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

a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

optimal control of 2 d systems two dimensional 2 d optimal control theory that parallels one dimensional 1 d optimal control is developed a generalized performance measure suited to 2 d systems is introduced the canonical equations associated with this performance measure and a general nonlinear model are obtained the 2 d linear quadratic regulator problem is formulated and its canonical equations are derived for the roesser model an earlier result by t kaczorek and j klamka 1986 for the solution of the minimum energy problem with fixed final local state is rederived using this approach a new problem minimum energy with fixed final pass local states is formulated and solved and a numerical example is given

complete decomposition of sub optimal regulators for singularly perturbed systems the linear regulator problem is studied for systems with slow and fast modes using the descriptor variable method the design of sub optimal regulators for such systems is completely decomposed it is shown that the sub optimal regulator is composed of fast and slow regulators and that both of these mutually decoupled can be designed independently a relevant computational procedure is also presented the procedure is applicable to designs of sub optimal regulators with either finite or infinite terminal time and for both singular and normal singularly perturbed systems

the linear quadratic regulator for periodic hybrid systems abstract the main objective of this paper is to characterize feedback control laws that are optimal with respect to a quadratic cost functional in the framework of linear hybrid systems undergoing time driven periodic jumps namely the so called hybrid linear quadratic regulator lqr problem the optimal solution to the hybrid lqr problem is determined both in the case of finite horizon and infinite horizon optimal control problems by introducing a hybrid periodic extension of the classic differential and difference riccati equations thus leading to the notion of monodromy riccati equation interestingly due to the periodic nature of the discrete time events the computation of the optimal feedback hinges upon the solution of a differential rather than algebraic riccati equation also in the infinite horizon case hence yielding a time varying periodic control law necessary and sufficient conditions that ensure asymptotic stability of the closed loop system are provided and discussed in detail in the case of infinite horizon optimal control problems

a lyapunov approach to analysis of discrete singular systems in this paper a new type generalized lyapunov equation for discrete singular systems is proposed then it is applied to study problems such as pole clustering controllability and observability for discrete singular systems first some necessary and sufficient conditions for pole clustering are derived via the solution of this new type lyapunov equation further the relationship between the solution of the lyapunov equation and structure properties of discrete singular systems will be investigated based on these results finally a type of generalized riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the finite poles of the closed loop systems are clustered into a specified disk

the linear quadratic optimal regulator for singular 2 d systems with variable coefficients the linear quadratic optimal regulator problem for singular 2 d linear systems with variable coefficients has been formulated sufficient conditions for the existence of a solution to the problem have been established an optimal sequence of input vectors can be found with slight modification the considerations can be extended for singular n d linear systems with variable coefficients

matrix assignment for 2 d systems with periodically variable coefficients by periodic feedback abstract the matrix assignment problem by periodic state feedback for 2 d linear systems with periodically variable coefficients is considered it is shown that an arbitrary matrix can be assigned by periodic state feedback if the system is locally controllable a procedure for finding a sequence of feedback gain matrices is presented and illustrated by a numerical example

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

kalman filtering and riccati equations for descriptor systems a general formulation of a discrete time filtering problem for descriptor systems is considered it is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems using a dual approach to estimation the authors derive a so called 3 block form for the optimal filter and a corresponding 3 block riccati equation for a general class of time varying descriptor models which need not represent a well posed system in that the dynamics may be either over or under constrained specializing in the time invariant case they examine the asymptotic properties of the 3 block filter and in particular analyze in detail the resulting 3 block algebraic riccati equation the noncausal nature of discrete time descriptor dynamics implies that future dynamics may provide some information about the present state a modified form for the descriptor kalman filter that takes this information into account is presented

general response formula and minimum energy control for the general singular model of 2 d systems the general response formula for the general singular model of 2 d linear systems is derived the concepts of local reachability and local controllability are extended to the singular model necessary and sufficient conditions for local reachability and local controllability are established the minimum energy control problem for the singular model is formulated and solved

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

new results and application of singular control we reconsider the singular control problem with marginal stability of the closed loop system when the transfer matrix from the input to the output can have linearly dependent columns and zeros on the extended imaginary axis we present a new theoretical existence result and a new numerical algorithm based on finding an orthogonal transformation of the matrix pencil associated to the euler lagrange differential equations into a block triangular form we present an application in linear quadratic control of descriptor systems under the constraints of physical realizability of the control and impulse free and marginally stable closed loop system

a discrete optimal control problem for descriptor systems a class of discrete optimization problems described by nonlinear descriptor systems is considered on the basis of nonlinear programming a discrete maximum principle is derived for a very general structural form of descriptor systems the necessary and sufficient conditions for optimality are given for a subclass of optimization problems which is represented by linear descriptor dynamics with quasiconvex inequality constraints and pseudoconvex cost functional for illustration an example of a county macroeconomic system is presented

analysis of stability robustness for generalized state space systems with structured perturbations abstract a new approach is proposed to analyze the stability robustness of generalized state space systems with structured perturbations the presented method is computationally simple to use and can easily be calculated by computer as far as we are aware this paper seems to be the first one to solve the robust stability problems for generalized state space systems with structured uncertainties the robust stability problem of generalized state space systems is more complicated than that of regular state space systems because it needs consideration of not only stability robustness but also system regularity and impulse elimination the latter two ones need not be considered in regular state space systems

lq control of descriptor systems by cancelling structure at infinity this paper presents results concerning lq control of arbitrary descriptor systems and a numerical algorithm for achieving it the algorithm consists of two steps the first step is using the silverman algorithm orthogonal version transformation of the problem of lq control of descriptor system into a problem of lq control of a strictly proper system with singular or non singular new matrix 22 if new 22 is singular we perform the second step i e solving the lq singular control problem using again the silverman algorithm regarding the existing results on the lq singular control problem we have new results a counter example that this problem may not have a stable solution most general solvability conditions with or without stability and a new numerical algorithm based on using orthogonal matrices when stabilizability conditions are satisfied the algorithm places simultaneously stable poles in the closed loop system we show that the traditional riccati equation associated to this proble

minimum energy control for general model of 2 d linear systems a new definition of the straight line controllability for the general model of 2 d linear systems is presented necessary and sufficient conditions for the straight line controllability of the general model are established the minimum energy control problem for the general model of 2 d systems with constant coefficients is formulated and solved

a design approach for constrained regulation in discrete singular systems abstract in the present study the problem of designing controllers for linear discrete time singular descriptor systems under initial state and control constraints is considered the aim is to eliminate any non causalities and make the closed loop system stable while the state feedback control vector must respect appropriate constraints the present approach constitutes an extension to the design of constrained linear state feedback controllers for linear regular discrete time systems

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

transformations between discrete time and continuous time algebraic riccati equations we introduce a transformation between the discrete time and continuous time algebraic riccati equations we show that under mild conditions the two algebraic riccati equations can be transformed from one to another and both algebraic riccati equations share common hermitian solutions the transformation also sets up the relations about the properties commonly in system and control setting that are imposed in parallel to the coefficient matrices and hermitian solutions of two algebraic riccati equations the transformation is simple and all the relations can be easily derived we also introduce a generalized transformation that requires weaker conditions the proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic riccati equations and the associated system and control problems

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

deadbeat control of discrete descriptor systems with inaccessible descriptor vector the problem of deadbeat compensator design is considered for a discrete descriptor system with inaccessible descriptor variables the full compensator design is reduced to separate designs of two regular deadbeat controllers of a lower order driven by this compensator the descriptor variables of the system starting from given initial conditions can reach the origin in a minimum of c 0 time steps where c and c are the largest right kronecker indices of the pencils ze a b and ze a c respectively

exact unidirectional perturbation bounds for robustness of uncertain generalized state space systems continuous time cases abstract the robustness of continuous time generalized state space systems with unidirectional perturbations is investigated in this paper by transforming the original problem to a robust rank problem we provide a simple method to compute the exact bounds on such perturbations so that the important features of regularity impulse immunity and stability of nominal systems are all preserved this approach can be viewed as a generalization of fu and barmish s result to generalized state space setups it also provides a tractable solution to many practical robustness problems this is clearly explained by a simple circuit example 1997 elsevier science ltd

lq optimal control for 2d roesser models of finite extent abstract this paper investigates several aspects of linear quadratic lq optimal control for roesser models over a two dimensional 2d signal index set of finite extent first we consider the characterisation and computation of open loop control laws when constraints on the system semi states are imposed at both the south west and north east boundaries of the frame i e signal index set of interest by virtue of the quarter plane causal structure of the roesser model the south west and north east boundary conditions are analogous to initial conditions and terminal constraints respectively a necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data in the second part of the paper the problem of optimal lq control via semi state feedback is considered a 2d riccati like difference equation is introduced to characterise in a sufficient sense a solution to this problem

a spectral factorization algorithm for discrete time descriptor systems via generalized eigenproblems abstract this paper considers a factorization of discrete time improper spectral matrices using the generalized eigenproblem gep approach we first derive a factorization algorithm for the spectral matrix formed by an rh transfer matrix based on the gep associated with a descriptor representation of the spectral matrix and a related linear equation we then extend the factorization algorithm to the case where the spectral matrix is formed by an improper rl matrix a simple example is included to show the applicability of the present algorithm

general model of n d system with variable coefficients and its reduction to 1 d system with variable structure a general model of an n d linear discrete system with variable coefficients and its solution are presented a method for reduction of this model to an equivalent 1 d system with a variable structure is given

a note on optimal control of generalized state space descriptor systems linear time invariant systems of the form e dx dt ax bu are considered where e is a square matrix which may be singular and b is a rectangular matrix having full column rank it is assumed that for any admissible initial state x 0 any control vector u t yields one and only one state vector x t the problem is this find a control vector u t that will drive the system from an admissible initial state x 0 to a fixed final state xf in a fixed time tf while minimizing some cost functional only elementary matrix and variational techniques are used necessary conditions are derived for the existence of minima of j the problem of finding sufficient conditions of the existence of minima of j is not considered it is shown that in many cases the necessary conditions for the existence of minima of j yield a two point boundary value problem consisting of a system of ordinary differential equations containing only elements of x t and the boundary conditions prescribed at 0 and tf if some

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

square root kalman filtering of descriptor systems abstract a novel square root information filter algorithm is proposed to solve the kalman filtering problem for discrete time descriptor systems unlike standard square root information filters the present approach does not impose restrictions on the state transition matrix or on the noise covariance matrix the resulting implementation exhibits a moderate computational burden and good numerical properties and has general applicability to the state estimation of discrete time linear dynamic descriptor systems

the quasi weierstras form for regular matrix pencils regular linear matrix pencils a e k n n where k q r or c and the associated differential algebraic equation dae e u x ax are studied the wong sequences of subspaces are investigate and invoked to decompose the k n into v w where any bases of the linear spaces v and wtransform the matrix pencil into the quasi weierstras form the quasi weierstras form of the matrix pencil decouples the original dae into the underlying ode and the pure dae or in other words decouples the set of initial values into the set of consistent initial values vand pure inconsistent initial values w 0 furthermore vand ware spanned by the general ized eigenvectors at the finite and infinite eigenvalues resp the quasi weierstras form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of a e lead to the well known weierstras form so the latter can be viewed as a generalized jordan form finally it is shown how eigenvector chains constitute a basis for the solution space of e u x ax

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

minimum energy control of 2 d linear systems with variable coefficients the definition of the state transition matrix for roesser s model of 2 d systems with variable coefficients is introduced and the general response formula is derived necessary and sufficient conditions for local controllability are formulated the minimum energy control problem for roesser s model of 2 d systems with variable coefficients is formulated and solved

variational calculus for descriptor problems the first order necessary condition for optimality is derived from a variational argument that involves an ad hoc modification of the bliss method resulting in a hamiltonian characterization in terms of edx dt rather than dx dt the former being smoother than the latter this approach sidesteps the regularity conditions of the lagrange multiplier theory under some mild assumptions the necessary condition for optimality is also sufficient and the optimal control exists the numerically relevant result is a generalized eigenvector inverse free characterization of optimality

deadbeat control of linear multivariable generalized state space systems the classic idea of deadbeat control is extended to linear multivariable discrete time generalized state space systems using algebraic methods the asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback the solution is constructed from a cheap control problem both semistate and output deadbeat control laws are considered the main design criteria are to drive the semistate and or outputs of the system to zero in minimum time and that the closed loop system be internally stable unique properties of these types of control laws are discussed for semistate deadbeat control all the dynamic poles including the ones at infinity are moved to the origin whereas for output deadbeat some of the finite transmission zeros are canceled numerically reliable algorithms are developed to solve both problems

the optimal regulation of generalized state space systems with quadratic cost abstract in this paper the optimal feedback control for regulating generalized state space systems with quadratic cost is presented by using an algebraic method and the weighting matrix q in the cost is allowed to be positive semi definite

robust analysis and control of parameter dependent uncertain descriptor systems in this paper we address the robust admissibility analysis and control of descriptor systems with parameter dependent possibly time varying uncertainties first we propose a new necessary and sufficient condition for the admissibility of linear time invariant lti descriptor systems this condition uses a strict linear matrix inequality lmi formulation and involves two additional matrix variables usually called slack variables then we present an extension of this condition to robust admissibility of time varying parameter dependent descriptor systems using parameter dependent lyapunov functions and employing two slack variables this extension states robust admissibility lmi conditions that are less restrictive as supported by our numerical examples than the existing ones our robust admissibility analysis result which can be directly used for state estimation cannot be so for state feedback control design this is due to the absence of duality between state estimation and state feedback control in the case of parameter dependent systems with uncertain parameters as recently reported in the literature in order to overcome this problem we also present in this paper a new robust admissibility analysis result involving only one slack variable which we apply to state feedback control design

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

linear quadratic zero sum differential games for generalized state space systems in this note we consider linear quadratic zero sum differential games for generalized state space systems it is well known that a unique linear feedback saddle point solution can exist in the game of state space systems however for the generalized state space system we show that the game admits uncountably many linear feedback saddle point solutions sufficient conditions for the existence of linear feedback saddlepoint solutions are found a constructive method is given to find these linear feedback saddle point solutions a simple example is included to illustrate the nonuniqueness of the linear feedback saddle point solutions

brief paper minimax state estimation for linear discrete time differential algebraic equations this paper presents a state estimation approach for an uncertain linear equation with a non invertible operator in hilbert space the approach addresses linear equations with uncertain deterministic input and noise in the measurements which belong to a given convex closed bounded set a new notion of a minimax observable subspace is introduced by means of the presented approach new equations describing the dynamics of a minimax recursive estimator for discrete time non causal differential algebraic equations daes are presented for the case of regular daes it is proved that the estimator s equation coincides with the equation describing the seminal kalman filter the properties of the estimator are illustrated by a numerical example

observers for singular systems the following three major aspects of the design of observers for linear time invariant singular systems are resolved the necessary and sufficient conditions for the problem to have a solution the order of the minimal observer and the properties of the closed loop system separation principle an algorithm is presented for the construction of the minimal order observer

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

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

numerical algorithms and existence results on lq control of descriptor systems with conditions on x 0 and stability we present a numerical algorithm based on using orthogonal matrices and sufficient existence conditions for an lq control problem of descriptor systems which generalise the existing results the associated hamiltonian pencil may be singular and may have infinite generalised eigenvalues iges of multiplicity 1 besides on x 0 we present optimality conditions on x 0 we present sufficient conditions for stability of the optimal closed loop descriptor system as well as a stabilisation algorithm

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

a review of 2 d implicit systems abstract the one dimensional notion of causality has no counterpart in the two dimensional plane where at best partial orderings of i j may be defined two dimensional implicit systems unlike the familiar state space 2 d models do not require any notion of causality or recursibility instead they require the milder notion of regularity thus the implicit models are more suited to the description of naturally occurring two dimensional systems such as are described by the hyperbolic equation and the heat equation as well as for applications in image processing this paper provides a brief summary of the current state of the theory of 2 d implicit systems

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

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

the singular general model of 2d systems and its solution a general singular model for 2d linear discrete systems with constant coefficients is defined necessary and sufficient conditions for the existence of a solution to the model are established the considerations of the author s previous work 1986 can be extended to singular models of 2d linear systems with variable coefficients

existence uniqueness and stability of solutions to singular linear quadratic optimal control problems we consider general continuous and discrete time linear quadratic optimal control problems where the dynamics of the system is a linear differential or difference algebraic equation with constant coefficient matrices and the cost functionals are defined by infinite integrals or sums these problems occur in numerous applications in many different fields such as economics operations research electrical engineering model and system design biology and chemistry since they form two of the simplest optimal control problems and they are also used as linear approximations for many nonlinear problems which are much harder to solve we extend some of the known results to these general problems

observers for descriptor systems abstract the design of observers is investigated for the estimation of the descriptor vector x t of descriptor systems of the form ex t ax t bu t y t cx t such systems are divided into two categories non causal impulsive systems and causal impulse free systems for non causal systems an observer of order rank e is proposed and termed a full order observer since the dynamical order of a descriptor system is regarded as rank e for causal systems it is shown that the number of independent outputs m does not exceed rank e and an observer of order rank e m is introduced and termed a minimal order observer the approach is based on the transformation of the system into particular design coordinates via a singular value decomposition of the matrix e illustrative examples are included

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

eigenvalue placement for generalized linear systems this paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by e ax bu where e is a singular map it is shown that controllability of the infinite eigenvalues of the pencil se a is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre specified complex numbers a procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed

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

feedback strategies for discrete time linear quadratic two player descriptor games this paper deals with discrete time linear quadratic descriptor two player games the two frameworks of nash and stackelberg strategies with feedback information structure are considered the aim is to provide sufficient conditions ensuring for any initial state the existence and the uniqueness of a tri trajectory gathering the state and the controls of the two players which reaches the equilibrium the provided results are mainly based on a discrete time matrix block formulation the provided algorithms consist in an iterative backward in time procedure issued from dynamic programming numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

on equivalence of pencils from discrete time and continuous time control we study two matrix pencils that arise respectively in discrete time and continuous time optimal and robust control we introduce a one to one transformation between these two pencils we show that for the pencils under the transformation their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related the eigen structures of the pencils under consideration have strong connections with the associated control problems our result may be applied to connect the discrete time and continuous time control problems and eventually lead to a unified treatment of these two types of control problems

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed

hamilton jacobi equation for descriptor systems abstract this paper is concerned with the extension of the hamilton jacobi equation from the state space system to the descriptor system the principle of optimality is interpreted in the sense of the strategy or closed loop control the hamilton jacobi equation for the descriptor system is derived an illustrative example is given to show the application of the hamilton jacobi equation in the optimal control problem

on the existence of geodesics on stationary lorentz manifolds with convex boundary abstract in this paper we consider the problem of the existence and multiplicity for geodesics not touching the boundary of a stationary lorentz manifold having convex boundary a physical example of a stationary and nonstatic lorentz manifold having convex boundary is the stationary axisymmetric asymptotically flat gravitational field outside a rotating massive object whenever its angular speed is small and its mean radius is close to the schwarzschild radius

optimal control problem for linear time varying descriptor systems abstract the linear quadratic optimal control problem is considered for time varying descriptor systems of the form e t 1 x t 1 a t x t b t u t c t f t where e is a square matrix with constant rank r which may be singular and f t is a known vector by using variational techniques necessary conditions are derived for the existence of optimal control by transforming to a coordinate system which can be computed by performing a non singular linear transformation necessary con ditions for the existence of optimal control can be changed to a two point boundary value problem which looks like the usual two point boundary problem of linear quadratic optimal control by applying the sweep method such a two point boundary value problem results in a matrix riccati equation and a linear difference equation at the same time the optimal control is uniquely determined in terms of the state x1 t which is the dynamic part of the descriptor vector x t the optimal cost is also given in terms of x1 t

new sufficient conditions for linear feedback closed loop stackelberg strategy of descriptor systems this paper is concerned with the derivation of linear feedback closed loop stackelberg lfcls strategies for a class of continuous time two person nonzero sum differential games characterized by linear descriptor systems and the quadratic cost functionals compared with existing results new sufficient conditions for the existence of a lfcls strategy are obtained under less restrictive conditions furthermore from the results of this paper the authors also arrive at some conclusions about the linear quadratic closed loop nash game for continuous time descriptor systems these conclusions are distinguished from those of the linear quadratic closed loop nash game for state space systems

on state feedback h control for discrete time singular systems this technical note deals with the state feedback h control problem for linear time invariant discrete time singular systems relied on the use of auxiliary matrices and a positive scalar a novel necessary and sufficient condition for the bounded real lemma is derived for discrete time singular systems the characterization is reduced to a strict linear matrix inequality lmi when the scalar is fixed and the resulting lmi is non conservative as long as the scalar is chosen sufficiently large moreover the result is further expanded to h controller design and a numerically efficient and reliable design procedure is given since no particular restriction is imposed on the auxiliary matrices the proposed result outperforms the existing methods in the literature numerical examples are included to illustrate the effectiveness of the present result

brief paper robust filtering for uncertain linear discrete time descriptor systems this paper is concerned with the problem of robust filtering for uncertain linear discrete time descriptor systems the matrices of the system state space model are uncertain belonging to a given polytope a linear matrix inequality based method is proposed for designing a linear stationary filter that guarantees the asymptotic stability of the estimation error and gives an optimized upper bound on the asymptotic error variance irrespective of the parameter uncertainty the proposed robust filter design is based on a parameter dependent lyapunov function which is shown to outperform parameter independent ones

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

dimensions of the simple restricted modules for the restricted contact lie algebra according to 2 the finite dimensional restricted simple lie algebras over an algebraically closed field of characteristic p 7 are either classical or of cartan type lusztig has a conjecture see 8 p 294 which provides a formula for the dimensions of the simple modules for the classical algebras in this paper modules are assumed to be left finitedimensional and restricted anderson jantzen and soergel 1 have recently shown that for p a 0 the lusztig conjecture follows from an analogous conjecture also due to lusztig for certain quantum groups they have also pointed out that this latter conjecture is implied by recent work of kazhdan and lusztig and a paper by casian although casian s paper has not yet convinced all of its readers the simple modules for the cartan type algebras fall into two classes there is a large class see 11 and 5 consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components constructed as mixed products in 11 the degree zero homogeneous components of cartan type algebras are either classical or nearly so therefore the lusztig conjecture can be used to get a formula for the dimensions of the modules in this class the small class consists of modules corresponding to exceptional weights in 11 the dimensions of these modules were determined for three of the four cartan type algebras namely the witt special and hamiltonian algebras there the exceptional weights turned out to be fundamental dominant weights in this paper we obtain the dimensions assuming p is not too small in the remaining case of the contact algebra in the process we determine the contragredients of the simple modules

cartan matrices for restricted lie algebras consider a finite dimensional restricted lie algebra over a field of prime characteristic each linear form on this lie algebra defines a finite dimensional quotient of its universal enveloping algebra called a reduced enveloping algebra this leads to a cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra in this paper we show how to compare such cartan matrices belonging to distinct linear forms as an application we rederive and generalise the reciprocity formula first discovered by humphreys for lie algebras of reductive groups for simple lie algebras of cartan type we see for example that the cartan matrices for linear forms of non positive height are submatrices of the cartan matrix for the zero linear form

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

solutions of differential algebraic equations as outputs of lti systems application to lq control problems abstract in this paper we propose a unified geometric framework for representing all solutions of a linear time invariant differential algebraic equation dae lti as outputs of classical linear time invariant ordinary differential equations ode lti the proposed framework is then used to solve an lq optimal control problem for dae ltis with rectangular matrices

brief paper feedback nash equilibria for linear quadratic descriptor differential games in this paper we consider the non cooperative linear feedback nash quadratic differential game with an infinite planning horizon for descriptor systems of index one the performance function is assumed to be indefinite we derive both necessary and sufficient conditions under which this game has a nash equilibrium

bounded real lemma and robust h control of 2 d singular roesser models abstract this paper discusses the problem of robust h control for linear discrete time two dimensional 2 d singular roesser models 2 d srm with time invariant norm bounded parameter uncertainties the purpose is the design of static output feedback controllers such that the resulting closed loop system is acceptable jump modes free stable and satisfies a prescribed h performance level for all admissible uncertainties a version of bounded realness of 2 d srm is established in terms of linear matrix inequalities based on this a sufficient condition for the solvability of the robust h control problem is solved and a desired output feedback controller can be constructed by solving a set of matrix inequalities a numerical example is provided to demonstrate the applicability of the proposed approach

numerical solution of differential algebraic riccati equations abstract we consider riccati matrix differential algebraic equations arising from singular or descriptor control problems we discuss the solvability of such equations under different conditions in order to apply numerical methods for differential algebraic systems one has to transform the equation unfortunately these equations then have a linear part which is described by a singular pencil and thus the usual integration methods do not apply under some conditions which we discuss these singularities can be removed by a preprocessing algorithm and the equation can then be solved by well known methods for differential algebraic systems like dassl of l petzold or limex of deuflhard hairer and zugck we discuss the numerical procedures and give some numerical examples

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

asymmetric sufficient condition for stability robustness of singular systems with structured parameter uncertainties in this paper under assumptions that the linear nominal continuous time or discrete time singular system is regular impulse free and asymptotically stable an asymmetric sufficient condition is proposed to preserve the assumed properties when the structured elemental parameter uncertainties are added into the nominal singular system the corresponding stable region in the parameter space that is obtained by using the proposed asymmetric sufficient condition is not necessarily symmetric with respect to the origin of the parameter space an illustrated example in which the signs and ranges of the parameters are known although their values are uncertain is given to show that the proposed sufficient condition may yield less conservative results than the existing ones reported recently

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

defect correction method for the solution of algebraic riccati equations the solution of discrete and continuous algebraic riccati equations is considered it is shown that if an approximate solution is obtained then the defect for this solution again solves an algebraic riccati equation of the same form and that the system properties of detectability and stabilizability are inherited by this defect equation on the basis of these results a general defect correction method is proposed and numerical examples are given for the use of this method in combination with a bunse gerstner and v mehrmann s 1986 sr method