the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

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

a new computational algorithm for functional inequality constrained optimization problems abstract in this note a computational algorithm is devised for solving a class of functional inequality constrained optimization problems based on a penalty function for illustration a numerical example is solved

alternative algorithms for solving nonlinear function and functional inequalities several sophisticated and efficient algorithms are now available in the literature for solving nonlinear function inequalities in a finite number of iterations this paper addresses an alternative approach to this class of problems the approach is relatively straightforward and easy to use essentially the nonlinear inequality constrained problem is reformulated as a standard unconstrained optimization problem via a differentiable transcription thus any existing efficient unconstrained optimization software packages can be used to solve the corresponding unconstrained optimization problem due to the special structure of the differentiable transcription a solution of the nonlinear inequality constraints can be obtained after a finite number of iterations in the process of solving the unconstrained optimization problem the main aim of this paper is however to extend the approach to solving nonlinear functional rather than function inequalities in a finite number of iterations for illustration several examples are computed to demonstrate the feasibility and versatility of the approach

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

a linearization algorithm for optimizing control systems subject to singular value inequalities the design of feedback linear multivariable systems subject to inequality constraints frequently leads to nonlinear programming problems involving singular values of matrix transfer functions we present a readily implementable and globally convergent algorithm for solving such problems it exploits the structure inherent in singular values differently than do other existing methods

a quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix abstract optimization involving eigenvalues arise in many engineering problems we propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices under certain assumptions it is shown that if started close enough to the minimizer x the proposed algorithm converges to x quadratically the proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of hermitian matrices also it has been extended to minimizing sums of the largest eigenvalues of a symmetric or hermitian matrix

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

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

when is the greatest eigenvalue of a parametrized symmetric matrix a convex function of the parameter let a aij n n be a real symmetric matrix valued function of u rm let u be the greatest eigenvalue of a u we are concerned with the convexity of when a does not depend affinely on u

a second order bundle method to minimize the maximum eigenvalue function in this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n n symmetric matrices we show how a simple bundle method the approximate eigenvalue method can be used to globalize the second order method developed by m l overton in the eighties and recently revisited in the framework of the lagrangian theory with no additional assumption the resulting algorithm generates a minimizing sequence a geometrical and constructive proof is given to prove that quadratic convergence is achieved asymptotically some strict complementarity and non degeneracy assumptions are needed we also introduce new variants of bundle methods for semidefinite programming

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

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

an iqc for uncertainty satisfying both norm bounded and passivity constraints this paper introduces a new integral quadratic constraint for describing a single block of dynamic linear time invariant uncertainty that simultaneously satisfies norm bound and passivity constraints for the associated matrix uncertainty problem we give an upper bound on the quantitative stability indicator and state conditions under which this bound is tight

nondifferentiable optimization algorithm for designing control systems having singular value inequalities it has been known for some time that many control system design requirements can be expressed as differentiable inequalities more recently it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix this paper presents an optimization algorithm which permits all these constraints to be considered

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

on the extension of constrained optimization algorithms from differentiable to nondifferentiable problems this paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems it is shown that the armijo gradient method phase i phase ii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable analogs for problems with semismooth functions the exact penalty method has required the development of a new optimality condition

robust local stability of multilayer recurrent neural networks we derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers the stability condition follows from linking theories about linearization robustness analysis of linear systems under nonlinear perturbation and matrix inequalities a characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic lyapunov function similar to the nl sub q theory the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion while the proven basin of attraction is relatively small due to conservatism of the criterion modification of the dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples

brief paper nonsmooth optimization for multiband frequency domain control design multiband frequency domain synthesis consists in the minimization of a finite family of closed loop transfer functions on prescribed frequency intervals this is an algorithmically difficult problem due to its inherent nonsmoothness and nonconvexity we extend our previous work on nonsmooth h synthesis to develop a nonsmooth optimization technique to compute local solutions to multiband synthesis problems the proposed method is shown to perform well on illustrative examples

on minimizing the largest eigenvalue of a symmetric matrix abstract optimization problems involving eigenvalues arise in many engineering problems in this paper we consider the problem of minimizing the largest eigenvalue over an affine family of symmetric matrices this problem has a variety of applications such as the stability analysis of dynamic systems or the computation of structured singular values given 0 we give an optimality condition which ensures that the largest eigenvalue is within error bound of the solution also a new line search rule is proposed and it is shown to have good descent properties when the multiplicity of the largest eigenvalue the solution is known a new algorithm for the optimization problem under consideration is proposed some numerical experiments on the proposed algorithm are presented

optimal l sub infinity bounds for disturbance robustness robust disturbance rejection bounds obtained recently by zhu et al are tight with respect to the disturbance set that has a specified l sub 2 norm bound it is shown that the bounds are also tight with respect to a weighted l sub 2 disturbance set that has a specified bound on its outer product by choosing a special weight matrix in the l sub 2 norm the matrix can computed by numerical iteration

majorization a computational complexity reduction technique in control system design a methodology is presented for transcribing worst case control system design specifications into semi infinite inequalities of low computational complexity which are tractable by semi infinite programming algorithms these algorithms can be used to find a design which satisfies specifications or to find a design which satisfies specifications and minimizes a cost a computational example is given

the linear limit of the dipole problem for the thin film equation we investigate self similar solutions of the dipole problem for the one dimensional thin film equation on the half line x ge 0 we study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem the similarity solutions are generally of the second kind given by the solution of a nonlinear eigenvalue problem although there are some notable cases where first kind solutions also arise we examine the conserved quantities connected to these first kind solutions difficulties associated with the lack of a maximum principle and the non self adjointness of the fundamental linear problem are also considered seeking similarity solutions that include sign changes yields a surprisingly rich set of coexisting stable solutions for the intermediate asymptotics of this problem our results include analysis of limiting cases and comparisons with numerical computations

proper invariant realizations of singular system problems the notion of proper invariant realization of a triple for a specially defined linear space is introduced and its properties and construction are investigated this notion of realization allows the reduction of the study of dynamic characterization of subspaces of singular systems to standard problems of linear systems theory the proof of the results is given

convergence rate for an approximation approach to h norm optimization problems with an application to controller order reduction in a recent paper by fisher et al 1989 a constrained h smooth optimization technique proc 28th cdc florida u s a a smooth approximation technique is proposed to solve a general class of constrained h norm optimization problems the aim of this paper is to show that such approximation has a convergence rate of o in p2p furthermore this technique is applied to handle a general class of frequency weighted controller order reduction problems

spectral bundle methods for non convex maximum eigenvalue functions second order methods we study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions we show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point we examine in which way standard and projected newton steps may be combined with a nonsmooth first order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence