- Email: [email protected]
Linear control guaranteeing stability of uncertain systems via orthogonal decomposition
0005-1098/91 $3.00 + 0.00 Pergamon Press plc ~) 1991 International Federation of Automatic Control
Automat/ca, Vol. 27, No. 5, pp. 873-876, 1991 Printed in Great Britain.
Brief Paper
Linear Control Guaranteeing Stability of Uncertain Systems via Orthogonal Decomposition* KEQIN GUi" and Y. H. CHEN$ Key Words--Robust control, stability, time-varying systems; computer-aided design; deterministic analysis; Lyapunov methods.
A b s t r a c t - - A class of uncertain systems with time-varying uncertainty is considered. The uncertainty is decomposed "orthogonaUy". Based on this, a design of linear feedback control is proposed for linear uncertain systems. The control always exists if a generalized matching condition is met.
2. Systems with generalized matching conditions satisfied Consider a class of uncertain linear systems described by Yc(t) = lA + AA(q(t))]x(t) + [B + AB(q(t))]u(t)
1. Introduction A CLASS OF linear uncertain systems with (possibly fast) time-varying uncertain parameters is under consideration. The problem proposed here is to design linear feedback control which will result in the exponentially stable closed-loop system. A thorough review of the major past works along this line can be found in Corless and Leitmann (1988). Barmish et al. (1983) proposed a Lyapunov approach for the linear feedback control design. If the uncertainty fits a certain characterization (often called the matching condition), then a class of linear feedback controls can be designed based only on the possible bound of the uncertainty. Petersen and Hollot (1986) later explored a new approach, namely the Riccati design. If the uncertainty can be decomposed "linearly", the existence of the control is then dependent on a Riccati-type equation. The formulation of this Riccati equation is based on the structure as well as the possible bound of the uncertainty. No matching conditions as mentioned above are required for the design. However, no further explorations on the existence of solution of this Rice,aft equation are made; that is, no demonstrated advantage is shown for the existence of the solution as matching condition is met. The major contributions of the present work are as follows. The uncertainty is decomposed into two parts which are "orthogonal" to each other. A linear feedback control can be designed via solving a Riccati-type equation. This Riccati equation can always be solved (and hence the control exists) if a generalized matching condition is met. This setup is then extended to incorporate the mismatched case (i.e. as the generalized matching condition is not met).
for a l l t ~ R
q(t)~
(la)
where fl c R k is a compact set. The following generalized matching conditions are assumed in this section. Assumption 1. There exist a constant matrix • e R "×t and continuous functions E ( . ) : ~ - - * R t×t and D(.):f~---, R t×" such that
AB(q)~ = B~E(q);
(2)
AA(q) = B~D(q);
(3)
l+½(E(q)+Er(q))->61,
6>0,
for a l l q e g 2 ;
Bq~ has full rank;
(4) (5)
and (A, B ~ )
is stabilizable.
(6)
Throughout this article, we refer to "mismatched uncertainty" as AA and AB do not meet (2) and (3). For the sake of simplicity, the argument q is sometimes omitted in this article when no confusions are likely to arise. Notice that does not have to be square. When ~ = I, conditions (2) and (3) turn to the well known matching conditions (Barmish et al., 1983). It should be noticed, however, that even ~ is restricted to be square and nonsingular, a proper choice of is often essential to satisfy (4). For example, consider
* Received 20 February 1990; revised 10 October 1990; received in final form 11 January 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Kimura under the direction of Editor H. Kwakernaak. t Department of Mechanical Engineering, Southern Illinois University at Edwardsville, Edwardsviile, IL 62026, U.S.A. S The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, G A 30332, U.S.A. AUTO 27:5-H
(1)
X(to) = x o
where x(t) • R" is the state, u(t) ~ R m is the control input, A and B are the nominal system matrices. The system is subjected to the uncertainty AA(.) and AB(.) which are continuous functions of an unknown, and possibly fast time-varying, vector q ( t ) • R k. Such an uncertainty may be due to uncertain factors as imprecise knowledge of system parameters, nonlinearity and time-varying parameters. A standing assumption about the uncertainty throughout this article is
AB = BN
(7)
where
If • is chosen as an identity matrix I, then E = N , clearly (4) is not satisfied. A proper choice is
873
and
874
Brief Paper
Then E = ~ - 1 N d p = O . 5 N , 6 = 0.25 in this case).
and (4) can be satisfied (with
will satisfy (12) and (13). This is so since yT(/31 -- S - ST)y >--(/3 -- yt)yTy -->0
Remark. A physical interpretation of (2) and (3) can be as
follows: One determines which part of the uncertainty in AA and AB is more influential via the choice of ep and anticipates a stronger control action will be enforced there. However, one can also show that this is in fact equivalent to transforming the input to u ( t ) = a P v ( t ) . This in turn is equivalent to choosing a " n e w " input matrix B,e w = B ~ and a "new" uncertain input part A B , , w = A B e . The condition (2) and (3) is in this sense reduced to Barmish et al. (1983). We decide to phrase the condition (2) and (3) in the present manner since now the matrix ~ will become an active member (shown later) in the control design. A class of linear feedback controls of the form (8)
u(t) = - Otcb~T BT p x ( t )
will be sought to stabilize the system (1). Here a is a scalar and P is an n × n matrix. The procedure of deciding P and o~ is as follows: Step 1. Choose Q > 0. Let P be the positive definite solution of the algebraic Riccati equation (ARE) P A + A T p - 6 P B ~ d p T B T p + Q = 0,
(9)
and yW[(1 - ~)Q - [_,S(fll - S - ST)- Is'rLT]y -->yT(1 -- ~)r3y --yVy2Oma,,[(fl I -- S - SV)-l]y2y y2
-.T[,1-0,..
=(1-~)r3 fl-r, /3-Yl
r~ 1 , (1-~)r3Jy y
-0. Therefore, step 3 is also applicable. To show exponential stability of the closed loop system, consider the Lyapunov function candidate V(x) = xTpx
where P is the positive definite solution of (9). It follows that the time derivative of V along a given trajectory is given by (I(x) = x T [ p A + A T p + PBePD + (PBdPD) T - otPBep(21 + E + ET)dpTBTp]x -< x T [ p A + A T p + PBaPD + (PBaPD) T
where 6 > 0 is as in (4). Step 2. Decompose D in (3) as
-- 2tSoIPBdp~ T BT p]x
DT(q) = ES(q) + ZS(q),
(10)
= x T [ ( p A + A T p -- 6 P B a p ~ T B T p ) -26(O:-2)PBdpapTBTp+
where
L = PB~,
(11)
and L (or S) is such that (L, L) is a square and nonsingular matrix. For example, thc columns of.L can be chosen as a sct of basisvectorsof the orthogonal subspace to the columns of L. Step 3. Choose a sufficiently large/3 > 0 such that /31 - S ( q ) - ST(q) > 0 ,
for all q e f~
PB~PD
+ (PB~D)T]x = xT[--Q -- flEE T + E ( E S
+/~)T
"J-i t S -}-/~S)LT]x x T [ ( - - Q + / ~ S ( f l l - S - sT) - I~TLT)
(12)
and £ S ( q ) ( f l l - S ( q ) - S T ( q ) ) - I s r ( q ) L T -< (1 - ~)Q,
0<~<1,
for a l l q e f l .
(13)
-< -~xTax.
Step 4. Now choose o~ to be
1
/3
(14)
"=~+~3"
In the last step, (13) has been used for the first two terms of the brackets, and the big matrix in the last term is positive semi-definite. Since Q > 0, the proof is complete.
Theorem 1. If the system (1) satisfies (la) and assumption 1, then the design procedure described above for the feedback control (8) is feasible, and the resulting closed-loop system is exponentially stable.
Remark. T h e A R E
Proof. Since (A, Bib) is stabilizable and Q > 0 ,
Remark. As shown in step 2, the uncertainty is decomposed
(9) has a unique positive definite solution P (Kawakernaak and Sivan, 1972). Step 1 is valid. Step 2 is also valid since P is nonsingular and B ~ is of full rank, and therefore L has full rank. Also, since D is bounded [implied by (la) and its continuity property], S and L S are also bounded. Choose constants Ya, )'2, and )'3 such that
)'1 > SUg O'max(S(q ) + ST(q)) ^^ )'2 -> su[) a ~ x ( L S ( q ) ) qe~* and
(9) is related to ~ 4 and AB only through • and 6. Here • comes from the generalized matching conditions (2) and (3) and 6 is dependent on the bound f~ [as shown in (4)]. "orthogonally". A special case of the above procedure is that D T in (10) happens to be within the range space of L. Then the second term in (10) vanishes, and (13) is automatically satisfied. In general cases, a proper manipulation of decomposition of D + into two parts [as in (10)1 will be beneficial in reducing feedback gain. 3. Systems with mismatched uncertainty in A A ( q )
The procedure described in the last section can be extended to the case with mismatch in AA matrix [i.e. (3) is not satisfied]. In such cases, we decompose AA as A A ( q ) = BCbO(q) + ft.(q).
0 < )'3 -< omi,(Q ) where Omax(') and o=i,(') are the maximum and minimum singular value of the designated matrix. The constant fl chosen by
)'I
fl -->)'1 q" (1 - ~))'3
(15)
Here A is in a complement space of the columns of B ~ . For example, we can choose A such that A T B ~ = 0.
(16)
The algebraic Riccati equation should be modified as PA+ ATp - 6PBOapTBTp + U(P) + Q = 0
(17)
Brief Paper where U(P) is a symmetric matrix satisfying
P.~(q)+filX(q)P<_U(P)
875
where
for all q e if2.
~=Bo.
(17a)
The corresponding design steps become:
Step 1. Decompose AA as (15). Step 2. Solve (17) for positive definite solution P. If such solution exists, continue with next step. If not, try different Qs. If no solutions can be found, the algorithm is failed.
Step 3. Decompose D as in (10) and (11). Step 4. Choose sufficiently large fl to satisfy (12) and (13), then compute o: by (14).
If the above equation has a solution for some E, then • and Q can be rescaled to write the equation in the form of (17). The design can then proceed as described above. An interesting result about (19) is that if it has a positive definite solution for e = Cl > 0 and Q = QI > 0 , then, for any other Q 2 > 0 , there always exists an E 2 > 0 such that (19) has a positive definite solution P. Therefore, one only needs to try one Q > 0 and gradually vary E to obtain a stabilizing solution or declare failure of the process.
4. Systems with mismatched uncertainty in both ~A (q ) and
AB(q)
Theorem 2. Suppose that AA(q) is mismatched but the decomposition (15) is performed. Suppose also that assumption 1 except (3) is satisfied. If (17) has a positive definite solution P for a given Q > 0, then the above design procedure for the feedback control u(t)=--orOO'rBTPx(t), as it is applied to (1), results in an exponentially stable closed-loop system.
Proof. This is similar to that of Theorem 1. There are various ways of choosing U(P). One explicit form of U(P) similar to Kosmidou and Bertrand (1987) is as follows: Let
As both AA and 6 B matrices are mismatched, be decomposed as:
they should
AA(q) = BOO(q) + .4(q) A B ( q ) ~ = BOE(q) + B(q)
PA + ATp - 6PBOOTBTp + &PWP + U(P) + Q = 0 (21) where W should be a constant matrix satisfying (22)
and & -> tr. The design procedure is:
i=l
where 1, 2,.
(20)
where E is assumed to satisfy (4). The algebraic Riccati equation to be satisfied is then
W + B(q)OTB T + BOBX(q) >-0 for all q ~ ~ ft(q) = ~ qiAi
(15)
q = ( q t , q2 . . . . . qk) T with Iqil <-oi, , k. Then U(P) can be chosen as
oi >0,
i=
Step 1. Decompose A,4 and AB as in (15) and (20). Step 2. Choose appropriate Q and 5: and solve (21) for
k
positive definite solution P. If such solution exists, continue with next step.
U(P) = r ~, A X~F-' A, + kPFP i=l
Step 3. Decompose D as (10) and (11).
where
Step 4. Choose f l > 0 to satisfy (12) and (13), compute or
k
r=~o/ and F is any symmetricpositive definite matrix. A special choice of F was suggested by Kosmidou and Betrand (1987) with k
F = ee -1,
e = ~ [Iaill i=1
which produces r
k
U(P) = - ~ A~PA, + ekP.
according to (14); if tr-< &, then an exponentially stabilizing controller has been obtained, otherwise, choose a larger & and repeat steps 2 to 4.
Theorem 3. Suppose that AA(q) and AB(q) are mismatched but the decomposition (15) and (20) is performed. Suppose also that assumption (1) except (2) and (3) is satisfied. If (21) has a positive definite solution P for a given Q > 0, then the above design procedure for the feedback control u(t)= -orOOXBxPx(t), as it is applied to (1), results in an exponentially stable closed-loop system. Proof. This is similar to that of Theorem 1.
An earlier form of U(P) was proposed by Chang and Peng (1972), which is much more complicated but generally has less magnitude. The solution of the equation of the form (17) can also be found in Kosmidou and Bertrand (19_87) and Chang and Peng (1972). Suppose the uncertainty A can be decomposed as (Petersen and Hollot, 1986; Schmitendorf, 1988):
Remark. As described above, the choice of & generally needs several iterations. If & is chosen to be too large, (21) may very well be unsolvable; if & is chosen to be too small, & -> or may not be achieved. There are a number of ways to choose W. For example, if k
= E qiBi,
k
-A = ~ qi(t)dieXi
(18)
i=1
where B i is a constant matrix, and Iqil -< ol, then we can choose
where d~ and el are column vectors. We can then choose
k
w = Y~ ~BiB~ + k~(I,(BO) T.
U(P) = PMP + N
i=1
where k
M=rE i=l
Other discussionson choosing U ( P ) and W can be found in Zhou and Khargonekar (1988).
k
di aT
N=rE
eieT 5. Conclusions
i=l
f-> Iq~(t)l for all i and t. To use the standard procedure of solution proposed in Petersen and Hoilot (1986), write the algebraic Riccati equation
PA + A T p - p[12BIBX- M]P + N + eQ =O
(19)
We consider linear uncertain systems with time-varying uncertain parameters. The parameters may vary fast in view of the description given in Section 2. A new method of decomposing the uncertainty "orth0gonally" is adopted. Linear state feedback controls for such systems are designed. Both matched and mismatched uncertainties are considered. The proposed design procedure takes full advantage of the situation as the uncertainty is matched. This method can
876
Brief Paper
generally design stabilizing controllers with less feedback gains, and often allows a larger class of uncertain systems to be stabilized.
Acknowledgement--The research of Y. H. Chen was supported in part by the National Science Foundation under grant MSS-9014714. References Barmish, B. R., M. Corless and G. Leitmann (1983). A new class of stabilizing controllers for uncertain dynamical systems. SlAM J. Control Optimiz., 21, 246-255. Chang, S. S. L. and T. K. C. Peng (1972). Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Aut. Control, AC-17, 474-483. Corless, M. and G. Leitmann (1988). Deterministic control
of uncertain systems. In Ch. I. Byrnes and A. Kurzhanski (Eds), Modelling and Adaptive Control. Springer, New York, pp. 108-133. Kosmidou, O. I. and P. Bertrand, (1987). Robust-controller design for systems with large parameter variations. Int. J. Control, 45, 927-938. Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems. Wiley-Interscience, New York. Petersen, I. R. and C. V. Hollot (1986). A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22, 397-411. Schmitendorf, W. E. (1988). Designing stabilizing controllers for uncertain systems using the Riccati equation approach. 1EEE Trans. Aut. Control, AC-33, 376-379. Zhou, K. and P. P. Khargonekar (1988). Robust stabilization of linear systems with norm bounded time varying uncertainty. Syst. Control Lett., 10, 17-20.
Automat/ca, Vol. 27, No. 5, pp. 873-876, 1991 Printed in Great Britain.
Brief Paper
Linear Control Guaranteeing Stability of Uncertain Systems via Orthogonal Decomposition* KEQIN GUi" and Y. H. CHEN$ Key Words--Robust control, stability, time-varying systems; computer-aided design; deterministic analysis; Lyapunov methods.
A b s t r a c t - - A class of uncertain systems with time-varying uncertainty is considered. The uncertainty is decomposed "orthogonaUy". Based on this, a design of linear feedback control is proposed for linear uncertain systems. The control always exists if a generalized matching condition is met.
2. Systems with generalized matching conditions satisfied Consider a class of uncertain linear systems described by Yc(t) = lA + AA(q(t))]x(t) + [B + AB(q(t))]u(t)
1. Introduction A CLASS OF linear uncertain systems with (possibly fast) time-varying uncertain parameters is under consideration. The problem proposed here is to design linear feedback control which will result in the exponentially stable closed-loop system. A thorough review of the major past works along this line can be found in Corless and Leitmann (1988). Barmish et al. (1983) proposed a Lyapunov approach for the linear feedback control design. If the uncertainty fits a certain characterization (often called the matching condition), then a class of linear feedback controls can be designed based only on the possible bound of the uncertainty. Petersen and Hollot (1986) later explored a new approach, namely the Riccati design. If the uncertainty can be decomposed "linearly", the existence of the control is then dependent on a Riccati-type equation. The formulation of this Riccati equation is based on the structure as well as the possible bound of the uncertainty. No matching conditions as mentioned above are required for the design. However, no further explorations on the existence of solution of this Rice,aft equation are made; that is, no demonstrated advantage is shown for the existence of the solution as matching condition is met. The major contributions of the present work are as follows. The uncertainty is decomposed into two parts which are "orthogonal" to each other. A linear feedback control can be designed via solving a Riccati-type equation. This Riccati equation can always be solved (and hence the control exists) if a generalized matching condition is met. This setup is then extended to incorporate the mismatched case (i.e. as the generalized matching condition is not met).
for a l l t ~ R
q(t)~
(la)
where fl c R k is a compact set. The following generalized matching conditions are assumed in this section. Assumption 1. There exist a constant matrix • e R "×t and continuous functions E ( . ) : ~ - - * R t×t and D(.):f~---, R t×" such that
AB(q)~ = B~E(q);
(2)
AA(q) = B~D(q);
(3)
l+½(E(q)+Er(q))->61,
6>0,
for a l l q e g 2 ;
Bq~ has full rank;
(4) (5)
and (A, B ~ )
is stabilizable.
(6)
Throughout this article, we refer to "mismatched uncertainty" as AA and AB do not meet (2) and (3). For the sake of simplicity, the argument q is sometimes omitted in this article when no confusions are likely to arise. Notice that does not have to be square. When ~ = I, conditions (2) and (3) turn to the well known matching conditions (Barmish et al., 1983). It should be noticed, however, that even ~ is restricted to be square and nonsingular, a proper choice of is often essential to satisfy (4). For example, consider
* Received 20 February 1990; revised 10 October 1990; received in final form 11 January 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Kimura under the direction of Editor H. Kwakernaak. t Department of Mechanical Engineering, Southern Illinois University at Edwardsville, Edwardsviile, IL 62026, U.S.A. S The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, G A 30332, U.S.A. AUTO 27:5-H
(1)
X(to) = x o
where x(t) • R" is the state, u(t) ~ R m is the control input, A and B are the nominal system matrices. The system is subjected to the uncertainty AA(.) and AB(.) which are continuous functions of an unknown, and possibly fast time-varying, vector q ( t ) • R k. Such an uncertainty may be due to uncertain factors as imprecise knowledge of system parameters, nonlinearity and time-varying parameters. A standing assumption about the uncertainty throughout this article is
AB = BN
(7)
where
If • is chosen as an identity matrix I, then E = N , clearly (4) is not satisfied. A proper choice is
873
and
874
Brief Paper
Then E = ~ - 1 N d p = O . 5 N , 6 = 0.25 in this case).
and (4) can be satisfied (with
will satisfy (12) and (13). This is so since yT(/31 -- S - ST)y >--(/3 -- yt)yTy -->0
Remark. A physical interpretation of (2) and (3) can be as
follows: One determines which part of the uncertainty in AA and AB is more influential via the choice of ep and anticipates a stronger control action will be enforced there. However, one can also show that this is in fact equivalent to transforming the input to u ( t ) = a P v ( t ) . This in turn is equivalent to choosing a " n e w " input matrix B,e w = B ~ and a "new" uncertain input part A B , , w = A B e . The condition (2) and (3) is in this sense reduced to Barmish et al. (1983). We decide to phrase the condition (2) and (3) in the present manner since now the matrix ~ will become an active member (shown later) in the control design. A class of linear feedback controls of the form (8)
u(t) = - Otcb~T BT p x ( t )
will be sought to stabilize the system (1). Here a is a scalar and P is an n × n matrix. The procedure of deciding P and o~ is as follows: Step 1. Choose Q > 0. Let P be the positive definite solution of the algebraic Riccati equation (ARE) P A + A T p - 6 P B ~ d p T B T p + Q = 0,
(9)
and yW[(1 - ~)Q - [_,S(fll - S - ST)- Is'rLT]y -->yT(1 -- ~)r3y --yVy2Oma,,[(fl I -- S - SV)-l]y2y y2
-.T[,1-0,..
=(1-~)r3 fl-r, /3-Yl
r~ 1 , (1-~)r3Jy y
-0. Therefore, step 3 is also applicable. To show exponential stability of the closed loop system, consider the Lyapunov function candidate V(x) = xTpx
where P is the positive definite solution of (9). It follows that the time derivative of V along a given trajectory is given by (I(x) = x T [ p A + A T p + PBePD + (PBdPD) T - otPBep(21 + E + ET)dpTBTp]x -< x T [ p A + A T p + PBaPD + (PBaPD) T
where 6 > 0 is as in (4). Step 2. Decompose D in (3) as
-- 2tSoIPBdp~ T BT p]x
DT(q) = ES(q) + ZS(q),
(10)
= x T [ ( p A + A T p -- 6 P B a p ~ T B T p ) -26(O:-2)PBdpapTBTp+
where
L = PB~,
(11)
and L (or S) is such that (L, L) is a square and nonsingular matrix. For example, thc columns of.L can be chosen as a sct of basisvectorsof the orthogonal subspace to the columns of L. Step 3. Choose a sufficiently large/3 > 0 such that /31 - S ( q ) - ST(q) > 0 ,
for all q e f~
PB~PD
+ (PB~D)T]x = xT[--Q -- flEE T + E ( E S
+/~)T
"J-i t S -}-/~S)LT]x x T [ ( - - Q + / ~ S ( f l l - S - sT) - I~TLT)
(12)
and £ S ( q ) ( f l l - S ( q ) - S T ( q ) ) - I s r ( q ) L T -< (1 - ~)Q,
0<~<1,
for a l l q e f l .
(13)
-< -~xTax.
Step 4. Now choose o~ to be
1
/3
(14)
"=~+~3"
In the last step, (13) has been used for the first two terms of the brackets, and the big matrix in the last term is positive semi-definite. Since Q > 0, the proof is complete.
Theorem 1. If the system (1) satisfies (la) and assumption 1, then the design procedure described above for the feedback control (8) is feasible, and the resulting closed-loop system is exponentially stable.
Remark. T h e A R E
Proof. Since (A, Bib) is stabilizable and Q > 0 ,
Remark. As shown in step 2, the uncertainty is decomposed
(9) has a unique positive definite solution P (Kawakernaak and Sivan, 1972). Step 1 is valid. Step 2 is also valid since P is nonsingular and B ~ is of full rank, and therefore L has full rank. Also, since D is bounded [implied by (la) and its continuity property], S and L S are also bounded. Choose constants Ya, )'2, and )'3 such that
)'1 > SUg O'max(S(q ) + ST(q)) ^^ )'2 -> su[) a ~ x ( L S ( q ) ) qe~* and
(9) is related to ~ 4 and AB only through • and 6. Here • comes from the generalized matching conditions (2) and (3) and 6 is dependent on the bound f~ [as shown in (4)]. "orthogonally". A special case of the above procedure is that D T in (10) happens to be within the range space of L. Then the second term in (10) vanishes, and (13) is automatically satisfied. In general cases, a proper manipulation of decomposition of D + into two parts [as in (10)1 will be beneficial in reducing feedback gain. 3. Systems with mismatched uncertainty in A A ( q )
The procedure described in the last section can be extended to the case with mismatch in AA matrix [i.e. (3) is not satisfied]. In such cases, we decompose AA as A A ( q ) = BCbO(q) + ft.(q).
0 < )'3 -< omi,(Q ) where Omax(') and o=i,(') are the maximum and minimum singular value of the designated matrix. The constant fl chosen by
)'I
fl -->)'1 q" (1 - ~))'3
(15)
Here A is in a complement space of the columns of B ~ . For example, we can choose A such that A T B ~ = 0.
(16)
The algebraic Riccati equation should be modified as PA+ ATp - 6PBOapTBTp + U(P) + Q = 0
(17)
Brief Paper where U(P) is a symmetric matrix satisfying
P.~(q)+filX(q)P<_U(P)
875
where
for all q e if2.
~=Bo.
(17a)
The corresponding design steps become:
Step 1. Decompose AA as (15). Step 2. Solve (17) for positive definite solution P. If such solution exists, continue with next step. If not, try different Qs. If no solutions can be found, the algorithm is failed.
Step 3. Decompose D as in (10) and (11). Step 4. Choose sufficiently large fl to satisfy (12) and (13), then compute o: by (14).
If the above equation has a solution for some E, then • and Q can be rescaled to write the equation in the form of (17). The design can then proceed as described above. An interesting result about (19) is that if it has a positive definite solution for e = Cl > 0 and Q = QI > 0 , then, for any other Q 2 > 0 , there always exists an E 2 > 0 such that (19) has a positive definite solution P. Therefore, one only needs to try one Q > 0 and gradually vary E to obtain a stabilizing solution or declare failure of the process.
4. Systems with mismatched uncertainty in both ~A (q ) and
AB(q)
Theorem 2. Suppose that AA(q) is mismatched but the decomposition (15) is performed. Suppose also that assumption 1 except (3) is satisfied. If (17) has a positive definite solution P for a given Q > 0, then the above design procedure for the feedback control u(t)=--orOO'rBTPx(t), as it is applied to (1), results in an exponentially stable closed-loop system.
Proof. This is similar to that of Theorem 1. There are various ways of choosing U(P). One explicit form of U(P) similar to Kosmidou and Bertrand (1987) is as follows: Let
As both AA and 6 B matrices are mismatched, be decomposed as:
they should
AA(q) = BOO(q) + .4(q) A B ( q ) ~ = BOE(q) + B(q)
PA + ATp - 6PBOOTBTp + &PWP + U(P) + Q = 0 (21) where W should be a constant matrix satisfying (22)
and & -> tr. The design procedure is:
i=l
where 1, 2,.
(20)
where E is assumed to satisfy (4). The algebraic Riccati equation to be satisfied is then
W + B(q)OTB T + BOBX(q) >-0 for all q ~ ~ ft(q) = ~ qiAi
(15)
q = ( q t , q2 . . . . . qk) T with Iqil <-oi, , k. Then U(P) can be chosen as
oi >0,
i=
Step 1. Decompose A,4 and AB as in (15) and (20). Step 2. Choose appropriate Q and 5: and solve (21) for
k
positive definite solution P. If such solution exists, continue with next step.
U(P) = r ~, A X~F-' A, + kPFP i=l
Step 3. Decompose D as (10) and (11).
where
Step 4. Choose f l > 0 to satisfy (12) and (13), compute or
k
r=~o/ and F is any symmetricpositive definite matrix. A special choice of F was suggested by Kosmidou and Betrand (1987) with k
F = ee -1,
e = ~ [Iaill i=1
which produces r
k
U(P) = - ~ A~PA, + ekP.
according to (14); if tr-< &, then an exponentially stabilizing controller has been obtained, otherwise, choose a larger & and repeat steps 2 to 4.
Theorem 3. Suppose that AA(q) and AB(q) are mismatched but the decomposition (15) and (20) is performed. Suppose also that assumption (1) except (2) and (3) is satisfied. If (21) has a positive definite solution P for a given Q > 0, then the above design procedure for the feedback control u(t)= -orOOXBxPx(t), as it is applied to (1), results in an exponentially stable closed-loop system. Proof. This is similar to that of Theorem 1.
An earlier form of U(P) was proposed by Chang and Peng (1972), which is much more complicated but generally has less magnitude. The solution of the equation of the form (17) can also be found in Kosmidou and Bertrand (19_87) and Chang and Peng (1972). Suppose the uncertainty A can be decomposed as (Petersen and Hollot, 1986; Schmitendorf, 1988):
Remark. As described above, the choice of & generally needs several iterations. If & is chosen to be too large, (21) may very well be unsolvable; if & is chosen to be too small, & -> or may not be achieved. There are a number of ways to choose W. For example, if k
= E qiBi,
k
-A = ~ qi(t)dieXi
(18)
i=1
where B i is a constant matrix, and Iqil -< ol, then we can choose
where d~ and el are column vectors. We can then choose
k
w = Y~ ~BiB~ + k~(I,(BO) T.
U(P) = PMP + N
i=1
where k
M=rE i=l
Other discussionson choosing U ( P ) and W can be found in Zhou and Khargonekar (1988).
k
di aT
N=rE
eieT 5. Conclusions
i=l
f-> Iq~(t)l for all i and t. To use the standard procedure of solution proposed in Petersen and Hoilot (1986), write the algebraic Riccati equation
PA + A T p - p[12BIBX- M]P + N + eQ =O
(19)
We consider linear uncertain systems with time-varying uncertain parameters. The parameters may vary fast in view of the description given in Section 2. A new method of decomposing the uncertainty "orth0gonally" is adopted. Linear state feedback controls for such systems are designed. Both matched and mismatched uncertainties are considered. The proposed design procedure takes full advantage of the situation as the uncertainty is matched. This method can
876
Brief Paper
generally design stabilizing controllers with less feedback gains, and often allows a larger class of uncertain systems to be stabilized.
Acknowledgement--The research of Y. H. Chen was supported in part by the National Science Foundation under grant MSS-9014714. References Barmish, B. R., M. Corless and G. Leitmann (1983). A new class of stabilizing controllers for uncertain dynamical systems. SlAM J. Control Optimiz., 21, 246-255. Chang, S. S. L. and T. K. C. Peng (1972). Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Aut. Control, AC-17, 474-483. Corless, M. and G. Leitmann (1988). Deterministic control
of uncertain systems. In Ch. I. Byrnes and A. Kurzhanski (Eds), Modelling and Adaptive Control. Springer, New York, pp. 108-133. Kosmidou, O. I. and P. Bertrand, (1987). Robust-controller design for systems with large parameter variations. Int. J. Control, 45, 927-938. Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems. Wiley-Interscience, New York. Petersen, I. R. and C. V. Hollot (1986). A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22, 397-411. Schmitendorf, W. E. (1988). Designing stabilizing controllers for uncertain systems using the Riccati equation approach. 1EEE Trans. Aut. Control, AC-33, 376-379. Zhou, K. and P. P. Khargonekar (1988). Robust stabilization of linear systems with norm bounded time varying uncertainty. Syst. Control Lett., 10, 17-20.