robustness of uncertain descriptor systems this paper considers a class of descriptor systems with perturbations it derives sufficient conditions for the uncertain systems to be impulse free regular and or asymptotically stable the problems of the impulse controllability and impulse observability and the so called robust impulsive controller ric of the uncertain system are considered numerical examples are presented to illustrate the results

a connection between state space and doubly coprime matrix fraction descriptions of multivariable systems abstract explicit formulas for doubly coprime matrix fraction descriptions mfds of the transfer matrix of a linear time invariant state space system are given in terms of a controllable and observable state space realization of the transfer matrix these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime mfds of multivariable 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

linear quadratic control with and without stability subject to general implicit continuous time systems coordinate free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality existence and uniqueness of optimal controls and state trajectories abstract we consider linear quadratic control problems with and without stability subject to an arbitrary implicit continuous time system in a simple distributional framework and it is shown that the associated optimal costs if existent are solutions of our dissipation inequality for implicit systems this concept is related to the linear matrix inequality which is expressed in original system coefficients only and the above mentioned optimal costs turn out to be characterizable uniquely by certain solutions of this inequality however these solutions need not be rank minimizing if the underlying system is not standard and we specify why this is the case our statements are valid for regular as well as for singular problems and the possible significance of the algebraic riccati equation is illustrated for both regular and singular problems furthermore we present necessary and sufficient conditions for solvability of our problems and for the existence of optimal controls and associated optimal state trajectories finally we elaborate on the uniqueness of these controls and state trajectories

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

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

lyapunov and riccati equations of discrete time descriptor systems in this paper we further develop the generalized lyapunov equations for discrete time descriptor systems given by bender we associate a stable discrete time descriptor system with a lyapunov equation which has a unique solution furthermore under the assumptions of reachability and observability the solutions are guaranteed to be positive definite all results are valid for causal and noncausal descriptor systems this provides a unification of lyapunov equations and theories established for both normal and descriptor systems based on the developed lyapunov equation a riccati equation is also obtained for solving the state feedback stabilization problem

the robustness properties of the linear quadratic regulators for singular systems the lqr linear quadratic regulator robustness properties of singular systems are investigated it is shown that the optimal regulators for singular systems with an appropriate selection of the feedback gain have the same robustness properties as those in regular systems namely at least 60 degrees guaranteed phase margin infinite gain margin and 50 gain reduction margin

myerson values for games with fuzzy communication structure in 1977 myerson considered cooperative games with communication structure a communication structure is an undirected graph describing the bilateral relationships among the players he introduced the concept of allocation rule for a game as a function obtaining an outcome for each communication structure among the players of the game the myerson value is a specific allocation rule extending the shapley value of the game more recently the authors studied games with fuzzy communication structures using fuzzy graph theoretic ideas now we propose a general framework in order to define fuzzy myerson values players in a coalition need to measure their profit using their real individual and communication capacities at every moment because these attributes are fuzzy when the game is proposed so they look for forming connected coalitions working at the same level the different ways to obtain these partitions by levels determine different myerson values for the game several interesting examples of these ways are studied in the paper following known models in games with fuzzy coalitions the proportional model and the choquet model

on the generic controllability of continuous generalized state space systems abstract in this paper using a duality property between regular and generalized state space systems the necessary and sufficient conditions for the generic controllability of continuous generalized state space systems are established the necessary and sufficient conditions for the generic observability of such systems are also derived

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

structural decomposition of linear singular systems the single input and single output case in this paper we present a structural decomposition for general single input and single output linear singular systems such a decomposition has a distinct feature of capturing and displaying all the structural properties such as the nite and in nite zero structures and redundant dynamics ofthe given system it is expected to be a powerful tool in solving control problems for singular systems such as h2 and h control model reduction and disturbance decoupling problems to name a few c 2002 elsevier science b v all rights reserved

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

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

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

necessary and sufficient conditions for the controllability of linear interval descriptor systems abstract in this paper we investigate a class of uncertain linear descriptor systems with interval plants and give necessary and or sufficient conditions for controllability examples are also given to illustrate the results the corresponding results for the dual observability robustness problems are straightforward extensions

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

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 feti based domain decomposition technique for time dependent first order systems based on a dae approach we present a novel partitioned coupling algorithm to solve first order time dependent non linear problems e g transient heat conduction the spatial domain is partitioned into a set of totally disconnected subdomains the continuity conditions at the interface are modeled using a dual schur formulation where the lagrange multipliers represent the interface fluxes or the reaction forces that are required to maintain the continuity conditions the interface equations along with the subdomain equations lead to a system of differential algebraic equations daes for the resulting equations a numerical algorithm is developed which includes choosing appropriate constraint stabilization techniques the algorithm first solves for the interface lagrange multipliers which are subsequently used to advance the solution in the subdomains the proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps i e it allows subcycling in each subdomain this implies that existing software and numerical techniques can be used to solve each subdomain separately the coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers we present examples showing the feasibility of the proposed coupling algorithm copyright 2008 john wiley sons ltd

time domain solution to descriptor variable systems the time domain solution to the fast subsystem of the descriptor variable system of the form ex spl dot ax bu with e singular is given by laplace inverse transformation a new and significant feature of the solution is that it contains impulse terms excited by the values of sufficiently smooth input u and its derivatives at the initial time point based on the solution the notions of consistent initial conditions classical solutions and impulse free solutions are discussed perfectly the impulse controllability is discussed also which admits the same interpretation as the controllability at infinity in frequency domain

descriptor systems expanded descriptor equation and markov parameters generalized concepts of solvability and conditionability are presented for descriptor systems and tests are given for these generalized properties an expression is derived for descriptor system markov parameters the notion of expanded descriptor equation is presented

a canonical form for controllable singular systems abstract a new canonical form for the action of restricted system equivalence on controllable singular systems is given the construction of this form is based on the weierstrass decomposition of the singular system into a slow and a fast subsystem both subsystems are transformed into hermite canonical form the resulting hermite canonical form for singular systems has a particularly simple structure and is expected to be useful for e g identification purposes continuity properties of the hermite form are investigated and the nonexistence of a globally defined continuous canonical form for controllable singular systems is shown

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

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

a jordan control canonical form for singular systems a canonical form for controllable singular systems modulo restricted system equivalence is presented the canonical form respects the decomposition of the singular system into a regular and an impulsive part the construction is based on popov s control canonical form and on the jordan canonical form recently developed by hinrichsen and pratzel wolters continuity properties are analysed and the cellular partition of the orbit space induced by the canonical form is briefly discussed

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

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

zero dynamics design and its application to the stabilization of implicit systems abstract we present a formula that computes the output of an r controllable regular single input linear time invariant implicit system in such a way that it has prescribed relative degree and zeros the formula is inspired on different generalizations of ackermann s formula a possible application is in the context of sliding mode control of implicit systems where as the first step one can use the proposed formula to design a sliding surface with desired dynamic characteristics and as the second step apply a higher order sliding mode controller to enforce a sliding motion along the resulting sliding surface

on a new class of realization formulas and their application abstract a new set of realization formulas is derived for a class of matrix valued functions w these include the standard realization formulas for rational w observability controllability and minimality are defined and characterized conditions for w to be j 1 j 2 isometric and or j 1 j 2 coisometric with respect to a pair of signature matrices j 1 and j 2 are given in terms of the realizations minimal factorizations for square w are considered and formulas for the factors are deduced associated reproducing kernel spaces of pairs too are discussed

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 generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

remarks on observability of implicit linear discrete time systems abstract some results concerning the observability of implicit linear discrete time systems of the form exk 1 fxk yk hxk are presented it is not assumed that the corresponding pencil se f is regular the concept of reachability with pulse introduced in this paper is shown to be dual to the notion of observability

cooperative fuzzy games we derive versions of known results dealing with core equilibria and shapley values of cooperative games in the case of cooperative fuzzy games i e games defined on fuzzy subsets of the set of n players a fuzzy coalition is an n vector i associating with each player i his rate of participation i 0 1 in the fuzzy coalition and the real number v is the coalition worth assumed to be positively homogeneous if also v is superadditive or equivalently concave the fuzzy game with side payments has a nonempty core associated with the coalition worth function for any ordinary game with side payments is a fuzzy extension thereof viz the fuzzy game whose coalition worth function is the least positively homogeneous superadditive function majorizing the coalition worth function of the original game the fuzzy extension always has a nonempty core moreover if the original game has a nonempty core it coincides with that of its fuzzy extension analogous results are established for games without side payments an axiomatization of values of fuzzy games with side payments is also given the results are applied to show that the set of walras equilibria coincides with the fuzzy core of an economy

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

generally deformed oscillator isospectral oscillator system and hermitian phase operator the generally deformed oscillator gdo and its multiphoton realization as well as the coherent and squeezed vacuum states are studied we discuss in particular the gdo depending on a complex parameter q therefore we call it q gdo together with the finite dimensional cyclic representations as a realistic physical system of a gdo the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator it is pointed out that starting from the q gdo with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically the new creation and annihilation operators of the pegg barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit

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

decoupling in the design and synthesis of singular systems abstract necessary and sufficient conditions for the decoupling of singular systems using p d proportional derivative feedback are determined given a system which satisfies these conditions the class of all feedback matrices which decouple the system is characterized this class is used to determine the number of closed loop pole zeros which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed loop pole zero configurations

design of observers for descriptor systems using a descriptor standard form methods are developed for constructing luenberger type observers for linear time invariant descriptor systems based on the descriptor standard form derived here by means of a simple matrix calculus full order observers reduced order observers and function observers of a general type are proposed some necessary and sufficient conditions for the function observers to estimate linear functions of the state are discussed then the so called separation property of combined observer controller systems is established

descriptor systems decomposition into forward and backward subsystems it is shown that a decomposition of the descriptor space with certain properties induces a decomposition of the descriptor system into forward and backward subsystems different decompositions correspond to different apportionments of the required boundary conditions between initial and final times applications include several different but equivalent methods for solving the descriptor system and a clarification of the relations between some previously known methods of solution

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

feedback canonical form for singular systems controllability and impulse controllability indices for singular systems are defined via the concept of minimal indices for modules over polynomial rings these indices are invariant under the transformations of restricted system equivalence and feedback equivalence they yield a characterization of controllability and impulse controllability of singular systems analogous to the state space situation and form a complete list of invariants for impulse controllable systems under feedback equivalence a feedback canonical form described by these invariants is constructed explicitly

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

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

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

robust c controllability and or c observability for uncertain descriptor systems with interval perturbations in all matrices for descriptor systems of the form ex spl dot t ax t bu t y t cx t existing results are mostly for systems with uncertainties in the a matrix only the case where the e matrix is perturbed renders its robust analysis quite involved and has seldom been touched by researchers this paper will show that if e a b and g are all interval matrices the robust controllability and or observability problems can be solved in terms of the structured singular value p of certain fixed matrices necessary and sufficient conditions are given using the structured singular value spl mu by changing the problem into a robust nonsingularity problem for a class of uncertain matrices