Linear Periodically Time-varying Systems Research Articles

On stabilization and spectrum assignment in periodically time-varying continuous-time systems

This note discusses the stabilization and spectrum assignment problems in linear periodically time-varying (LPTV) continuous-time systems with sampled state or output feedback. The hybrid nature of the overall feedback system in this case imposes some carefulness in handling classical concepts related to purely LPTV continuous-time systems, In particular, this note points out the fact that the stabilization of such systems by periodic feedback gains with sampled state or output does not imply the relocation of the original characteristic exponents of the LPTV systems. It is also shown that the concept of monodromy matrix as extended to LPTV hybrid systems has not all the features of a true monodromy matrix.

On alias-component matrices of discrete-time linear periodically time-varying systems

We first use the Fourier series to find the steady-state response of discrete-time linear periodically time-varying (LPTV) systems to periodic inputs. Then, in order to obtain the response of LPTV systems to general inputs, we use the Fourier transform (FT) and establish a direct link between alias-component matrices and LPTV systems modeled by periodically time varying difference equations. This is given in terms of the Fourier series of the parameters in the model.

Existence of Right and Left Representations of the Graph for Linear Periodically Time-Varying Systems

The graph representation of a system (the set of all input-output pairs) has gained considerable attention in the control literature in view of its usefulness for the analysis of feedback systems. In this paper it is shown that the graph of any stabilizable, linear, periodically time-varying (LPTV), continuous-time system can be expressed as the range and kernel of bounded, causal, LPTV systems that are, respectively, left and right invertible by bounded, causal, LPTV systems. These so-called strong-right and strong-left representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat characterization of closed-loop stability is obtained in terms of strong-right and strong-left representations of the plant and controller graphs. This in turn leads to a Youla-style parametrization of stabilizing controllers. All of the results obtained accommodate possibly infinite-dimensional input and output spaces and apply, as a special case, to sampled-data control-systems. Furthermore, they are particularly useful for robustness analysis in terms of the gap metric.

Gap-Metric Robustness Analysis of Linear Periodically Time-Varying Feedback Systems

Quantitative robust stability results are established in this paper for feedback systems that evolve in continuous-time and exhibit linear, periodically time-varying (LPTV) behavior. The results presented are analogous to results known to hold for linear, time-invariant (LTI) systems, although they do not follow directly from these. System uncertainty is measured using the gap metric, which quantifies the distance between systems in terms of the aperture between their graphs (the subspaces corresponding to all input-output pairs for each system). The main robustness result characterizes the largest gap-ball of LPTV plants stabilized by a nominal LPTV feedback controller known to stabilize a nominal LPTV plant at the center of this ball. A key step in the proof of this result makes use of a formula derived for the directed gap between LPTV systems. This formula is essentially a generalization of Georgiou's for LTI systems. Importantly, all of the results presented apply to a class of sampled-data control systems, as a special case.

Stabilizing Predictive Control For Linear Periodically Time Varying Systems

The results of (Mosca and Zhang, 1992) for LTI systems is generalized to linear periodically time varying (LPTV) systems. A simple periodic predictive control law is obtained for the control of LPTV systems which possesses a guaranteed stabilizing property.

Linear periodically time-varying discrete-time systems: Aliasing and LTI approximations

Linear periodically time-varying (LPTV) systems are abundant in control and signal processing; examples include multirate sampled-data control systems and multirate filter-bank systems. In this paper, several ways are proposed to quantify aliasing effect in discrete-time LPTV systems; these are associated with optimal time-invariant approximations of LPTV systems using operator norms.