic controller design yields the standard static measurement feedback problem, as a special case. Special Case: Full Order Dynamic Con~oller Consider the following strictly proper full order dynamic controller x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x (k).
(4.23)
By defining the closed loop system state vector x, output vector y and input vector w as (3.17), and defining the closed loop system matrix A, D and C as (3.18). The discrete time E O L problem with full order dynamic controller can be stated as follows f A~.B,.C~.W:,O min trace N W - ' . trace RC u XC ur; C,,= [0,C c] subject to (2.1) and
(4.24)
"o[C, XC 7 ]. trace NW -t < e.~, i = 1, 2 .... m. Theorem 4.4 Partition W, K and X as in (3.20). Suppose that the triple ( C , A c, B ) is an optimal solution o f the EOL problem defined in (4.24), and that the EOL problem is regular. Then there exists a matrix Q=blockdiag[Q~, Q2 ..... Q, ] > 0 such that Q{block diag[C~ XC r, C 2XC r ..... C XCr., ]. traceNW-' - F} = 0,
(4.25)
118
GUOMING G. ZHU AND ROBERT E. SKELTON
and A = Ap + B,,C - B Mp,
(4.26a)
) -'. B c = (ApX,,M pr +D,W~:)(W22 + MpX,, M T ,, C = -(R + BrK22Bp
-l
T
n K22A ,
where X,, and K~2 satisfy Xl, = ApX,,A r + DpWi,D r P
(4.26b)
P
- ( A p X, , M rp +OpW~ 2)(W22 + g p x , ] Mr)-l(ap X , Mrt, +DpW~2) r r r K22 = ApK22A p - ApK22B,, (-~ + B r K22Bp )_, BpK22A r + C Pr QC, P P m
and R and Q are defined in (4.19). In addition, DTKD =W-'NW-' (traceRC XC[ +traceQC XC T ) II2 ,
(4.27)
where K is defined in (3.20) with K,2 = K22 and K n satisfying (K,, - K 2 2 ) = ( A p - B Mp) ~(K,, -K2:)(A p - B M p ) + C [ ' R C ,
(4.28)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying X22 = ( A p + B p C ) X : : ( A p + B p C ) r +B (W:2 + MpX,,M~)B[.
(4.29)
The proof is similar to that of Theorem 3.4.
C. NUMERICAL ALGORITHM Step I
Given system matrices A, Bp ~ Cp ~ Dp.M p , initial weighting matrices Wo,Qo,R,
outer
product
bound
N,
output
L
bounds
e,, (i = 1,2 ..... ny), error tolerance e , and free parameters 0<13 < 1
and 1I >0, define R~ = R . trace lVW~-' ~ = Q trace NW. -~ Let i Step 2
Controller design step: Case 1: static state feedback
O.
GUARANTEEING OUTPUT L CONSTRAINTS
119
Compute a i -" --(R-~
"~
Brr, Bp) -' Brr, Ap
(4.30a)
by solving K, = ArK, A - A r K , B , ( ~ +BrK, Bp)-'BrK, A +Cr~Cp P
P
"
P
P
Pi
(4.30b)
P
and then solve for X, by (4.30c)
X, =(Ap+BpG,)X,(Ap+BpG,) r +DpWD r. Let D~ = Dp, Cy = Cp, and C'u = G,. Go to Step 3. Case 2: Static measurement feedback
Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G, for given Q~ and W G, =-(R~ + Brr, Bp) -' BrK, ApX, Mr (MpX, Mr)-';
(4.31a)
K, = (Ap + BpG, Mp) r K (Ap + BpG, Mp)
(4.31b)
+ MrG , ( ~ + BrKiBp)G, Mp +Cr~Cpp r X,=(Ap+BpG, Mp)X,(Ap +B pG,M p)r +D pWD , p.
; (4.31c)
Let D i = Dp, Cy = Cp, and C~u= G~M r. Go to Step 3. Case 3: Full order dynamic controller
Solve for controller matrices A i B ~ and C ~ by c ~
c p
c
(4.32a)
A: = Ap + BpC: - B: Mp ; , pr + DpW~;)(W:': + MpX,, M~) p -' 9 B: = ( ApXllM T
i
-1
T
i
C: = - ( R + BpK~2Bp) BpK22Ap, and
120
GUOMING G. ZHU AND ROBERT E. SKELTON
'
XI!
-- A p X 11 i A pr
-~-
D p W l ,' D pr
(4.32b)
X,,' M Pr + DpW/t)(W:'2 + M P X,,, M Pr)-I (ApX,,, M Pr + DpWl2, ) r . , s2 , B p ( ~ + B r K22B , r , p K n' A p - A p K p) -, BpK~sA +C ;~ C
- (A P
Kn'
-A"
P
i
P
i
"
)
i
Then solve for K,, - Ks2 and X22 by
(K[,- K~2 ) = (Ap - B~ Mp) r(K[,- K~s)(Ap - B: M) + C r ( ~ + B p KT s 2, B p ) C c
Xss' = (Ap + BpC;)Xs:(A i
(4.32c) )
p + BpC~ ) r + V~ (W~'2 + M p X , ,' M rp) ( B 'c) r ,
where
[o ,:]
D,= o
B~ ; r , = (K;,) ~ K;, ; x =
ix: o] o
(4.32d)
x~,
Let C = [Cp , Cp ] and C~ = [0, C~ ]. Go to Step 3. Step 3
Compute ,
D r
)us;
(4.33)
k', = traceRC~ X, (C~ )r + traceQ, C X,C r , Then solve for
w,., = x, (K;)-' (K; NK;)"' (K',)-'l~ + (~- 13)W,;
(4.34)
Qj(i + 1) = E-~Y j j~ (i)Qj(i) Y~" j , Y = Cj XC r, j = 1,2 , .. . ,m, Q,§ = block diag[Q, (i), Qs (i) ..... Q, (i)]. Step 4
,f
and 2
F =blockdiag[e,l,,
,op,
where
2
e212 ..... e S l ] . Else, i =i + 1 and go to Step 2. !"1
GUARANTEEING OUTPUT L CONSTRAINTS
121
Lemma 4.5
Suppose that the discrete time EOL algorithm converges to some weight matrix W>O. Then the resulting controller is feasible, i.e., the closed loop system (4.2) with the resulting controller will satisfy the inequality constraints (4.14). V.
EXAMPLE
In this section we will present an example to illustrate the EOI_, algorithm and to make comparisons with the O/., design. The example is a simply-supported Euler-Bernoulli beam of length L with a force and torque applied at the 0.45L point from left end. Using the beam model in chapter 3 of [8] and taking the first 5 modes of the model one can put the plant into the following state space model .;c
=
A x
yp
=
CpXp
z
=
M xp
P
P
+B
P
p
P
u+D
P
w
P
(5.1) +v,
where the output yp is the vertical displacement and rotation angle at the point of force and torque application which is at the 0.45L point from left end. The measurements are the vertical velocity and angular velocity at the same point (i.e., the sensors and actuators are collocated). Also, we assume that wp and v are L2 disturbances and wp is applied at the same position as that of actuators (hence, Dp = B )p . The design results for the continuous and discrete time controllers are similar, so we only present the discrete design results. For the discrete time controller designs, we choose the sampling time T = 0. 05 second and discretize the continuous model (5.1) to obtain the following discrete time model. x (k+l)
=
A xp(k)+Bu(k)+Dpw(k)
y,(k)
=
Cpxp(k)
z(k)
=
M , x (k)+v(k),
(5.2)
where system matrices Ap, Bp, Dp, C and Mp for discrete system (5.2) are presented in Appendix A. Note that the system matrices Ap, Bp, Dp, C and
122
GUOMING G. ZHU AND ROBERT E. SKELTON
Mp (5.2) are different from the system matrices in (5.1) .Under the assumption that the closed loop system input w(.) is constant during the sample period we have
IoW(t)w r (,)dt = s fj~i*"rw(t)w r (,)dt = T E w ( k ) w r (k). k~O
(5.3)
k=O
Equation (5.3) provides the connection between continuous and discrete outer products. A.
DISCRETE TIME STATE FEEDBACK CONTROLLER DESIGN
For the state feedback controller design, we assume that v = 0. Note that the dimension of wp is 2. Suppose that the components of wp satisfy
[[wp,(')~i = ~oWp' (t)wr~ (t)dt < 1~ [[wp2(')~i = Io wp2 (t)w;~ (t)dt < 1 ,
(5.4)
and
IoW ~(t)w;2 (t)dt = 0 . Using (5.4) we can have
ol
wp (k)w r (k) <
ol
=
1/T
= N.
(5.5)
20
Note that N is the upper bound matrix for the closed loop input disturbance w, and in the state feedback case w = wP . We choose, (5.6)
R =diag[1,1]; e = 10 -~', a = 1.5; [3= 1.0; F =diag[O.l,O.1]
as design parameters. The outer product bound on the input L2 disturbance is defined in (5.6). See Table 5.1 for results, Table 5.1 State Feedback Controller Design Controllers
"OL - 1 .a
0.10
[ 0.10
OL - 2
0.10
0.10
EOL
0.10
[ 0.10
GUARANTEEING OUTPUT L CONSTRAINTS
"-2
-'2
""2
"-2
U,
0.0999
0.2380
123
0.10
0.10
0.10
0.10
i.21x 103 1.72x 10~ 1.57 x 102 1.16x 102il.54x 102 1.17 xlO 2
U2 W
:4.55•
Q
I
~ 1
0
Jot,. or J cot.
%
N(see (5.5)) 82.9 0 1.27x109
0
1.73x10 s
"81.2
0
0
186
184
2.72x102
2.72x102
|
.
,,
17 Iteration # 10 16 The algorithm has not yet converged. With the choice W = I the best output g bounds that can be reached by state feedback are [0.0028, 0.2375]. Hence the algorithm would not converge since the solution does not exist. .
.
.
.
.
where
~',
m
2
(5.7)
= c, xc~, ., u, =[cxc~],,., i= 1,2,
Pole Plactm~at of EOL..inf and OL-inf 1 0.8 ...................... 0.6 .......... 0.4
~.2
~.6
..........
~.8 -I -I
......................
~.5
0
O~
l
Figure 1 Closed Loop Pole Positions for State Feedback
124
GUOMING G. ZHU AND ROBERT E. SKELTON
and the optimal weighting matrix
2.4436 -0.03721 Wsr = -0.0372
(5.8)
0.2513"
The resulting E O L controller guarantees that for any s 2 disturbance satisfying (5.5), the square of the output t ' norm of each output will be bounded by 0.1, and Table 5.1 shows that the E O L designs keep the outputs within their bounds using smaller control efforts (lower design cost E O L ) than the O L designs. If one chooses W = N in the O L design, the optimal cost of the O L design is close to that of the E O L design, and Figure 1 shows that the closed loop poles of the two design cases are very close. On the basis of our examples, W = N is a good choice for W in O L designs. To appreciate the disadvantage (conservativeness) in the O/., design, a choice W = 1 for the O L design yields unachievable design goals, (in this case when MF = 1 the lower bounds
on
the
e,,2 (i = 1,2)
achievable
in
(4.17)
is
[CDWDrCr],, p p p p
9traceNW-', but if one changes W to N for the O L design, the same design goals are achievable.
4000
C o n v e r g e n t Process of q I & q2
Solid line: q l D a s h e d line: q2 3000 ................ i.................. .................................... I
9 I
9
. 2 0 0 0 ....... i ......',~. 9
.
......................
.,.,,,,w
I
1
~
'
~
.....................................
0
:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Fig. 2
10
15
20
Iteration No.
Figure 2 Convergence Process of Q for State Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
125
Hence, in the OL design the choice of W is very important and the EOL| design provides the best choice of W. Also, the cost of the OL design is very high, which implies the bound of IIu,
+11",
is large. Figures 2 and 3
provide convergence information for the discrete time
EOL| problem. Even
though this is a 10th order system, we can see that matrices Q and W converge in only 16 iterations.
Convergent Process of W
3 2.5
.... 1.5
-..~
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii . . . .
.
.................
., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
,,
0
5
1. . . .
10
15
20
Fig. 3 Iteration No.
Figure 3 Convergence Process of W for State Feedback B.
DISCRETE TIME FULL ORDER DYNAMIC CONTROLLER DESIGN
For the full order dynamic design, we added disturbances in the measurement channels. We assume that the L: norm of the continuous time measurement disturbance for each channel is 0.1, and all the disturbances are independent, i.e., E
w, (t)wj7 (t)dt = 0; ~' i ~: j, i, j = 1,2,3,4.
Then using the same method as in the state feedback case, we obtain the matrix bound on the outer product of closed loop input g 2 disturbance as
126
GUOMING G. ZHU AND ROBERT E. SKELTON
N =diag[200,20,O.2,0.2].
(5.9)
r =[2,2].
(5.10)
The design goals are
We use parameters defined in (5.6) (except F) to design the EOL| full order dynamic controller, as in the state feedback case, we designed two OL| controllers with W = I and W = N for comparisons, see Table 5.2 for results.
Controllers
Table 5.2 Dynamic Feedback Controller Design "OL - 1 OIL - 2 2.00 10.9
F --"2
"--2
--'2
"--2
U2
U,
.o
2.00 2.00
2.00 4.57
2.00 2.00
2.00 2.00
2.00 2.00
1.68 x 10 S 1.25 x 105 1.60x 102 5.25x 10 4.14x10 1.98x 10
!
W
Q
EOL
.0
r2.49x 10"
L
o
0
!
7.14X 10'.
N(see (5.9)) "141 0" 0
43.2
(5.11)) 28.1
0
0
19.0
Jo,~.or J eo,_~
*
2.12x102 2.90x 105 6.12x10 Iteration # 10 35 25 See the notion for Table 5.1 but the bounds becomes [ 10.9192 3.6787].
The Wor in Table 5.2 is
I 2.3039 -0. 0298 W~ = / 0 . 0 3 6 4
L-0.0032
-0.0298
0.0364
-0.0032]
0.4297
-0. 0007
0.0162 | .
-0.0007
0.0146
-0.0044 /
0.0162
-0.0044
0.0694.]
(5.11)
As in the state feedback case, the full order dynamic E O L design provides the lower design cost, see Tables 5.2. Unlike the state feedback case, the results of the O/., design with W = N are no longer close to the EOI., design, around twice the difference in the design costs J o,.. and J~.o,... Also Figure 4 shows that the pole placements for the different designs are quite different.
GUARANTEEING
OUTPUT
L
CONSTRAINTS
127
Pole Placement of EOL-inf and OL-inf 1 0.8 0.6 0.4
0"20
'
~
I e
:
-0.2 -0.4 ............................ i.............................. !.............................. i................... .t ....... -0.6 -0.8
9
-1 -1
-0.5
t
0
.
x
0.5
1
Figure 4 Closed Loop Pole Positions for Full Order Feedback Hence, the example shows that in the full order dynamic controller design, the choice of weight matrix W by the E O L algorithm is much more important than in state feedback case. In the full order dynamic OL| design the maximal accuracy bounds are available. In the discrete case, X~ satisfies
X,x = ApX,, Ar -
pX, , Mr
+ M pX, , M r- )' , M X,,A r + DpWDr9
(5.12)
and the achievable e: (i = 1 2) in (4.24) are i
"8[C X,x Cr, ]. traceNW -'" i - 1,2 9
9
(5.13)
where W = block diag[W,V]. Hence the choice of W in the OL| design will directly effect the feasibility of the O L design. When we chose W = I in the O L design in Table 5.2, it turns out that the design goal is not achievable (there are no feasible solutions). While the EOI., design does not have this problem because the algorithm will choose the weight matrix W to make constraints feasible. Also from numerical experience, the controller "gain" (cost of the E O L ) will increase as the g bounds decrease, which is consistent with the OL case. .a
128
G U O M I N G G. ZHU AND ROBERT E. SKELTON
Table 5.2 shows that the convergence rate of the OL and EOL design is very close (even though EOL. is iterating on two Riccati, while OL is iterating on one). From Figures 5 and 6 we can see that, even though this is a 20th order system, the algorithm converges in only 26 iterations.
Convergent Process of q I & q 2 _
80
.~.... ]---]
60 e,q o"
Solid line: .(:11
TD he
.....
nei q2 ...........
40
o"
.....::ii:!iiiil
20 0
ii
10
0
20
30
Fig. 5 Iteration No. Figure 5 Convergence Process of Q for Full Order Feedback
Convergent Process of W
2.6 2.4 2.2
2
1.8 ................. ,
1.6 0
i ..................................... |
10
20
30
Fig. 6 Iteration No.
Figure 6 Convergence Process of W for Full Order Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
129
VI. C O N C L U S I O N S The paper presents a methodology to design measurement or dynamic controllers that guarantee output L bounds while the input L2 disturbances have an outer product which is bounded by a given constant matrix N, while minimizing a weighted summation of the L upper bound for each control channel. The feasibility of the E O L algorithm has been shown under the assumption of convergence. Finally, an example is presented for 2 special cases: state feedback and full order dynamic controller design, those example show that the EOL| design uses much less control than O L designs, given similar L2 bounds on inputs. Also the algorithms converge fast and smoothly (converges in 16 iterations for state feedback design and 26 for full order dynamic design for a 20th order system). In fact, the new E O L algorithm converges as fast as the old OL| algorithm even though the E O L algorithm iterates on two Riccati equations instead of one. Finally, inequality constraints which are not achievable using an O L algorithm might be achievable with the new E O L algorithm. The basic reason for this phenomenon is that more information about the disturbance environment is used in the E O L , where an upper bound on the L2 norm of each disturbance w~(.) is known. Only the upper bound of the L~ norm of the D
a
me ~
o
vector w(.)
utilized in the O L design, making it too qv~
-
-
i=o
conservative.
VII. A P P E N D I X A The system matrices for discrete system (5.2) in the example are Ap = block diag[ A~ , A 2 .... , A~ ],
(A.1)
where
=[0.99880.0500], [0.9801
-0.7944];,
A~ L-0-0500
0.9987j" A~ = -0.7944
0.9794 J
(A.2a)
130
A3 =
GUOMING G. ZHU AND ROBERT E. SKELTON
[0.9905
-3.9101 =[ 0.3186 A, 1-0.0377
Mr= P
-3.9101]; A, = [0.6973 0.8983 -23.5535] 0.3076 jr
[.-11.4410
0
0.9877
0.1564
0
0
O.3090
- 1.9021
0
0
=.08910
-1.3620
0
0
-.5878
3.2361
0
0
0.7071
3.5355
P
=D
P
=
(A.2b)
0.6916 J" (A.2c)
0
n
-11.4410],,
"C
7p
0.9877
0.1564
0.3090
-1.9021
-0.8910
-1.3620
-0.5878
3.2361
0.7071
3.5355
=
0.0012
0.0002
0.0494
0.0078
0.0004
-0.0024
0.0153
-0.0944
-0.0011
-0.0017
-0.0430
--0.0567
-0.0007
0.0038
-0.0263
0.1446
0.0008
0.0039
0.0266
0.1332
(A.3)
(A.4)
GUARANTEEING OUTPUT L CONSTRAINTS
131
VIII. R E F E R E N C E S 1.
C. Hsieh, J. H. Kim, G. Zhu, K. Liu, and R. E. Skelton, "An Iterative Algorithm Combining Model Reduction and Control Design," Proceeding of American Control Conference, 1990. 2. C. Hsieh, "Control of Second Order Information for Linear Systems," Ph.D. Dissertation, Purdue University, W. Lafayatte, IN, 1990. 3. G. Zhu, and R. E. Skelton, "Mixed L2 and L Problems by Weight Selection in Quadratic Optimal Control," Int. J. Contr., Vol. 33, No. 5. J. Cont. 1991. 4. D . A . Wilson, "Convolution and Hankel Operator Norms for Linear System," IEEE Trans. Auto. Contr., Vol. AC-34, No. 1, 1989. 5. G. Zhu, M. Corless, and R. E. Skelton, "Robusmess of Covariance Controllers," Allerton Conf., Monticello, IL, 1989. 6. R . E . Skelton, and G. Zhu, "Optimal L Bounds for Disturbance Robustness," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 7. G. Zhu, and R. E. Skelton, '~Robust Discrete Controllers Guaranteeing g 2 and g| performances," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 8. R.E. Skelton, Dynamics System Control, Wiley, New York, 1988. 9. M. Corless, G. Zhu, and R. E. Skelton, "Robust Properties of Covariance Controllers," In Proceedings of the IEEE Control and Decision Conference, Tampa, 1989. 10. G. Zhu, M. A. Rotea, and R. E. Skelton, "A Feasible Convergent Algorithm for Output Covariance Constraint Problem," In the Proceedings of 1993 American Control Conference, San Francisco, CA., 1993.