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

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

pole assignment and observer design for continuous descriptor systems the problem of pole assignment and asymptotic observer design for multi input multi output continuous descriptor generalized state space systems is studied the problem is solved by introducing a new canonical form for controllable descriptor systems the internal stability of the descriptor variable feedback and observer configuration is also considered i e the extension of the separation principle to descriptor systems is given

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

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

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

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

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

tree shifts of finite type a one sided resp two sided shift of finite type of dimension one can be described as the set of infinite resp bi infinite sequences of consecutive edges in a finite state automaton while the conjugacy of shifts of finite type is decidable for one sided shifts of finite type of dimension one the result is unknown in the two sided case in this paper we study the shifts of finite type defined by infinite ranked trees indeed infinite ranked trees have a natural structure of symbolic dynamical systems we prove a decomposition theorem for these tree shifts i e we show that a conjugacy between two tree shifts can be broken down into a finite sequence of elementary transformations called in splittings and in amalgamations we prove that the conjugacy problem is decidable for tree shifts of finite type this result makes the class of tree shifts closer to the class of one sided shifts of sequences than to the class of two sided ones our proof uses the notion of bottom up tree automata

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

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

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

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

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

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

a nonlinear luenberger like observer for nonlinear singular systems this paper investigates observer design problem for a large class of nonlinear singular systems with multiple outputs we firstly regularize the singular system by injecting the derivative of outputs into the system then differential geometric method is applied to transform the regularized system into a simple normal form for which a luenberger like observer is proposed

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

approximating the hard square entropy constant with probabilistic methods for any two dimensional nearest neighbor shift of finite type x and any integer n 1 one can define the horizontal strip shift hn x to be the set of configurations on z 1 n which do not contain any forbidden pairs of adjacent letters for x it is always the case that the sequence htop hn x n of normalized topological entropies of the strip shifts converges to htop x the topological entropy of x in this paper we combine ergodic theoretic techniques with methods from percolation theory and interacting particle systems to show that for the two dimensional hard square shift h the sequence htop hn 1 h htop hn h also converges to htop h and that the rate of convergence is at least exponential as a corollary we show that htop h is computable to any tolerance e in time polynomial in 1 e we also show that this phenomenon is not true in general by defining a block gluing two dimensional nearest neighbor shift of finite type y for which htop hn 1 y htop hn y does not even approach a limit

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

a geometric theory for derivative feedback the authors use a singular system setting to provide a geometric theory for dynamical systems under derivative feedback they define the relevant subspace and provide computational design techniques in terms of a generalized sylvester or lyapunov equation for which efficient solution techniques are well known the authors provide both geometric and algebraic characterizations of the effects of derivative feedback drawing connections with previous work in state variable systems as well as extending that work to singular systems

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

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

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

sequential cavity method for computing free energy and surface pressure we propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice d our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of d then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest the method works under the assumption of strong spatial mixing ssp which is a form of a correlation decay

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

fundamental 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

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 a conjecture on bidimensional words we prove that given a double sequence w over the alphabet a i e a mapping from z2 to a if there exists a pair n0 m0 z2 such that pw n0 m0 1 100n0m0 then w has a periodicity vector where pw is the complexity function in rectangles of w

a numerical method for deadbeat control of generalized state space systems abstract in this paper we give a new numerical method for constructing a rank m correction bf to an n n matrix a such that the generalized eigenvalues of e a bf are all at 0 in the control literature this problem is known as deadbeat control of a generalized state space system ex i 1 ax i bu i whereby the matrix f is the feedback matrix to be constructed

on the solutions of linear differential equations with singular coefficients abstract equations of the form e ax u with e and a square matrices and e singular are considered the controversy that exists in the literature concerning the solutions of such equations is investigated solutions are arrived at through an application of singular perturbation theory

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

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

entropy dimension of shift spaces on monoids we consider the entropy dimension of g shifts of finite type for the case where g is a non abelian monoid entropy dimension tells us whether a shift space has zero topological entropy suppose the cayley graph cg of g has a finite representation that is cgg g g is a finite set up to graph isomorphism and relations among generators of g are determined by a matrix a we reveal an association between the characteristic polynomial of a and the finite representation of the cayley graph after introducing an algorithm for the computation of the entropy dimension the set of entropy dimensions is related to a collection of matrices in which the sum of each row of every matrix is bounded by the number of leaves of the graph furthermore the algorithm extends to g having finitely many free followers

chained eigenstructure assignment for constant ratio proportional and derivative crpd control law in controllable singular systems this paper deals with chained eigenstructure assignment for controllable singular systems of the form ex t ax t bu t with constant ratio proportional and derivative crpd control of the form u t kx t kx t w t the closed loop system exhibits the following features regularity impulse free response and arbitrary eigenvalue except open loop poles assignment this parametric characterization conveniently chooses the nonunique gain matrix k to modify the dynamic response of the system an illustrative example is included to demonstrate our approach

on the solution to the sylvester matrix equation av bw evf a simple direct complete and explicit parametric solution of the generalized sylvester matrix equation av bw evf is proposed for matrix f in the jordan form with arbitrary eigenvalues using the smith canonical form of the matrix a se b the obtained solution is in a clear neat form and may have important applications in descriptor linear system theory

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

tree shift topological entropy abstract we give a definition of topological entropy for tree shifts prove that the limit in the definition exists and show that it dominates the topological entropy of the associated one dimensional shift of finite type when the labeling of the tree shares the same restrictions

solution to matrix equation av bw evf and eigenstructure assignment for descriptor systems a simple complete analytical restriction free parametric solution with complete and explicit freedom of matrix equation av bw evf is presented based on the presented solution of this equation an approach for eigenstructure assignment for continuous descriptor system ex ax bu via descriptor variable feedback u kx is proposed parametric forms of the gain matrix and the eigenvectors associated with the assigned finite closed loop eigenvalues are given the approach is shown to be better than that of fahmy and o reilly 1989 int j contr 49 129 143 in the sense that it not only possesses the functions of the approach of fahmy and o reilly 1989 int j contr 49 129 143 but also has the following three advantages 1 it eliminates the requirements on the assigned finite closed loop poles 2 it provides more design parameters and 3 it needs less computational work

infinite labeled trees from rational to sturmian trees this paper studies infinite unordered d ary trees with nodes labeled by 0 1 we introduce the notions of rational and sturmian trees along with the definitions of strongly balanced trees and mechanical trees and study the relations among them in particular we show that strongly balanced trees exist and coincide with mechanical trees in the irrational case providing an effective construction such trees also have a minimal factor complexity hence are sturmian we also give several examples illustrating the inclusion relations between these classes of trees

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

boundary value problems and cauchy problems for the second order euler operator differential equation abstract explicit solutions of boundary value problems and cauchy problems for the euler operator differential equation are given in terms of solutions of algebraic operator equations

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

boundary value problems for second order differential operator equations abstract it is proved that the resolution problem of an operator boundary value problem for a second order differential operator equation with constant coefficients is solved in terms of solutions of certain algebraic operator equations explicit expressions of solutions are given

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

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

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

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

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

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

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

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

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

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

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

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

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

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

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

brief paper observability analysis and sensor location study for structured linear systems in descriptor form with unknown inputs this paper deals with the state and input right hand side observability and the sensor location problem to ensure this property for structured linear systems in the descriptor form the proposed method 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 underdetermined overdetermined or square after defining a bipartite graph which represents such systems we deduce some graphic necessary and sufficient conditions to achieve the right hand side observability of the state and the input next we propose a sensor placement procedure to recover this property more precisely we provide the minimal number of the sensors required and either their pertinent location or a necessary and sufficient condition to check if a given sensor configuration is adequate or not to ensure state and input right hand side observability our approach uses algorithms with polynomial complexity orders and assumes only the knowledge of the system s structure and so is suited for large scale systems

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

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

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

design of a minimal order observer for singular systems this paper considers the design of an observer for generalized state space systems using the concept of generalized inverses in contrast to the luenberger observer theory a new method is proposed which does not presuppose the observer structure it is shown that under certain conditions it is possible to construct a minimal order observer for this class of systems illustrative examples are included

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

mixing properties of tree shifts the uncertainty of multidimensional shift spaces draws attracted attention of researchers for example the emptiness problem is undecidable there exist aperiodic shifts of finite type there is a nonempty shift of finite type exhibiting nonextensible local patterns this paper investigates symbolic dynamics on cayley trees and gives affirmative answers to the above questions in tree shifts beyond that with introducing block gluing tree shift a sufficient condition of exhibiting positive topological entropy is revealed