Discrete-time Periodic Systems Research Articles

An LMI characterization of the class of stabilizing controllers for periodic discrete-time systems

In this paper we consider the problem of stabilizing periodic linear systems via state feedback. With reference to the discrete time case, we show that it is possible to characterize the class of stabilizing gains in terms of a suitable Linear Matrix Inequality. This also includes the case of systems subject to norm-bounded parameter uncertainties.

Stabilization of Periodic Discrete Systemsby Output Sample Hold Control

In linear ω-periodic discrete-time systems, eigenvalues of the state transition matrix over the time interval nω dominate stability, where n is the dimension of the state vector. The periodic discrete-time system is called to be sample observable when the state can be determined from observation of output at only one time within one period. Under the assumption that eigenvalues of the monodromy matrix in an open-loop system are distinct, it is shown that the closed-loop system can be stabilized by applying the sampled output nω-periodic hold control if and only if the open-loop system is stabilizable and sample detectable at some time t0.

Robust disturbance-rejection problems for linear ω-periodic discrete-time systems

Two robust disturbance-rejection problems with state feedback and with incomplete-state feedback for linear /spl omega/-periodic discrete-time systems are studied in the framework of the so-called geometric approach. Also, some necessary conditions and/or sufficient conditions for the problems to be solvable are presented.

Output Regulation, Tracking and Nominal Decoupling with Stability for Uncertain Linear Periodic Systems

In this paper uncertain linear periodic discrete-time systems are considered, for obtaining the row-by-row reference-output decoupling with stability at the nominal parameters, together with the asymptotic tracking and output regulation in a neighbourhood of the nominal parameters, for classes of exponential-polynomial reference signals and exponential disturbance functions. By-using the periodic version of the internal model principle, the necessary and sufficient conditions for the existence of a solution are given, and a design procedure for the dynamic compensator is proposed.

From singular to nonsingular filtering of periodic systems: filling the gap with the spectral interactor matrix

With reference to periodic discrete-time systems, the concept of the periodic spectral interactor matrix operator (PSIMO) is introduced. Such an interactor enables one to transform the singular system into a nonsingular one with identical spectral properties. This concept is useful to characterize the family of periodic solutions of the singular periodic Riccati equation arising from the singular filtering problem. The PSIMO is also a synthetic tool to represent the invariant zeros structure at infinity (delays) of periodic systems in an operational fashion.

Module Theory for Linear Periodic Systems: Structural Properties and Realization

Reachability, observability, controllability and reconstructibibty of linear Periodic Discrete-Time (LPDT) systems are defined here in the context of module theory. We describe after, the two possible structures of the generalized state space representation according to the LPDT system to be regular or descrptor. The no-equivalence between canonical and minimal realization for LPDT (and consequently for discrete time-varying) systems is clearly proved. Periodic structural invariants of LPDT system (such as poles, zeros…) and their link with invariants factors (when they exist) of periodic matrices is sketched in the appendix.

Stabilization of Linear Discrete-Time Periodic Systems

The problem of stabilization of linear discrete-time periodic systems is considered. LMI based stabilization conditions via static periodic state feedback as well as via static periodic output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient whereas for output feedback the result is only sufficient and depends on the particular state-space representation used to describe the system. The problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. As an application of the output feedback stabilization technique, we consider the problem of designing a stabilizing (respectively, quadraticaliy stabilizing) static periodic output feedback controller for linear time-invariant discrete-time systems which are not stabilizable (respectively, quadraticaliy stabilizable) by static constant output feedback.

Periodic-polynomial interpretation for structural properties of linear periodic discrete-time systems

The structural properties of linear periodic discrete-time systems are analyzed in the periodic polynomial representation. It is shown that the classical polynomial approach for linear time-invariant systems can be extended to periodic systems. New definitions and properties are given in terms of skew polynomial rings and periodic difference algebra. Necessary and sufficient conditions for the characterization of reachability, controllability, observability and reconstructibility are given in this framework.

Model Matching with Stability for Periodic Discrete-Time Systems

In this paper the model matching problem with stability is studied for discrete-time periodic systems. It is shown that, under some zeros matching conditions and stabilizability/detectability assumptions, it is possible to assign a target input-output periodic system by means of a periodic controller. Most remarkably, this allows to find conditions under which a periodic system can be converted into a closed-loop time invariant system via periodic control.

Disturbance localization for left invertible linear periodic discrete-time systems

The disturbance localization problem for left invertible linear periodic discrete-time systems is solved using periodic state feedback controllers. The proposed technique is of algebraic nature and has the following two main characteristics: (i) It yields simple algebraic criteria for testing the solvability of the problem, as compared to known geometric criteria, which may not be so easy to check. (ii) It derives analytically the general expressions of all periodic controllers admissible for disturbance localization, as compared to known techniques, which lead to nonanalytic parametrizations of the admissible controllers via constructive procedures. Moreover, for the aforementioned class of periodic systems, the state feedback simultaneous disturbance localization and stabilization or pole placement problem is treated, and conditions for its solvability are established, on the basis of a decentralized control approach, that makes use of the equivalence between the above problem and the stabilization or pole placement problem of a general proper multichannel system by decentralized static output feedback.

A stability problem for discrete-time linear periodic systems

Conditions to construct a periodic output feedback of a periodic system are given in terms of properties of rational matrices. It turns out that those conditions are analogous to the ones involved in the existence of periodic realization of a periodic collection of rational matrices. The obtained periodic output feedback is used in the stabilization problem.

Periodic Lyapunov equations: Some applications and new algorithms

The discrete-time periodic Lyapunov equation has several important applications in the analysis and design of linear periodic control systems. Specific applications considered are the solution of state- and output-feedback optimal periodic control problems, the stabilization by periodic state feedback and the square-root balancing of discrete-time periodic systems. Efficient and numerically reliable algorithms based on the periodic Schur decomposition are proposed for the solution of periodic Lyapunov equations. The proposed algorithms are extensions of the direct solution methods for standard discrete-time Lyapunov equations for the case of indefinite as well as non-negative definite solution.

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.

A note on robust pole assignment for periodic systems

In this note a robust pole assignment algorithm is proposed for linear periodic discrete-time systems with time-varying dimensions of the state and/or input spaces. The algorithm deduces a periodic state feedback law by the minimization of the condition numbers of the eigenvector matrices of the closed-loop system. Numerical examples are provided to show the performances of the algorithm.

Robust output regulation and tracking for linear periodic systems under structured uncertainties

The problem of robust output regulation and tracking is studied for a linear periodic discrete-time system whose state-space description depends on some unknown parameters. The preservation of asymptotic regulation and tracking under perturbations of such parameters is obtained for classes of exponential disturbance functions and reference signals. Both the conditions for the existence of a solution and a constructive design procedure for the periodic compensator are given.

Model Reduction of Linear Discrete-Time Periodic Systems Using Hankel-Norm Approximations

In this paper we study model reduction of discrete-time periodic systems using Hankel-norm approximation. We review some ancillary results from the linear discrete time-invariant and discrete-time periodic systems. Then, we introduce a direct approach of Hankel-norm approximations for linear discrete time-invariant systems. Further, we extend the results to the linear discrete-time periodic systems. Simulation results show that the Hankel-norm approximation for discrete-time periodic systems has the desired error bounds. The balanced periodic system is obtained by solving the Lyapunov equations using recently proposed methods.

Decoupling with stability for linear periodic systems

In this paper, the classical row-by-row decoupling problem with stability through the use of a dynamic compensator is studied for linear periodic discrete-time systems. The necessary and sufficient conditions for the existence of a solution are found, and a design procedure of the dynamic compensator is provided by the sufficiency proof, and also for the case of a dead-beat requirement for the state free response.

Ripple-free dead-beat tracking for multirate sampled-data systems

In this paper, a solution is given to the problem of dead-beat ripple-free tracking for a multivariable multirate sampled-data system. The solution of such a problem is reduced to the solution of a dead-beat tracking problem for the corresponding discrete-time periodic system by introducing an additional requirement of dead-beat control for the control input of the multirate sampled-data system. The conditions for the problem solvability include the existence of a continuous-time internal model of all reference signals in the continuous-time plant. The design procedure of the periodic discrete-time compensator reduces to the design of a dead-beat periodic controller.

The realization problem for linear periodic systems

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.

Pole assignment for linear periodic systems by memoryless output feedback

We consider linear periodic discrete-time systems. We are interested in the problem of placing the poles of the monodromy map by means of periodic output feedback of the same or multiple period. It is well known that, in general, the poles of time-invariant systems cannot be assigned by constant output feedback. This is in contrast with what can be obtained in the context of time-variant systems. The main contribution of this paper is that periodic output feedback suffices for pole placement of periodic systems. >