Dept. of Aeronautics and Astronautics. because the structure of the closed-loop system has not been

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1 LMI Synthesis of Parametric Robust H 1 Controllers 1 David Banjerdpongchai Durand Bldg., Room 110 Dept. of Electrical Engineering banjerd@isl.stanford.edu Jonathan P. How Durand Bldg., Room Dept. of Aeronautics and Astronautics howjo@sun-valley.stanford.edu Stanford University, Stanford CA 905 Abstract his paper presents a new algorithm for designing full order LI controllers for systems with real parametric uncertainty. he approach is based on the robust L gain analysis of the Lur'e system using Popov analysis and multipliers. he core algorithm, previously applied to the robust H performance synthesis problem, is shown to be applicable to the robust controller design with the H 1 cost. Although the performance metrics are dierent, we demonstrate that the same solution algorithm based on LMI synthesis leads to a very effective and ecient technique for real parametric robust H 1 control design. Furthermore, it is dicult to compare robust H controllers to =K m designs, but in this work we provide insights into the issue of conservatism for various robust H 1 control approaches, in particular, the Popov controller synthesis, the robust H 1 design, and the =K m synthesis. he detailed analysis of these approaches is demonstrated on a exible structure benchmark problem. Keywords: Lur'e system, real parametric uncertainty; L gain; Popov controller synthesis; bilinear matrix inequality; linear matrix inequality. 1 Introduction Robust H 1 control problems for complex/real parametric uncertainty have been studied in detail since the introduction of the structured singular value,, bydoyle [1] and the multivariable stability margin, K m,by Safonov []. It was shown that the optimally scaled singular values produce a nonconservative estimate of the structured singular value. A hybrid of the H 1 control theory and the diagonal scaling techniques for the =K m synthesis has proven to be eective for designing robust controllers for systems with complex uncertainty. he synthesis based on the D{K iteration was originally devised by Dolye [] and Safonov []. One major drawback with the D{K iteration approach is that it requires curve tting approximations after each D iteration which can signicantly increase the designer's input during the controller synthesis. Due to the diculties in curve tting for the complex and real parametric uncertainty case, Safonov and Chiang [5] showed that the curve-tting from the design procedure can be eliminated. he full =K m synthesis problem using Bilinear Matrix Inequalities (BMI's) is 1 his research was supported by Ananda Mahidol Foundation and in part by AFOSR under F Author to whom all correspondence should be addressed. el: (15) -98; Fax: (15) -8. formulated in Refs. [, ]. A key benet of the BMI approach is that the compensator architecture (reduced order, decentralized control) can be included in the design framework. However, the problem size in this formulation is quite large because the structure of the closed-loop system has not been fully exploited. Moreover, the problem of how to optimally select the basis of the scaling multipliers is still unresolved. In this paper we investigate the =K m synthesis problem by applying the Popov absolute stability analysis to the Lur'e system [8]. he design objective of minimizing an upper bound of the L gain, i.e., robust H 1 performance for real parametric uncertain systems, naturally leads to BMI's. El Ghaoui and Balakrishnan [9] propose an iterative solution procedure for these BMI's using a two-stage optimization process, called the V{K iteration. We have already successfully applied an extension of this algorithm to parametric robust H control design problem with Popov multipliers [10] and generalized multipliers [11]. his paper presents a similar algorithm for the =K m synthesis problem. A unique feature of this work is that we use the same core algorithm to solve both the parametric robust H and H 1 synthesis problems. Our problem statement is similar to the one in Refs. [, ], but we restrict our attention to the system subject to sector bounded nonlinear uncertainty. Moreover, we take advantage of the closed-loop system structure to eliminate some design parameters from the problem formulation using a simple algebraic technique [1]. his well-known approach signicantly reduces the problem size and the number of design parameters. However, the coupling in the BMI is not completely removed when the multipliers are added to the design problem. Hence, an iterative algorithm is still required, but it is quite distinct from the D{K iteration for the =K m synthesis. For example, some variables are shared between the two main stages of our iterative solution and we conjecture that this plays an important role in the eciency and robustness of the solution approach. A slight computational advantage in our design framework is that the overbound of the robust performance can be simultaneously minimized over the design parameters. his allows us to bypass the - iteration in the previous =K m design approach. hus, while the multipliers in this paper are not as general as the ones in =K m synthesis, this technique oers an alternative to the well-known D{K iteration based synthesis algorithms. Our procedure also eliminates the curve-tting of the real structured singular value. In addition, the problem size is smaller than the BMI's of the original formulation of the =K m synthesis. With further investigation, these combined benets p. 1

2 could lead to a more robust solution algorithm for the control synthesis of mixed uncertain systems and mixed performance objectives. Problem Statement We consider an LI system subject to sector bounded nonlinear uncertainty, i.e., a Lur'e system (see Figure 1), described by _x = Ax + B pp + B ww + B uu q = C qx + D qpp + D qww + D quu z = C zx + D zpp + D zww + D zuu (1) y = C yx + D ypp + D yww + D yuu p = (q); where x : R +! R n is the state, u : R +! R nu is the control input, w : R +! R nw is the disturbance input, y : R +! R ny is the measured output and z : R +! R nz is the performance output. p : R +! R np are the input/output of the nonlinear uncertainty. he nonlinear perturbation is assumed to satisfy the sector bound [0; 1], i.e., where := f : R np! R np ; (q) = [ 1(q 1);:::; np (q np )] ; where 0 i()= 1; 8 i = 1;:::;n pg. As discussed in Ref. [1, page 19], a loop transformation can be used to handle the more general sector condition i i()= i. he description of the Lur'e system also includes an important class of uncertain systems described by _x =(A +A)x + B ww + B uu; A U; where U := fa R nn : A = B pdc q; D = diag( 1;:::; np ); where i [0; 1]; 8 i =1;:::;n pg. In control theory, this is referred to as the system subject to real parametric uncertainty [1, 1]. his special case of the Lur'e system (1) occurs when the functions i are linear, i.e., i() = i, where i [0; 1]; 8 i =1;:::;n p. o signicantly simplify the analysis and synthesis, we assume D zp, D zw, D qp, D qw, and D qu are identically zero. he objective of this paper is to design a strictly proper full order LI controller using Popov absolute stability theory for the system (1) such that the robust stability of the system is achieved and an overbound of the L gain is minimized. p w u G K q z y Figure 1: Elements of the robust synthesis problem Let U R np. U? is dened as an orthogonal complement of U, i.e., U U? = 0 and [U U?] is of maximum rank. L n is the Hilbert space of square-integrable signals R dened over R + with n components, i.e., w L n 1 satisfying w wdt<1. 0 L n is often abbreviated as L. A causal n-input n-output operator F : R n! R n is said to be L stable if there exist 0 and such that kfwk kwk + ; 8w L ; () where kk is dened as the L norm. he L gain of F is dened as the smallest such that () holds for some..1 Popov Robust H 1 Performance Analysis he Popov robust stability analysis is based on Lyapunov functions of the form V (x) =x Px+ n p X i=1 Z Ci;q x i i() d () where C i;q denotes the i th row ofc q. hus the data describing the Lyapunov function are the matrix P and the scalars i, i =1;:::;n p. We require P > 0 and i 0, which implies that V (x) x Px > 0 for nonzero x. For the case when i() = i, i.e., linear or real parametric uncertainty, the Lyapunov function will have the form V (x) = x (P + C q DC q)x, where D = diag( 1;:::; np ) and = diag( i;:::; np ). his Lyapunov function is referred to as a parameter-dependent Lyapunov function [1, 1]. For the nonlinear system (1), the robust performance is derived from the L gain, i.e., the RMS gain. While the exact L gain of the system (1) is dicult to compute, its upper bound can be easily computed as shown in the following theorem. heorem 1 ([1]) If there exists a Lyapunov function of the form (), :=diag np i=1 (i) 0, := diagnp i=1 (i) 0, and > 0 satisfying A P + PA+ PB p+ PB w Cz C z A Cq +Cq Bp P + C qb p+ C qb w C qa + C q Bp Cq, B wp B w C q, I 0 5 0; () then the upper bound on the L gain is nite and can be obtained by solving the optimization problem of minimizing over the variables, P,, and, i.e., minimize subject to (); P>0; 0; 0: Proof. See [1, page 1]. Note that while Ref. [1] states the convex optimization technique for analyzing the robust H 1 performance, it provides no insight onhow to solve the synthesis problem. In the following subsection, we will use this Popov robust H 1 performance analysis as a tool to design robust compensators.. Popov Controller Synthesis Our design goal is to nd a strictly proper full order LI controller that minimizes the upper bound of the L gain derived in the preceding subsection. he controller is of the form _x c = A cx c + B cy; u = C cx c; () where x c : R +! R n is the controller state; A c, B c, and C c are constant matrices of appropriate size. he closed-loop system of the Lur'e system (1) and the LI controller (), shown in Figure 1, is described by where _~x = A~x ~ + Bpp ~ + Bww ~ q = C ~ q ~x + D ~ qpp + D ~ qww z = C ~ z ~x + D ~ zpp + D ~ zww p = (q); ~A ~ Bp ~ Bw ~C q ~ Dqp ~ Dqw ~C z ~ Dzp ~ Dzw 5 = (5) () p.

3 A B uc c B p B w B cc y A c + B cd yuc c B cd yp B cd yw C q D quc c D qp D qw C z D zuc c D zp D zw 5 ; and ~x =[ x x c ]. hen it is straightforward to compute the upper bound of the L gain for the closed-loop system (). We note that () is equivalent to ~A P ~ + P ~ A+ ~ P ~ Bp+ ~ P ~ Bw ~ ~C z C ~ z A ~ C ~ q + C ~ q ~B p P ~ + Cq ~ Bp+ ~ Cq ~ Bw ~ Cq ~ A ~ + Cq ~ B ~ p C ~ q, ~B w ~ P ~ B w ~ C q, I 5 0: (8) In summary, the design objective is to solve the non-convex optimization problem over the parameters, ~ P,,, Ac, B c, and C c. minimize subject to (8); ~ P>0; 0; 0: (9) Design Procedure his section closely parallel the developments in Refs. [15, 10, 11]. As will be shown, controllers are developed in two main steps. Observing the structure of the compensator parameters in (8), the rst step is to eliminate some controller parameters from the problem formulation (9). We then solve for the remaining variables, and use these results to construct the controllers. An iterative algorithm is required to calculate the controllers, but in the process the procedure capitalizes on the very ecient design tools that are available for solving Linear Matrix Inequalities (LMI's) [1, 1]. he resulting compensators are full-order, and cannot include architecture constraints. However, the solution procedure is very robust, which signicantly reduces the user workload. Furthermore, this approach is easily expandable to include other sophisticated analysis tests such as analysis for systems with mixed uncertainty or linear time-invariant uncertainty..1 Controller Elimination We rst note that the controller matrix A c only appears in (8). hus it is possible to reduce the number of variables in the problem by eliminating A c.o proceed, we dene A B ~A 0 := uc c ; ~ 0 J := : B cc y B cd yuc c I hen ~ A can be written as ~ A = ~ A0 + ~ JAc ~ J and we rewrite (8) as ~G + VA c V + UA cu < 0; (10) where ~ G, V, and U are dened as ~G = ~A ~ 0 P + P ~ A0+ ~ P ~ Bp+ ~ P ~ Bw ~ ~C z C ~ z A ~ 0 C ~ q + C ~ q B w ~ P ~ B w ~ C q ~B p P ~ + Cq ~ Bp+ ~ Cq ~ Bw ~ Cq ~ A ~ + Cq ~ B ~ p C ~ q, 5 ; V := J ~ 0 0 ; U := J ~ P ~ 0 0 : Applying the Elimination Lemma [1, page ], we rst note that the orthogonal complements of V and U are as following. V? = ~ J? I I 5 ; U? = ~ P,1 ~ J? I I 5 : hen, it follows that (10) holds if and only if V? ~ GV? < 0; U? ~ GU? < 0: (11) o proceed, we partition P ~ and its inverse Q ~ as P M ~P = M ; Q ~ = P ~ Q N,1 = R N S ; (1) where P and Q R nn. N is related to P, Q, and M in the form satisfying N =(I, QP )M,.We dene Y := C cn and Z := MB c. hen, after some algebra, it can be shown that (11) is equivalent to F11 F1 F1 F 1 F F F 1 F I 5 < 0; H 11 H 1 H 1 H 1 H 1 H H 0 H 1 H, I 0 H 1 0 0,I 5 < 0; (1) where F 11 = PA + ZC y +(PA + ZC y) + C z C z; F 1 = PB p + ZD yp + A C q +C q ; F 1 = PB w + ZD yw; F = C qb p +(C qb p), ; F = C qb w; H 11 = AQ + B uy +(AQ + B uy ) ; H 1 = B p +(AQ + B uy ) C q + QC q ; H 1 = B w;h 1 =(C zq+d zuy ) ;H =C qb p + (C qb p), ; and H =C qb w. By the Completion Lemma [18], for every Q>0;P Q,1, the lower right n n block of P ~ and that of Q ~ in (1) can be shown to satisfy R = M (P, Q,1 ),1 M and S = N (Q, P,1 ),1 N respectively. he conditions P ~ > 0 and P ~ Q ~ = I with P ~ written in (1) imply P I 0: (1) I Q Restricting (1) to be positive denite, we are eectively searching for full-order controllers (i.e., of order n) [15]. We observe that the second matrix inequality in (1) is BMI, i.e., there are product terms involving (Q, Y ) and (, ). his is a direct consequence of optimizing both the compensator parameters (related to Q and Y ) and the analysis multiplier (, ) simultaneously. Note that if and are xed, then (1) is an LMI in Q and Y. Similarly, ifq and Y are xed, then (1) is an LMI in and. he positive denite constraint of P ~ and Q ~ is implied from the existence of symmetric matrices W and X such that X Z 0 0 Z P I 0 0 I Q Y 0 0 Y W 5 > 0: (15) In summary, after eliminating A c from the formulation the optimization problem (9) is equivalent to: minimize subject to (1); (15); 0; 0: (1). Controller Reconstruction Given that there exist, P, Q, Y, Z, W, X, and satisfying (1), we can construct a controller as follows. First we construct the quadratic term of the Lyapunov function, i.e., P ~, such that the condition (8) holds. he set of the p.

4 quadratic part of the closed-loop Lyapunov functions is parameterized by Eq. (1), where M is an arbitrary invertible matrix. Because M corresponds to a change of coordinates in the controller states x c, the choice of M has no eect on the controller transfer function [15]. After constructing the Lyapunov function, the set of input/output controller matrices (B c and C c) can be parameterized by B c = M,1 Z and C c = Y (I, PQ),1 M: With, ~ P,,, Bc, and C c determined, it suces to nd A c satisfying the condition (10), which can then be formulated as an LMI problem in A c.. Algorithm It has already been shown that BMI problems are NP-hard, and it is thought to be rather unlikely that there is a polynomial time algorithm to compute the optimal solutions [19]. Since there are product terms involving compensator parameters and the Popov parameters, our approach to solving the non-convex optimization problem is based on an iterative procedure. he proposed algorithm, which we call the V{K iteration, basically alternates between three dierent LMI problems, i.e., (9) with xed compensator parameters, (1) with xed multiplier parameters, and (10). he rst LMI problem, considered as the V step or analysis step, is to solve (9) with xed compensator parameters (A c, B c, and C c) which yields Popov multiplier parameters ( and ). For the K step or synthesis step, the second and third LMI problems are solved. he solution parameters of the second LMI problem, i.e., (1) with xed multiplier parameters, implicitly includes the input/output compensator matrices (B c and C c)asvariables. After obtaining B c and C c, the dynamics of the compensator A c can be computed by solving the third LMI problem (10). At this point a robust compensator, which guarantees the robust stability and satises the upper bound of the L gain, is completely calculated. We then repeat the procedure until the decrease in the upper bound of the L gain is suciently small. he solution algorithm to design a set of controllers with increasing robustness is briey summarized as the following: 1. Initialize the sector bound nonlinearities to be zero (a nominal system) and design the controller via H 1 controller synthesis.. Initialize and by by solving (9) where (A c;b c;c c) are xed.. Repeatf [Outer Loop] (a) Repeatf [Inner Loop] i. Solve the optimization problem (1) for (, P, Q, Y, Z, W, X) where (, ) are xed. hen compute P;Bc, ~ and C c by the Completion Lemma. ii. Compute A c by solving a feasibility LMI problem (10). iii. Compute and by by solving (9) where (A c;b c;c c) are xed. g [Inner Loop] Until stopping criterion satised. (b) Increase the sector bound nonlinearity to the next desired size and initialize and by the most recent values. g [Outer Loop] Until the desired robustness is achieved or the problem is infeasible. Remark 1. We note that this algorithm has been successfully applied to the parametric robust H control design problem with Popov multipliers [10] and generalized multipliers [11]. Although the robust performance metrics are dierent, the same solution procedure based on LMI synthesis is very eective and ecient for parametric robust H 1 control design. o be specic, the core algorithm is developed in two key steps. First, we eliminate the controller matrix A c from the problem formulation (9). hen, we solve for the remaining variables which are subsequently used to reconstruct the controller parameters. he V{K iteration is then used to compute the controller parameters. his shows a unique versatility of our solution algorithm for designing robust controllers, and may eventually lead to better insight into the relationship between these synthesis approaches. Remark. he procedure of alternating between the LMI problems is an iterative approach of solving a non-convex optimization problem, which is not guaranteed to converge in general. However, the same algorithm solving the parametric robust H synthesis has been analyzed for several examples in Ref. [0]. hese results show that, to the best of our knowledge, the algorithm does converge to the global optimal solution for the simple examples considered. However, much further analysis is required to generalize this statement. Each step of the iteration can be solved very eciently by a previously developed semidenite programming algorithm sp [1] and very easily coded using a user-friendly interface sdpsol [1]. Remark. An important distinction between the V{K iteration and the D{K iteration of the =K m synthesis is that in our approach there are shared variables between each iteration: specically, and ~ P are the common variables between the V step and K step (where ~ P appears as P, Q, Y, Z, W and X). However, for the D{K iteration, the D step (the robust analysis with or without curve tting) is entirely separate from the K step (the H 1 synthesis). We conjecture that these shared variables play a key role in the eciency and robustness of the convergence of this new algorithm to a local optimum, and are currently investigating this point further. Numerical Example he parametric H 1 control design algorithm is performed on the exible structural benchmark problem, which was previously considered for robust H control design [1, 10, 11]. he system is a cantilevered Bernoulli Euler beam with unit length and mass density, and stiness scaled so that the fundamental frequency is 1 rad/sec. he innite order dynamics of the beam are truncated at four modes, where w 1 =1 rad/sec, w =: rad/sec, w =1:55 rad/sec, w =:9 rad/sec and damping = 0:01. he changes in the system dynamics due to perturbations in the frequency of the third mode cause substantial variations in the system gain and phase in the 1, 5 rad/sec frequency range [1, 10]. he disturbance input, control input, sensor output and performance output are all collocated at the tip of the beam, and the frequency of the third mode of the system is considered to be uncertain. Note that this problem was chosen so that the realness of the parametric uncertainty in the design problem is accentuated. o proceed, a frequency dependent weighting function, W (s), was placed on the disturbance to the performance loop. For this study, wechoose W (s) = 10(s + 0)=(s + 00). Several controllers were designed by the Popov controller synthesis (PCS) approach described in x.. he controllers were p.

5 H1 Cost 1 0:9 0:8 0: H 1;0 PCS1 PCS PCS able : Achieved robust stability and H 1 performance for the robust H 1 control, =K m synthesis, and Popov controller synthesis with % guaranteed robust bounds. ype of % Change Lower Upper Control of Nominal Stability Stability Design H1 Cost Bound % Bound % H1 :8,5 > 00 =Km 1:5,5 5 PCS :0, 80, Percentage Change in the Uncertain Frequency Figure : Robust performance plots for Popov controllers with the guaranteed stability bounds 1,, and % and the nominal H 1 controller. designed for dierent levels of the frequency uncertainty (i.e., 1%, %, and %). With this reliable design technique, it is now feasible to undertake a comparison of the Popov controllers with the standard H 1 control technique. he curves in Figure are developed by computing the H 1 cost for the system with the given percentage change in the mode frequency. he controllers were designed using symmetric sector bounds, with the sizes (i.e., 1%) given in the gure legend. We rst note that the standard H 1 controller, labeled by H 1;0 in the gure, performs extremely well at the nominal frequency. However, it is clearly not robust to changes in the uncertain frequency because the H 1 controller was designed without a robust guarantee on the frequency change. As expected, for the Popov controllers the system is robustied to the parameter changes with a slight trade-o on the nominal performance. he plot illustrates that the guaranteed stability boundaries for Popov controllers are obtained and that the actual performance is quite at and asymmetric about the nominal frequency. his asymmetry was also observed in robust H control designs (see Refs. [10, 11]). able 1 summarizes the key points for the robust performance analysis of this plot: the percentage change of the nominal H 1 cost (i.e., the H 1 cost evaluated on the nominal system) for the Popov controllers compared with the nominal H 1 cost for the H 1 controller, and the lower (upper) achieved and guaranteed stability bounds. he achieved robust stability was determined by analyzing the closed-loop eigenvalues. Similar results are presented in Ref. [10] for the robust H case able 1: Robust stability and H 1 performance for Popov controllers with various robustness bounds. ype of % Change Lower Bound Upper Bound Control of Nominal of Stability % of Stability % Design H1 Cost Ach. Guar. Guar. Ach. H1;0 0, 0 0 PCS1 :,,1 1 PCS :0,, 80 PCS :8,1, 100 using weighting values that yield very similar loop properties (see detailed discussion in Ref. [1]). A thorough comparison of the H and H 1 controllers is quite dicult, but a comparison of the H and H 1 performance plots illustrates some interesting relationships. he rst observation is that the robust performance results are very similar with very at bottomed asymmetric curves. Furthermore, the changes in nominal performance are quite similar (i.e., relatively small), and both approaches show quite large gaps between the guaranteed and achieved stability bounds. he last characteristic is considered as a function of the conservatism in the Popov robust H 1 performance analysis, which has been improved for the H case in Ref. [11]. hus, using the core algorithm in x. we can design parametric robust H or H 1 controllers that yield a consistent closed-loop robust performance. We continue this discussion by directly comparing controllers from three dierent techniques: the robust H 1 control design, the =K m-synthesis via the D{K iteration with rst order scaling transfer function, and the Popov controller synthesis (PCS) using LMI synthesis. hese controllers are designed to guarantee the robust stability within % changes of the third mode frequency. he H 1 cost of the system with various percentage changes in the mode frequency for these controllers is shown in Figure. We rst note that the robust H 1 compensator assumes a full complex block of combined uncertainty and performance in the design methodology and its controller order is equal to the order of the nominal LI system plus that of the weighting function (8 + 1 = 9). On the other hand, the =K m synthesis exploits the structure of the uncertainty and performance blocks. As a consequence, it produces a higher order of the controller, i.e., the order of the nominal LI system augmented with the weighting function plus the order of rst order scaling function and its inverse (9 + ( ) = 15). he gure shows that all robust controllers achieve the desired stability bounds and that in the desired uncertainty region the Popov synthesis yields an improved H 1 performance, which ismuch lower than that of other approaches. Moreover, the actual performance for all controllers is quite at in this region of guaranteed robustness. able summarizes the key points for the achieved robust stability bounds and performance for this gure. For the Popov synthesis, the achieved lower stability bound is much closer to the lower guaranteed robustness bound and much smaller than that of the other designs. However, the achieved upper robustness bound is slightly larger than that of the =K m synthesis and much smaller comparing to the upper bound of the robust H 1 design. his robust stability and H 1 performance analysis provide insights into the issue of conservatism for these design techniques. Although the p. 5

6 H1 Cost 1: 1:1 0:9 0: H 1;0 H 1 =K m PCS, Percentage Change in the Uncertain Frequency Figure : Robust performance plots for the robust H 1 control, =K m synthesis, and Popov controller synthesis with the guaranteed stability bounds equal %. he performance analysis for the nominal H 1 controller is given as a reference. =K m controller is appropriate for systems with structured complex uncertainty, itobviously becomes conservative for the case of real parametric uncertainty. hus, the Popov controller synthesis provides a means to capture the realness of the uncertainty which results in a reduction of the conservatism in the control design. 5 Conclusions his paper presents an ecient and eective design technique for the real parametric =K m synthesis problem by applying the Popov robust H 1 performance analysis to a Lur'e system. As discussed, this approach oers several potential benets over the current D{K iteration and the BMI synthesis procedure. A unique feature of our approach is that the core solution algorithm can be used to solve both the parametric robust H and H 1 problems. We rst illustrate this approach by using the algorithm to design robust controllers for a simple system with real parametric uncertainty. A direct comparison of this approach with other robust H 1 control design techniques for systems with mixed uncertainty, such as=k m synthesis, indicates that the Popov controller synthesis yields less conservative designs. References [1] J. Doyle, \Analysis of Feedback Systems with Structured Uncertainties," IEE Proc., vol. 19-D, pp. {50, Nov [] M. G. Safonov, \Stability Margins of Diagonally Perturbed Multivariable Feedback Systems," IEE Proc., vol. 19-D, pp. 51{5, Nov [] J. Doyle, \Synthesis of Robust Controllers and Filters," in Proc. IEEE Conf. on Decision and Control, pp. 109{ 11, Dec [] M. G. Safonov, \L 1 Optimal Sensitivity vs. Stability Margin," in Proc. IEEE Conf. on Decision and Control, 198. [5] M. G. Safonov and R. Y. Chiang, \Real/Complex K m- Synthesis without Curve Fitting," in Control and Dynamic Systems (C.. Leondes, ed.), vol. 5, pp. 0{, New York: Academic Press, 199. [] M. G. Safonov, K. C. Goh, and J. H. Ly, \Control System Synthesis via Bilinear Matrix Inequalities," in Proc. American Control Conf., pp. 5{9, 199. [] K. C. Goh, J. H. Ly, L.urand, and M. G. Safonov, \=k m-synthesis via Bilinear Matrix Inequalities," in Proc. IEEE Conf. on Decision and Control, pp. 0{0, Dec [8] V. M. Popov, \Absolute Stability of Nonlinear Systems of Automatic Control," Automation and Remote Control, vol., pp. 85{85, 19. [9] L. El Ghaoui and V. Balakrishnan, \Synthesis of Fixed-structure Controllers via Numerical Optimization," in Proc. IEEE Conf. on Decision and Control, pp. 8{8, Dec [10] D. Banjerdpongchai and J. P. How, \Parametric Robust H Control Design Using LMI Synthesis," in he 199 AIAA Guidance, Navigation, and Control Conference, AIAA-9-, July 199. [11] D. Banjerdpongchai and J. P. How, \Parametric Robust H Control Design with Generalized Multipliers via LMI Synthesis," in Proc. IEEE Conf. on Decision and Control, pp. 5{0, Dec [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control heory, vol. 15 of Studies in Applied Mathematics. Philadelphia, PA: SIAM, June 199. [1] W. Haddad and D. Bernstein, \Parameter-Dependent Lyapunov Functions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis," in Proc. IEEE Conf. on Decision and Control, pp. {9, {, Dec [1] J. P. How, Robust Control Design with Real Parameter Uncertainty using Absolute Stability heory. PhD thesis, Massachusetts Institute of echnology, Cambridge, MA 019, Feb [15] L. El Ghaoui and J. P.Folcher, \Multiobjective Robust Control of LI Control Design for Systems with Unstructured Perturbations," Syst. Control Letters, vol. 8, pp. { 0, June 199. [1] L. Vandenberghe and S. Boyd, sp: Software for Semidenite Programming. User's Guide, Beta Version. K.U. Leuven and Stanford University, Oct [1] S.-P. Wu and S. Boyd, sdpsol: A Parser/Solver for Semidenite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide, Beta Version. Stanford University, June 199. [18] A. Packard, K. Zhou, P. Pandey, and G. Becker, \A Collection of Robust Control Problems Leading to LMI's," in Proc. IEEE Conf. on Decision and Control, pp. 15{150, [19] O. oker and H. Ozbay, \On the NP-Hardness of Solving Bilinear Matrix Inequalities and Simultaneous Stabilization with Static Output Feedback," in Proc. American Control Conf., pp. 55{5, June [0] D. Banjerdpongchai and J. P. How, \Convergence Analysis of Parametric Robust H Synthesis Algorithm," in Proc. IEEE Conf. on Decision and Control, Dec Submitted. [1] S. C. O. Grocott, J. P. How, and D. W. Miller, \Comparison of Robust Control echniques for Uncertain Structural Systems," in AIAA Guidance, Navigation, and Control Conference, pp. 1{1, Aug p.