near optimal control of nonstandard singularly perturbed systems abstract near optimal control of nonstandard multiparameter multitime scale singularly perturbed systems is studied using the descriptor variable approach in particular the slow models of the considered systems are allowed to exhibit impulses in the case of multiparameter singular perturbations the near optimal control is a cascade connection of separately designed slow and fast subregulators in the case of multitime scale singular perturbations the near optimal control is hierarchically composed of n 1 subregulators a parallel algorithm is provided for designing the subregulators separately the knowledge of the small singular perturbation parameters is not required in the overall design furthermore the frequency responses of the near optimal systems are analyzed and several design examples are given

observability of structured linear systems in descriptor form a graph theoretic approach this paper deals with the right hand side observability of structured linear systems in descriptor form the method we propose is based on a graph theoretic approach it is applicable to a large class of descriptor systems including regular and nonregular systems even if they are square underdetermined or overdetermined after defining digraphs which represent such systems we express in graphic terms necessary and sufficient conditions to achieve the observability of the whole or of a given part of the state these conditions are easy to check because they are based on comparison of integers and on finding edges and subgraphs in a digraph therefore our approach is suited to study large scale and or uncertain systems

inner and outer geometry for singular systems with computation of subspaces abstract output nulling and unknown input subspaces are defined for singular systems with recursions given for their computation inner i e domain subspaces are distinguished from outer i e codomain subspaces a general duality theory is presented and the composite image composite pre image and composite subspaces are introduced it is shown how output nulling and unknown input subspaces may be computed via the singular system structure algorithm and a new square root algorithm the latter algorithm is shown to be related to the solution of a riccati equation for singular systems a simplified riccati solution procedure is introduced that only requires the solution of a non square lyapunov equation and also applies in the state variable case

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

a frequency domain approach to control of singular systems in this paper singular control systems are studied from the frequency domain point of view the proper stable factorization of a singular system is derived the stability of the closed loop system consisting of a singular system and controller is addressed most significantly all proper stabilizing controllers are parameterized for a given singular system

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

h control for descriptor systems a matrix inequalities approach abstract this paper considers the h control problem for descriptor systems that possibly have impulsive modes and or j axis zeros first we propose matrix inequalities that give a generalized stability condition and an h norm condition for descriptor systems using these matrix inequalities we show that the solvability of a set of matrix inequalities is necessary and sufficient to the existence of a proper controller that satisfies a prescribed h norm condition as well as stabilizing the closed loop system and eliminating all impulsive modes these inequalities are equivalent to certain linear matrix inequalities to which we can get solutions whenever they exist using efficient polynomial time algorithms

design of a class of luenberger observers for descriptor systems a new approach to the design of a class of luenberger observers for descriptor systems is presented the design is performed in a straightforward manner the conditions required in the design are given in terms of the original system matrices the relationship among these and other well known conditions is highlighted

topological aspects of controllability and observability on the manifold of singular and regular systems abstract in a previous article j d cobb j math anal appl nov 1986 we considered the class of all singular and regular linear time invariant systems and proved some basic topological properties of that set in this paper we examine specific implications of those results to control theory and demonstrate among other things that controllability and observability are generic properties even when singular systems are included in the construction we also derive related results for other important subclasses of systems proving that only some of the remaining fundamental system properties are generic finally we extend existing results on connectedness and show that the number of connected components of the controllable and observable sets is diminished whenever singular systems are brought into the picture

analysis on generalized superspaces the analysis over commutative algebras generalized supercommutative algebras that is differentiation and integration for functions defined on superspace over a commutative algebra is studied

open loop optimal control of linear time invariant systems containing the first derivative of the input abstract linear time invariant systems of the form are considered the problem is to find a control vector u t that will drive the state of this system from a given z 0 to some not necessarily fixed final state z t f in some not necessarily fixed time t f while minimizing a cost functional of the form the original system is re written in the form of a descriptor variable system consisting of the equations diag it is shown that there is a matrix such that thus the original optimal control problem is equivalent to the following problem find a control vector u t that drives the descriptor vector x t from a given x 0 to some x t f while minimizing using the results of an earlier paper on optimal control of descriptor systems ibrahim et al 1988 necessary conditions are derived for the existence of minima of the problem of finding sufficient conditions for the existence of minima of is not considered various formulae are obtained for designing open loop controls corresponding to three diffe

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

statistical inference method for liner impedance eduction with a shear grazing flow this paper presents a statistical inference method for impedance eduction in a flow duct facility the acoustic impedance is recast into a random variable and bayes s theorem is used to obtain the

geometric structure and feedback in singular systems the output nulling a e r b invariant subspaces are defined for singular systems rigorously justifying the name and demonstrating that special cases of these geometric objects are the familiar subspace of admissible conditions and the supremal a e r b invariant subspace a novel singular system structure algorithm is used to compute them by numerically efficient means their importance for describing the possible closed loop geometric structure in terms of the open loop geometric structure is shown an approach to spectrum assignment in singular systems that is based on a generalized lyapunov equation is introduced the equation is used to compute feedback gains to place poles and assign various closed loop invariant subspaces while guaranteeing closed loop regularity

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

on the lyapunov theorem for singular systems in this paper we revisit the lyapunov theory for singular systems there are basically two well known generalized lyapunov equations used to characterize stability for singular systems we start with the lyapunov theorem of the work by lewis we show that the lyapunov equation of that theorem can lead to incorrect conclusion about stability some cases where that equation can be used are clarified we also show that an attempt to correct that theorem with a generalized lyapunov equation similar to the original one leads naturally to the generalized equation of takaba et al

the linear quadratic optimal regulator for descriptor systems discrete time case abstract in this paper the linear quadratic optimal regulator problem for the discrete time descriptor system ex k 1 ax k bu k where e is in general a singular matrix is investigated the continuous time version of this problem was treated elsewhere the discrete time theory is largely parallel to the continuous time theory but there are several important differences one is that in the discrete time case a fairly general problem can be solved with singular cost matrices the key to the solution as for the continuous time problem is a coordinate system which can be computed by performing a singular value decomposition of e this is called an svd coordinate system the optimization problem is solved by applying hamiltonian minimization to compute the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors in the general singular cost case this trajectory is computed by recursively solving a discrete time riccati equation from the optimal trajectory the optimal descriptor variable feedback gain relating the control to the descriptor variable can be computed several riccati difference equations are derived all of which have the same solution this solution can be used to compute the optimal trajectory and cost for non singular problems even in the finite horizon case the optimal feedback gain can be computed non recursively by solving an eigenvalue eigenvector problem formulated from the system parameters in the svd coordinate system the optimal gain for the infinite horizon steady state problem on the other hand can be computed from the solution of an eigenvector problem involving the untransformed system parameters

solutions to the output regulation problem of linear singular systems in this paper we revisit the output regulation problem for linear singular systems and identify an important case of the general regulation problem where the measurement output is identical to the vector to be regulated we derive a necessary and sufficient condition for the regulation problem to be solvable via either full information feedback or error feedback then we show how full information feedback and error feedback controllers can be constructed explicitly

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

direct controllability and observability time domain conditions of singular systems controllability and observability criteria for singular systems using direct formulas in the time domain are introduced the method used is based on a direct expansion of the transfer function matrix via a leverrier type algorithm hence no decomposition of the state space model is needed and the criteria are implementable on a digital computer the compact form of the matrices might be helpful in other analysis as well as synthesis problems in singular systems

an algebraic condition for controllability at infinity abstract a necessary and sufficient algebraic condition is given for a generalized state space system to be controllable at infinity a dual condition is given for observability at infinity

adaptive discontinuous galerkin methods for eigenvalue problems arising in incompressible fluid flows in this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible navier stokes equations particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries here computable a posteriori error bounds are derived based on employing the generalization of the standard dual weighted residual approach originally developed for the estimation of target functionals of the solution to eigenvalue stability problems the underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution in this way errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented

robustness of regional pole placement for uncertain continuous time implicit systems in this note under assumptions that the nominal linear implicit system is regular impulse free with all its generalized eigenvalues lying inside some specified regions sufficient conditions are proposed to preserve the assumed properties when structured perturbations are added into the nominal system explicit bounds on perturbations with different structures are given in terms of the spectral radii of some nonnegative matrices

descriptor variable systems and optimal state regulation linear systems of the form e dot x ax bu with e singular are treated it is desired to find a control which drives the system asymptotically to the origin minimizing a quadratic cost functional no restrictions are placed on initial conditions the cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order riccati equation

fundamental problems with the model of uniform flow over acoustic linings abstract the walls of ducts containing flow are often acoustically lined in order to reduce sound many simple models of acoustic linings assume the lining to be linear and locally reacting examples considered here include the three parameter mass spring damper helmholtz resonator and enhanced helmholtz resonator models all of these models have been found to have stability issues with uniform mean grazing flow and there has been some confusion over the existence of hydrodynamic instability surface waves over such linings mathematically the standard proven briggs bers stability analysis is not applicable computationally the hydrodynamic modes are routinely ignored in the frequency domain and instabilities filtered out in the time domain this confusion also causes significant problems for mode matching green functions and scattering analyses in this paper it is shown that any situation not capable of being analysed using the briggs bers criterion is illposed and that this is the root cause of the confusion over hydrodynamic instabilities a large class of lining models including all those mentioned above are shown to be illposed with uniform mean flow an explanation is given of the effects of this illposedness in practice and it is argued that illposed models should not be used an alternative stability criterion to briggs bers has been occasionally used in the literature this alternative criterion involving analytic continuation to purely imaginary frequencies was recently christened the crighton leppington stability criterion however this stability criterion is incorrect and several counter examples are given here

some recent results in combinatorial approaches to dynamical systems abstract this paper is a unified presentation of recent results obtained by the author on the structural analysis of dynamical systems by matroid theoretic combinatorial methods the first part consists of the results on combinatorial and algorithmic characterizations of control theoretic generic properties the analysis is based on a physically reasonable mathematical model which assumes that the coefficients in the equations are classified into independent physical parameters and dimensionless fixed constants the concept of mixed matrix and its canonical form turn out to be convenient mathematical tools the second part describes a combinatorial analogue of the dynamical system theory developed in the matroid theoretic framework the matrices a and b in the state space equations ax bu are replaced by bimatroids or linking systems

controllability and observability of second order descriptor systems we analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case we show that performing a classical transformation to first order form may destroy some controllability and observability properties as an example we demonstrate that the loss of impulse controllability in constrained multibody systems is due to the representation as a first order system to avoid this problem we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties

fundamental properties of the manifold of singular and regular linear systems abstract we construct a real analytic manifold l of systems of the form e x ax bu y cx and show that l is the completion with respect to solutions of the set of regular state space systems i e those systems with nonsingular e other geometric and analytic properties of l are established including genericity of the regular systems

generalized complex superspace involutions of superfields distinguished involutions called transposition and conjugation in the algebra of complex supernumbers are introduced on the base of these involutions an analysis is developed over complex superspaces consistency of integration under supertransformations between two complex superspaces is shown

observers and dynamic controllers for linear differential algebraic systems in the present paper we study state estimation and stabilization by dynamic feedback for linear differential algebraic systems which are not necessarily regular we show that the observer synthesis approach for behavioral systems in m e valcher and j c willems ieee trans automat control 44 1999 pp 2297 2307 can be applied to differential algebraic systems in a closed form i e the observers and dynamic controllers are again differential algebraic systems the concept of an asymptotic exact observer is introduced and existence is characterized since initialization of the observer is an important issue we investigate regular and freely initializable observers whose existence can be guaranteed by impulse observability of the plant the observers are then exploited for the construction of dynamic controllers we show that there exists a stabilizing controller if and only if the given system is both behaviorally stabilizable and behaviorally detectable

stability analysis and stabilization of 2 d singular roesser models abstract this paper addresses the problem of controller design for 2 d singular roesser models sufficient stability conditions are formulated in terms of strict linear matrix inequalities lmis for 2 d singular systems described by the roesser models then a set of sufficient conditions for existence of desired a state feedback controller are obtained numerical examples are provided to demonstrate the effectiveness of the proposed method

spectral characterization and enforcement of negative imaginariness for descriptor systems abstract systems with counterclockwise input output dynamics or negative imaginary transfer functions arise in various applications such as the modeling of flexible mechanical structures or electrical circuits when certain kinds of measurements are taken in this paper we introduce descriptor systems with such an additional structure we state various of their properties and prove algebraic characterizations of negative imaginariness in terms of spectral conditions of certain structured matrix pencils for this purpose we also analyze particular boundary cases which are characterized by properties of a structured kronecker canonical form finally we describe a method which can be used to restore the negative imaginary property in case that it is lost this happens e g when a system with theoretically negative imaginary transfer function is obtained by e g model order reduction methods linearization or other approximations the method is illustrated by numerical examples

computation of the impulsive behavior of multivariable linear systems using a division algorithm a new method is proposed which allows the evaluation of the impulsive behavior of a linear multivariable system of arbitrary order the proposed method is in closed matrix form that is only constant real matrix arithmetic i e multiplication and addition is involved among the polynomial matrices and therefore the algorithm can be easily programmed in a digital computer

on pole structure assignment in linear systems the problem of pole structure assignment psa by state feedback in implicit linear and uncontrollable systems is discussed in the article it is shown that the problem is solvable if the system is regularisable then necessary and sufficient conditions for characteristic polynomial assignment are established in the case of psa invariant polynomials assignment just necessary conditions have been obtained but it turns out that these conditions are also sufficient in some special cases this happens for example when the system does not possess any non proper controllability indexes a possible application of the achieved results to modelling a constrained movement of a robot arm is outlined too

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

controllability and observability at infinity of multivariable linear second order models in this note we give in the spirit of 4 numerically attractive tests of controllability and observability of linear multivariable second order systems in which the leading matrix is singular

discontinuous galerkin method for the computation of acoustic modes in lined flow ducts with rigid splices a discontinuous galerkin method for the discretisation of the linearised euler equations is derived for the computation of acoustic modes in an infinite duct of arbitrary geometric shape in presence of axial sheared flow validations are performed with analytical solutions in the case of a uniformly lined duct and comparisons are done with the literature in the case of a lined cylindrical duct with rigid splices we take advantage of the discontinuous formulation to study the behaviour of the acoustic eigenfunctions in the vicinity of the liner splice transition where singular behaviours of the acoustic perturbation are found

h2 output feedback control for descriptor systems this paper solves the h 2 output feedback control problem for continuous time descriptor systems an existence condition of the h 2 optimal controller is derived based on the standard model matching technique and a characterization of the optimal controller is given in terms of the solutions of two generalized algebraic riccati equations it turns out that the optimal controller has an observer based structure well known in the state space case so that the results in this paper are natural extensions of the state space results to the descriptor systems a numerical example is also included to show the applicability of the proposed results

technical communique impulse observability and impulse controllability of linear time varying singular systems impulse controllability and impulse observability of linear time varying singular systems in a standard canonical form are examined we focus on the fast subsystems with constant rank n matrices sufficient and necessary conditions are derived and presented directly in terms of system parameters conditions obtained for these two properties are algebraically dual to each other

sufficient conditions for impulse uncontrollability and impulse unobservability of singular systems simple sufficient conditions for impulse uncontrollability and impulse unobservability of completely singular systems are established it is well know that a singular linear system can be decomposed into a regular subsystem and a completely singular subsystem from previously published results it follows that a singular system is impulse uncontrollable unobservable if its singular subsystem is impulse uncontrollable unobservable it is shown that in a particular case the results can be also applied to the controllability and observability of regular linear systems

initial and transient response improvement for singular systems this paper deals with a generalized lq regulator problem for singular systems with random initial states the idea of the generalization is to take care of instantaneous jumps and the problem is solved using the gradient flow approach the designed output feedback will ensure that the closed loop system has not only a satisfactory transient response but also significantly low instantaneous jumps with respect to any random initial state with given expectation and covariance

some generalized notions of observability generalized state space systems are a more natural way of describing dynamical behavior than standard state space systems such systems exhibit a much richer observability structure than standard state space systems three notions of observability are defined for generalized state space systems in contrast to previous work regularity is not assumed geometric characterization of each type of observability and matrix rank tests for each type of observability are derived

filtering and lqg problems for discrete time stochastic singular systems discrete time stochastic singular systems are discussed filtering and linear quadratic gaussian lqg problems are considered the problems are first transformed into the normal form via state augmentation and then solved by utilizing standard results for nonsingular systems the linear unbiased least squares state estimation algorithm and the optimal control law for the lqg problem are given the order of the filtering algorithm obtained in this way is not much increased moreover this algorithm allows the presence of some kinds of control inputs

well posed boundary condition for acoustic liners in straight ducts with flow doi 10 2514 1 j050723 the myers boundary condition for acoustics within flow over an acoustic lining has been shown to be ill posed leading to numerical stability issues in the time domain and mathematical problems with stability analyses this paper gives a modification for flat or cylindrical straight ducts to make the myers boundary condition well posed and indeed more accurate by accounting for a thin inviscid boundary layer over the lining and correctly deriving the boundary condition to first order in the boundary layer thickness the modification involves two integraltermsovertheboundarylayer the firstmaybewrittenintermsofthemass momentum andkinetic energy thicknessesoftheboundarylayer whichareshownto physicallycorrespondtoamodifiedboundarymass modified grazingvelocity andatensionalongtheboundary thesecondintegral termisrelatedtothecriticallayerwithinthe boundary layer a time domain version of the new boundary condition is proposed although not implemented the modified boundary condition is validated against high fidelity numerical solutions of the pridmore brown equation for sheared inviscid flow in a cylinder absolute instability boundaries are given for certain examples though convective instabilities appear to always be present at certain frequencies for any boundary layer thickness

unknown input functional observability of descriptor systems with neutral and distributed delay effects in this paper a general class of linear systems with time delays is considered which includes linear classical systems linear systems with commensurate delays neutral systems and singular systems with delays after given a formal definition of functional backward observability bo an easily testable condition is found the fulfillment of the obtained condition allows for the reconstruction of the trajectories of the system under consideration using the actual and past values of the system output and some of its derivatives the methodology we follow consists in an iterative algorithm based upon the classical silverman algorithm used for inversion of linear systems by using basic module theory we manage to prove that the proposed algorithm is convergent a direct application of studying functional observability is that a condition can be derived for systems with distributed delays also we do this as a case of study the obtained results are illustrated by two examples one is merely academic but illustrates clearly the kind of systems which the proposed methodology works for and the other is a practical system with distributed delays

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

linear transformations which leave controllable multiinput descriptor systems controllable abstract a general linear descriptor system e itx ax bu where e a c n n b c n m x c n u c m e singular is called controllable if rank e a b n for all 0 0 c let f c n n m c n n m be a linear transformation form f x cxd we characterize all such linear transformations that leave the set c e a b rank e a b n 0 0 c invariant and show that only the well known transformations that leave controllability invariant can occur

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

the full information and state feedback h2 optimal controllers for descriptor systems in this paper we consider the full information and the state feedback h 2 control problems for continuous time descriptor systems an explicit characterization of all optimal controllers is given for these cases as a consequence we are able to present a parameterization of all optimal static and dynamic controllers for the lqr problem via state feedback we also show that for the general state feedback h 2 optimization the usual gain matrix defined as a linear function of a solution of a generalized riccati equation can be non optimal

an sdr algorithm for the solution of the generalized algebraic riccati equation an algorithm is presented for solving certain classes of the algebraic riccati equation by using the mdr transformation and extending the symplectic qr like or sr method of a bunse gerstner linear alg applic vol 83 p 49 78 1986 and v mehrmann ph d diss univ bielefeld w germany to the generalized eigenvalue problem the riccati equations arise from linear systems of the form edx dt ax bu where e is nonsingular but for numerical reasons e sup 1 a and e sup 1 b cannot be computed reliably

lie groups and lie algebras with generalized supersymmetric parameters matrices with symmetric parameters the most general extension of supersymmetric parameters are investigated the superdeterminants of such matrices are defined lie groups consisting of these matrices and their lie algebras are studied

composite nonlinear feedback control for linear singular systems with input saturation abstract this paper investigates the composite nonlinear feedback cnf control technique for linear singular systems with input saturation first a linear feedback control law is designed for the step tracking control problem of linear singular systems subject to input saturation then based on this linear feedback gain a cnf control law is constructed to improve the transient performance of the closed loop system by introducing a generalized lyapunov equation this paper develops a design procedure for constructing the cnf control law for linear singular systems with input saturation after decomposing the closed loop system into fast subsystem and slow subsystem it can be shown that the nonlinear part of the cnf control law only relies on slow subsystem the improvement of transient performance by the proposed design method is demonstrated by an illustrative example

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers

comments on on the robust popov criterion for interval lur e systems this paper supplements the remarks made in a paper by dahleh et al ieee trans automat contr vol 38 p 1400 5 1993 regarding the popov sector of lur e systems with interval plants a numerical example is provided to show that strict inequalities hold among three sectors those obtained by the theorem of the paper by applying popov theorem to each member system and by applying that to each of the systems with kharitonov plants this counterexample negates a tempting conjecture on strong robust popov criterion

convergence of singular perturbations in singular linear systems abstract a convergence criterion for singular perturbations in linear systems is established the criterion is useful for regular and singular systems it is assumed that the matrices of the system depend analytically on e it is proved that if the real parts of the divergent eigenvalues tend to the solution of the system converges to the solution for e 0 uniformly in compact subsets of 0 on the other hand if there is a divergent eigenvalue whose real part does not tend to then there are initial values such that the solution does not converge at some points the criterion is valid both for the homogeneous equation and for the nonhomogeneous equation if functions are sufficiently smooth the use of distributions allows the analysis with inconsistent initial conditions

a geometric approach to proportional plus derivative feedback using quotient and partitioned subspaces abstract in this paper a new characterization of invariant subspaces is presented using the notions of partitioned and quotient subspaces this classification is based on a feedback approach that exhibits the importance of these subspaces for the problem of proportional plus derivative feedback it also provides the ability to decompose the closed loop system into two subsystems that completely characterize the closed loop behavior a feedback free formulation is also considered which opens new horizons for the problem of general semistate feedback by utilizing nonsquare pencils for spectrum assignability the importance of this technique arises from the fact that we deal with reduced order pencils therefore the computational methods are more stable in order to accomplish this we use two generalized lyapunov equations and exploit the generalized hessenberg form

preview control for impulse free continuous time descriptor systems this paper studies the preview control problem of impulse free linear continuous time descriptor systems the system is first decomposed into a normal system i e slow subsystem and an algebraic equation set by restricted equivalent linear transformation then applying the method of preview control theory to the slow subsystem by taking derivatives on both the error vector and the state function and with the error vector being a part of the new state vector the augmented system is constructed and the tracking problem is transformed into a regulation problem according to preview control theory the controller of the augmented system can be obtained and the control input of the original descriptor system with preview function can be acquired by integrating on the controller of the augmented system both the stabilisability and detectability of the augmented system are discussed numerical simulation verifies the presented results

formulae for the distance between controllable and uncontrollable linear systems in this paper under some special well defined norms the formulae for the distance between controllable and uncontrollable linear systems of the general form eucx ax bu are presented where e a and b are constant complex matrices and e may be singular these results generalize and improve the existing ones