asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

a generalized reaching law with different convergence rates traditionally the convergence rate of the sliding variable given by a discrete reaching law has been fixed in nature this paper presents a generalized algorithm for discrete time sliding mode control systems which offers a flexible convergence rate for the sliding variable for analysis we use the recently developed band approach method which gives us the values of the parameters in the algorithm for the desired convergence to take place once the ultimate band is chosen appropriately

spectral factorizations with unmixed pole and zero sets minimal spectral factors of spectral density matrices are constructed where poles and zeros lie in prescribed unmixed subsets of the complex plane

adaptive sliding mode control in discrete time systems abstract discrete time sliding mode control is considered the case of known parameters is first analyzed leading to a new definition of the so called equivalent control as the piecewise constant control that reduces to zero in finite time the distance of the system state from the sliding manifold in the presence of bounded parametric uncertainties an adaptive control scheme that guarantees the asymptotic satisfaction of the same control objective is presented the main feature of this approach is the reduction of the order of the relevant error equation and the possibility of dealing with the nonmatched uncertainties introduced by the sampling process

smoothed hat functions in subdivision following the general idea of smoothing through convolution we discuss schemes for smoothing hat functions on the three directional and the four directional mesh the smoothing masks which in the four directional case are matrix masks have the same support as the refinement mask for the hat functions several smoothing schemes are presented and in some cases the smoothness properties of the smoothed functions are analysed in particular the results for the four directional case follow new and original lines of research

observers for lipschitz non linear systems in an interesting paper r rajamani and y m cho have proposed a systematic methodology to design observers they have introduced a new problem relation between distance to unobservability and convergence of luenberger like observers a result for the convergence of the observer has also been given they have presented a quantity denoted by i claimed as the distance to unobservability we show that this number is not actually the distance from the system to the set of unobservable systems moreover the result for the observer is incorrect we provide a counterexample for the result of convergence in this paper results for convergence are obtained with additional strengthened hypothesis to correct rajamani and cho s result the results are used to design an observer for a now classical single link flexible robot joint the behaviour of the observer to noisy output is quite satisfactory

stability of invariant maximal semidefinite subspaces ii applications self adjoint rational matrix functions algebraic riccati equations abstract the stability of various factorizations of self adjoint rational matrix functions and matrix polynomials as well as of hermitian solutions of symmetric matrix algebraic riccati equations is studied in the first part of this paper results on stability of certain classes of invariant subspaces of a matrix which is self adjoint in an indefinite inner product were obtained these results serve as the main tools in the investigation

generalization of gao s reaching law for higher relative degree sliding variables in this paper a generalized discrete time reaching law for sliding variables with arbitrary relative degree is proposed the reaching law is based on gao s strategy and ensures higher order switching type sliding motion which is a novel concept in the field of discrete time plants the controller synthesis process is simplified by performing a specific system state transformation it is demonstrated that the width of the layer to which the representative point of the system is driven decreases as the relative degree of the sliding variable gets bigger it has further been proven that achieving a narrower quasi sliding mode band width is reflected in reduced state error finally the reaching phase elimination for variables with arbitrary relative degree is considered this is obtained by implementing a time varying sliding surface in a finite number of initial time instants uniform ultimate boundedness of all state variables is guaranteed

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

a nested computational approach to the discrete time finite horizon lq control problem a new algorithmic setting is proposed for the discrete time finite horizon linear quadratic lq optimal control problem with constrained or unconstrained final state no matter whether the problem is cheap singular or regular the proposed solution based on matrix pseudoinversion is completed and made practically implementable by a nesting procedure for welding optimal subarcs that enables arbitrary enlargement of the control time interval

application of vibrational control to linear quadratic control problems abstract the application of the theory of vibrational control to linear quadratic control problems is developed the solution of the matrix riccati differential equation rde and the optimal trajectory are found approximately as power series of a small parameter the minimizing effect of vibrations on the maximal solution of rde and the cost functional is studied the region of attraction of the maximal solution of rde for the case of hamiltonian matrix with imaginary axis eigenvalues is found special attention is paid to the application of vibrations to the linear quadratic problem of stabilization with respect to a part of variables and transfer of the other variables to a given position a problem of vibrational stabilization and optimal control of a carriage with an inverted pendulum is solved as an example

a generalized reaching law for discrete time sliding mode control the paper presents a generalized algorithm for the reaching and boundedness of the sliding variable in case of discrete time sliding mode control systems the algorithm is general as it not only comprises functions in the sliding variable but also functions in other variables which are known with the help of the proposed generalized reaching law an important problem of a reaching law with uncertainty bounded by known functions is studied the convergence of the sliding variable to an ultimate bounding function in presence of such an uncertainty is shown by the band approach method this is achieved by appropriately selecting the functions in the proposed generalized reaching law

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

perturbation bounds and characterisation of the solution of the associated algebraic riccati equation abstract the paper deals with the associated algebraic matrix riccati equation aamre closely related to the standard algebraic matrix riccati equation arising in the theory of linear quadratic optimisation and filtering the sensitivity of the aamre relative to perturbations in its coefficients is studied both linear local norm wise and component wise and non linear non local perturbation bounds are obtained the conditioning of the aamre is determined in particular a full characterisation of the solution of aamre in terms of neutral subspaces of certain hermitian matrix is given which is a counterpart of the characterisation of the solutions to the standard riccati equation in terms of the invariant subspaces of the corresponding hamiltonian matrix a reliable method to obtain all solutions to aamre is briefly outlined

yet another characterization of solutions of the algebraic riccati equation abstract this paper deals with a characterization of the solution set of algebraic riccati equation are over reals for both controllable and uncontrollable systems we characterize all solutions using simple linear algebraic arguments it turns out that solutions of are of maximal rank have lower rank solutions encoded within it we demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution we characterize situations where there are no full rank solutions to the are we also characterize situations when the number of solutions to the are is finite when they are infinite and when they are bounded we also explore the poset structure on the solution set of are which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix we provide several examples that bring out the essence of these results

implicit euler numerical scheme and chattering free implementation of sliding mode systems in this paper it is shown that the implicit euler time discretization of some classes of switching systems with sliding modes yields a very good stabilization of the trajectory and of its derivative on the sliding surface therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used are avoided moreover the method an event capturing or time stepping algorithm allows for multiple switching surfaces i e a sliding surface of codimension 2 the details of the implementation are given and numerical examples illustrate the developments this method may be an alternative method for chattering suppression keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces links with discrete time sliding mode controllers are studied

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

invariant manifolds and projective combinations of solutions of the riccati differential equation abstract in this paper we show how families of solutions of the general riccati differential equation rde can be generated via projective combinations of a given number of reference solutions our approach is based upon the extension of the domain of the equation to the grassmannian manifold and the application of the radon lemma in this context we briefly discuss the relevance of our results to the study of the invariant manifolds of the equation and compare them to existing results concerning representation formulas for solutions of rde the results of the paper have been motivated by the recent characterization of solutions of the rde given in m pavon d d alessandro families of solution of matrix riccati differential equations siam j control optim 35 1 1997 194 204 which extends the classical results on the algebraic riccati equation due to willems coppel and shayman

totally isotropic subspaces for pairs of hermitian forms and applications to riccati equations abstract we prove a result on the existence of a common totally isotropic subspace for a pair of forms on v v where v is a finite dimensional space real complex or quaternionic and where one of the forms is hermitian while the other is skew hermitian applications to algebraic matrix riccati equations are given

uncontrollable dissipative systems observability and embeddability the theory of dissipativity is well developed for controllable systems a more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function namely a function such that along every system trajectory its rate of change at each time instant is at most the power supplied to the system at that time however even when the supplied power is expressible in terms of just the external variables the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable hidden from the external variables this paper investigates the key aspects of both cases and also proposes another intuitive definition of dissipativity these three definitions are compared we show that drawbacks of one definition are addressed by another dealing first with observable storage functions under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as f

reaching law for dsmc systems with relative degree 2 switching variable abstractthis paper presents the first attempt to design a reaching law based discrete time sliding mode controller with a relative degree 2 switching variable the current value of this variable is only affected by the control signal and disturbance generated two time instants ago it is demonstrated that the new reaching law based strategy offers a smaller quasi sliding mode band width when compared to a similar control scheme with relative degree 1 switching variable this in turn leads to reduced system output error in the control system proposed in the paper

unitary solutions of a class of algebraic riccati equations and factorization abstract we consider the algebraic riccati equation xd 1 x xd 2 d 3 x d 1 0 where d 2 and d 3 are self adjoint we are interested in contractive and unitary solutions x of this equation stability properties of such solutions under perturbations of the coefficients are studied further the connection between unitary solutions of such equations and symmetric factorizations of nonnegative rational matrix functions is investigated

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

families of solutions of algebraic riccati equations abstract we consider homogeneous algebraic riccati equations in the general situation when the matrix of the dynamics can be mixed we show that in this case the equation may have infinitely many families of solutions an analysis of these families is carried over and an explicit formula is derived when the matrix of the dynamics is unmixed this formula provides an explicit parametrization of the whole set of the solutions we finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which on the contrary the union of the families covers the whole set of solutions

perturbation of invariant subspaces the questions of how perturbation of an operator affects its invariant subspaces have raised a substantial amount of interest especially in the w x finite dimensions see e g 5 10 12 and the references given there it turns out that linear perturbation of the operator gives rise to an analytic perturbation of the projection on its invariant subspace under some reasonable conditions although the question of estimating the norms of the terms of the taylor series expansion of this projection valued function does have some applications in numerical analysis it seems to be studied w x less frequently for instance in 12 the matrix algebra approach combined with complex analysis in several variables was used to get existence and analyticity of the perturbed projection the terms were obtained as well using specially developed formulas for their direct computation however we find it hard to see that these estimates are good enough to ensure the convergence of this series let alone to give an estimate of its radius of convergence in this article we propose a different approach to the subject which achieves the two goals at one stroke and we believe with better results namely we study the problem in terms of operator valued analytic functions thus reducing the problem to solving an algebraic riccati type equation in order to solve it we extend in sections 2 and 3 somewhat the usual methods to our case see the comments about the meaning of usual in the last paragraph of the introduction the main result given in section 4 is the estimates of the radius of convergence and of the terms the estimates are clearly good enough to yield an estimated radius of

on the geometry of symplectic pencils arising from discrete time matrix equations abstract in this paper general symplectic matrix pencils are considered disregarding the particular matrix equations from which they arise a parameterization of the lagrangian deflating subspaces is given with the only assumption of regularity of the matrix pencil

on the necessity of diffusive couplings in linear synchronization problems with quadratic cost we show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals this holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left half plane whilst for eigenvalues in the open right half plane we present a counterexample in which the strong solution to the associated algebraic riccati equation is not diffusive for nonidentical systems satisfying the internal model principle for synchronization we show that a certain part of the coupling must be diffusive for equally chosen weights in the cost functional we show that the dimension of the associated algebraic riccati equation can be reduced significantly

on a connection between spectral factorization and geometric control theory we investigate here how the geometric control theory of basile marro and wonham can be obtained in a hilbert space context as the byproduct of the factorization of a spectral density with no zeros on the imaginary axis we show how controlled invariant subspaces can be obtained as images of orthogonal projections of co invariant subspaces onto a semi invariant markovian subspace of the hardy space of square integrable functions analytic in the right half plane output nulling subspaces are then related to a particular spectral factorization problem a similar construction is presented for controllability subspaces and a new algorithm for the computation of these subspaces is presented

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

existence of unmixed solutions of the discrete time algebraic riccati equation and a nonstrictly bounded real lemma abstract the existence of a solution of the discrete time algebraic riccati equation is established assuming modulus controllability and positive semidefiniteness on the unit circle of the popov function as an application a nonstrictly bounded real lemma is obtained

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

existence and uniqueness of unmixed solutions of the discrete time algebraic riccati equation a solution x of a discrete time algebraic riccati equation is called unmixed if the corresponding closed loop matrix x has the property that the common roots of det si x and det i s x if any are on the unit circle a necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of x lie in a prescribed subset of c

multivariate matrix refinable functions with arbitrary matrix dilation characterizations of the stability and orthonormality of a multivariate matrix refinable function d with arbitrary matrix dilation m are provided in terms of the eigenvalue and 1 eigenvector properties of the restricted transition operator under mild conditions it is shown that the approximation order of d is equivalent to the order of the vanishing moment conditions of the matrix refinement mask pa the restricted transition operator associated with the matrix refinement mask pa is represented by a finite matrix ami j i j with a idet m i 1 e p j q3 p e and p j q3 p being the kronecker product of matrices p j and p the spectral properties of the transition operator are studied the sobolev regularity estimate of a matrix refinable function 4 is given in terms of the spectral radius of the restricted transition operator to an invariant subspace this estimate is analyzed in an example

the regular indefinite linear quadratic problem with linear endpoint constraints this paper deals with the infinite horizon linear quadratic problem with indefinite cost given a linear system a quadratic cost functional and a subspace of the state space we consider the problem of minimizing the cost functional over all inputs for which the state trajectory converges to that subspace our results generalize classical results on the zero endpoint version of the linear quadratic problem and more recent results on the free endpoint version of this problem

vibrational control of nonlinear time lag systems vibrational stabilization and transient behavior abstract this paper addresses two problems of control of nonlinear time lag systems i an existence and a synthesis of parametric vibrations for their stabilization and ii the transient behavior analysis of the vibrationally controlled nonlinear systems with time lags in this work stabilizability conditions for two vibration types are formulated and procedures for the synthesis of the corresponding stabilizing vibrations are proposed the method for the transient behavior analysis of vibrationally controlled systems on a finite time interval is developed as well several examples are given to support the theory presented

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

a new subdivision method for bivariate splines on the four directional mesh we present a new bivariate subdivision scheme based on two generators of a four directional spline space in particular we are dealing with piecewise linear continuous splines and with piecewise cubic continuously differentiable splines the subdivision schemes rely on a matrix mask of small support in addition we present some results on related quasi interpolating operators and on approximation order of the underlying shift invariant spaces

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

the regular free endpoint quadratic problem with indefinite cost this paper studies an open problem in the context of linear quadratic optimal control the free endpoint regular linear quadratic problem with indefinite cost functional it is shown that the optimal cost for this problem is given by a particular solution of the algebraic riccati equation this solution is characterized in terms of the geometry on the lattice of all real symmetric solutions of the algebraic riccati equation as developed by willems ieee traps automat control 16 1971 pp 621 634 and coppel bull austral math soc 10 1974 pp 377 401 a necessary and sufficient condition is established for the existence of optimal controls this condition is stated in terms of a subspace inclusion involving the extremal solutions of the algebraic riccati equation the optimal controls are shown to be generated by a feedback control law finally the results obtained are compared with classical results on the linear quadratic regulator problem

a parametrization of solutions of the discrete time algebraic riccati equation based on pairs of opposite unmixed solutions the paper describes the set of solutions of the discrete time algebraic riccati equation it is shown that each solution is a combination of a pair of opposite unmixed solutions there is a one to one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution the results of the paper provide an extended counterpart of the parametrization theory of continuous time algebraic riccati equations by willems coppel and shayman

a symmetry group for the matrix riccati equation we describe a symmetry group for the matrix riccati differential equation and prove a reduction theorem which simplifies the calculation of the elements of this group we also find necessary and sufficient conditions for two equilibrium points to be symmetry equivalent