brief paper identification and data driven model reduction of state space representations of lossless and dissipative systems from noise free data we illustrate procedures to identify a state space representation of a lossless or dissipative system from a given noise free trajectory important special cases are passive systems and bounded real systems computing a rank revealing factorization of a gramian like matrix constructed from the data a state sequence can be obtained the state space equations are then computed by solving a system of linear equations this idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced order model

stability of switched linear differential systems we study the stability of switched systems whose dynamic modes are described by systems of higher order linear differential equations not necessarily sharing the same state space concatenability of trajectories at the switching instants is specified by gluing conditions i e algebraic conditions on the trajectories and their derivatives at the switching instants we provide sufficient conditions for stability based on lmis for systems with general gluing conditions we also analyse the role of positive realness in providing sufficient polynomial algebraic conditions for stability of two modes switched systems with special gluing conditions

a two variable approach to solve the polynomial lyapunov equation abstract a two variable polynomial approach to solve the one variable polynomial lyapunov equation is proposed lifting the problem from the one variable to the two variable context allows to use faddeev type recursions in order to solve the polynomial lyapunov equation in an iterative fashion the method is especially suitable for applications requiring exact or symbolic computation

a behavioral characterization of the positive real and bounded real characteristic values in balancing in passivity preserving and bounded realness preserving model reduction by balanced truncation an important role is played by the so called positive real pr and bounded real br characteristic values both for the positive real as well as the bounded real case these values are defined in terms of the extremal solutions of the algebraic riccati associated with the system more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the riccati equation in this paper we will establish a representation free characterization of these values in terms of the behavior of the system we will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior we will then show that both for the pr and the br case the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input output behavior its unique maximal supply extracting future continuation we will explain that the term singular values should be interpreted here in a generalized sense since in our setup the future behavior is only an indefinite inner product space c 2010 elsevier b v all rights reserved

canonical conservative state signal shift realizations of passive discrete time behaviors a passive linear discrete time invariant s s state signal system v x w consists of a hilbert state space x a kre n signal space w a maximal nonnegative generating subspace v of the kre n space k x x w the sets of trajectories x w generated by v on the discrete time intervals i z are defined by x n 1 x n w n v n i this system is forward conservative or backward conservative or conservative if v v v v or v v respectively the set w of all signal components w of trajectories x w of on i z with x 0 0 and w l2 z w is called the future time domain behavior of the fourier transform w of w is called the future frequency domain behavior of this set is a maximal nonnegative right shift invariant subspace in the kre n space k2 d w that as a topological vector space coincides with the usual hardy space h2 d w but has the indefinite kre n space inner product inherited from w a subspace of k2 d w with the above properties is called a passive future frequency domain behavior on w it has been shown earlier by the present authors that every passive future frequency domain behavior w on w may be realized as the future frequency domain behavior of some passive s s system and that it is possible to require in addition that is a controllable and forward conservative b observable and backward conservative or c simple and conservative these three types of realizations are determined by w up to unitary similarity canonical functional shift realizations of the types a and b have been obtained earlier by the present authors and their connection to the classical de branges rovnyak models have been discussed here we present analogous results for a realization of the type c

nevanlinna test of stability abstract in this paper a new test of stability for scalar autonomous systems known as nevanlinna test of stability is introduced this test can be derived using the bezoutian of the characteristic polynomial of a given system it is shown in this paper that an associated quadratic differential form called the bezoutian functional is positive if and only if the given system is asymptotically stable nevanlinna test of stability proceeds by checking if the bezoutian functional associated with a given system is positive

a behavioral approach to the h optimal control problem this paper considers the general h optimal control problem from a behavioral perspective a formalization of this problem is given that departs from the usual h control paradigm in the sense that system variables of the plant are treated in a symmetric way without distinguishing control inputs measurements exogenous inputs and to be controlled variables interconnection variables are introduced and controllers are allowed to constrain the interconnection variables of the plant necessary and sufficient conditions are given for the existence of controllers that achieve an h control objective for a given linear time invariant plant the set of all such controllers is parametrized by means of j spectral factorizations

optimal robust stabilization and dissipativity synthesis by behavioral interconnection given a nominal plant together with a fixed neighborhood of this plant the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood in an appropriate sense if a controller achieves this design objective we say that it robustly stabilizes the nominal plant in this paper we formulate the robust stabilization problem in a behavioral framework with control as interconnection we also formulate a relevant behavioral mathcal h infty synthesis problem which will be instrumental in solving the robust stabilization problem we use both rational and polynomial representations for the behaviors under consideration necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems we will also find the optimal stability radius i e the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller this smallest upper bound is expressed in terms of certain storage functions associated with nominal control system

bifurcation in granular materials an attempt for a unified framework sudden collapse mechanisms strictly inside the mohr coulomb plastic limit condition have been observed in granular materials in the laboratory as well as in the field the purpose of this paper is to show that the theoretical framework of loss of sustainability is convenient to describe such mechanisms in this context the notions of both loading and response variables which characterize the loading path applied to the specimen considered and its response path are fundamental moreover by investigating the relation between loading and response parameters it is established that this framework also embeds the notions of loss of constitutive uniqueness and loss of controllability therefore a unified approach is attempted finally by highlighting the basic role played by the loading parameters the vanishing of the second order work is shown to be a proper criterion to detect the occurrence of a bifurcation from a quasi static regime to a dynamic regime leading to the collapse of the material

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

distance between behaviors and rational representations in this paper we study notions of distance between behaviors of linear differential systems we introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input output systems represented by transfer matrices three of these are defined in terms of gaps between closed subspaces of the hilbert space mathcal l 2 mathbb r in particular we generalize the classical gap metric we express these metrics in terms of rational representations of behaviors in order to do so we establish a precise relation between rational representations of behaviors and multiplication operators on mathcal l 2 mathbb r we introduce a fourth behavioral metric as a generalization of the well known nu metric as in the input output framework this definition is given in terms of rational representations for this metric however we establish a representation free behavioral characterization as well we make a comparison between the four metrics and compar

a search for positive storage functions abstract dissipative systems is a widely studied subject and the concept of storage functions are closely associated with dissipative systems these storage functions for 1 d systems systems evolving over time are more or less well understood positive storage functions have a special place in the theory of 1 d dissipative systems through behavioural theory there has been an attempt at unifying the theory of n d systems for n 1 with that of 1 d systems dissipative n d systems also have a associated concept of storage functions this is an effort at finding positive storage functions or components for n d systems

a bilinear differential forms approach to parametric structured state space modelling we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

algorithms for multidimensional spectral factorization and sum of squares in this paper algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems these algorithms are based on the calculus of 2n variable polynomial matrices and their associated quadratic differential forms and share the common feature that the problems are lifted from the original n variable polynomial context to a 2n variable polynomial context this allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities lmi s to the feasibility of a semialgebraic set or to a linear eigenvalue problem

deterministic least squares filtering a deterministic interpretation of the kalman filtering formulas is given using the principle of least squares estimation the observed signal and the to be estimated signal are modeled as being generated as outputs of a finite dimensional linear system driven by an input disturbance postulating that the observed signal is generated by the input disturbance that has minimal least squares norm leads to a method of computing an estimate of the to be estimated output the derivation of the resulting filter is carried out in a completely self contained way the analogous approach to least squares control is also discussed

stability criteria for first order dynamical systems in a behavioral framework abstract this paper addresses the stability of dynamical systems described by first order differential equations in the behavioral framework some algebraic stability criteria are derived under the assumption that the behavior does not include distributions the derivation of our criteria is based on quadratic differential forms and two variable polynomial matrices cf willems and trentelman siam j optim control 36 1998 1703 1749 willems proceedings of the 30th ieee conference on decision and control 1991 pp 900 904 which are appropriate tools for system synthesis and analysis in the behavioral approach thus our result and approach will contribute to the developement of the behavioral system theory furthermore our criteria are represented in terms of linear matrix inequalities and algebraic matrix equations which leads to easy examination of the stability with the help of appropriate software packages

improper cal l _ infty optimal suboptimal controllers we consider l control of mimo systems and address solvability of the problem over all finite dimensional lti controllers i e controllers whose transfer functions can be proper or improper we show that improper controllers are easily dealt with using the behavioral approach unlike the standard state space transfer matrix methods and argue that there are cases where an improper controller can outperform a proper controller in this setting we next formulate and prove necessary and sufficient conditions for suboptimal l control problem solvability and relate this to existing results about system invariant zeros further we infer that in our formulation assuming suboptimal solvability conditions on the system an optimal controller always exists possibly with an improper transfer function in other words the infimum l norm of the closed loop system is achievable when dealing with both proper and improper controller transfer functions we illustrate these results through an example for which the optimal l controller has an improper transfer function

tensored polynomial models the theme of the present paper is the study of two different versions of tensor products of functional models and present some applications to various problems related to system theory among those are stability of higher order systems tangent spaces spaces of intertwining maps invariant factors of tensor products of linear transformations and the solvability of sylvester equations

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

the kalman yakubovich popov lemma in a behavioural framework the classical kalman yakubovich popov lemma provides a link between dissipativity of a system in state space form and the solution to a linear matrix inequality in this paper we derive the kyp lemma for linear systems described by higher order differential equations the result is an lmi in terms of the original coefficients in which the dissipativity problem is posed subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices this enables us to derive a new algorithm for polynomial spectral factorization in terms of an lmi in the coefficients of a polynomial matrix

closed form solutions of singular kyp lemma strongly passive systems and fast lossless trajectories in this paper we deal with a special class of passive systems which possess the characteristic property of having no finite spectral zeros we call these systems strongly passive it is well know

exploration exploitation tradeoff using variance estimates in multi armed bandits algorithms based on upper confidence bounds for balancing exploration and exploitation are gaining popularity since they are easy to implement efficient and effective this paper considers a variant of the basic algorithm for the stochastic multi armed bandit problem that takes into account the empirical variance of the different arms in earlier experimental works such algorithms were found to outperform the competing algorithms we provide the first analysis of the expected regret for such algorithms as expected our results show that the algorithm that uses the variance estimates has a major advantage over its alternatives that do not use such estimates provided that the variances of the payoffs of the suboptimal arms are low we also prove that the regret concentrates only at a polynomial rate this holds for all the upper confidence bound based algorithms and for all bandit problems except those special ones where with probability one the payoff obtained by pulling the optimal arm is larger than the expected payoff for the second best arm hence although upper confidence bound bandit algorithms achieve logarithmic expected regret rates they might not be suitable for a risk averse decision maker we illustrate some of the results by computer simulations

canonical realizations by factorization of constant matrices in van der schaft and rapisarda 2011 3 we showed that a state variable for a lti system can be computed factorizing a two variable polynomial matrix computed directly from the system equations different factorizations of this matrix yield different state maps and consequently different state equations based on this fact we present a unifying point of view on some classical canonical forms for linear systems

dissipativity of pseudorational behaviors this paper studies dissipativity for a class of infinite dimensional systems called pseudorational in the behavioral context extending the finite dimensional counterpart we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function for its proof we derive a new necessary and sufficient condition for entire functions of exponential type in the so called paley wiener class to allow a symmetric factorization characterization of dissipative behaviors and linear quadratic optimal behaviors are also given

the strict dissipativity synthesis problem and the rank of the coupling qdf the problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate is studied the relation between strictness and the rank of a suitable coupling condition that combines the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed

fast modes in the set of minimal dissipation trajectories abstract in this paper we study the set of trajectories satisfying both a given lti system s laws and also laws of the corresponding adjoint system in other words trajectories in the intersection of the system s behavior and that of the adjoint system this intersection has important system theoretic significance for example it is known that the trajectories in this intersection are the ones with minimal dissipation underlying the notion of adjoint is that of a power supply it is with respect to this supply rate that the trajectories in the intersection are known to be stationary in this paper we deal with half line solutions to the differential equations governing both the system and its adjoint analysis of half line solutions plays a central role for example in initial value problems and in well posedness studies of an interconnection we interpret the set of half line trajectories allowed by a system and its adjoint as an interconnection of these two systems and thus address issues about well posedness ill posedness of the interconnection we formulate necessary and sufficient conditions for this intersection to be autonomous for the case of an ill posed interconnection and resulting autonomous system we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system we link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system we show that absence of impulsive initial conditions is equivalent to the well known subspace iteration algorithms for these subspaces converging in one step

singular lq control optimal pd controller and inadmissible initial conditions we consider a classical control problem the infinite horizon singular lq problem i e some inputs are unpenalized in the quadratic performance index in this case it is known that the slow dynamics is constrained to be in a proper subspace of the state space with the optimal input for the slow dynamics implementable by feedback in this technical note we show that both the fast dynamics and the slow dynamics can be implemented by a feedback controller moreover we show that the feedback controller cannot be a static feedback controller but can be pd i e proportional differentiate exactly once in the state we show that the closed loop system is a singular descriptor state space system and we also characterize the conditions on the system performance index for existence of inadmissible initial conditions i e initial conditions that cause impulsive solutions there are no inadmissible initial conditions in the controlled system if and only if in the strictly proper transfer matrix from the unpenalized inputs to the penalized states there exists at least one maximal minor of relative degree equal to the number of unpenalized inputs in addition to the above we prove solvability of the infinite horizon singular lq problem under milder assumptions than in the literature

an efficient algorithm for computing the cal h _ infty norm this technical note addresses the computation of the h norm by directly computing the isolated common zeros of two bivariate polynomials unlike the iteration algorithm that is currently used to find the h norm the proposed method to h norm calculation is compared with the existing method by bruinsma and steinbuch 1990 using numerical experiments on random transfer functions of orders upto 240 the time taken is better by 15 to 30 times in addition to improved accuracy the proposed method uses techniques involving structured linearization of the bezoutian matrix constructed from two bivariate polynomials

time relevant stability of 2d systems for many 2d systems one of the independent variables plays a distinct role in the evolution of the trajectories since often this special independent variable is time we call such systems time relevant in this paper we introduce a stability notion for time relevant systems described by higher order difference equations we give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system we also give an lmi characterization of time relevant stability involving only constant matrices

systems with sector bound nonlinearities a behavioral approach the class of nonlinear dynamical systems obtained by interconnection of a linear time invariant dynamical system and a memoryless time invariant sector bound nonlinearity has been widely studied in literature in this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems systems with slope restricted nonlinearities are also considered

h sub spl infin control in a behavioral context the full information case the authors formulate the h sub spl infin control problem in a behavioral setting given a mathematical model say a set of higher order differential equations together with some static equations the vector of manifest variables is partitioned into yet to be controlled variables unknown exogenous variables and interconnection variables the interconnection variables are available for interconnection in the sense that they can be made to obey certain differential or static equations to be specified by the designer such a system of differential equations and static equations is called a controller the design problem that we consider is to find controllers such that the size of the to be controlled variables is less than a given tolerance for all disturbances in the unit ball and such that the interconnection is a stable system we find necessary and sufficient conditions for the existence of suitable controllers under the hypothesis that we have a full information problem these conditions involve indefinite factorizations of polynomial matrices and a test on a given pick matrix

dissipative systems synthesis a linear algebraic approach abstract in this paper we consider the problem of synthesis of dissipative systems for the case that first and higher order derivatives of the concerned variables also appear in the weighting function the problem is formulated and solved using the behavioral approach to systems and control we relate the problem of weighted h control as a special case of this synthesis problem the synthesis problem and its solution can be systematically understood when one notices that it is similar to finding a non negative subspace non negative with respect to a given constant matrix within a finite dimensional vector space satisfying certain inclusion and dimension constraints

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

a characterization of solutions of the discrete time algebraic riccati equation based on quadratic difference forms this paper is concerned with a characterization of all symmetric solutions to the discrete time algebraic riccati equation dare dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper along the line of the continuous time results due to trentelman and rapisarda h l trentelman p rapisarda pick matrix conditions for sign definite solutions of the algebraic riccati equation siam j contr optim 40 3 2001 969 991 we show that the solvability of the dare is equivalent to a certain dissipativity of the associated discrete time state space system as a main result we characterize all unmixed solutions of the dare using the pick matrix obtained from the quadratic difference forms this characterization leads to a necessary and sufficient condition for the existence of a non negative definite solution it should be noted that when we study the dare and the dissipativity of the discrete time system there exist two difficulties which are not seen in the continuous time case one is the existence of a storage function which is not a quadratic function of state another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the dare due to the infinite generalized eigenvalues of the associated hamiltonian pencil one of the main contributions of this paper is to demonstrate how to resolve these difficulties

on positive realness and lyapunov functions for switched linear differential systems we show new results about lyapunov stability of switched linear differential systems slds using the concept of positive realness the main results include stability conditions for a class of slds with augmented banks and the parametrization of families of asymptotically stable slds with three modes such conditions can be verified using lmis that can be directly set up from the higher order differential equations describing the modes

the subspace nevanlinna interpolation problem and the most powerful unfalsified model a generalization of the tangential nevanlinna interpolation problem will be studied from a behavioral point of view necessary and sufficient conditions for its solvability and a characterization of all its solutions are derived these results are obtained by associating to the interpolation data a behavior that enjoys a special structure

dissipative switched linear differential systems the authors develop a dissipativity theory for switched systems whose dynamical modes are described by systems of higher order linear differential equations they give necessary and sufficient conditions for dissipativity based on systems of linear matrix inequalities lmis constructed from the coefficient matrices of the differential equations describing the modes the relationship between dissipativity and stability is also discussed and an application to the stabilization of power converters is provided

a theory of passive linear systems with no assumptions we present two linked theorems on passivity the passive behavior theorem parts 1 and 2 part 1 provides necessary and sufficient conditions for a general linear system described by a set of high order differential equations to be passive part 2 extends the positive real lemma to include uncontrollable and unobservable state space systems

a parametrization for dissipative behaviors the behavioral equivalent of single input single output siso systems are behaviors with two manifest variables passive siso systems can therefore be viewed as j dissipative behaviors with two manifest variables here the special matrix j defines a qdf that captures the passivity property of siso systems in this paper we investigate more general qdfs q s induced by some operator these qdfs define some relation between the input the output and their derivatives of a siso system we characterize all behaviors that are dissipative with respect to the prescribed qdf q in fact we parametrize all the behaviors dissipative with respect to q in terms of those dissipative with respect to the special qdf qj induced by the matrix j similar results can also be given for lossless systems

totally dissipative systems in a totally dissipative behavior all non trivial trajectories dissipate energy a characterization of such behaviors is given in terms of properties of the one and two polynomial matrices associated with the supply rate and with their kernel and image representation

uncontrollable dissipative systems observability and embeddability the theory of dissipativity is well developed for controllable systems a more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function namely a function such that along every system trajectory its rate of change at each time instant is at most the power supplied to the system at that time however even when the supplied power is expressible in terms of just the external variables the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable hidden from the external variables this paper investigates the key aspects of both cases and also proposes another intuitive definition of dissipativity these three definitions are compared we show that drawbacks of one definition are addressed by another dealing first with observable storage functions under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as f

equivalence of rational representations of behaviors this article deals with the equivalence of representations of behaviors of linear differential systems in general the behavior of a given linear differential system has many different representations in this paper we restrict ourselves to kernel and image representations two kernel representations are called equivalent if they represent one and the same behavior for kernel representations defined by polynomial matrices necessary and sufficient conditions for equivalence are well known in this paper we deal with the equivalence of rational representations i e kernel and image representations that are defined in terms of rational matrices as the first main result of this paper we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices we will also establish conditions for the equivalence of rational image representations finally we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation

canonical forms for polynomial and quadratic differential operators we define the canonical representative for the equivalence class consistingof all polynomial and quadratic differential operators that take the same values on a given subspaceof c

interconnection of j lossless behaviours in this paper motivated by the phenomenon of the interconnection of lossless electrical networks a class of behaviours known as j lossless behaviours is introduced where j is a symmetric two variable polynomial matrix it is shown that for certain values of j interconnection of j lossless behaviours leads to an oscillatory behaviour physically this translates to the fact that interconnection of two multi port lossless electrical networks results in an autonomous lossless electrical network finally the problem of decomposition of an oscillatory behaviour with a given characteristic polynomial as an interconnection of two single input single output behaviours such that one has a lossless positive real transfer function and the other has a lossless negative real transfer function is also considered this problem can be viewed as an inverse problem to the one of interconnection of j lossless behaviours

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

reprint of a bilinear differential forms approach to parametric structured state space modelling abstract we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

deterministic kalman filtering in a behavioral framework the purpose of this paper is to obtain a deterministic version of the kalman filtering equations we will use a behavioral description of the plant specifically an image representation the resulting algorithm requires a matrix spectral factorization we also show that the filter can be implemented recursively

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

linear hamiltonian behaviors and bilinear differential forms we study linear hamiltonian systems using bilinear and quadratic differential forms such a representation free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher order differential equations thus dispensing with the usual point of view in classical mechanics of considering first and second order differential equations only

higher order linear lossless systems we define a lossless autonomous system as one having a quadratic differential form associated with it called the total energy which obeys the property of positivity and which is conserved in this paper we show that an autonomous system is lossless if and only if it is oscillatory next we discuss a suitable way of splitting the total energy function into its kinetic and potential energy components we also extend the investigation to the case of open systems

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

linear quadratic control and quadratic differential forms for multidimensional behaviors this paper deals with systems described by constant coefficient linear partial differential equations nd systems from a behavioral point of view in this context we treat the linear quadratic control problem where the performance functional is the integral of a quadratic differential form we look for characterizations of the set of stationary trajectories and of the set of local minimal trajectories with respect to compact support variations turning out that they are equal if the system is dissipative finally we provide conditions for regular implementability of this set of trajectories and give an explicit representation of an optimal controller

an algebraic approach to the realization of lossless negative imaginary behaviors we consider a behavior with an equal number of inputs and outputs for which there exists a quadratic differential form which is positive for nonzero trajectories of the behavior and whose derivative is equal to the scalar product of the input vector and the derivative of the output vector such systems occur for example when considering conservative mechanical systems we give a method of computation of a state space realization from an observable image representation of such a behavior we apply the insights derived from this realization procedure to the synthesis of lossless mechanical systems

realization of lossless systems via constant matrix factorizations we study the realization problem for linear time invariant systems described by higher order differential equations which are j lossless with constant j our approach is based on the factorization of a constant matrix obtained in a straightforward way from the storage function of the system state equations are obtained directly from this factorization and from the original system representation

lyapunov stability of 2d finite dimensional behaviours in this article we investigate a lyapunov approach to the stability of finite dimensional 2d systems we use the behavioural framework and consider a notion of stability following the ideas in pillai and shankar h pillai and s shankar 1998 a behavioral approach to control of distributed systems siam journal of control and optimization 37 388 408 rocha p rocha 2008 stabilization of multidimensional behaviors multidimensional systems and signal processing 19 273 286 valcher m valcher 2000 characteristic cones and stability properties of two dimensional autonomous behaviors ieee transactions on circuits and systems part i cas 47 290 302 we characterise stability in terms of the existence of a quadratic lyapunov function and provide a constructive algorithm for the computation of all such lyapunov functions

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

a minimal state map for strictly lossless positive real behaviors in this paper behaviors with equal input and output cardinalities and with lossless positive real transfer functions corresponding to certain input output partitions are considered based on cauer synthesis method an algorithm to obtain a minimal state map of such behaviors starting from an observable image representation is given this algorithm makes use of polynomial algebraic methods to arrive at a minimal state map it is shown that the algorithm has application in synthesis and analysis of lossless electrical networks

novel insights on the stabilising solution to the continuous time algebraic riccati equation in the present paper we present a closed form solution as a function of the closed loop poles for the continuous time algebraic riccati equations cares related to single input single output systems with non repeated poles the proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument offering potential insight into any control technique based on a care and its solution as an example we present the equivalence of two fairly recent control over network results furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k or equivalently the luenberger s observer gain l and present an example finally we conclude by discussing the possible extension of the proposed closed form solution to the repeated eigenvalues case and to the case when the care is related to multiple input multiple output systems

state space approach to behavioral systems theory the dirac bergmann algorithm this paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised this representation is a first order hybrid representation of behaviors called pencil representation an algorithm well known after dirac and bergmann db is shown to be central in obtaining a constraint free and observable cfo state space representation of a behavior results and criteria for asymptotic stability controllability inclusions and markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

stabilization lyapunov functions and dissipation for linear time invariant systems any stabilizing controller for a given plant can be associated with a supply rate with respect to which the plant in open loop is half line dissipative we also prove the equivalence between the stability of the interconnection of two systems and their half line dissipativity with respect to a supply rate and its negative

on the takagi interpolation problem in this paper we approach the interpolation problem of takagi from the perspective of time series modeling the framework of exact identification and the notion of mpum are used in order to generalize some results on the nevanlinna interpolation problem these results are of independent interest and concern the relation between the roots of the determinant of the denominator of an unfalsified model and the signature of the pick matrix of the data

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

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

nyquist plots finite gain phase margins dissipativity abstract the relation between the small gain theorem and infinite phase margin is classical in this paper we formulate a novel supply rate called the not out of phase supply rate to first prove that infinite gain margin i e non intersection of the nyquist plot of a transfer function and the negative half of the real axis is equivalent to dissipativity with respect to this supply rate capturing non intersection of half line makes the supply rate system dependent a novel feature unobserved in the supply rates considered in the literature so far we then show that the traditional finite and positive gain phase margin conditions for stability are equivalent to dissipativity with respect to a frequency weighted convex combination of the not out of phase supply rate and the small gain supply rate both frequency weightings and combining two supply rates performance indices have been investigated in the literature in different contexts but only as sufficient conditions

kalman yakubovich lemma in the behavioural setting in this paper the kalman yakubovich lemma is formulated and proved in the behavioural theoretic setting the behavioural framework is used to generalize the lemma for behaviours with two manifest variables with respect to supply rates that are quadratic differential forms qdfs

controllability of conservative behaviours in this article we first define the class of j conservative behaviours with observable storage functions where j is a symmetric two variable polynomial matrix we then provide two main results the first result states that if j is nonsingular the input cardinality of a j conservative behaviour with an observable storage function is always less than or equal to its output cardinality the second result states that if j is constant and nonsingular a j conservative behaviour with an observable storage function and equal input and output cardinalities is always controllable physically the second result implies that a class of multiport lossless electrical networks is controllable