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

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

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

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

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

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

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

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

reduced markov chains at stochastic spin models abstract from glauber s stochastic spin model in discrete time reduced markov chain models are constructed the transition matrices of the reduced models utilize equilibrium correlation functions of the full n spin system however the reduced models involve the time dependent behavior of only a cluster of spins the reduced models have as an invariant vector the exact marginal equilibrium probability for the spins in the cluster in that sense the reduced models have the same equilibrium as the n spin glauber model but will in general display a different time dependence one of the reduced models is solved exactly here for a one dimensional lattice a square lattice and a simple cubic lattice

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

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

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

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

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

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

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