the model matching problem for periodic discrete time systems in this paper the model matching problem via state feedback is studied for discrete time periodic systems it is shown that under some zero matching conditions it is possible to assign a target input output periodic system by means of a periodic state feedback control law furthermore this entails that a periodic system can be converted into a closed loop time invariant system via state feedback

on the assignment of invariant factors by time varying feedback strategies abstract in this paper the realization of arbitrary invariant factors or cyclic components of the closed loop system matrix is discussed it is shown that the use of time varying feedback strategies enables one to realize arbitrary invariant factors in contrast with time invariant feedback control

disturbance localization by measurement feedback for linear periodic discrete time systems abstract the recent notions of inner controllable outer controllable and outer reconstructible subspaces together with the extension of the notion of c a b pair of subspaces to the class of linear periodic discrete time systems are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems by means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases i e with and without an additional requirement of output or state dead beat control

computing the zeros of periodic descriptor systems in this paper we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system the algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time invariant state space system and extracts a low order pencil which contains the zeros both finite and infinite as well as the kronecker structure of the periodic descriptor system the proposed algorithm is efficient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations which ensures some form of numerical stability

brief paper assigning closed loop invariant polynomials over polytopes for linear periodic systems this paper provides a parameterization of all output feedback periodic controllers providing closed loop invariant polynomials varying over some assigned polytopes for linear discrete time periodic plants using the notion of time invariant system associated to a periodic system the parameterization is provided in terms of the general solution of a suitably defined matrix diophantine equation

brief minimal representations of mimo time varying systems and realization of cyclostationary models this paper describes a family of minimally parameterized state space models for completely n step observable multi input multi output time varying systems and their algebraical links with equivalent input output models these results are then used to develop an algorithm for the realization of generic input output sequences generated by a periodic system a complete numerical example is also developed to illustrate all steps of the algorithm described in the paper

positive n periodic descriptor control systems this paper establishes the relationship between the positive n periodic descriptor system in discrete time and its associated invariant systems reachability and controllability properties of these kinds of systems are analyzed transmission of structural properties from the n periodic system to their associated invariant systems is studied finally some comments on the stability property are made

fault detection and isolation of linear discrete time periodic systems using the geometric approach this technical note solves the problem to detect and isolate faults in linear periodic discrete time systems a residual generator is designed through a suitable unknown input observer with the requirement that each residual should be sensitive only to one fault and simultaneously insensitive to the other faults that can affect the system and with the additional requirement of detectability that is the transition matrix of its relative error system has all eigenvalues smaller than in modulus for a given positive p 1 the solution is based on geometric tools and it makes use of the outer observable subspace notion

on the zeros of blocked time invariant systems abstract this paper studies the zero properties of blocked linear systems resulting from blocking of linear time invariant systems the main idea is to establish a relation between the zero properties of blocked systems and the zero properties of their corresponding unblocked systems in particular it is shown that the blocked system has a zero if and only if the associated unblocked system has a zero furthermore the zero properties of blocked systems under a generic setting i e a setting which parameter matrices a b c d assume generic values are examined it is demonstrated that nonsquare blocked systems i e blocked systems with number of outputs unequal to the number of inputs generically have no zeros however square blocked systems i e blocked systems with equal number of inputs and outputs generically have only finite zeros and these finite zeros have geometric multiplicity one

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

balancing related methods for minimal realization of periodic systems we propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time varying dimensions the first method belongs to the family of square root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations an alternative balancing free square root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems the key numerical computation in both methods is the solution of nonnegative periodic lyapunov equations directly for the cholesky factors of the solutions for this purpose a numerically reliable computational algorithm is proposed to solve nonnegative periodic lyapunov equations with time varying dimensions

optimal and robust controllers for periodic and multirate systems the problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete time periodic systems the solution consists of solving an equivalent time invariant standard l sup 1 optimization problem subject to an additional constraint this constraint assures the causality of the resulting periodic controller by the duality theory the problem is shown to be equivalent to a linear programming problem which is no harder than the standard l sup 1 problem also it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems finally the results are applied to the problem of robust stabilization of periodic and multirate systems

output stabilization via pole placement of discrete time linear periodic systems the output stabilization problem for discrete time linear periodic systems is solved both the state feedback control law and the state predictor are based on a suitable time invariant state sampled reformulation associated with a periodic system preliminary concepts of periodic system theory are briefly recalled in particular the structural properties of a linear discrete time periodic system are properly related to those of a time invariant system associated with it by resorting to such a time invariant reformulation the output stabilization problem via pole placement is solved the stabilizing controller is constituted by a control law and an asymptotic state predictor both of which are shown to be periodic

the realization problem for linear periodic systems abstract the realization problem for discrete time linear periodic systems is considered deep insight on the various complicated aspects of minimal quasi minimal and uniform realization is provided based on the knowledge of the structural properties of discrete time periodic systems the results hinge on the lifting isomorphism between such systems and time invariant ones

computational issues for linear periodic systems paradigms algorithms open problems linear periodic systems originate in various control fields involving periodic phenomena in the beginnings of algorithmic developments for periodic systems computational detours have been often employed to reduce the computational problems for periodic systems to those for standard systems well known techniques are the employment of lifted representations in discrete time or the use of periodic generators in continuous time new computational paradigms evolved later whose main ingredients are numerically reliable and efficient algorithms for manipulating matrix products without forming explicitly them reduction of large scale structured matrix pencils without building the underlying pencils computing with discrete time periodic system models with time varying state dimensions or solving periodic matrix differential equations using multiple shooting techniques the paper will give a succinct overview of all these new developments and will point out some of still existing open computational problems

an algebraic riccati equation for the discrete time periodic prediction problem abstract in this paper the periodic problem is reformulated as a prediction problem relative to a suitable time invariant system this enables one to establish a one to one correspondence between the solutions of the discrete time periodic riccati equation dpre and the solutions of a suitable discrete time and time invariant riccati equation in particular the constant solutions of this last equation are the periodic generators of the original periodic riccati equation this enables one to easily derive existence uniqueness and convergence results for the drpe from known results relative to the time invariant case

a new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location this work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body beyond the obvious merits of its applications this problem also presents significant mathematical challenges the near incompressibility inherent in the model of linear elasticity in the body gives rise to the locking effect and necessitates a unique treatment of both the direct and inverse problems a general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least squares mols objective functional the mols functional is shown to be convex thus overcoming the nonconvexity of the classical output least squares ols functional and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters generalized derivative formulas for the coefficient to solution map are also given along with a complete convergence analysis

brief a periodically time varying minimal partial realization algorithm based on twisting the 1975 paper of meyer and burrus gave necessary conditions on the coefficients of a periodically time varying difference equation to generate a prespecified set of impulse responses the main contribution of the present paper consists of an algorithm that constructs the coefficients of such a periodically time varying realization of minimal lag thus the algorithm solves the minimal partial realization problem for the periodically time varying scalar case we use a general technique twisting for associating a time invariant system with a periodically time varying system in such a way that the time step is preserved the technique plays an essential role in the ideas underlying the algorithm

connectedness of refined goetschel voxman fuzzy matroids this paper studies connectedness of goetschel voxman fuzzy matroids briefly g v fuzzy matroids an analog of connectedness of crisp finite matroids based on the results of fuzzy circuits given by goetschel and voxman the transitivity theorem concerning fuzzy circuits of g v fuzzy matroids is established and thus the useful notion of refined g v fuzzy matroid is introduced the connectedness of refined g v fuzzy matroids is then defined by using an equivalence relation on the set of all fuzzy points on the ground set and some expected properties of these g v fuzzy matroids are presented additionally five kinds of fuzzy matroids are compared

robust tracking and regulation of linear periodic discrete time systems in this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete time systems and for general classes of disturbance functions and reference signals consisting in free responses of periodic processes the necessary and sufficient conditions for the existence of a solution which involve the periodic notion of invariant zero are found a constructive design procedure of the periodic compensator is given it is based on the internal model principle

zeros of discrete time linear periodic systems in this note the concept of zero for linear discrete time siso periodic systems is discussed the definition is based on a time invariant mimo equivalent representation an input output characterization of the zeros is provided which extends a well known property for time invariant systems

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

parametrized solution of the periodic dead beat control problem the dead beat control problem for linear periodic discrete time systems in presence of unmeasurable disturbances and external reference signals is considered using a polynomial method a parametrization of all causal output feedback controllers solving the problem is provided

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

minimal periodic realizations of transfer matrices periodic controllers designed based on the so called lifting technique are usually represented by transfer matrices real operations require that the controllers be implemented as periodic systems the problem of realizing an nn sub o nn sub i proper rational transfer matrix as an n sub i input n sub o output n periodic discrete time system is studied an algorithm for obtaining periodic realizations which have a minimal number of states is proposed the result can also be used to remove any redundant states that exist in a periodic system

a new class of shift varying operators their shift invariant equivalents and multirate digital systems a class of linear shift varying operators that generalize the notion of n periodicity is defined it is shown that shift invariant equivalents for these operators exist and that the equivalence is in a strong sense preserving both algebraic and analytic system properties it is shown that multirate sampled data systems although not generally periodic fall into this class kranc vector switch decomposition and block filter implementations for single input single output multirate systems are connected under the unifying framework of shift invariant equivalents and this framework provides a way to extend them both to multi input multi output systems

brief paper mpc of constrained discrete time linear periodic systems a framework for asynchronous control strong feasibility stability and optimality via periodic invariance state feedback model predictive control mpc of discrete time linear periodic systems with time dependent state and input dimensions is considered the states and inputs are subject to periodically time dependent hard convex polyhedral constraints first periodic controlled and positively invariant sets are characterized and a method to determine the maximum periodic controlled and positively invariant sets is derived the proposed periodic controlled invariant sets are then employed in the design of least restrictive strongly feasible reference tracking mpc problems the proposed periodic positively invariant sets are employed in combination with well known results on optimal unconstrained periodic linear quadratic regulation lqr to yield constrained periodic lqr control laws that are stabilizing and optimal one motivation for systems with time dependent dimensions is efficient control law synthesis for discrete time systems with asynchronous inputs for which a novel modeling framework resulting in low dimensional models is proposed the presented methods are applied to a multirate nano positioning system

properties of blocked linear systems this paper presents a systematic study on the properties of blocked linear systems that have resulted from blocking discrete time linear time invariant systems the main idea is to explore the relationship between the blocked and the unblocked systems existing results are reviewed and a number of important new results are derived focus is given particularly on the zero properties of the blocked system as no such study has been found in the literature

a period specific realization of linear continuous time systems it is a well known fact that a weighting pattern matrix has a periodic realization if and only if the matrix is separable and periodic this fact however does not cope with a reasonable question of period specific realization problem for a given weighting pattern matrix and a given positive real number find a periodic realization corresponding to the given matrix and with a period equal to the given number this paper answers the question by showing that the period specific realization problem has a solution if and only if the given weighting pattern matrix is separable and has a period equal to the given positive real number to this end two types of period specific realizations are constructed one is with a constant a matrix whose dimension is not necessarily minimal among all possible period specific realizations the other is with a non constant and periodically time varying a matrix whose dimension turned out minimal among all possible period specific realizations