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

wold decomposition prediction and parameterization of stationary processes with infinite variance a discrete time stochastic process t is said to be a p stationary process 1 p 2 if e left sum limits k 1 n b k x tk h right p e left sum limits k 1 n b k x tk right p for all integers n 1 t 1 t n h and scalars b 1 b n the class of p stationary processes includes the class of second order weakly stationary stochastic processes harmonizable stable processes of order 1 2 and p thorder strictly stationary processes for any nondeterministic process in this class a finite wold decomposition moving average representation and a finite predictive decomposition autoregressive representation are given without alluding to any notion of covariance or spectrum these decompositions produce two unique interrelated sequences of scalar which are used as parameters of the process t it is shown that the finite wold and predictive decomposition are all that one needs in developing a kolmogorov wiener type prediction theory for such processes

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

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

synthesis of proportional plus derivative feedbacks for descriptor systems a construction for proportional plus derivative pd feedback for descriptor or singular systems is given it is shown that under the assumption of controllability of the open loop finite and dynamic infinite modes feedback employing states and state derivatives can be constructed to shift all open loop modes finite as well as infinite to desired finite points while ensuring the regularity of the closed loop pencils the construction reveals that the required state derivative feedback is specified completely by the open loop dynamic infinite mode structures of the descriptor systems this class of pd feedback circumvents certain difficulties associated with the construction for constant state feedback

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

controllability analysis and eigenvalue assignment for generalized state space systems abstract an extended hessenberg triangular form is introduced as a tool for the controllability analysis of linear descriptor generalized state space systems the extended hessenberg triangular form may be applied to the multiple input eigenvalue placement via state feedback with the advantage that the number of eigenvalues that have to be assigned with each input may be decided only unitary transformations that lead to numerical stability are applied to the systems the assignment of the eigenvalues to both finite or infinite positions is considered

generalized proper inverse of polynomial matrices and the existence of infinite decoupling zeros a necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices the condition is then proved to be equivalent to the absence of infinite decoupling zeros this condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form it can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system and to provide a new look of the unimodular matrix operation effect at infinity for the left and right polynomial fraction forms

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

matrix pencil of closed loop descriptor systems infinite eigenvalue assignment abstract the regular matrix pencil se a bk is considered that characterizes a closed loop descriptor system u t kx t t a carefully contrived and efficient method is given for the determination of a set of matrices k such that the determinant se a bk is a constant value independent of s the problem is formulated as an infinite eigenvalue assignment by finite gain descriptor variable feedback via a singular value decomposition of the matrix e the result is interesting in its own right and finds application in controller and observer design

eigenstructure assignment for descriptor systems by output feedback compensation abstract a method for output feedback compensator design that assigns the eigenstructure of a descriptor system is presented the adopted approach is based on transforming the problem into that of eigenstructure assignment for a linear state space system using output feedback

satisfiability in many valued sentential logic is np complete abstract cook s np completeness theorem is extended to all many valued sentential logics including the infinite valued calculus of ukasiewicz

stationary lqg control of singular systems the stationary linear quadratic gaussian control problem is formulated and solved for single input single output singular systems the control system is required to be internally proper and stable in order to avoid both impulsive and unstable exponential behavior the set of all controllers resulting in such a control system is specified in parametric form all controllers that yield finite cost are identified once again in parametric form within this set necessary and sufficient conditions are then established for an optimal controller to exist all optimal controllers are shown to possess the same transfer function the problem is analyzed in the complex domain the transfer functions are expressed as quotients of proper strict hurwitz rational functions by means of this maneuver the powerful tools of algebra are made available the synthesis of the optimal controller is reduced to the solution of two linear diophantine equations whose coefficients are obtained by spectral factorization

pole placement for singular systems abstract in this paper we present a pole placement design for singular systems in polynomial fraction form the design can also be used for the role placement of singular systems in generalized state space form and requires no decomposition

brief structural properties and poles assignability of lti singular systems under output feedback this paper studies the structural properties of both finite poles and the infinite pole of linear time invariant singular systems under output feedback three main problems are studied namely 1 the algebraic structures of both finite poles and the infinite pole 2 the assignability of finite poles and the elimination of the infinite pole by output feedback and 3 the controllability and observability of the system with minimal number of inputs and outputs new generic solutions to these problems are presented in terms of some new concepts defined in this paper including the geometric multiplicity of the infinite pole the finite and impulsive output feedback cycle indices of the system determination of these multiplicities and indices are discussed an assignability equivalence is established between the variable finite poles and the poles of a controllable and observable non singular system the number of the independent infinite poles that can be reduced is given in terms of the system matrices the minimal number of inputs and outputs that guarantee controllability and observability are shown to be the output feedback cycle indices

robust stabilization of discrete generalized systems the robust stabilization problem for generalized discrete time systems described by polynomial or improper transfer function matrices subject to perturbations acting on the normalized coprime factors is solved the maximum achievable stability margin and the robust stabilizing controller are given in terms of realizations and solutions to appropriate riccati equations

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

partial algebras and complexity of satisfiability and universal theory for distributive lattices boolean algebras and heyting algebras characterizations are given for the classes of partial subalgebras of distributive lattices boolean algebras and heyting algebras thereby complexity results are obtained for the satisfiability of quantifier free first order sentences in these classes satisfiability is np complete for distributive lattices and boolean algebras and for heyting algebras is pspace complete consequently the universal theory of distributive lattices and of boolean algebras is co np complete and the universal theory of heyting algebras is pspace complete

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

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

inner outer factorization for strictly proper functions with j w axis zeros abstract this paper discussed the inner outer factorization for square real rational matrices which may have zeros on the j axis including infinity a factorization method is given in terms of the descriptor form representation of the rational matrix this method uses the solution of a generalized riccati equation we also propose an efficient computation algorithm for solving the generalized riccati equation

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

pole placement and compensator design of generalized systems abstract the system considered is a generalized system e ax bu y cx with matrix e singular when the impulsive modes of the system are both controllable and observable it is shown in this paper that these modes can be eliminated by almost any constant output feedback for a strongly controllable and observable generalized system after its impulsive modes are eliminated the poles of the system can be assigned arbitrarily by state feedback even if the states are not available the paper adopts an efficient method to design a compensator for the system

dynamical order assignment for linear descriptor systems this paper considers the problem of dynamical order assignment for linear descriptor systems by derivative feedback control the approach is based on an ability to adjust the number of infinite non dynamical modes of the system within a permissible range increasing the dynamical order enables the designer to implement specific otherwise unrealizable control strategies on the other hand dynamical order reduction simplifies the design procedures for a large scale system the solution is obtained by imbedding the descriptor space problem in an equivalent state space problem where a zero value eigenvalue with a prescribed geometric multiplicity is assigned to a linear state system by proportional feedback control

analysis of descriptor systems using numerical algorithms in this paper we analyze numerical methods for the solution of the large scale dynamical system e dot y t ay t g t y t 0 y 0 where e and a are matrices possibly singular systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables special cases of such systems are algebraic differential systems we discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure detect nonsolvable systems resolve initial value consistency problems and are applicable to stiff descriptor systems we also present an algorithm for the control of the local truncation error on only the state variables

brief bezout identity related to reduced order observer based controllers for singular systems a design method of causal reduced order observer based controllers for singular systems is presented a new state space representation of the bezout identity for singular systems is developed a parameterization of all causal reduced order controllers is also given the general results given here can be extended to various cases including minimal and full order observer based controllers

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

state feedback and transfer matrix in singular systems abstract the influence of state feedback on the transfer matrix in singular systems is considered we first discuss the problem for siso systems then by utilizing the time scale transformation the same problem is solved for mimo systems we also consider realizations of the feedback control strategies given here it is pointed out that the singular system is distinguished from the normal one by an important feature we can design a pure input prediction system for it finally the results are applied to the model matching problem

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

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

a descriptor system approach to singular perturbation of linear regulators the relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system is investigated meanwhile a decomposition approach is provided for solution of the linear quadratic regulator problem for singularly perturbed systems this approach is a significant advance over the two stage method

fundamental theorem of state feedback for singular systems abstract the limits of state feedback in altering the dynamics of singular systems are studied a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system the condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials the development is based on the properties of polynomial matrices a procedure is given for the calculation of a feedback gain which achieves the desired dynamics

controllability of descriptor systems different concepts of controllability for descriptor systems ex ax bu have been proposed and investigated in this paper their relationship their interpretation in terms of the kronecker canonical form of e a and the implications in terms of pole assignment regularity and robustness of the closed loop system are discussed

on the design of observers for generalized state space systems using singular value decomposition in this paper a method is developed for the design of luenberger type observers for linear time invariant control systems whose state equation is of the form ex ax bu where e is a singular square matrix the method is based on the singular value decomposition of the matrix e and on the reduction of the equation ex ax bu to a system consisting of a differential equation of the form w1 f1w1 f2 w2 g1u and an algebraic equation of the form h1w1 h2w2 g1u 0 if w2 can be eliminated from the differential equation by the aid of the algebraic equation and the original output equation of the system the method yields a reduced order observer for the generalized state space system

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

on minimal order inverses of discrete time descriptor systems in this paper linear discrete time control systems of the form e k x k 1 a k x k b k u k y k c k x k d k u k are considered where e k and a k are not necessarily square matrices the problem to be solved is this find a system of the form e k v k 1 k v k bcirc k k u k kvk dcirc k k where k is a partitioned column vector whose blocks are linear combinations of some of the output vectors of the original system and which possesses either or both of the following properties i if the outputs of the original system are known then the output equation of the second system uniquely specifies the same inputs that generated the known outputs of the original system ii if the output vectors of the original system are desired outputs then the equation uk kvk dcirc k k yields one or more sequences of inputs for the original system that can generate the desired outputs we shall furthermore seek the system which among all systems having property i or ii or both has the sm

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

on series representations for linear predictors the series expressions for the linear predictors of a stationary process have been known for a long time but necessary and sufficient conditions for the mean square convergence of these series are still not available it is shown that an equivalent problem is to find necessary and sufficient conditions for the invertibility of the infinite moving average representation of the process two known sufficient conditions are discussed and a more general condition that includes both as special cases is given the process that arises from fractional differencing of a random walk is discussed as an example

convergence in the boundary layer for singularly perturbed equations a singularly perturbed linear finite dimensional ordinary differential equation is considered on the half line 0 the reduced system is assumed to have a unique solution in the sense of distributions it is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense and the order of the limiting distribution is computed an application to the optimal linear quadratic regulator with cheap control is described

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

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

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

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

decoupling of square singular systems via proportional state feedback necessary and sufficient conditions for the decoupling of a solvable square singular system ex t ax t bu t with output y t dx t through an admissible control law of the form u t kx t hr t where h is a square nonsingular matrix it has been shown that for a given singular system that satisfies these conditions a propagational state feedback exists for which the system s transfer function is a diagonal nonsingular and proper rational matrix the proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback

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

a simple procedure for the stabilization of a class of uncontrollable generalized systems this paper gives a simple procedure for the stabilization with a prescribed rate of convergence of the free motions of a class of possibly single input uncontrollable generalized continuous time time invariant linear systems preserving the impulse free property if any

a general doubly coprime factorization for descriptor systems doubly coprime factorization plays an important role in dealing with multivariable control problems using a factorization approach in this study based on an arbitrary and properly stabilizing controller a general formula of doubly coprime factorization is obtained for use in descriptor systems detailed results have been derived on bridging the factorization approaches previously available in the literature

the generalized ackermann s formula for singular systems abstract being an elegant algorithm for state feedback pole placement ackermann s 1972 formula had been widely quoted in control texts in this paper the formula is extended to solve the root assignment problem for singular systems without loss of generality it is both well known and convenient that any regular generalized system can be transformed into the standard form e x ax t bu t where e a i and is a real constant in the derivation of the generalized ackermann s formula the closed loop characteristic polynomial det se a bk is simplified due to the relationship of e and a if e is nonsingular the feedback gain k can be computed from the generalized ackermann s formula directly in this case only the desired closed loop characteristic polynomial is required if e is singular the feedback algorithm needs both closed loop and open loop characteristic polynomials two numerical examples are presented to demonstrate our algorithms

the algorithm for computing nonmonic second order multivariable systems abstract an efficient method for computing impulse response matrices of nonmonic multivariable second order systems is proposed a procedure has been developed which uses some properties of regular matrix polynomials p s s 2 a 2 sa 1 a 0 with a 2 identically singular an algorithm suitable for computer use is established and its applications can be effectively controlled via the newly introduced special graphical scheme it produces simultaneously the adjoint and determinant of the matrix polynomial p s and avoids the inversion of any matrix moreover all computations can be performed on constant matrices and coefficients only it is shown that substantial economy may be achieved in computation time and storage when solving large order problems characteristic of many physical plants and engineering structures

h infinity kalman estimation for rectangular descriptor systems with unknown inputs this note considers h filtering and prediction for rectangular descriptor systems with unknown inputs that affect both the system and the output an optimal h descriptor kalman estimator hdke which can simultaneously solve the h filtering and prediction problems for rectangular descriptor systems with unknown inputs is developed based on the maximum likelihood descriptor kalman filtering method the hdke serves as a unified solution to solve h and kalman filtering for descriptor systems and standard systems with or without unknown inputs to reduce the computational complexity problem some efficient alternatives to the developed hdke are further proposed the relationship between the hdke and the existing literature results is also addressed an illustrative example is given to show the usefulness of the proposed results

on the computation of the fundamental subspaces for descriptor systems abstractin this paper we investigate several theoretical and computational aspects of fundamental subspaces for linear time invariant descriptor systems which appear in the solution of many control and estimation problems different types of reachability and controllability for descriptor systems are described and discussed the rosenbrock system matrix pencil is employed for the computation of supremal output nulling subspaces and supremal output nulling reachability subspaces for descriptor systems

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

geometric aspects of reduced order compensators for disturbance rejection the disturbance decoupling problem with stability constraint and pole placement is examined in a geometric framework two types of reduced order compensators are proposed one based on the observation scheme the other on the dual observation scheme the novelty lies mainly in these points 1 the avoidance of eigenspace computations in the syntheses as they rely on the determination of the maximum controlled invariant contained in a given subspace and its dual i e the minimum conditioned invariant containing a given subspace and pole assignability and 2 the concerted use of controlled and conditioned invariants which leads to a straightforward synthesis and also allows a clear interpretation of the dual compensation

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

a geometric approach to eigenstructure assignment for singular systems moore s well known result on the feedback assignment of eigenstructure in linear state space systems 5 was generalized to descriptor systems in 9 in this note we make some useful extensions to fill out the theory to begin with we do not assume controllability of the plant it is shown that the maximum number of finite closed loop eigenvalues is equal to dim el where l is the supremal a e r b invariant subspace in order to relate the possible closed loop structure to the open loop plant structure we define a set of indexes for descriptor systems which are in many respects similar to the controllability indexes of state space systems in terms of these controllability indexes we give a systematic approach to selecting the closed loop eigenvectors a major contribution of this note is the fact that it is geometric in nature and hence avoids any decomposition of the system matrices into a special form e g weierstrass form 1 the geometric setting introduced here should constitute a basis for further research in generalizing the well known results in geometric system theory disturbance decoupling input output decoupling etc to singular systems

autoregressive representations of multivariate stationary stochastic processes consider a q variate weakly stationary stochastic process xn with the spectral density w the problem of autoregressive representation of xn or equivalently the autoregressive representation of the linear least squares predictor of xn based on the infinite past is studied it is shown that for every w in a large class of densities the corresponding process has a mean convergent autoregressive representation this class includes as special subclasses the densities studied by masani 1960 and pourahmadi 1985 as a consequence it is shown that the condition w 1 lqxq1 or minimality of xn is dispensable for this problem when w is not in this class or when w has zeros of order 2 or more it is shown that xn has a mean abel summable or mean compounded cesaro summable autoregressive representation

geometric characterization of complete controllability indices for singular systems abstract we show in this short note that our recent geometric definitions for singular systems 12 provide a natural characterization of the complete controllability indices introduced by kucera and zagalak 11 13

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

robust control analysis and design for discrete time singular systems abstract in this paper we propose a simple approach to analyse stability robustness of discrete time singular systems under structured perturbations the developed robustness criteria are then applied to solve robust regional pole assignment problems of singular systems a robust control design algorithm via state feedback is also given the robust stability problem of singular systems is more complicated than that of regular systems not only stability robustness but system regularity and impulse elimination should be considered simultaneously since the results of robust control and analysis for singular systems is not available in the literature as much as other fields the paper may be viewed as a complementary result in this field although only discrete time case is discussed several results can be directly applied to continuous time systems as well

term satisfiability in flew algebras fl ew algebras form the algebraic semantics of the full lambek calculus with exchange and weakening we investigate two relations called satisfiability and positive satisfiability between fl ew terms and fl ew algebras for each fl ew algebra the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory we identify and investigate the complements as fragments of its universal theory we offer characterizations of those algebras that positively satisfy just those terms that are satisfiable in the two element boolean algebra providing its semantics to classical propositional logic in case of positive satisfiability these algebras are just the nontrivial weakly contractive fl ew algebras in case of satisfiability we give a characterization by means of another property of the algebra the existence of a two element congruence further we argue that positive satisfiability problems in fl ew algebras are computationally hard some previous results in the area of term satisfiability in mv algebras or bl algebras are thus brought to a common footing with known facts on satisfiability in heyting algebras

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

parametric eigenstructure assignment for continuous time descriptor systems abstract eigenstructure assignment for continuous time descriptor systems ex t ax t bu t via descriptor variable feedback control u t kx t is treated a new parametric form for the gain matrix k which assigns rank e arbitrary finite eigenvalues to the closed loop system is developed this form embraces two sets of design parameter vectors characterizing respectively the assignable eigenvectors associated with the finite eigenvalues and a necessary and sufficient condition for closed loop system regularity in addition a certain submatrix of k turns out to be totally free whenever nullity e exceeds rank b unlike earlier methods the eigenvectors associated with the infinite eigenvalues are dispensed with in the solution algorithm

a sufficient condition for arbitrary eigenvalue assignment in linear descriptor systems by output feedback the problem of eigenvalue assignment in the linear descriptor system e mathdotx ax bu y cx via output feedback is considered it is shown that mp rank e m and p are respectively the numbers of inputs and outputs of the system is a sufficient condition for generic real output feedback eigenvalue assignability the result is the best possible for general m p rank e in general the assignment is achieved not in an exact sense but in arbitrarily small neighborhoods around a given set of eigenvalues

an approach for pole assignment in singular systems pole assignment in a singular system edx dt ax bu is discussed it is shown that the problem of assigning the roots of det se a bf by applying a proportional feedback u fx r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system an immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems the approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system

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

exact pole assignment with regularity by output feedback in descriptor systems part 2 in this paper the first of two we examine the problem of modal control of linear time invariant multivariable descriptor systems in particular for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix possibly complex which exactly assigns an arbitrary set of r the number of independent semi state variables of the system distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition furthermore it is shown that if the set of proposed poles contains only real elements the assignment can be carried out using real feedback next we present conditions that are necessary and sufficient for pole assignability by real feedback of a self conjugate set of complex numbers provided that the system satisfies a further inequality

a controllability condensed form and a state feedback pole assignment algorithm for descriptor systems a direct algorithm for the pole assignment problem of a time invariant linear multivariable descriptor system with linear state feedback is presented the system is assumed to be completely controllable for both finite and infinite poles the algorithm is based on the controllability condensed form which is derived