energy for systems subject to sector bounded nonlinear uncertainty ë15ë. An extension of this synthesis that incorporates generalized multipliers to c

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1 Convergence Analysis of A Parametric Robust H Controller Synthesis Algorithm 1 David Banjerdpongchai Durand Bldg., Room 110 Dept. of Electrical Engineering banjerd@isl.stanford.edu Jonathan P. How Durand Bldg., Room 77 Dept. of Aeronautics and Astronautics howjo@sun-valley.stanford.edu Stanford University, Stanford CA 9405 Abstract his paper presents an iterative algorithm for solving the parametric robust H controller synthesis problem and analyzes the convergence properties of the algorithm on several examples. Iterative procedures are normally applied to a large class of robust control design problems in which the formulation naturally leads to bilinear matrix inequalities èbmisè. It is diæcult to make concrete statements about the behavior of these iterative algorithms, except that it is often conjectured that the cost in each step of the solution procedure is reduced, which implies that the algorithms should converge to a local minimum. Similar diæculties exist for the new LMIbased iterative algorithm that wehave recently proposed to solve the BMIs that occur in robust H control design. he eæectiveness of the new algorithm has already been demonstrated on several numerical examples. his paper adds an important component tothediscussion on the convergence of the new algorithm by verifying that it eæciently converges to the optimal solution. In the process, we provide some new key insights on the proposed design technique which indicate that it exhibits properties similar to the DíK iteration of the complex =K m -synthesis. 1 Introduction he bilinear matrix inequality èbmiè approach has been demonstrated to be eæective for solving the complexèreal =K m synthesis problems ë1, ë. he global optimization of BMIs is NP-hard, and it is unlikely that there is a polynomial time algorithm to compute the optimal solutions ë, 4ë. However, Goh et al. ë5, 6ë devised algorithms for solving the BMI problem via a straightforward method, i.e., using the currently available LMI tools, such as ë7, 8, 9, 10ë. he approach is to alternately minimize the performance cost subject to BMI constraints with respect to some variables where the other variables 1 his research was supported by Ananda Mahidol Foundation and in part by AFOSR under grant F Author to whom all correspondence should be addressed. el: è650è 7-44; Fax: è650è are æxed, and vice versa. hey have demonstrated that the iterative technique based on the BMI framework improves the guaranteed lower bounds to multivariable stability margins of the closed-loop system by 10è over the corresponding results from the DíK iteration with no increase in the controller order. Akey advantage of the BMI technique is that it enables control engineers to address several open problems of the robust control synthesis, namely the complexèreal =K m synthesis via dynamical scalings or multipliers, the æxed order control synthesis, and the decentralized controller architecture. El Ghaoui and Balakrishnan ë11ë have concurrently proposed a similar approach using the numerical optimization in the synthesis of æxed-structure controllers and it has been demonstrated to work well on simple examples. However, on complicated objectives, such as control designs to minimize an H cost function, the iterative technique has been found to converge very slowly, ifat all. Furthermore, a thorough study of the convergence behavior of the iterative algorithms in Refs. ë5, 6, 11ë remains to be done in order to better understand the eæectiveness and eæciency of these approaches. he robust controller synthesis algorithm was improved by El Ghaoui and Folcher leading to a more systematic design approach for systems with unstructured uncertainty ë1ë and structured uncertainty ë1ë. he design objectives considered include robust stability, robust H performance, robust settling time, and robust input peak bound. A heuristic algorithm ë14ë, which is a local optimization, is used to solve these BMI problems. he algorithm produces dynamic, output-feedback controllers of the order equal to or smaller than the order of the nominal system. However, the single quadratic Lyapunov function used in Refs. ë1, 1, 14ë can be a source of signiæcant conservatism in the robust control design for systems with real parametric uncertainty. We have extended the design procedure in Refs. ë1, 1ë to develop a new synthesis algorithm for systems with parametric uncertainty. his controller synthesis combines the robust stability analysis with Popov multipliers and robust performance bounds on the total output

2 energy for systems subject to sector bounded nonlinear uncertainty ë15ë. An extension of this synthesis that incorporates generalized multipliers to capture real parametric uncertainty is discussed in ë16ë. We also take advantage of the closed-loop system structure to eliminate some design parameters from the problem formulation and use an iterative procedure to calculate the remaining variables. Exploiting the structure of design parameters in the synthesis formulation appears to signiæcantly improve the convergence of the algorithm. he main result of this paper is to numerically verify that the iterative algorithm presented in Ref. ë15ë eæciently converges to the optimal solution. In the process, we compute a global optimum for this BMI problem by an exhaustive search. While the exhaustive search is not particularly eæcient, this process is similar in many ways to the proposed global BMI optimizations ë5, 6ë and provides a simple check of the iterative solution. Problem Statement We consider the Lur'e system described by _x = Ax + B p p + B w w + B u u q = C q x + D qp p + D qw w + D qu u z = C z x + D zp p + D zw w + D zu u y = C y x + D yp p + D yw w + D yu u p = èqè; è1è 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, z : R +! R nz is the performance output, q : R +! R np and p : R +! R np are the inputèoutput of the nonlinear uncertainty. he nonlinear perturbation is assumed to satisfy the sector bound ë0; 1ë, i.e., æ where 8 9 é : R np! R np ; = æ := èqè =ë 1 èq 1 è;:::; np èq np èë ; : ; : where 0 i èè= 1; 8 i =1;:::;n p : he description of the Lur'e system also includes an important class of uncertain systems described by where 8 é U := : _x = èa +æaèx + B w w + B u u z = C z x + D zw w + D zu u y = C y x + D yw w + D yu u æa U; æa R næn :æa = B p DC q ; D = diagèæ 1 ;:::;æ np è; where æ i ë0; 1ë; 8 i =1;:::;n p : 9 = ; : èè In control theory, èè is referred to as the system subject to real parametric uncertainty ë17, 18ë. his special case of the Lur'e system occurs when the functions i are linear, i.e., i èè =æ i, where æ i ë0; 1ë; 8 i =1;:::;n p. For well-posedness, we will assume that D zw is identically zero, and to signiæcantly simplify the analysis and synthesis, we assume D zp, D qp, D qw, and D qu are identically zero. We are interested in ænding a strictly proper full order LI controller of the form _x c = A c x c + B c y; u = C c x 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 design objective is to guarantee the robust stability and minimize an upper bound of the worst case H performance. As discussed in Ref. ë15ë, by applying the Popov robust H performance analysis to the closed-loop Lur'e system, the control design problem can be formulated as a non-convex optimization over the variables ~ P, æ,, A c, B c and C c, i.e., minimize r ~ B w h ~P + ~ C q æ ~ Cq i ~Bw subject to ~ P é 0; æ 0; 0; ~A ~ P + ~ P ~ A + ~ C z ~ Cz ~ P ~ Bp + ~ A ~ C q æ+ ~ C q ~B p ~ P +æ~ Cq ~ A + ~ Cq æ ~ Cq ~ Bp + ~ B p ~ C q æ, where 6 4 ~A ~ Bp ~ Bw ~C q ~ Dqp ~ Dqw 7 5 = ~C z Dzp ~ Dzw ~ A B u C c B p B w 6 B c C y A c + B c D yu C c B c D yp B c D yw 7 4 C q : C z D zu C c 0 0 Solution Procedure 0; èè We note that èè is a BMI problem, i.e., there are product terms involving the analysis variables è ~ P,æ,and è and compensator variables èa c, B c, and C c è in the performance objective and constraints. Observing the structure of the compensator parameters in the last matrix inequality in èè, the ærst step of the design procedure is to eliminate some controller parameters from the problem formulation. We then solve for the remaining variables, and use these results to construct the controllers. An iterative algorithm is used to calculate the compensators, but in the process the procedure capitalizes on the very eæcient design tools ë7, 8, 9, 10ë that are available for solving LMI problems. he resulting compensators are full-order, and cannot include architecture constraints. However, the solution procedure is very robust which reduces the user workload. Furthermore, this approach is easily expandable to include other sophisticated analysis tests such as one in Ref. ë16ë..1 he VíK Iteration Since the controller matrix A c only appears in the last matrix constraint in èè, it is possible to reduce the number of variables in the problem by eliminating A c. Applying the Elimination Lemma ë19, page ë, it is not

3 diæcult to show that èè is equivalent to minimize r subject to 4 where F11 F 1 F 1 F Bw D yw X Z 0 Z P I 0 I Q é 0; P + C q æc q 6 4 Z Z X 5 é 0; æ 0; 0; H 11 H 1 H 1 H 1 H 0 H 1 0,I Bw D yw 7 5 é 0; è4è F 11 = PA+ A P + ZC y + C y Z + C z C z ; F 1 = PB p + ZD yp + A C q æ+c q ; F =æc q B p + B p C q æ, ; H 11 = AQ + QA + B u Y + Y B u ; H 1 = B p +èqa + Y B u èc q æ+qc q ; H 1 = QC z + Y D zu ; H =æc q B p + B p C q æ, : Given that there exist P; Q; Y; Z; X; æ and satisfying è4è, we can construct a controller by the following steps. First, by the Completion Lemma ë0ë, we compute P ~, which is parameterized by P ~ = P M M M, where èp, Q,1 è,1 M is an arbitrary M invertible matrix. Because M corresponds to a change of coordinates in the controller states, the choice of M has no eæect on the controller transfer function ë1, 1ë. After P ~ is constructed, we proceed by computing the inputèoutput controller matrices èb c and C c è, which are parameterized by B c = M,1 Z and C c = Y èi,pqè,1 M. With P ~,æ,, Bc, and C c determined, it suæces to ænd A c satisfying the last matrix constraint in èè which can then be formulated as an LMI feasibility problem in A c, i.e., ænd A c satisfying ~ G + VA c U + UA c V é 0; è5è where ~ G, U, and V are deæned as ~G := " ~A 0 ~ P + ~ P ~ A0 + ~ C z ~ Cz ~ P ~ Bp + ~ C q + ~ A 0 ~ C q æ ~B p ~ P + ~ Cq +æ~ Cq ~ A0 æ ~ Cq ~ Bp + ~ B p ~ C q æ, V := æ J ~ 0 æ æ æ ; U := ~J P ~ 0 ; A B ~A 0 := u C c ; ~ 0 J := : B c C y B c D yu C c I Since there are product terms involving compensator parameters and the Popov parameters in èè and è4è, our approach to solving the non-convex optimization problems is based on an iterative procedure. he proposed algorithm, which we call the VíK iteration, basically alternates between three diæerent LMI problems, i.e., èè with æxed compensator parameters, è4è with æxed multiplier parameters, and è5è. he ærst LMI problem, considered as the V iteration or analysis iteration, is to solve èè with æxed compensator parameters èa c, B c, and C c è è ; which yields Popov multiplier parameters èæ and è. For the K iteration or synthesis iteration, the second and third LMI problem are solved. he solution parameters of the second LMI problem, i.e., è4è with æxed multiplier parameters, implicitly includes the inputèoutput compensator matrices èb c and C c è as variables. After obtaining B c and C c, the dynamics of the compensator A c can be computed by solving the third LMI problem è5è. At this point a robust compensator, which guarantees the robust stability and satisæes the upper bound of the worst case H performance, is completely calculated. We then repeat the procedure until satisfying the stopping criterion, such as the decrease in the upper bound of the worst case H performance is less than a given absolute and relative accuracy. Remark.1 his solution procedure is not guaranteed to converge globally, but our experience is that it eæciently converges to a local optimum. We will demonstrate the convergence of the alogithm and compare the iterative and optimal solution in the numerical example in x4. Note that each step of the VíK iteration can be solved very eæciently by a previously developed semidefinite programming algorithm sp ë8ë and very easily coded using a user-friendly interface sdpsol ë10ë. Remark. here are two important distinctions between the VíK iteration and the DíK iteration of the complex =K m synthesis. First, in our approach there are shared variables between each iteration: speciæcally, ~P is the common variable between the V iteration and K iteration èwhere ~ P appears as P; Q; X; Y and Zè. However, for the DíK iteration, the D iteration èthe robust analysis with or without curve ættingè is entirely separate from the K iteration èthe H1 synthesisè. We conjecture that these shared variables play a key role in the eæciency of the convergence of this new algorithm to a local optimum. Second, the new solution procedure also eliminates the curve-ætting of the structured singular value because the multiplier can be parameterized in the synthesis formulation and these multiplier parameters are solved simultaneously with the controller parameters. Remark. As a consequence of the systematic solution procedure and concise representation of the robust H control design, this technique can easily be extended to consider the H1 norm, instead of the H cost, for designing parametric robust controllers ë1ë. In contrast, there appears to be no direct simple extension for the current robust H1 control approaches to include the H cost. his shows a unique versatility of our iterative algorithm for designing robust controllers.. he Global BMI Optimization In this subsection, we present a simple global optimization technique for computing the optimal solution of è4è. We ærst note that when the Popov multiplier parameters èæ and è are æxed, the optimization problem è4è

4 Upper bound of H Cost is simply a semideænite program in the remaining variables P, Z, Q, Y, and X. Hence, we can ænd the optimal solution by taking a æne grid of nodes in some closed bounded space of the multiplier parameters, i.e., ææ ëæ L ; æ U ëæë L ; U ë, where 0 æ L;i é æ U;i é 1, and 0 L;i é U;i é 1; i= 1;:::;n p. his restriction on the multiplier space is consistent with the assumption required for global optimization techniques of the general BMI problems ë6ë. he initial space of the multiplier parameters can be speciæed by letting èæ L, L è equal to zero, and èæ U, U è be the values computed from the Popov robust H performance analysis of the closed-loop Lur'e system in which controller parameters are given and æxed. hen, the search is reæned to regions of interest that contain local minima. At each node, we can solve the corresponding semideænite program and obtain the upper bound of the worst case H performance. As discussed in Ref. ë6ë, the solution of è4è is a locally Lipschitz continuous function over a bounded set of the multiplier parameters. herefore, the optimal robust performance is the least upper bound of the worst case H performance over these nodes. Since the accuracy of the optimal value depends on the grid size, we must increase the number of nodes in a particular region to obtain a more precise solution. In general, the exhaustive search is not a particular eæcient nor elegant method to calculate the optimal solution since the dimension èi.e., the number of nodesè grows exponentially with the number of the nonlinearities. However, in practice this approach provides the simplest available method of solving the global optimization for BMI problems, and thus is very useful for investigating the convergence of the VíK iteration on simple systems. Remark.4 Goh et al. ë5,6ëhave attempted to achieve global solutions of BMI problems via branch and bound algorithms. he lower bounds are computed by using local optimization algorithms and the upper bounds are calculated from a relaxed version of the BMI problems, i.e., byintroducing new variables to represent the products of the original variables. his approach has been demonstrated to converge for non-control examples, but to the best of our knowledge, it has not been applied to the very complex robust control design problem discussed in this paper. 4 Numerical Examples We demonstrate the convergence of the iterative algorithm for the Popov H controller synthesis on the three mass-spring benchmark system ëë èsee Figure 1è. We ærst consider the case that the second spring constant has æ5è uncertainty, i.e., k = k ;nom è1 + æè, where k ;nom is the nominal value and æ ë,0:05; 0:05ë. he system parameters are m 1 = m = m =1,k 1 = 1, and k ;nom =1. o apply this Popov H synthesis technique, we eæectively approximate the uncertainty in the spring stiæness as k èxè = k ;nom ëx + èxèë, where = 0:05 x 1 x x k 1 k m 1 m m w Figure 1: Conæguration of the three mass-spring system. is a measure of the guaranteed uncertainty bound, and èxè is a ë,1; 1ë sector-bounded memoryless nonlinear function of the spring displacement, x. he iterative algorithm in x.1 is used to compute the robust controller, where the stopping criterion is based on 1è absolute and relative accuracy. he history of the VíK iteration is summarized in able 1. As shown in the table, the VíK iteration yields the Popov H controller after only four iterations. able 1: Summary of the VíK iteration of the Popov H controller designed for the three massspring system with æ5è guaranteed uncertainty of the second spring constant. Iter. Up bnd è Error Multiplier parameters è H Cost H Cost æ Opt Optimum 0 Figure : Mesh plots of the H cost overbounds as a function of the multiplier parameters. As discussed in x., the optimal solution can be computed by ærst coarsely gridding a closed bounded space of the multiplier parameters, i.e., the set of total 9 æ 47 nodes over the space æ æ ë0; 9ë æ ë0; 46ë with an equal grid size. his region was selected to include the constraints on èæ;è and the initial value of the VíK iteration. he surface of the H cost overbound is shown in Figure. We then reæned the region, particularly near the local optimum to obtain a new set of 41 æ 41 nodes 10 0 æ u 0 40

5 Upper bound of H Cost over the subspace æ æ ë0:7; :ë æ ë1:; :6ë with an even grid size. he contour plots shown in Figure and the mesh plots shown in Figure 4 illustrate the worst case H cost overbounds over this reæned multiplier space. Note that these plots display only the reæned region of the search grid. he ærst surprising observation seen in these ægures is that there is only one local minimum over the entire grid. Furthermore, the worst case H cost overbound behaves like a convex function in this region of the multiplier parameters. In three dimensions, the cost surface shown in Figure 4 looks like smooth paraboloid which has a local minimum at æ = 1:474 and =:78. :5 Iter 1 Iter H HHj Iter H HHHj - Optimum - 0:01è 0:05è 0:1è 1 1:5 æ 0:5è Figure : Contour plots of the H cost overbounds as a function of the multiplier parameters over a reæned region. We continue the discussion of the convergence of the Ví K iteration by drawing the path connecting the multiplier of each iteration marked by ëæ" in Figure and 4. Note that the optimal Popov multiplier computed by the exhaustive search is marked by ëæ" in these ægures and also shown in able 1. Labels next to the contour lines of Figure represent the percentage cost increment with respect to the optimal H cost overbound. he initial multiplier, as shown in the table, lies far outside the region of the contour plot depicted in these ægures. However, after the ærst iteration, the algorithm results in a multiplier for which the worst case H cost overbound is within 0:5è of the optimal value. As shown in this ægure, the algorithm stops after the third iteration which corresponds to a worst case H cost overbound within 0:01è of the optimal value. his result clearly shows that in this case, the algorithm converges to the optimal solution. o further demonstrate that the algorithm converges locally for systems with multiple nonlinearities, we consider the second case of the three mass-spring system where the stiænesses of both springs are uncertain with æ5è guaranteed bounds. As before, we compute the Popov H controller using the VíK iteration in x.1. For the stopping criterion of 1è accuracy, four itera- 17:9 17:8 Optimum 17:7 :5 Iter Iter?? Iter 1 Figure 4: Mesh plots of the H cost overbounds as a function of the multiplier parameters over a reæned region. tions are required to solve for this robust controller. able summarizes the history of the VíK iteration for the case with two uncertainties. he exhaustive search procedure described in x. was used to calculate the optimal solution for the BMI problem è4è. he coarse grid è11 4 nodesè was uniformly taken over the parameter space æ æ = diagè 1 ; è æ diagè 1 ; è, 1 ë1; ë; ë1; ë; 1 ë1:; :8ë, and ë1; ë. After carefully observing the location of a local minimum, we reæned the search to9 4 nodes with an even grid size over the parameter region æ æ, 1 ë1:8; :ë; ë1:7; :1ë; 1 ë:; :8ë, and ë1:8; :ë. Of course, there is no convenient visualization tool, so only numerical results are presented here in able. As shown previously in the one uncertainty case, the overbound of the worst case H performance is monotonically decreasing during the iterative solution. Furthermore, at the third iteration, the algorithm yields a solution that is very close to the optimum, i.e., to within 0:01è. Note however that the multiplier parameters from the VíK iteration shown in the table are not as close to the optimal ones in this case. hese convergence results are typical of the trends seen for each Popov H control design presented in Ref. ë15ë. o the best of our knowledge, this analysis indicates that, for these numerical examples, the VíK iteration has in fact converged to the globally optimal solution. able : Summary of the VíK iteration of the Popov H controller designed for the three massspring system with æ5è guaranteed uncertainty of both spring constants. Iter. Up bnd è Error Multiplier parameters è H Cost H Cost 1 ; 1 ; , , , , , , , , Opt , , 1.985? :5 1:5 æ 1

6 5 Conclusions his paper presents convergence results of the LMIbased iterative algorithm for solving the Popov H controller synthesis problem. Our approach has already been shown to yield consistent Popov H controllers, when compared with compensators designed by quasi- Newton search technique. he numerical results presented here indicate that the VíK iteration converges to the optimal solution in a few iterations and demonstrates the eæciency of the proposed algorithm in practice. In the process, we use a simple exhaustive search to compute the BMI global optimization, but this technique is limited to BMI problems with a small number of uncertainties. he convergence of the VíK iteration conærms our previous design experience and, more importantly, it oæers key insights into the design methodology and the beneæts of exploiting the structure of the controller matrices. Because of the low overhead associated with developing and implementing the LMI optimization, the VíK iteration can be easily extended to other sophisticated analysis tests and controller constraints. References ë1ë M. G. Safonov, K. C. Goh, and J. H. Ly, ëcontrol System Synthesis via Bilinear Matrix Inequalities," in Proc. American Control Conf., pp. 45í49, ëë K. C. Goh, J. H. Ly,L.urand, and M. G. Safonov, ë=k m -Synthesis via Bilinear Matrix Inequalities," in Proc. IEEE Conf. on Decision and Control, pp. 0í 07, Dec ëë O. oker and H. íozbay, ëon the NP-Hardness of Purely Complex Computation, AnalysisèSynthesis, and Some Related Problems in Multidimensional Systems," in Proc. American Control Conf., pp. 447í451, June ë4ë O. oker and H. íozbay, ëon the NP-Hardness of Solving Bilinear Matrix Inequalities and Simultaneous Stabilization with Static Output Feedback," in Proc. American Control Conf., pp. 55í56, June ë5ë K. C. Goh, L. uran, M. G. Safonov, G. P. Papavassilopoulos, and J. H. Ly, ëbiaæne Matrix Inequality Properties and Computational Methods," in Proc. American Control Conf., pp. 850í855, ë6ë K. C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, ëa Global Optimization Approach for the BMI Problem," in Proc. IEEE Conf. on Decision and Control, pp. 009í014, Dec ë7ë P. Gahinet and A. Nemirovsky, LMILab: A Package for Manipulating and Solving LMIs. INRIA, 199. ë8ë L. Vandenberghe and S. Boyd, sp: Software for Semideænite Programming. User's Guide, Beta Version. K.U. Leuven and Stanford University, Oct ë9ë L. El Ghaoui, F. Delebecque, and R. Nikoukhah, lmitool: A User-Friendly Interface for LMI Optimization, Beta Version. ENSA, Feb ë10ë S.-P. Wu and S. Boyd, sdpsol: A ParserèSolver for Semideænite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide, Beta Version. Stanford University, June ë11ë L. El Ghaoui and V. Balakrishnan, ësynthesis of Fixed-structure Controllers via Numerical Optimization," in Proc. IEEE Conf. on Decision and Control, pp. 678í68, Dec ë1ë L. El Ghaoui and J. P. Folcher, ëmultiobjective Robust Control of LI Control Design for Systems with Unstructured Perturbations," Syst. Control Letters, vol. 8, pp. í0, June ë1ë L. El Ghaoui and J. P. Folcher, ëmultiobjective Robust Control of LI Systems Subject to Structured Perturbations," in Int. Federation of Automatic Control, pp. 179í184, June ë14ë L. El Ghaoui, F. Oustry, and M. AitRami, ëa Cone Complementarity Linearization Algorithm for Static Ouput Feedback and Related Problems," IEEE rans. Aut. Control, o appear. ë15ë D. Banjerdpongchai and J. P. How, ëparametric Robust H Control Design Using LMI Synthesis," in he 1996 AIAA Guidance, Navigation, and Control Conference, AIAA-96-7, July ë16ë D. Banjerdpongchai and J. P. How, ëparametric Robust H Control Design with Generalized Multipliers via LMI Synthesis," in Proc. IEEE Conf. on Decision and Control, pp. 65í70, Dec ë17ë W. Haddad and D. Bernstein, ëparameter Dependent LyapunovFunctions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis," in Proc. IEEE Conf. on Decision and Control, pp. 74í79, 6í6, Dec ë18ë J. P. How, Robust Control Design with Real Parameter Uncertainty using Absolute Stability heory. PhD thesis, Massachusetts Institute of echnology, Cambridge, MA 019, Feb ë19ë 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 ë0ë A. Packard, K. Zhou, P. Pandey, and G. Becker, ëa Collection of Robust Control Problems Leading to LMI's," in Proc. IEEE Conf. on Decision and Control, pp. 145í150, ë1ë D. Banjerdpongchai and J. P. How, ëlmi Synthesis of Parametric Robust H1 Controllers," in Proc. American Control Conf., pp. 49í498, June ëë A. G. Sparks and D. S. Bernstein, ëreal Structured Singular Value Synthesis Using the Scaled Popov Criterion," AIAA J. of Guidance, Control, and Dynamics, vol. 18, no. 6, pp. 144í15, 1995.