numerically reliable computation of optimal performance in singular h_ inf control a numerically stable algorithm is described for the computation of the optimal h inf performance gamma rm opt when the feedthrough matrices d12 or d21 are rank deficient using only orthogonal transformations the singular h inf problem is reduced to a regular subproblem and a standard riccati based gamma iteration is applied to this subproblem to compute gamma rm opt various interpretations of this scheme are given in terms of the infinite zero structure of the plant and the deflation of hamiltonian pencils the implementation of this algorithm is straightforward and its performance and reliability are confirmed by extensive numerical testing

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

robust pole placement design a new technique is presented for the robust design of linear feedback controllers such that the system closed loop poles stay inside a prescribed region in the complex plane despite bounded plant parameter variations the method is based on the definition of a robust measure for any given controller and on the search by an iterative method for the controller which maximizes that measure

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

a survey of viability theory some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state dependent control constraints are motivated and surveyed they all deal with viable solutions to nonlinear control problems i e solutions satisfying at each instant given state constraints of a general and diverse nature some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case including inequality constraints bearing on the state and state dependent constraints on the controls for instance existence of a viability kernel of a closed set corresponding to the largest controlled invariant manifold is provided under general conditions even when the zero dynamics algorithm does not converge the concepts of slow and heavy viable solutions are introduced providing concrete ways of regulating viable solutions by closed loop feedbacks and closed loop dynamical feedbacks viability theorems also allow the extension of lyapunov s second meth

analysis of a class of probability preserving measure algebras on compact intervals the measure algebras of the title are those which are also hypergroups with some regularity conditions examples include the convolutions associated with jacobi polynomial series and fourier bessel series it is shown here that there is a one to one correspondence between these hypergroups and a class of sturm liouville problems which have the characters of the hypergroup as eigenfunctions the interplay between these two characterizations allows a detailed analysis which includes a hilb type formula for the characters and asymptotic estimates for the plancherel measure and the eigenvalues of the associated sturm liouville problem

between controllable and uncontrollable the distance between a system a b and the set of uncontrollable systems is the minimum of the smallest singular value of i a b with respect to

a structure preserving algorithm for the minimum h norm computation of finite time state feedback control problem the algorithm being developed here is based on the generating function approach for finite time h control and application of canonical transformation of linear hamiltonian system first an equivalent finite time h control law in terms of the third type generating function is derived then by using symplectic structure of the hamiltonian system s state transition matrix a group of matrix recursive formulae are deduced for the evaluation of the finite time h control law combining with a matrix singularity testing procedure this recursive algorithm verifies the existence condition of sub optimal h controllers and gives the minimum h norm of finite time control systems inherited from the canonical transformation the matrix recursive formulae have a standard symplectic form this structure preserving property helps facilitate reliable and effective computation numerical results show the effectiveness of the proposed algorithm

on the selection of poles in the single input pole placement problem abstract it is well known that the single input pole placement problem can be quite sensitive to perturbations in the data recent results by mehrmann and xu show how this sensitivity depends on the location of the poles in many applications it suffices to prescribe a set k in the complex plane that contains the poles mehrmann and xu formulated a minimization problem for allocating the poles in a given set k so that the sensitivity to perturbations is reduced the present paper uses methods of potential theory to derive simple algorithms that yield approximate solutions of this minimization problem

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

convolution and hypergroup structures associated with a class of sturm liouville systems product formulas of the type u k u k 0 u k d are obtained for the eigenfunctions of a class of second order regular and regular singular sturm liouville problems on 0 by using the riemann integration method to solve a cauchy problem for an associated hyperbolic differential equation when d is nonnegative which can be guaranteed by a simple restriction on the differential operator of the sturm liouville problem it is possible to define a convolution with respect to which m 0 becomes a banach algebra with the functions u k u 0 as its characters

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

robust stability and robust pole assignment of linear systems with structured uncertainty sufficient conditions for the robust stability of linear time varying systems are presented the stability criteria are extended to the problem of the robust pole assignment of linear time invariant systems to a specified region these criteria both for robust stability and robust pole assignment improve some results in the literature examples are given to demonstrate the new criteria

pole assignment via sylvester s equation it is shown that the pole assignment problem can be reduced to solving the linear matrix equations ax xa bg fx g successively for x and then f for almost any choice of g the result is a new pole assignment procedure and proof of the pole assignment theorem that should play an important role in both theoretical and practical applications

structure preserving model reduction of port hamiltonian systems by moment matching at infinity model reduction of port hamiltonian systems by means of the krylov methods is considered aiming at port hamiltonian structure preservation it is shown how to employ the arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the markov parameters but also the port hamiltonian structure for the reduced order model furthermore it is shown how the lanczos method can be applied in a structure preserving manner to a subclass of port hamiltonian systems which is characterized by an algebraic condition in fact for the same subclass of port hamiltonian systems the arnoldi method and the lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

product formulas and convolutions for angular and radial spheroidal wave functions product formulas for angular spheroidal wave functions on 0 and for radial spheroidal wave functions on 0 are presented which generalize results for the ultraspherical polynomials and functions as well as for the mathieu functions although these functions cannot be given in closed form the kernels of the product formulas are represented in an explicit and surprisingly simple way in terms of bessel functions so that the exact range of positivity can easily be read off the formulas are used to introduce two families of convolution structures on 0 and 0 many of which provide new hypergroups we proceed from the fact that the spheroidal wave functions are eigenfunctions of sturm liouville equations of confluent heun type and employ a partial differential equation technique based on riemann s integration method

an efficient schur method to solve the stabilizing problem an efficient solution to the stabilizing problem for both continuous and discrete constant linear systems is presented the real schur form reduction of the system state matrix is used

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

globally convergent algorithms for robust pole assignment by state feedback it is observed that an algorithm proposed in 1985 by kautsky nichols and van dooren knv amounts to maximizing at each iteration the determinant of the candidate closed loop eigenvector matrix x with respect to one of its columns with unit length constraint subject to the constraint that it remains an achievable closed loop eigenvector matrix this interpretation is used to prove convergence of the knv algorithm it is then shown that a more efficient algorithm is obtained if det x is concurrently maximized with respect to two columns of x and such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs variations exploiting the availability of multiple processors are suggested convergence properties of the proposed algorithms are established their superiority is demonstrated numerically

robust and minimum norm pole assignment with periodic state feedback a computational approach is proposed to solve the minimum norm or robust pole assignment problem for linear periodic discrete time systems the proposed approach uses a periodic sylvester equation based parametrization of the periodic pole assignment problem and exploits the nonuniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback or on the condition numbers of the periodic eigenvector matrices of the closed loop system the solution method relies on using gradient search methods on suitably defined cost functions explicit expressions of the gradients of cost functions are derived and the efficient evaluation of the cost functions and gradients is discussed numerical examples illustrate the effectiveness of the proposed approach

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

on the computation of set induced control lyapunov functions for continuous time systems the paper presents a computation oriented method for characterizing and obtaining local control lyapunov functions induced by particular star shaped nonconvex sets for continuous time nonlinear systems with bounded inputs for a given set the necessary and sufficient conditions for the induced function to be a nonconvex local control lyapunov function are provided the related convex problems for computing the exact region in which the function is decreasing and the optimal control input are presented the results are applied to the brockett integrator

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

efficient eigenvalue assignments for general linear mimo systems abstract this paper deals with the transformation of linear multi input multi output mimo systems into frobenius canonical form with the ultimate objective of developing a new computationally efficient methodology for a poleplacement procedure both time invariant and time varying systems are considered the conventional pole placement steps for time invariant siso single input single output systems are generalized for both classes this is a unique study of the expansion of the pole placement capability in particular for time varying mimo systems this depth of generalization has been neglected in the past due to its complex formulation the practical advantage of the proposed technique is that it does not require the coefficients of the characteristic polynomial the eigenvalues of the original system or the coefficients of the characteristic polynomial of the transformed system the repercussions of such a development are expected in nonlinear systems theory as well considering the fact that some recently developed techniques yield exact i o linearization with time varying coefficients

algorithmic analysis of polygonal hybrid systems part ii phase portrait and tools polygonal differential inclusion systems spdi are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions the reachability problem as well as the computation of certain objects of the phase portrait is decidable in this paper we show how to compute the viability controllability and invariance kernels as well as semi separatrix curves for spdis we also present the tool speedi which implements a reachability algorithm and computes phase portraits of spdis

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

dampening controllers via a riccati equation approach an algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes but also dampens the closed loop system dynamics in other words a feedback gain matrix is computed such that the eigenvalues of the closed loop state matrix are within the region of the left half plane where the magnitude of the real part of each eigenvalue is greater than that of the imaginary part this may be accomplished by solving a damped algebraic riccati equation and a degenerate riccati equation the solution to these equations are computed using numerically robust algorithms damped riccati equations are unusual in that they may be formulated as an invariant subspace problem of a related periodic hamiltonian system this periodic hamiltonian system induces two damped riccati equations one with a symmetric solution and another with a skew symmetric solution these two solutions result in two different state feedbacks both of which dampen the system dynamics but produce different closed loop eigenvalues thus giving the controller designer greater freedom in choosing a desired feedback

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

a direct algorithm for pole assignment of time invariant multi input linear systems using state feedback abstract a direct algorithm is suggested for the computation of the linear state feedback for multi input systems such that the resultant closed loop system matrix has specified eigenvalues the extra freedom can be used in different ways for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed loop matrix the algorithm uses unitary transformations for numerical reliability and is based on ideas from the qr algorithm for solving the eigenproblem the stability of the algorithm is proven by doing a backward rounding error analysis and numerical examples are given as well

robust pole assignment abstract this paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots and applies them to the problem of robust pole placement we formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed loop poles within desired intervals for plants with unknown but bounded parameter uncertainties the algorithms are computationally reasonable and provide a useful addition to currently existing control cad tools

decay rate estimations for linear quadratic optimal regulators let u t fx t be the optimal control of the open loop system x t ax t bu t in a linear quadratic optimization problem by using different complex variable arguments we give several lower and upper estimates of the exponential decay rate of the closed loop system x t a bf x t main attention is given to the case of a skew hermitian matrix a given an operator a for a class of cases we find a matrix b that provides an almost optimal decay rate we show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems a bj j 1 n and illustrate this on an example of a concrete mechanical system at the end of the article we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

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

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

differential inclusions and target problems where and how solutions associated to a differential inclusion can or cannot enter a given target is studied for this purpose partitions of the target boundary are associated with the dynamic of the system the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets these kernels determine points such that there exist respectively all solutions starting at these points remain in a given set of constraints the sets that are reached in finite time by viable solutions to the system are also studied finally some applications to control systems with one target are provided and the concept of semipermeable barriers will be generalized