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Robust State Feedback Design of Parametric Uncertain Systems via Linear Matrix Inequality
Copyright © IFAC Control of Industrial Systems, Belfort, France, 1997
ROBUST STATE FEEDBACK DESIGN OF PARAMETRIC UNCERTAIN SYSTEMS VIA LINEAR MATRIX INEQUALITY
Cheng Wen, I-Kong Fong, and Shin-Hao Lu
Department of Electrical Engineering National Taiwan University Taipei, Taiwan 10617 Republic of China E-mail: [email protected] Fax: 886-2-3660449
Abstract: This paper is devoted to the design ofrobust state feedback controllers for a class of linear uncertain systems. It is assumed that in the system and input matrices there are linear uncertain parameters, which satisfy no special restrictions, such as the matching condition or norm-bounded constraints. Through the matrix nonsingularity analysis, it is known that systems stabilized by a state feedback controller have robust stability bounds on the uncertain parameters, which can be computed from the structured singular value of a composite matrix. Based on this result and linear matrix inequalities, we form a convex optimization problem so that stabilizing controllers can be found to maximize the robust stability bounds. Keywords: Uncertain linear systems, Robust control, Convex programming, State feedback, Structured singular value
1. INTRODUCTION
ion are considered. In the literature, it can be seen that the stabilization problem of this category of uncertain models has attracted a considerable amount of attention. A long existing approach assumes that uncertain matrices in the system to be stabilized satisfy the so-called matching conditions (Gutman and Palmor, 1982). However, these conditions are quite restrictive to the structure of uncertain matrices, and limit the positions where uncertain parameters can appear as well as how those parameters are related. Attempts have also been made to relieve the matching conditions (Petersen and Hollot, 1986), but the resulting controller design method depends on a trial and error procedure, and the considered norm-bounded uncertainties are still subject to some constraints.
For many years, standard linear control schemes have been successfully applied 10 different kinds of practical engineering problems. Often the plant characteristics are described by approximate mathematical models, because it is difficult, if not impossible, to accommodate all conceivable uncertainties in a reasonably simple model. However, to design a robust and precise system, it becomes clear that effects of various unavoidable uncertainties in the plant must be taken into account. This paper deals with the problem of controller design for system model containing parametric uncertainties. To be more specific, linear state-space models with uncertain parameters entering system and input matrices in a linear fash313
nal system is controllable, but there are no special restrictions on A; and'B;, which may be zero if k; does not appear in the state or input matrix. To avoid trivial situations, we assume that at least one of A; and B; is nonzero for each i. Suppose the static state feedback controller
This paper is devoted to the design of robust state feedback controllers for a class of linear uncertain systems with uncertain parameters satisfying no special restrictions, such as the above-mentioned ones. Through the matrix nonsingularity analysis (Tseng et al., 1994), it is known that the system stabilized by a state feedback controller has a robust stability bound on the uncertain parameters, which can be computed from the structured singular value of a composite matrix formed by known system and uncertainty structural information. Based on this result, a set of linear matrix inequalities (LMI) are formulated to meet the requirement of obtaining a stabilizing state feedback gain for the uncertain system, and a convex objective function is established for the purpose of maximizing the robust stability bound. The overall optimization problem, with the welcome convex property, can be solved easily with many different efficient numerical procedures.
u(t) = -Kx(t)
(2) mxn
is to be used, where K E R is the constant gain matrix. Then the closed-loop system can be written as p
x(t) = [Aa - BaK +
L k;(A; -
B;K)]x(t)
(3)
;=1
Stability robustness analysis of this kind of systems have been addressed by many researchers (Tesi and Vicino, 1990; Su and Fong, 1993; Tseng et al., 1994) to obtain diverse uncertainty bounds for preserving stability. Through the method of matrix nonsingularity analysis (Tseng et ai., 1994), we can derive the following lemma:
This paper is organized as follows. In Section 2, the general problem to be studied is formulated. Section 3 derives the convex optimization problem for finding a state feedback gain with the largest robust stability bound. In Section 4, an example is provided to illustrate the design scheme. Finally, this paper ends with a conclusion in Section 5.
Lemma 1: System (3) is asymptotically stable for Ik;1 < 'Y, i = 1,'" ,p, ifthere exists P > 0 and X = -K P such that
5 = -(AaP + PA6
Throughout this paper, we shall use the following notations. J.1.(M) stands for the structured singular value of the matrix M with respect to a set of block diagonal uncertainty matrices of a fixed structure (Doyle, 1982). For an n x n matrix A, .A(A) represents its eigenvalue set, with .A;(A) denoting the ith member (in any order), and i1(A) denotes the maximal singular value, i.e., i1(A) = Jmax; .A;(ATA). The symbol ® refers to the Kronecker product (Weinmann, 1991). Finally, for a symmetric matrix A we use A > 0 « 0) to indicate that A is positive (negative) definite. Similar notations are used for semi-definiteness.
+ BoX + X T Bn > 0
(4)
and
(5) where
[M M... M] (p
(6)
blocks)
2. PROBLEM FORMULATION Consider an uncertain plant described by the linear state-space model
and J.1.( Ms) stands for the structured singular value of the composite matrix Ms with respect to a set of uncertainty structure matrices in diagonal form (Doyle et al., 1991; Tseng et al., 1994). Proof: Robust asymptotic stability of (3) can be guaranteed by the existence of a P > 0 such that the Lyapunov inequality
p
x(t) = (Aa
+ L k;A;)x(t) ;=1 p
+ (Ba + L
k;B;)u(t)
(1)
;=1 p
m where x(t) E Rn, u(t) E R are the state
[Aa - BaK +
L k;(A; -
B;K)] P
+
;=1
and input vectors of the plant, respectively, Aa, Ba represent the nominal plant dynamics, and k;, i 1, ... ,p, are real uncertain parameters with known structural information matrices Ai, B; of suitable dimensions. It is assumed that the nomi-
p
P [Aa - BaK
=
+ L k;(A; - B;K)]T < 0 (8) ;=1
holds for a range of k;'s. To use the matrix nonsingularity analysis method, it is helpful to rearrange 314
the above equation as
T/>O P> 0
(A o - BoK)P + P(A o - BoK)T +
5>0
p
L k;[(A; -
B;K)P
+ P(A; -
B;K)TJ < 0
(9)
In particular,
;=1
Substitution of X = - K P leads to (AoP+ PA6
(14)
+ BoX +XT B6)+
p
L k;(A;P + PAT + B;X + X T BT) < 0
(10)
which is a block matrix with identical blocks. To further simplify the first inequality constraint, which involves (14), we give another lemma.
;=1
If AaP + P A6 + BoX + x T BB < 0 for some P > o and X, then the parametric uncertain matrix (10) is nonsingular when all k;'s are zero. The proof follows by a straightforward application of the method developed in (Tseng et al., 1994).
Z
7]
-I - Z p
(pxp blocks)
>0
(16)
(17)
With the help of the eigenvalue property of the Kronecker product, we obtain
A;j
(11)
([: . • :]) A;
which is adopted here. To indirectly maximize the robust stability bound in (5), it is our choice to minimize o'. Thus the original problem is changed to
([
1
1
]) Aj(Z)
(18)
1
1
Furthermore, since
i]
(12)
= 1
subject to P > 0 and 5 > 0
(19)
and
By the definition of 0'( Ms), the above optimization problem can be transformed into the problem of minimizing the maximal eigenvalue of the matrix M; Ms subject to some matrix inequality constraints:
i1,.,)
J
P,X
(15)
Proof: Obviously,
There are many upper bounds for f.l(M s ), but some are too complicated to be useful for our purpose. By the definition of the structured singular value, it is well known that
mm T/
>0
,
is equivalent to
3. MAIN RESULTS
P,X
Z] :
,1- [ :
Note that when a pair of matrices P > 0 and X satisfying (4) and (10) can be found for Ik;1 < " i = 1, .. " p, with some, > 0, the state feedback gain J{ = _Xp- 1 can be used. Moreover, if, in (5) can be maximized by a proper selection of P and X, then the robust stability bound, i.e., the allowable variation range of k;'s can be maximized. However, because the quantity in (5) is not concave with respect to P and X, the maximization is usually difficult to achieve. Below we focus on forming a convex minimization problem which lets us maximize a lower bound of (5).
min O'(Ms )
nxn
R and Z E R
Lemma 2: For T/, pE
(13)
subject to T/1 - M; Ms > 0 315
(20)
:]
Lemma 4: IfY E R if and only if
(21)
) ={p,O"O}
nxn
is symmetric, then Y
TTYT> 0
pxp
>0 (28)
whenever TERn xn is nonsingular. which implies With the help of Lemma 3 and Lemma 4, we can construct a series of suitable t and T to obtain a sufficient condition for the inequality constraint (25). The result is stated in the following theorem.
= {pAi(Z), i = 1,···, n} U {O,···, O}
(22)
Theorem 1:
The proof of this lemma is completed by using the definition of positive definiteess of symmetric matrices.
25- ]
T1 T2
0
25- ]
0 T1
Utilizing the special form of (14) and applying Lemma 2, we see that the first inequality constraint in (13) can be transformed into
>0
(29)
>0
(30)
25- ] Tp
T2
1]
Tp
p
implies p
"1] - 2)A i P + PAT + BiX + X T BT). P
52
i=l
5- 2 . (AiP
+ PAT + BiX + X T BT) > 0
0
T1 T2
52
Tp
Tp
1]
52
(23)
0 T1
To simplify expressions, we define
T2
p
(24) Proof: Using Lemma 4 and the following transformation matrix
such that (23) can be rewritten as TJ - [T T -] 1 2
P
~]
]
T,].
...
]
p
o we see that
> 0 (25)
]
0 T1
-]
25
By using the fact that 52 > 0 for any 5 > 0 and the Schur complement formula, (25) can be further rewritten as 52
0
52
>0 0
T1
T2
-]
Tp
Tp
1]
nxn
(26)
tTYt> 0 whenever
nxm
25
Tp
Tp
1]
5
p
0 0
0 0
>0
has rank m. 316
(33)
5
0
(27)
t ER
., .
]
p
is symmetric, then Y
T1
(32)
is equivalent to (29). Again by using Lemma 4 and the transformation matrix
Now, to have a linear term in 5 rather than the nonlinear term 52 in (26), we resort to two additional well-known lemmas: Lemma 3: IfY E R implies
>0
0
T1 T2
52
(31)
]
0
0
]
we see that (32) is equivalent to
S-1
mal control method (Anderson and Moore, 1990), an optimal gain matrix J{opt = [-0.11 -1.78] can be obtained with identity weighting matrices. The robust stability bound is 0.34 from Lemma 1. In comparison, we obtain a robust controller J{robu.t = [1.11 -5.01] using the proposed method, and the corresponding robust stability bound is 0.41. Obviously, the stability robustness is improved.
0
1
0
S-1
0
S-1
0
0
S-1
S~
0
T1
0
...
0
T1
S2
Tp
Tp
!J.1 p
>0
(34)
Now, using Lemma 3 and the transformation matrix
5. CONCLUSION In this paper, we formulate an LMI optimization problem for finding robust stabilizing state feedback gains for parametric uncertain systems. The uncertainties do not need to satisfy special restrictions, and the proposed method tries to find a controller to maximize the robust stability bound. With a little modification, it can be shown that the same method can handle time-varying uncertain parameters.
(35)
we see that (34) implies (30). This completes our proof. Since
25- I
0
25- I
T1 T2
R=
REFERENCES
(36)
0 T1
T2
25- I Tp !LI Tp p
Anderson, B.D.O. and J .B. Moore (1990). Optimal Control-Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs, NJ. Doyle, J .C. (1982). Analysis of feedback systems with structured uncertainties. lEE Proc. Pt. D, 129, 242-250. Doyle, J.C., A. Packard and K. Zhou (1991). Review of LFTs, LMls, and J.L. Conference on Decision and Control, Brighton, 1227-1232. Gutman, S. and Z. Palmor (1982). Property of min-max controllers in uncertain dynamical systems. SIAM J. Control Opt., 20, 850. Petersen, I.R. and C.V. Hollot (1986). A Riccati equation approach to the stabilization of uncertain systems. Automatica, 22, 397-411. Su, J.H. and I K. Fong (1993). Robust stability analysis of linear continuous/discrete-time systems with output feedback controllers. IEEE Trans. Automat. Contr., 38, 11541158. Tesi, A. and A. Vicino (1990). Robust stability of state-space models with structured uncertainties. IEEE Trans. Automat. Contr., 35, 191-195. Tseng, C.L., I K. Fong and J.H. Su (1994). Analysis and applications of robust nonsingularity problem using the structured singular value. IEEE Trans. Automat. Contr., 39, 2118-2122. Weinmann, A. (1991). Uncertain Models and Robust Control. Springer-Verlag, New York.
is a symmetric matrix that depends affinelyon the optimization variables P and X, (29) is a convex constraint. Thus we have converted the original robust controller design problem into a standard LMI optimization problem: mm TJ
(37)
P,X
subject to
TJ>O R>O P>O
(38)
5>0
(40)
(39)
From the optimal solution p. and X·, a robust state feedback gain J{. = -X· . (p.)-l can be obtained. Note that the constraint (40) is a necessary condition of (38), thus it can be removed without altering the solution.
4. AN EXAMPLE Consider the following uncertain system with static state feedback control:
X=[-1+k 1 +k 2 1 + 2k 1
U
=
-J{x
-1 ]x+[1-k 2 ]U(41) 2 -- 2k 2 -3
(42)
If we use the standard linear quadratic opti317
ROBUST STATE FEEDBACK DESIGN OF PARAMETRIC UNCERTAIN SYSTEMS VIA LINEAR MATRIX INEQUALITY
Cheng Wen, I-Kong Fong, and Shin-Hao Lu
Department of Electrical Engineering National Taiwan University Taipei, Taiwan 10617 Republic of China E-mail: [email protected] Fax: 886-2-3660449
Abstract: This paper is devoted to the design ofrobust state feedback controllers for a class of linear uncertain systems. It is assumed that in the system and input matrices there are linear uncertain parameters, which satisfy no special restrictions, such as the matching condition or norm-bounded constraints. Through the matrix nonsingularity analysis, it is known that systems stabilized by a state feedback controller have robust stability bounds on the uncertain parameters, which can be computed from the structured singular value of a composite matrix. Based on this result and linear matrix inequalities, we form a convex optimization problem so that stabilizing controllers can be found to maximize the robust stability bounds. Keywords: Uncertain linear systems, Robust control, Convex programming, State feedback, Structured singular value
1. INTRODUCTION
ion are considered. In the literature, it can be seen that the stabilization problem of this category of uncertain models has attracted a considerable amount of attention. A long existing approach assumes that uncertain matrices in the system to be stabilized satisfy the so-called matching conditions (Gutman and Palmor, 1982). However, these conditions are quite restrictive to the structure of uncertain matrices, and limit the positions where uncertain parameters can appear as well as how those parameters are related. Attempts have also been made to relieve the matching conditions (Petersen and Hollot, 1986), but the resulting controller design method depends on a trial and error procedure, and the considered norm-bounded uncertainties are still subject to some constraints.
For many years, standard linear control schemes have been successfully applied 10 different kinds of practical engineering problems. Often the plant characteristics are described by approximate mathematical models, because it is difficult, if not impossible, to accommodate all conceivable uncertainties in a reasonably simple model. However, to design a robust and precise system, it becomes clear that effects of various unavoidable uncertainties in the plant must be taken into account. This paper deals with the problem of controller design for system model containing parametric uncertainties. To be more specific, linear state-space models with uncertain parameters entering system and input matrices in a linear fash313
nal system is controllable, but there are no special restrictions on A; and'B;, which may be zero if k; does not appear in the state or input matrix. To avoid trivial situations, we assume that at least one of A; and B; is nonzero for each i. Suppose the static state feedback controller
This paper is devoted to the design of robust state feedback controllers for a class of linear uncertain systems with uncertain parameters satisfying no special restrictions, such as the above-mentioned ones. Through the matrix nonsingularity analysis (Tseng et al., 1994), it is known that the system stabilized by a state feedback controller has a robust stability bound on the uncertain parameters, which can be computed from the structured singular value of a composite matrix formed by known system and uncertainty structural information. Based on this result, a set of linear matrix inequalities (LMI) are formulated to meet the requirement of obtaining a stabilizing state feedback gain for the uncertain system, and a convex objective function is established for the purpose of maximizing the robust stability bound. The overall optimization problem, with the welcome convex property, can be solved easily with many different efficient numerical procedures.
u(t) = -Kx(t)
(2) mxn
is to be used, where K E R is the constant gain matrix. Then the closed-loop system can be written as p
x(t) = [Aa - BaK +
L k;(A; -
B;K)]x(t)
(3)
;=1
Stability robustness analysis of this kind of systems have been addressed by many researchers (Tesi and Vicino, 1990; Su and Fong, 1993; Tseng et al., 1994) to obtain diverse uncertainty bounds for preserving stability. Through the method of matrix nonsingularity analysis (Tseng et ai., 1994), we can derive the following lemma:
This paper is organized as follows. In Section 2, the general problem to be studied is formulated. Section 3 derives the convex optimization problem for finding a state feedback gain with the largest robust stability bound. In Section 4, an example is provided to illustrate the design scheme. Finally, this paper ends with a conclusion in Section 5.
Lemma 1: System (3) is asymptotically stable for Ik;1 < 'Y, i = 1,'" ,p, ifthere exists P > 0 and X = -K P such that
5 = -(AaP + PA6
Throughout this paper, we shall use the following notations. J.1.(M) stands for the structured singular value of the matrix M with respect to a set of block diagonal uncertainty matrices of a fixed structure (Doyle, 1982). For an n x n matrix A, .A(A) represents its eigenvalue set, with .A;(A) denoting the ith member (in any order), and i1(A) denotes the maximal singular value, i.e., i1(A) = Jmax; .A;(ATA). The symbol ® refers to the Kronecker product (Weinmann, 1991). Finally, for a symmetric matrix A we use A > 0 « 0) to indicate that A is positive (negative) definite. Similar notations are used for semi-definiteness.
+ BoX + X T Bn > 0
(4)
and
(5) where
[M M... M] (p
(6)
blocks)
2. PROBLEM FORMULATION Consider an uncertain plant described by the linear state-space model
and J.1.( Ms) stands for the structured singular value of the composite matrix Ms with respect to a set of uncertainty structure matrices in diagonal form (Doyle et al., 1991; Tseng et al., 1994). Proof: Robust asymptotic stability of (3) can be guaranteed by the existence of a P > 0 such that the Lyapunov inequality
p
x(t) = (Aa
+ L k;A;)x(t) ;=1 p
+ (Ba + L
k;B;)u(t)
(1)
;=1 p
m where x(t) E Rn, u(t) E R are the state
[Aa - BaK +
L k;(A; -
B;K)] P
+
;=1
and input vectors of the plant, respectively, Aa, Ba represent the nominal plant dynamics, and k;, i 1, ... ,p, are real uncertain parameters with known structural information matrices Ai, B; of suitable dimensions. It is assumed that the nomi-
p
P [Aa - BaK
=
+ L k;(A; - B;K)]T < 0 (8) ;=1
holds for a range of k;'s. To use the matrix nonsingularity analysis method, it is helpful to rearrange 314
the above equation as
T/>O P> 0
(A o - BoK)P + P(A o - BoK)T +
5>0
p
L k;[(A; -
B;K)P
+ P(A; -
B;K)TJ < 0
(9)
In particular,
;=1
Substitution of X = - K P leads to (AoP+ PA6
(14)
+ BoX +XT B6)+
p
L k;(A;P + PAT + B;X + X T BT) < 0
(10)
which is a block matrix with identical blocks. To further simplify the first inequality constraint, which involves (14), we give another lemma.
;=1
If AaP + P A6 + BoX + x T BB < 0 for some P > o and X, then the parametric uncertain matrix (10) is nonsingular when all k;'s are zero. The proof follows by a straightforward application of the method developed in (Tseng et al., 1994).
Z
7]
-I - Z p
(pxp blocks)
>0
(16)
(17)
With the help of the eigenvalue property of the Kronecker product, we obtain
A;j
(11)
([: . • :]) A;
which is adopted here. To indirectly maximize the robust stability bound in (5), it is our choice to minimize o'. Thus the original problem is changed to
([
1
1
]) Aj(Z)
(18)
1
1
Furthermore, since
i]
(12)
= 1
subject to P > 0 and 5 > 0
(19)
and
By the definition of 0'( Ms), the above optimization problem can be transformed into the problem of minimizing the maximal eigenvalue of the matrix M; Ms subject to some matrix inequality constraints:
i1,.,)
J
P,X
(15)
Proof: Obviously,
There are many upper bounds for f.l(M s ), but some are too complicated to be useful for our purpose. By the definition of the structured singular value, it is well known that
mm T/
>0
,
is equivalent to
3. MAIN RESULTS
P,X
Z] :
,1- [ :
Note that when a pair of matrices P > 0 and X satisfying (4) and (10) can be found for Ik;1 < " i = 1, .. " p, with some, > 0, the state feedback gain J{ = _Xp- 1 can be used. Moreover, if, in (5) can be maximized by a proper selection of P and X, then the robust stability bound, i.e., the allowable variation range of k;'s can be maximized. However, because the quantity in (5) is not concave with respect to P and X, the maximization is usually difficult to achieve. Below we focus on forming a convex minimization problem which lets us maximize a lower bound of (5).
min O'(Ms )
nxn
R and Z E R
Lemma 2: For T/, pE
(13)
subject to T/1 - M; Ms > 0 315
(20)
:]
Lemma 4: IfY E R if and only if
(21)
) ={p,O"O}
nxn
is symmetric, then Y
TTYT> 0
pxp
>0 (28)
whenever TERn xn is nonsingular. which implies With the help of Lemma 3 and Lemma 4, we can construct a series of suitable t and T to obtain a sufficient condition for the inequality constraint (25). The result is stated in the following theorem.
= {pAi(Z), i = 1,···, n} U {O,···, O}
(22)
Theorem 1:
The proof of this lemma is completed by using the definition of positive definiteess of symmetric matrices.
25- ]
T1 T2
0
25- ]
0 T1
Utilizing the special form of (14) and applying Lemma 2, we see that the first inequality constraint in (13) can be transformed into
>0
(29)
>0
(30)
25- ] Tp
T2
1]
Tp
p
implies p
"1] - 2)A i P + PAT + BiX + X T BT). P
52
i=l
5- 2 . (AiP
+ PAT + BiX + X T BT) > 0
0
T1 T2
52
Tp
Tp
1]
52
(23)
0 T1
To simplify expressions, we define
T2
p
(24) Proof: Using Lemma 4 and the following transformation matrix
such that (23) can be rewritten as TJ - [T T -] 1 2
P
~]
]
T,].
...
]
p
o we see that
> 0 (25)
]
0 T1
-]
25
By using the fact that 52 > 0 for any 5 > 0 and the Schur complement formula, (25) can be further rewritten as 52
0
52
>0 0
T1
T2
-]
Tp
Tp
1]
nxn
(26)
tTYt> 0 whenever
nxm
25
Tp
Tp
1]
5
p
0 0
0 0
>0
has rank m. 316
(33)
5
0
(27)
t ER
., .
]
p
is symmetric, then Y
T1
(32)
is equivalent to (29). Again by using Lemma 4 and the transformation matrix
Now, to have a linear term in 5 rather than the nonlinear term 52 in (26), we resort to two additional well-known lemmas: Lemma 3: IfY E R implies
>0
0
T1 T2
52
(31)
]
0
0
]
we see that (32) is equivalent to
S-1
mal control method (Anderson and Moore, 1990), an optimal gain matrix J{opt = [-0.11 -1.78] can be obtained with identity weighting matrices. The robust stability bound is 0.34 from Lemma 1. In comparison, we obtain a robust controller J{robu.t = [1.11 -5.01] using the proposed method, and the corresponding robust stability bound is 0.41. Obviously, the stability robustness is improved.
0
1
0
S-1
0
S-1
0
0
S-1
S~
0
T1
0
...
0
T1
S2
Tp
Tp
!J.1 p
>0
(34)
Now, using Lemma 3 and the transformation matrix
5. CONCLUSION In this paper, we formulate an LMI optimization problem for finding robust stabilizing state feedback gains for parametric uncertain systems. The uncertainties do not need to satisfy special restrictions, and the proposed method tries to find a controller to maximize the robust stability bound. With a little modification, it can be shown that the same method can handle time-varying uncertain parameters.
(35)
we see that (34) implies (30). This completes our proof. Since
25- I
0
25- I
T1 T2
R=
REFERENCES
(36)
0 T1
T2
25- I Tp !LI Tp p
Anderson, B.D.O. and J .B. Moore (1990). Optimal Control-Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs, NJ. Doyle, J .C. (1982). Analysis of feedback systems with structured uncertainties. lEE Proc. Pt. D, 129, 242-250. Doyle, J.C., A. Packard and K. Zhou (1991). Review of LFTs, LMls, and J.L. Conference on Decision and Control, Brighton, 1227-1232. Gutman, S. and Z. Palmor (1982). Property of min-max controllers in uncertain dynamical systems. SIAM J. Control Opt., 20, 850. Petersen, I.R. and C.V. Hollot (1986). A Riccati equation approach to the stabilization of uncertain systems. Automatica, 22, 397-411. Su, J.H. and I K. Fong (1993). Robust stability analysis of linear continuous/discrete-time systems with output feedback controllers. IEEE Trans. Automat. Contr., 38, 11541158. Tesi, A. and A. Vicino (1990). Robust stability of state-space models with structured uncertainties. IEEE Trans. Automat. Contr., 35, 191-195. Tseng, C.L., I K. Fong and J.H. Su (1994). Analysis and applications of robust nonsingularity problem using the structured singular value. IEEE Trans. Automat. Contr., 39, 2118-2122. Weinmann, A. (1991). Uncertain Models and Robust Control. Springer-Verlag, New York.
is a symmetric matrix that depends affinelyon the optimization variables P and X, (29) is a convex constraint. Thus we have converted the original robust controller design problem into a standard LMI optimization problem: mm TJ
(37)
P,X
subject to
TJ>O R>O P>O
(38)
5>0
(40)
(39)
From the optimal solution p. and X·, a robust state feedback gain J{. = -X· . (p.)-l can be obtained. Note that the constraint (40) is a necessary condition of (38), thus it can be removed without altering the solution.
4. AN EXAMPLE Consider the following uncertain system with static state feedback control:
X=[-1+k 1 +k 2 1 + 2k 1
U
=
-J{x
-1 ]x+[1-k 2 ]U(41) 2 -- 2k 2 -3
(42)
If we use the standard linear quadratic opti317