REFERENCES. Analysis and Synthesis of Robust Control Systems Using Linear Parameter Dependent Lyapunov Functions

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1 1984 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER 26 While several algorithms have been developed for computing robust feedback matrices for complete pole assignment in standard first-order state-space and also for the matrix second-order systems, such algorithms for partial pole-placement (which is more practical for large and sparse systems) are rare. In this note, a computationally simple least-squares based algorithm for robust partial pole placement in a cubic matrix pencil arising from modelling of vibrating structures with aerodynamics effects is proposed. It should be emphasized that here we formulate the rank-two update method (for complex poles) into a least square problem with linear constraints which can be solved easily by elementary linear algebra techniques. Moreover, our new algorithm is not an iterative method. For instance, if we want to assign k pairs of complex poles, i.e., 2k number of eigenvalues then we only need to do exact k steps in our method. Clearly, this new algorithm is much simpler and more efficient than the only other algorithm proposed so far for this problem. REFERECES [1] S. P. Bhattacharyya and E. desouza, Pole assignment via Sylvester s equation, Syst. Control Lett., no. 4, pp , [2] R. Byers and S. G. ash, Approaches to robust pole assignment, Int. J. Control, no. 49, pp , [3] E. K. Chu and B.. Datta, umerical robust pole assignment for second-order systems, Int. J. Control, no. 64, pp , [4] B.. Datta, umerical Methods for Linear Control Systems Design and Analysis. ew York: Elsevier, 23. [5] B.. Datta, S. Elhay, and Y. M. Ram, Orthogonality and partial pole assignment for the symmetric definite quadratic pencil, Linear Alg. Appl., no. 257, pp , [6] B.. Datta, S. Elhay, Y. M. Ram, and D. R. Sarkissian, partial eigenstructure assignment for the quadratic pencil, J. Sound Vibrat., vol. 23, pp , 2. [7] B. Datta, W.-W. Lin, and J.-. Wang, Robust and minimum gain partial pole assignment for a third order system, in Proc. IEEE Conf. Dec. Control, 23, pp [8] B. Datta, W.-W. Lin, and J.-. Wang, Robust partial pole assignment for vibrating systems with aerodynamic effects, in Proc. IEEE Conf. Dec. Control, 24. [9] B.. Datta and D. R. Sarkissian, Theory and computation of some inverse problems, Contemp. Math., vol. 28, pp , 21. [1] B.. Datta and D. R. Sarkissian, Partial eigenvalue assignment in linear systems; Existence, uniqueness and numerical methods, in Proc. Math. Theory etworks Syst., 22. [11] B.. Datta, umerical Linear Algebra and Applications. CA: Brooks/Cole, 25. [12] W. R. Ferng, W. W. Lin, D. J. Pierce, and C.-S. Wang, onequivalence transformation of lambda-matrix eigenproblems and model embedding approach to model tuning, umer. Lin. Alg. Appl., vol. 8, pp. 53 7, 21. [13] G. H. Golub and C. F. Van Laon, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, [14] F. Hoblit, Gust Loads on Aircraft: Concepts and Applications 1988, AIAA Education Series. [15] D. Inman, Vibration with Control, Measurement, and Stability. Englewood Cliffs, J: Prentice-Hall, [16] J. Kautsky,. K. ichols, and P. Van Dooren, Robust pole assignment in linear state feedback, Int. J. Control, vol. 41, pp , [17] J. Kautsky and. K. ichols, Robust multiple eigenvalue assignment by state feedback in linear systems, in Linear Algebra and Its Role in Systems Theory. Providence, RI: AMS, 1985, pp [18] L. H. Keel, J. A. Fleming, and S. P. Bhattacharyya, Minimum norm pole assignment via Sylvester s equation, Contemp. Math., vol. 47, pp , [19] R. K. Kevin and S. P. Bhattacharyya, Robust and well conditioned eigenstructure assignment via Sylvester s equation, Optm. Control Appl. Meth., vol. 4, pp , [2] W.-W. Lin and J.-. Wang, Partial pole assignment for the vibrating system with aerodynamic effect, umer. Linear Algebra Appl., vol. 11, pp , 24. [21]. K. ichols, Robustness in partial pole placement, IEEE Trans. Autom. Control, vol. AC-32, no. 8, pp , Aug [22]. K. ichols and J. Kautsky, Robust eigenstructure assignment in quadratic matrix polynomials: onsingular case, SIAM J. Matrix Anal. Appl., vol. 23, pp , 21. [23] J. Qian and S. Xu, Robust partial eigenvalue assignment problem for the second-order system, J. Sound Vibrat., vol. 282, pp , 25. [24] Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback, IEEE Trans. Autom. Control, vol. 33, no. 2, pp , Feb [25] A. L. Tits and Y. Yang, Globally convergent algorithms for robust pole assignment by state feedback, IEEE Trans. Autom. Control, vol. 41, no. 1, pp , Oct [26] A. Varga, Robust and minimum norm pole assignment with periodic state feedback, in Proc. IEEE Conf. Dec. Control, 1998, pp [27] A. Varga, Robust pole assignment techniques via state feedback, in Proc. IEEE Conf. Dec. Control, 2, pp [28] J.-. Wang, S.-H. Chou, Y.-C. Chen, and W.-W. Lin, Pole assignment for a vibrating system with aerodynamic effect, SIAM J. Control Optim., pp , 24. Analysis and Synthesis of Robust Control Systems Using Linear Parameter Dependent Lyapunov Functions José C. Geromel and Rubens H. Korogui Abstract This note provides sufficient robust stability conditions for continuous time polytopic systems. They are obtained from the Frobenius Perron Theorem applied to the time derivative of a linear parameter dependent Lyapunov function and are expressed in terms of linear matrix inequalities (LMI). They contain as special cases, various sufficient stability conditions available in the literature to date. As a natural generalization, the determination of a guaranteed cost is addressed. A new gain parametrization is introduced in order to make possible the state feedback robust control synthesis using parameter dependent Lyapunov functions through linear matrix inequalities. umerical examples are included for illustration. Index Terms Linear matrix inequalities (LMIs), linear systems, robust control design, robust stability and performance. I. ITRODUCTIO Robust stability and performance of polytopic systems has received special attention during the last decades. The main motivation for its development was the possibility to analyze and to design control strategies to cope with uncertain parameters, that is, parameters that can not be considered precisely known. Classes of linear systems denominated affine and polytopic came to light in order to put in evidence the linear dependence of the model with respect to the uncertain parameters. The first stability criterion denominated quadratic stability condition was established using a quadratic in the state Lyapunov function and independent of the unknown parameters, see [3] for a discussion on this point. aturally, this condition provides conservative results but it is Manuscript received May 25, 25; revised ovember 14, 25 and April 18, 26. Recommended by Associate Editor A. Garulli. This work was supported by Conselho acional de Desenvolvimento Científico e Tecnológico (CPq) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil. The authors are with DSCE/School of Electrical and Computer Engineering, UICAMP, Campinas, SP, Brazil ( geromel@dsce.fee.unicamp.br). Digital Object Identifier 1.119/TAC /$2. 26 IEEE

2 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER very simple to be checked numerically, for instance, with the help of a linear matrix inequality (LMI) solver. Besides, it is amendable for robust control synthesis. The paper [6] was one the first to deal with quadratic in the state and affine in the parameters Lyapunov function. It has been followed by [1] to cope with polytopic continuous time systems specifically. Stability conditions using quadratic in the state and polytopic in the parameters Lyapunov functions have been developed using multipliers [5] and structural properties [7], [12]. The common characteristic of these conditions is that they provide better results than the quadratic stability condition but at expense of more computational burden. The result of [14] that apply to polytopic continuous time systems performs, as claimed by the authors, better than the methods mentioned so far. However, the involved formulas are extremely complicated and due to this, synthesis appears very difficult to be done preserving convexity. Finally, very recently, [2], [4], [9], and the references therein have introduced Lyapunov functions which are quadratic in the state but polynomial (of fixed order) in the parameters. For the important class of quadratic in the state and quadratic in the parameters Lyapunov functions, the reader is requested to see [15]. Once again, more accurate results are obtained at expense of more important computational effort. In this area, one of the most important theoretical result certainly is that reported in [2] which is necessary and sufficient for robust stability. This note aims at providing stability conditions for continuous time polytopic systems using quadratic in the state and linear in the parameters Lyapunov functions. Even though similar results can be obtained for discrete time polytopic systems (see [8]) and robust D-stability [1], [13] they will not be presented in this note. The conditions are simple to be established and contains as special cases the ones available in the literature, in special those provided in [1], [14], and related papers. In addition, numerical examples put in evidence that the new stability conditions have a good tradeoff between numerical complexity and precision. Finally, in this context, the robust state feedback synthesis design problem is addressed and it is shown how to get the gain matrix from a new parametrization that allows to express the stability conditions by means of LMIs. We believe that the gain parametrization proposed is general enough to be applied to many other design problems, including robust state feedback control design based on polynomially dependent Lyapunov functions. The notation is standard. Capital letters denote matrices, small letters denote vectors and small Greek letters denote scalars. For real matrices or vectors ( ) indicates transpose. For symmetric matrices, X> ( ) indicates that X is positive definite (positive semidefinite). The sets of real numbers is denoted by. For square matrices trace(x) denotes the trace function of X being equal to the sum of its eigenvalues and, for the sake of easing the notation of partitioned symmetric matrices, the symbol () denotes generically each of its symmetric blocks. Finally, we exhaustively use the matrix inequality introduced in [12], namely G P 1 G G + G P (1) which holds for all square matrices G and P > of compatible dimensions. II. PRELIMIARIES AD MOTIVATIO Given matrices A i 2 n2n with i = 1;...; let us call A the convex hull A := cofa1;...;a g: (2) In this note, our main concern is to provide conditions under which all matrices of A are stable in the continuous time sense, that is conditions which assure that all matrices belonging to A have n eigenvalues with negative real part. For short, in the affirmative case, we say that A is Hurwitz stable. Since any A 2 A can be written as A() := A i i for some := [1... ] 2 3 where 3 is the simplex 3:= 2 : i i =1 (3) it is natural to introduce the linear parameter dependent Lyapunov matrix P () := Pii with Pi being symmetric and positive definite matrices for all i = 1;...;. oticing that P () > for all 2 3 the following result is an immediate consequence of the continuous-time Lyapunov Lemma. Lemma 1: Let V ij be symmetric matrices and consider the quadratic inequality V ij i j 8 2 3: (4) j=1 If there exist symmetric matrices P j > 2 n2n and V ij 2 n2n satisfying the LMIs A ip j + P j A i V ij < ; i;j =1;...; (5) then A is Hurwitz stable whenever (4) holds. Proof: From (5), we have A() P () +P ()A() < V ij i j (6) j=1 for all 2 3. Hence, using (4) the result follows. Clearly, the condition (5) of Lemma 1 is expressed by means of 2 LMIs and 2 + symmetric matrix variables. Since the 2 LMIs are very simple, the complexity of the stability test strongly depends on how (4) is verified. In the sequel, we will show that many of the results available in the literature stems from a particular choice of matrices V ij which, of course, reduces the total number of matrix variables to.it is important to keep in mind that in the previous lemma, matrices V ij for i; j =1;...; are not restrict to exhibit any definiteness property. III. MATRIX QUADRATIC POLYOMIALS In this section, our goal is to determine conditions under which (4) holds. The main limitation to be kept in mind is that they must be expressed linearly with respect to the symmetric matrices V ij making possible the use of LMI solvers to test for robust stability. The result of the next theorem is one of the main contribution of this note and follows from the use of the well-known Frobenius Perron Theorem, applied to Metzler matrices, see [11] for details. Theorem 1: If the symmetric matrices V ij 2 n2n are such that then (4) holds. V ij + V ji ; 1 j<i (7) (V ij + V ji ) ; j =1;...; (8)

3 1986 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER 26 Proof: Consider x 2 n an arbitrary but fixed vector. Simple algebraic manipulations enable us to write x V ij i j x = 1 2 5(x) (9) j=1 where the elements of the matrix 5(x) 2 2 denoted by ij(x) are given by are apparent. The first is that we have imposed 5(1) to be a Metzler matrix in order to use the Frobenius Perron Theorem to get the conditions for the feasibility of the quadratic inequality (4) and the second one is that the nonnegativity of (1) was imposed through the determination of an upper bound as indicated in (11) and not on its exact value. As it is shown in Theorem 2 when V ij + V ji for all i; j =1;...; are constrained to be diagonal matrices these two undesirable aspects can be eliminated yielding a less conservative condition. Theorem 2: Consider the symmetric matrices V ij be such that ij (x) =x (V ij + V ji )x: (1) Hence, matrix 5(x) is symmetric and due to (7) all its off diagonal elements are nonnegative. Matrix 5(x) is in fact a symmetric Metzler matrix. The celebrated Frobenius Perron Theorem applies and allows the conclusion that the maximum eigenvalue of 5(x) denoted (x) satisfies 5(x)(x) = (x)(x) for some (x) 2 3 implying that the quadratic inequality (4) holds if and only if (x) ; 8x 2 n. ow, taking into account (8), we get (x) = ij (x)j(x) j=1 max ij (x) j=1;...; max j=1;...; x (V ij + V ji ) x 8x 2 n (11) proving thus the proposed theorem. Denoting the set of all symmetric matrices V ij satisfying the conditions (7) and (8) of Theorem 1 as V F, the following remarks are appropriate. First, it is obvious that the conditions V ij + V ji for all i; j =1;...; are sufficient to assure that (4) holds and they are not recovered by Theorem 1. However, taking into account that the simplex 3 contains nonnegative vectors only, it suffices to consider the less conservative conditions expressed by the equality constraints V ij +V ji = for all i; j = 1;...;, which obviously are included in the set V F. Second, Theorem 1 imposes constraints only on the sum V ij + V ji and not on each isolated matrix V ij. Since reversing the index i and j these matrices are the same, then the number of linear constraints can be reduced from 2 to ( +1)=2 by simple addition of the ijth to the jith inequality. In principle, this may introduce some additional difficulty to develop an algorithm for control synthesis. On the other hand, certainly at expense of some conservatism, matrices V ij can be a priori fixed. From Theorem 1, two possibilities are the most evident. I) Consider V ij + V ji = for all 1 j i. The set of all matrices V ij satisfying these linear equality constraints is denoted by V. II) Consider V ii = I for i =1;...; and V ij +V ji =2I=( 1) for all 1 j<i. The set of all matrices V ij satisfying these linear equality constraints is denoted by V1. It is important to notice that both particular choices indicated on items I and II are actually special cases of a more general condition which follows immediately from Theorem 1. III) Consider V ij + V ji = ij I for all 1 j i where ij for all 1 j < i and ij for j =1;...;. A careful reading of Theorem 1 puts in evidence that for the special set of matrices considered in cases I) III), two sources of conservatism V ij + V ji = ij I; 1 j i : (12) If the symmetric matrix with elements ij for i; j = 1;...; is negative semidefinite then inequality (4) holds. Proof: From (12), it is verified that for any x 2 n such that kxk 2 =1we get x V ij i j x = (13) j=1 and the result follows from the assumption 5. Denoting as V P the set of all symmetric matrices V ij satisfying the conditions of Theorem 2, taking into account that cases I) and II) are contained in case III and that conditions ij for all i 6= j = 1;...; and ij clearly imply that 5 (but the converse is not true in general) then it follows that V V P and V1 V P. From the numerical viewpoint the result of Theorem 2 does not increase significatively the computation burden required. Comparing with cases I) and II) only one additional LMI, corresponding to 5, has to be included. On the other hand, since case III satisfies Theorem 1 as well, for the same reason, we conclude that V V F and V1 V F. The sets V F and V P are not comparable. Indeed, the matrices of the set V F are not constrained to be diagonal but they must satisfy the mentioned nonnegative constraints (7). On the contrary, the matrices of the set V P must be diagonal but they do not need to satisfy the nonnegative constraints. These aspects are numerically illustrated in the next section. Under this framework, one of the best compromise between precision and numerical complexity is given in the next corollary of Theorem 1. Corollary 1: If there exist symmetric matrices P j > 2 Q i 2 n2n satisfying the LMIs n2n and A ip j + P j A i <Q i Q j ; i;j =1;...; (14) then A is Hurwitz stable. Proof: Defining V ij := Q i Q j then we have V ij + V ji =for all i; j =1;...;. Hence, the proof follows from Theorem 1. Clearly, the result of Corollary 1 is more conservative than the one provided by Theorem 1. However, its simplicity is remarkable since it is expressed by means of 2 linear constraints and only 2 matrix variables. The matrices Q i for all i =1;...; can be interpreted as slack variables which contribute to reduce conservatism. As already mentioned, similar results can be obtained for discrete time systems (for more details, see [8]) and robust D-stability [1], [13] provided the set V F is adequately defined with matrices of higher dimension V ij 2 2n22n for all i; j =1;...;. IV. EXAMPLES AD DISCUSSIO In this section, we compare our previous results using some examples borrowed from the literature. The comparison is done for polytopic systems using the stability conditions obtained from Lemma 1 with matrices V ij belonging to the set V where V is one of the sets V F, V P,

4 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER V and V 1 previously defined. In the same context, it is immediate to determine upper bounds to the H 2 and H 1 norms of transfer functions defined by the uncertain matrices (A();B();C();D()) with 2 3. otice that in all cases, the robust stability test and the guaranteed H 2 or H 1 performances can be evaluated with an LMI solver. We compare our results with those appearing in [1], [2], [4], [9], [14]. To this end, it is important to mention that from Lemma 1 and V = V the stability conditions of [1] are recovered, that is A ip j + P j A i + A jp i + P i A j < for all 1 j i is less restrictive than the stability conditions appearing in [1]. On the other hand, using again Lemma 1 and V = V 1 the result of [14] is obtained. In fact the stability conditions reduce to A ip i + P i A i < I for all i = 1;...; and A ip j + PjA i + A j P i + PiA j < 2I=( 1) for all 1 j<i which equal those given in [14]. The result on robust stability provided in [2] is important since, for the first time, necessary, and sufficient robust stability conditions derived with polynomially parameter dependent Lyapunov functions of arbitrary order are expressed by LMIs. In this section, the first problem solved concerns on the determination, for each given in a real domain, of the smallest H 2 guaranteed cost () such that k(si (A + r 1A 1 + r 2A 2 I)) 1 k 2 2 <() (15) for all (r 1;r 2) belonging to the box [1 1]2 [1 1]. Two cases have been considered, one with the matrices A, A 1 and A 2 provided in [2] and the other with these matrices given by A = A 2 = A 1 = : (16) 1 For each value of, the uncertain matrix in (15) is written as (2) with =4vertices. The following numerical experiments have been performed. We have determined the minimum value of such that robust stability is preserved. This amounts to determine the minimum value of such that the LMIs of Lemma 1 together with fv 11;...;V g2vare feasible. From the data provided in [2], in the four cases we have obtained the same value min = 5:24 which actually is the exact value of the robust stability margin. This value equals the one given in [2] but it is obtained with a quadratic in the parameters Lyapunov function. The same reference indicates that for the linear in the parameters Lyapunov function proposed, the minimum value of is 3:17. Considering (16), the four methods have given once again the exact stability margin min = 3:87, obtained by gridding. It can be verified that () defined in (15) satisfies () inf W ftrace(w ):W>P j > ; j =1;...;g (17) where matrices P 1 ;...;P are such that the linear matrix inequalities A ip j + P j A i + I V ij < for all i; j =1;...; hold with fv 11;...;V g 2 V. In this framework, the minimization of the right-hand side of (17) with respect to P 1 ;...;P gives (). Table I provides () calculated with matrices given in (16) and shows that the set VF provides the best figures, followed by VP, V and V 1 in this order. For comparison purpose, the last column of Table I provides, for each, the exact guaranteed H 2 cost which corresponds to the maximization of the left TABLE I GUARATEED H COST () hand side of (15) with respect to (r 1;r 2) 2 [1 1]2 [1 1]. The calculations have been performed by gridding. For both examples, the stability margin has been exactly calculated by the four methods. However, when the H 2 guaranteed cost is concerned, as expected, from Table I it is clear that Theorem 1 provides the best result but at expense of a more expressive computational effort needed to handle the set VF. Among the three other methods considered the one that presents a good compromise between the precision of the final solution and the computational effort need is the one of Theorem 2 which, in addition, contains the other two as particular cases. It is clear that the proposed stability conditions are not able to determine the exact stability margins in all cases. To illustrate this point, from the first example of [4] we have determined ^ =:463 for VF, VP, V and V 1 which is the exact value of the stability margin. However, for the second example of the same reference we have obtained ^ = 1:53 for VF and ^ = 1:47 for VP, V and V 1 while the exact value is ^ = 2:22. Finally, for the first example proposed in [9] the exact value of the stability margin =1:75 such that the uncertain parameters are confined in the box [; ] 2 [; ] has been determined by all stability conditions with VF, VP, V, and V 1, respectively. V. STATE FEEDBACK DESIG In this section, our main purpose is to show how to linearize a quite general nonlinear matrix inequality that appears in many control design problems including H 2 and H 1 guaranteed cost minimization. In particular, the proposed gain parametrization is well adapted to deal with the robust stability conditions provided in the previous section. The problem to be dealt with is stated as follows: Given real matrices A i 2 n2n and B i 2 n2m with i = 1;...; determine, if one exists, a state feedback gain matrix K 2 m2n such that the set A(K) :=cofa i + B i K : i =1;...;g (18) is Hurwitz stable. Theorem 3: Let the symmetric matrix W 2 n2n and matrix B 2 be given. If there exist matrices L 2 m2n, U 2 n2n and positive definite symmetric matrices P 2 n2n and J 2 n2n such that the LMI W + BL + L B + J BL U U P U J < (19) holds, then U is nonsingular and K = LU 1 satisfies the nonlinear matrix inequality W + BKP + PK B < : (2) Proof: Since (19) implies that U +U > then U is a nonsingular matrix. Furthermore, the same inequality also implies that J> and as a consequence, perturbing it slightly if necessary, one can assume with no loss of generality that the matrix P U is nonsingular. Hence, using the inequality (1) we get U [(P U)J 1 (P U )] 1 U U +U (P U)J 1 (P U ) (21)

5 1988 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER 26 which together with the Schur Complement applied to the third column and third row of the LMI (19) yields W +BL+L B +J BL U [(P U )J 1 (P U )] 1 U < (22) and from the Schur Complement applied to the second column and second row of this inequality, we get W+BL+L B +J+BLU 1 (P U )J 1 (P U )U 1 L B <: (23) Finally, using again the inequality (1) and taking into account the parametrization K = LU 1 we have BLU 1 (P U )J 1 (P U )U 1 L B BLU 1 (P U )+(P U )U 1 L B J BKP BL + PK B L B J (24) which together with (23) gives (2) proving thus the proposed theorem. At this point some remarks are important. The first one is related to the conservativeness introduced by Theorem 3. To analyze this point, let us consider the particular case U = P > for which the matrix inequality (2) reduces to the linear matrix inequality W +BL+L B <. Imposing this condition to (19) and noticing that J can be set arbitrarily close to the null matrix we get W + BL + L B BLP 1 L B < (25) which approximates arbitrarily the former one for all P > with sufficiently large norm. Hence, for W fixed and P > a matrix variable to be determined, both inequalities are equivalent. However, when W depends on P which is precisely the case to be analyzed afterwards, the following corollary is useful to reduce conservatism. Corollary 2: The result of Theorem 3 remains valid if in the LMI (19), matrix P is replaced by P and K is replaced by LU 1 where is any positive scalar. Proof: The proof follows the same pattern as that of Theorem 3, being thus omitted. Making large enough the inequality (19) reduces to the inequality (2). Considering the parameter >in Theorem 3 we can say that it does not introduce any kind of conservatism when compared to the classical LMI arising for U = P > and allows us, in the general case, to linearize the inequality (2) through a parametrization that does not explicitly depend on the Lyapunov matrix P >. We are now in position to propose a solution to the state feedback control design problem introduced before. It is an immediate consequence of the matrix parametrization given by Theorem 3 and its Corollary. To ease the notation, we define the matrix functions 8 i (P; L) :=A i P + PA i + B i L + L B i (26) for all i =1;...;. Theorem 4: If there exist positive definite matrices fj 11;...;J g, fp 1;...;P g and matrices U and L of compatible dimensions such that the LMIs 8 i (P j ;L) V ij + J ij B i L U U P j U < (27) J ij for i; j = 1;...; are feasible for some symmetric matrices fv 11 ;...;V g 2 V F then, for K = LU 1 the set A(K) is Hurwitz stable. Proof: Applying the result of Theorem 3 to the LMI (27) we get (A i+b ik)p j +P j (A i+b ik) V ij < ; i;j =1;...; (28) for some symmetric matrices fv 11;...;V g 2 V F and the proof follows from Lemma 1. This theorem gives a sufficient condition for the existence of a state feedback matrix such that A(K) is Hurwitz stable. From the numerical viewpoint, the most important feature of Theorem 4 is that only linear matrix inequalities have to be handled and consequently a possible solution can be determined in polynomial time. The main computational effort focus on the 2 linear constraints (27) with matrix variables which is approximately twice the number of matrix variables present in the 2 bilinear constraints (28). Hence, the price to be payed is not prohibitive mainly because solving directly the bilinear inequalities (28) is a matter of great difficulty. Certainly at expense of some conservatism, some simplifications can be further introduced in the previous result. Indeed, imposing V ij = Q i Q j as in Corollary 1 and J ij = J j, the number of LMIs in Theorem 4 decreases, since the constraint fv 11;...;V g2v F is automatically satisfied, and the number of matrix variables also decreases to The number of matrix variables can be further reduced to 2 +2by simply imposing Q i = J i > for all i =1;...;.Of course, this new set of linear constraints makes the result more conservative but, at the same time, the basic features of the robust stability condition remain. Furthermore, a supplementary degree of freedom may be introduced in the linear inequalities (27), with a positive parameter 2 as indicated in Theorem 3 and its corollary. Doing this, the determination of >needs an additional line search procedure but Theorem 4 contains as a particular case the quadratic stability condition. Let us compare the previous results to the well known quadratic stability condition which states that if there exists a pair of matrices (P; L) with P > such that 8 i(p; L) < for all i = 1;... then for K = LP 1 the set A(K) is Hurwitz stable. Since the robust stability conditions presented before contain the quadratic one as a particular case, our main purpose is to give a measure of the improvement. To this end, let us consider a simple example characterized by =2and by the extreme matrices and A 1 = A 2 = d 12 3d d d 25 1 B 1 = B 2 = 6 6 (29) 6 (3) where it is noticed that the matrices A 1 and A 2 depend on a free parameter d 2. For each value of d the robust stability condition of Theorem 4 with J ij = J j and = 1 as indicated in Corollary 2 has been solved, yielding a state feedback matrix K assuring that A(K) is Hurwitz stable. Fig. 1 shows the degree of stability index (d) defined as the maximum real part of the eigenvalues of all matrices belonging to A(K). As indicated, the dashed line corresponds to the quadratic stability condition. It is verified that for d 4 the quadratic stability constraints are feasible providing thus a stabilizing state feedback

6 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 51, O. 12, DECEMBER Fig. 1. Degree of stability. gain K. However, for d slightly greater than four the quadratic stability constraints become infeasible. In the same figure, the solid line shows that the conditions provided by Theorem 4 are always feasible for all d 11:62. Hence, in this example, the stability conditions provided by Theorem 4 perform significantly better than the quadratic stability conditions. For d =11:62, the state feedback gain K =[17: :5487 1:9946 4:178 ] (31) assures that A(K) is Hurwitz stable but this fact can not be tested by means of a common Lyapunov function. [8] J. C. Geromel and R. H. Korogui, Matrix quadratic polynomials with application to robust stability analysis, in Proc. 5th IFAC Symp. Robust Control Design, Toulouse, France, 26. [9] D. Henrion, D. Arzelier, D. Peaucelle, and J. B. Lasserre, On parameter-dependent Lyapunov functions for robust stability of linear systems, in Proc. 43rd IEEE Conf. Decision Control, Paradise Island, Bahamas, 24, pp [1] V. J. S. Leite and P. L. D. Peres, An improved LMI condition for robust D-stability of uncertain polytopic systems, IEEE Trans. Autom. Control, vol. 48, no. 3, pp. 5 54, Mar. 23. [11] D. G. Luenberger, Introduction to Dynamic Systems : Theory, Models and Applications. ew York: Wiley, [12] M. C. de Oliveira, J. Bernussou, and J. C. Geromel, A new discretetime robust stability condition, Syst. Control Lett., vol. 37, no. 4, pp , [13] D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, A new robust D-stability condition for real convex polytopic uncertainty, Syst. Control Lett., vol. 4, no. 5, pp. 21 3, 2. [14] D. Ramos and P. L. D. Peres, An LMI condition for the robust stability of uncertain continuous-time linear systems, IEEE Trans. Autom. Control, vol. 47, no. 4, pp , Apr. 22. [15] A. Trofino and C. E. de Souza, Biquadratic stability of uncertain linear systems, IEEE Trans. Autom. Control, vol. 46, no. 8, pp , Aug. 21. Policy Iterations on the Hamilton Jacobi Isaacs Equation for State Feedback Control With Input Saturation Murad Abu-Khalaf, Frank L. Lewis, and Jie Huang VI. COCLUSIO In this note, we have provided sufficient conditions to guarantee that a matrix quadratic polynomial with an arbitrary although finite number of variables has invariant sign in a simplex. We believe that the reported conditions are useful in matrix analysis and in particular on the determination of robust performance bounds for problems involving polytopic parameter uncertainties. We have shown theoretically that several sufficient robust stability conditions available in the literature to date are special cases of the proposed ones. In our opinion, one of the main contributions of this note is a new matrix gain parametrization which is used to design stabilizing state feedback controllers from the proposed stability conditions and which is sufficiently general to cope with other classes of robust control design problems. This aspect and others including discrete time systems stability, performance analysis and control design are now under investigation. REFERECES [1] P. Apkarian and H. D. Tuan, Parameterized LMIs in control theory, SIAM Control Optim., vol. 38, no. 4, pp , 2. [2] P. A. Bliman, A convex approach to robust stability for linear systems with uncertain scalar parameters, SIAM Control Optim., vol. 42, no. 46, pp , 24. [3] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, [4] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach, IEEE Trans. Autom. Control, vol. 5, no. 3, pp , Mar. 25. [5] M. Dettori and C. W. Scherer, ew robust stability and performance conditions based on parameter dependent multipliers, in Proc. IEEE Conf. Decision Control, Sydney, Australia, 2, pp [6] P. Gahinet, P. Apkarian, and M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Autom. Control, vol. 41, no. 3, pp , Mar [7] J. C. Geromel, M. C. de Oliveira, and L. Hsu, LMI characterization of structural and robust stability, Linear Alg. Appl., vol. 285, pp. 69 8, Abstract An H suboptimal state feedback controller for constrained input systems is derived using the Hamilton Jacobi Isaacs (HJI) equation of a corresponding zero-sum game that uses a special quasi-norm to encode the constraints on the input. The unique saddle point in feedback strategy form is derived. Using policy iterations on both players, the HJI equation is broken into a sequence of differential equations linear in the cost for which closed-form solutions are easier to obtain. Policy iterations on the disturbance are shown to converge to the available storage function of the associated L -gain dissipative dynamics. The resulting constrained optimal control feedback strategy has the largest domain of validity within which L -performance for a given is guaranteed. Index Terms Controller saturation, H control, policy iterations, zero-sum games. I. ITRODUCTIO In this note, we derive the Hamilton Jacobi Isaacs (HJI) equation for systems with input constraints and then develop an algorithm based on policy iterations to solve the obtained HJI equation. Although the formulation of the nonlinear H1 control theory has been well developed, [4], [5], [7], [11], [17], and [19], solving the corresponding HJI equation remains a challenge. Several methods have been proposed to solve the HJI equation. When its solution is smooth, it can be deter- Manuscript received May 25, 25; revised December 4, 25, May 2, 26, and June 2, 26. Research supported by the ational Science Foundation under Grant ECS and by the Army Research Office under Grant W91F M. Abu-Khalaf and F. L. Lewis are with the Automation and Robotics Research Institute, The University of Texas at Arlington, Fort Worth, TX USA ( abukhalaf@arri.uta.edu; lewis@uta.edu). J. Huang is with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin,.T., Hong Kong ( jhuang@acae.cuhk.edu.hk). Digital Object Identifier 1.119/TAC /$2. 26 IEEE