the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

cal l _ infty norm computation for continuous time descriptor systems using structured matrix pencils in this technical note we discuss an algorithm for the computation of the l norm of transfer functions related to descriptor systems we show how one can achieve this goal by computing the eigenvalues of certain skew hamiltonian hamiltonian matrix pencils and analyze arising problems we also formulate and prove a theoretical result which serves as a basis for testing a transfer function matrix for properness finally we illustrate our results using a descriptor system related to mechanical engineering

a trust region spectral bundle method for nonconvex eigenvalue optimization we present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions we prove global convergence of our method in the sense that for an arbitrary starting point every accumulation point of the sequence of iterates is critical the method is tested on several problems in feedback control synthesis

solution of a nonsymmetric algebraic riccati equation from a two dimensional transport model abstract for the steady state solution of an integral differential equation from a two dimensional model in transport theory we shall derive and study a nonsymmetric algebraic riccati equation b xf f x xb x 0 where f i s pd b b i s p d and b b i s pd with a nonnegative matrix p positive diagonal matrices d and nonnegative parameters f b b 1 f and s s 1 f we prove the existence of the minimal nonnegative solution x under the physically reasonable assumption f b s p d d 1 and study its numerical computation by fixed point iteration newton s method and doubling we shall also study several special cases e g when b 0 and p is low ranked then x s 2 uv is low ranked and can be computed using more efficient iterative processes in u and v numerical examples will be given to illustrate our theoretical results

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

a globally convergent method to compute the real stability radius for time delay systems abstract this paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays the real stability radius is the distance to instability measured as the minimal real valued perturbation that renders the system unstable our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function we develop a criss cross type algorithm that globally converges to this maximum and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments the algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems dep with positive and negative delays this is an extension of the well known linear case without delays where this results in algebraic eigenvalue problems in addition a novel numerical solver to compute all the imaginary eigenvalues of this dep is also presented it combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points finally due to the presence of multiple eigenvalues at the maximum these approximations are corrected with a block newton algorithm for nonlinear eigenvalue problems

large scale discrete time algebraic riccati equations doubling algorithm and error analysis we consider the numerical solution of large scale discrete time algebraic riccati equations the doubling algorithm is adapted with the iterates for a not computed explicitly but recursively the resulting algorithm is efficient with computational complexity and memory requirement proportional to the size of the problem and essentially converges quadratically an error analysis on the truncation of iterates and some numerical results are presented

a new class of nonsymmetric algebraic riccati equations abstract motivated by the study of linear quadratic differential games we introduce a new class of nonsymmetric algebraic riccati equations it is shown that every equation in this class has a unique stabilizing solution which is the solution required to find the open loop nash equilibrium for the differential game we show that the doubling algorithm can be used to find this solution efficiently the solution may also be found by the schur method and under further assumptions by newton s method and a basic fixed point iteration

mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation abstract in this paper the mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation are considered the explicit expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed some simple numerical experiments show that the mixed and componentwise condition numbers can be sharp

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

iqc analysis and synthesis via nonsmooth optimization integral quadratic constraints iqcs are used in system theory to model nonlinear phenomena within the framework of linear feedback control iqc theory addresses parametric robustness saturation effects sector nonlinearities passivity and much else in iqc analysis specially structured linear matrix inequalities lmis arise and are currently addressed by structure exploiting lmi solvers controller synthesis under iqc constraints is nonconvex and much harder and has been attempted sporadically by global optimization techniques such as branch and bound cutting plane or d k type coordinate descent ideas here we revisit iqc theory and propose a completely different algorithmic solution based on local and nonsmooth optimization methods this is less ambitious than global methods but is very promising in practice our approach while aiming high at iqc synthesis offers new answers even for iqc analysis because we optimize without lyapunov variables for high order systems this leads to a significant reduction of the number of unknowns 2006 elsevier b v all rights reserved

on the real stability radius of sparse systems abstract in this paper we study robust stability of sparse lti systems using the stability radius sr as a robustness measure we consider real perturbations with an arbitrary and pre specified sparsity pattern of the system matrix and measure their size using the frobenius norm we formulate the sr problem as an equality constrained minimization problem using the lagrangian method for optimization we characterize the optimality conditions of the sr problem thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system further we use the sylvester equation based parametrization to develop a penalty based gradient newton descent algorithm which converges to the local minima of the optimization problem finally we illustrate how our framework provides structural insights into the robust stability of sparse networks

computing the distance to continuous time instability of quadratic matrix polynomials a bisection method is used to compute lower and upper bounds on the distance from a quadratic matrix polynomial to the set of quadratic matrix polynomials having an eigenvalue on the imaginary axis each bisection step requires to check whether an even quadratic matrix polynomial has a purely imaginary eigenvalue first an upper bound is obtained using frobenius type linearizations it takes into account rounding errors but does not use the even structure then lower and upper bounds are obtained by reducing the quadratic matrix polynomial to a linear palindromic pencil the bounds obtained this way also take into account rounding errors numerical illustrations are presented

convergence rates of some iterative methods for nonsymmetric algebraic riccati equations arising in transport theory we determine and compare the convergence rates of various fixed point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic riccati equations arising in transport theory

a newton based method for the calculation of the distance to instability in this paper a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided the method uses newton s method to find a two dimensional jordan block corresponding to a pure imaginary eigenvalue in a certain two parameter hamiltonian eigenvalue problem introduced by byers r byers a bisection method for measuring the distance of a stable matrix to the unstable matrices siam j sci statist comput 9 1988 875 881 this local method is augmented by a test step previously used by other authors to produce a global method numerical results are presented for several examples and comparison is made with the methods of boyd and balakrishnan s boyd v balakrishnan a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm systems control lett 15 1990 1 7 and he and watson c he g a watson an algorithm for computing the distance to instability siam j matrix anal appl 20 1999 101 116

on computing the infinity norm an iterative algorithm is given for computing the infinity norm of the transfer matrix of a bounded linear time invariant finite dimensional system the procedure which is based on the numerically stable qz algorithm enables the infinite norm to be computed with a guaranteed accuracy no ad hoc procedures such as those for selecting a suitable grid of frequencies for example are used the degree of accuracy is under the direct control of the user with each additional decimal place of accuracy requiring approximately 3 3 calls to the qz algorithm experience has shown the algorithm to function reliably and in reasonable time for models of order as large as 56

h norm computation for ltv systems using nonlinear optimal control algorithms we discuss the computation for 2 induced norms of finite horizon linear time varying systems it is well known that 2 induced norm calculating problems can be converted into optimal control problems so with the help of optimal control algorithms we propose a numerical technique for 2 induced norm computation our approach is as follows first we present a new computational method for nonlinear optimal control problems that is numerically reliable stable and robust analysing the convergence property under some assumptions next by modifying the presented method we propose an algorithm for the 2 induced norm calculation of finite horizon linear time varying systems which is essentially different from all other existing methods finally the effectiveness of our algorithm is demonstrated by simulation

computation of extremum singular values and the strong h infinity norm of siso time delay systems we consider the computation of h infinity norms for single input single output siso time delay systems which are described by delay differential algebraic equations unlike the iterative level set methods in the literature we present a novel numerical method to compute the h infinity norm this method requires solving one eigenvalue problem of at most twice the size of the eigenvalue problem in every iteration of a level set method but in practice often considerably lower we first show that the computation of extrema of the transfer function can be turned into the computation of the imaginary axis zeros of a transcendental function we compute these zeros by a predictor corrector type algorithm it is known that the h infinity norm of delay differential algebraic systems which can model both retarded and neutral type systems might be sensitive with respect to arbitrarily small delay perturbations this recently led to the concept of strong h infinity norms which explicitly take into account such small delay perturbations we present a direct numerical method to compute the strong h infinity norm of siso time delay systems our algorithm is applicable to the closed loop system of interconnections series parallel feedback junctions of time delay systems and or controllers

parametric robust structured control design we present a new approach to parametric robust controller design where we compute controllers of arbitrary order and structure which minimize the worst case h infty norm over a pre specified set of uncertain parameters at the core of our method is a nonsmooth minimization method tailored to functions which are semi infinite minima of smooth functions a rich test bench and a more detailed example illustrate the potential of the technique which can deal with complex problems involving multiple possibly repeated uncertain parameters

low rank approximate solutions to large sylvester matrix equations in the present paper we propose an arnoldi based method for solving large and sparse sylvester matrix equations with low rank right hand sides we will show how to extract low rank approximations via a matrix krylov subspace method we give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual numerical experiments will also be given to show the effectiveness of the proposed method

solving large scale continuous time algebraic riccati equations by doubling we consider the solution of large scale algebraic riccati equations with numerically low ranked solutions for the discrete time case the structure preserving doubling algorithm has been adapted with the iterates for a not explicitly computed but in the recursive form a k a k 1 2 d k 1 s k 1 d k 2 with d k 1 and d k 2 being low ranked and s k 1 being small in dimension for the continuous time case the algebraic riccati equation will be first treated with the cayley transform before doubling is applied with n being the dimension of the algebraic equations the resulting algorithms are of an efficient o n computational complexity per iteration without the need for any inner iterations and essentially converge quadratically some numerical results will be presented for instance in section 5 2 example 3 of dimension n 20209 with 204 million variables in the solution x was solved using matlab on a macbook pro within 45 s to a machine accuracy of o 10 1 6

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

brief a fast algorithm for the computation of an upper bound on the µ norm a fast algorithm for the computation of the optimally frequency dependent scaled h norm of a finite dimensional lti system is presented it is well known that this quantity is an upper bound to the m norm furthermore it was recently shown to play a special role in the context of slowly time varying uncertainty numerical experimentation suggests that the algorithm generally converges quadratically

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

simple covariance approach to mathcal h _ infty analysis we revisit the classical mathcal h infty analysis problem of computing the ell 2 induced norm and the mathcal l 2 induced norm of a linear time invariant system for the discrete time case and the continuous time case respectively we follow an approach based on converting the problem of maximization over signals to that of maximization over a sort of deterministic covariance matrices the reformulation in terms of these covariance matrices greatly simplifies the dynamic analysis problem and converts the computation to a convex constrained matrix maximization problem furthermore the equivalence is for the actual mathcal h infty norm of the system rather than a bound and thus does not require the typical gamma iterations we argue that this approach is both attractive elementary and constructive in that the worst case disturbance is also easily obtained as a state feedback constructed from the solution of the matrix problem

pseudospectra and stability radii for analytic matrix functions with application to time delay systems definitions for pseudospectra and stability radii of an analytic matrix function are given where the structure of the function is exploited various perturbation measures are considered and computationally tractable formulae are derived the results are applied to a class of retarded delay differential equations special properties of the pseudospectra of such equations are determined and illustrated 2006 elsevier inc all rights reserved

distributed control in multiple dimensions a structure preserving computational technique we consider the problem of analysis and control of discretely distributed systems in multidimensional arrays for spatially invariant systems in one spatial dimension we build an efficient arithmetic that preserves the rational laurent operator structure leading to fast iterative methods of solving lyapunov and riccati equations and block diagonalizations and thus arbitrarily non conservative stability analysis and h2 and h distributed controller synthesis these one dimensional results are then used to build an efficient arithmetic and controller synthesis procedure in n dimensions by induction the extension of these techniques from laurent operators with rational symbols to sequentially semi separable matrices yields a procedure for linear computational complexity analysis and controller synthesis for finite extent heterogeneous multidimensional systems with boundary conditions the procedures are demonstrated on two computational examples

an eigenvalue based approach for the robust stabilization of linear time delay systems we present a new numerical procedure for the robust stabilization of linear delay differential equations based on the position of the eigenvalues in the complex plane we assume static perturbations on the system matrices and express the robustness of the stability in terms of complex stability radii in the numerical procedure these stability radii are maximized as a function of the controller parameters this corresponds to a synthesis problem which is solved by a quasi continuous shaping of some frequency response plots the structure of the algorithm is analogous to the continuous pole placement algorithm for the non robust stabilization of linear delay differential equations

fast iterative schemes for nonsymmetric algebraic riccati equations arising from transport theory for a class of nonsymmetric algebraic riccati equations arising from transport theory lu siam j matrix anal appl 26 2005 pp 679 685 reformulated them into a couple of fixed point equations in a vector form and constructed an iterative method for finding their minimal positive solutions in this paper based on this fixed point equation we propose several fast and effective iterative methods for solving the above mentioned special type of nonsymmetric algebraic riccati equations these iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed point iteration according to the principle of using the currently available information as promptly as possible the monotone convergence theorems about these new iterative methods are established and the sharper bounds about the solutions of the nonsymmetric algebraic riccati equations are derived numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic riccati equations arising from transport theory

level set methods for finding critical points of mountain pass type abstract computing mountain passes is a standard way of finding critical points we describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case we apply these techniques to describe a strategy for addressing the wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues finally we relate critical points of mountain pass type to nonsmooth and metric critical point theory

total linear least squares and the algebraic riccati equation abstract it is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation which turns into an algebraic riccati equation when the matrix of unknowns is square if there is an additional symmetry constraint on the solution the optimal solution is given by the anti stabilizing solution of this riccati equation

robust stability of linear systems described by higher order dynamic equations the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied different perturbation norms are considered the aim is to derive robustness criteria that are expressed directly in terms of the original data previous results on robust stability of hurwitz and schur polynomials are extended to monic matrix polynomials for disturbances acting via a uniform input matrix computable formulas are obtained whereas for perturbations with multiple input matrices structured singular values are involved

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

low rank approximation to the solution of a nonsymmetric algebraic riccati equation from transport theory abstract we consider the solution of the large scale nonsymmetric algebraic riccati equation xcx xd ax b 0 from transport theory juang 1995 with m d c b a r 2 n 2 n being a nonsingular m matrix in addition a d are rank 1 updates of diagonal matrices with the products a 1 u a u d 1 v and d v computable in o n complexity for some vectors u and v and b c are rank 1 the structure preserving doubling algorithm by guo et al 2006 is adapted with the appropriate applications of the sherman morrison woodbury formula and the sparse plus low rank representations of various iterates the resulting large scale doubling algorithm has an o n computational complexity and memory requirement per iteration and converges essentially quadratically as illustrated by the numerical examples

large scale algebraic riccati equations with high rank constant terms abstract we consider the numerical solution of large scale algebraic riccati equations with high rank constant terms the solutions are not numerically low rank so the previously successful methods based on low rank representations are not directly applicable we modify the doubling algorithm making use of the low rank in the input matrix b we also solve the challenging problems in the estimation of residuals and relative errors convergence control and the output of the modified algorithm illustrative numerical examples are presented

existence of algebraic matrix riccati equations arising in transport theory we consider the existence of positive solutions of a certain class of algebraic matrix riccati equations with two parameters c 0 c 1 and 0 1 here c denotes the fraction of scattering per collision and is an angular shift equations of this class are induced via invariant imbedding and the shifted gauss legendre quadrature formula from a simple transport model by establishing the existence of positive solutions of such equations the problem of the convergence of some iterative schemes for solving them can be completely solved

on the tripling algorithm for large scale nonlinear matrix equations with low rank structure for the large scale nonlinear matrix equations with low rank structure the well developed doubling algorithm in low rank form da lr is known as an efficient method to compute the stabilizing solution by further analyzing the global efficiency index constructed in this paper we propose a tripling algorithm in low rank form ta lr from two points of view the cyclic reduction and the symplectic structure preservation the new presented algorithm shares the same pre processing complexity with that of da lr but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset under the solvability condition the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order numerical experiments including some from nano research show that the ta lr is highly efficient to compute the stabilizing solution of large scale nonlinear matrix equations with low rank structure

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

on computing the worst case peak gain of linear systems based on the bounds due to doyle and boyd we present simple upper and lower bounds for the l1 norm of the tail of the impulse response of finite dimensional discrete time linear time invariant systems using these bounds we may in turn compute the l gain of these systems to any desired accuracy by combining these bounds with results due to khammash and pearson we derive upper and lower bounds for the worst case l gain of discrete time systems with diagonal perturbations

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

analytic properties of the singular values of a rational matrix abstract it is shown that the singular values of a rational matrix g s where s x jy are locally a set of real analytic functions of x and y a brief discussion is given of their global properties

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

on the computation of the optimal h norms for two feedback control problems abstract numerical computation of the optimal h norms for the state feedback and the output feedback control problems is considered in 1989 doyle et al derived for a given parameter r 0 necessary and sufficient conditions under which an admissible output feedback controller exists as well as conditions for the existence of a state feedback controller we explore numerical methods to find the infimum of parameter r that satisfies those conditions

fast computation of the largest stability radius for a two parameter linear system the author describes a fast algorithm to compute the minimum distance to instability or parameter margin for a linear time invariant system subject to rational variations in two real parameters both the l sub infinity and the euclidean norms are considered for measuring parameter deviations

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

on the computation of the real hurwitz stability radius recently qiu et al 1995 obtained a computationally attractive formula for the evaluation of the real stability radius this formula involves a global maximization over frequency here for the hurwitz stability case we show that the frequency range can be limited to a certain finite interval numerical experimentation suggests that this interval is often reasonably small

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

an iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems we consider the following class of nonlinear eigenvalue problems left sum i 1 m a i p i lambda right v 0 where a 1 ldots a m are given n times n matrices and the functions p 1 ldots p m are assumed to be entire this does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations our aim is to compute the epsilon pseudospectral abscissa i e the supremum of the real parts of the points in the epsilon pseudospectrum which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations delta a i i 1 m of norm at most epsilon of the matrices a i i 1 m under mild assumptions guaranteeing the existence of a globally rightmost point of the epsilon pseudospectrum we prove that it is sufficient to restrict the analysis to rank one perturbations of the form delta a i beta i u v where u in mathbb c n and v in mathbb c n wit

nonsmooth h_infty synthesis we develop nonsmooth optimization techniques to solve h infty synthesis problems under additional structural constraints on the controller our approach avoids the use of lyapunov variables and therefore leads to moderate size optimization programs even for very large systems the proposed framework is versatile and can accommodate a number of challenging design problems including static fixed order fixed structure decentralized control design of pid controllers and simultaneous design and stabilization problems our algorithmic strategy uses generalized gradients and bundling techniques suited for the h infty norm and other nonsmooth performance criteria we compute descent directions by solving quadratic programs and generate steps via line search convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods the proposed approach proves to be efficient even for systems with several hundreds of states

homotopy for rational riccati equations arising in stochastic optimal control we consider the numerical solution of the rational algebraic riccati equations in mathbb r n arising from stochastic optimal control in continuous and discrete time applying the homotopy method we continue from the stabilizing solutions of the deterministic algebraic riccati equations which are readily available the associated differential equations require the solutions of some generalized lyapunov or stein equations which can be solved by the generalized smith methods of o n 3 computational complexity and o n 2 memory requirement for large scale problems the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms in comparison the alternative modified newton s methods require a difficult initial stabilization step some illustrative numerical examples are provided