Linear Periodically Time-varying Research Articles

NONCAUSAL LINEAR PERIODICALLY TIME-VARYING SCALING FOR DISCRETE-TIME SYSTEMS

This paper is concerned with robust stability analysis of discrete-time systems. We first consider linear periodically time-varying (LPTV) nominal systems, for which we apply the discrete-time lifting to have their equivalent linear time-invariant (LTI) representations. Applying the conventional but general scaling approach to the LTI representations leads to the notion of noncausal LPTV scaling when the scaling is interpreted in the original time axis without lifting. Regarding this discrete-time noncausal LPTV scaling, we confirm its effectiveness over causal LPTV scaling and (causal) LTI scaling theoretically as well as with a numerical example. We next consider LTI nominal systems, for which we again apply noncausal LPTV scaling by regarding the LTI systems as a special case of LPTV systems and thus applying the discrete-time lifting in the same way as in the LPTV nominal systems. We then study the relationship of such an approach with the conventional LTI scaling applied directly to the LTI nominal systems without lifting treatment. In particular, we show that even static noncausal LPTV scaling yields dynamic (frequency-dependent) LTI scaling if it is interpreted in the context of lifting-free treatment, and an advantage of noncausal LPTV scaling for LTI nominal systems is investigated from this viewpoint. A numerical example is also provided that supports the advantage.

Efficient model reduction of linear periodically time-varying systems via compressed transient system function

This paper presents a new algorithm to obtain reduced-order model of linear periodically time-varying (LPTV) systems. The proposed algorithm introduces the approach of projection-based reduction algorithms, which have been used for reduction of linear time-invariant (LTI) systems, to the domain of LPTV systems. The key idea in the proposed approach is the utilization of integrated congruence transform to project the original LPTV system matrices onto the Hilbert subspace spanned by the time-dependent derivatives (or moments) of the transfer function. We prove that the transfer function of the resulting reduced-order model has the same derivatives as that of the original system in the Laplace-domain. The new approach presents a computationally efficient method to generate the orthogonal transformation operator (used in the integrated congruence transform) through expanding the time-varying transfer function in the right-half plane of the Laplace-domain. This enables using numerical time-domain integration for a very short transient period to generate the orthogonal transformation operator.

Analysis of periodically time-varying discrete-time systems in spectral domain

It is well known the concept of the steady-state response in the theory of linear time invariant systems. In this paper, the method called spectral analysis is newly adopted for periodically time varying (PTV) discrete time systems from the theory of the spectral analysis of linear PTV continuous time systems. Shortly, it describes the steady-state analysis of PTV discrete time systems. It leads to a simple algebraic relations in the frequency domain that minimizes calculations in the time domain. In order to show the generality and usefulness of this method, an illustrative examples are chosen. The comparison of the results with the exact solution concludes that the systems having smooth variation-less number of harmonics with respect to strongly varying systems are rapidly convergent with the high accuracy. The error analysis has being done by computing the root-mean-square error.

THE PERFORMANCE OF DISCRETE LINEAR TIME VARYING CONTROL OF LINEAR PERIODIC PLANTS

ABSTRACTThis paper analyzes the performance of discrete linear time varying (LTV) control of discrete linear periodically time varying (LPTV) plants for l2 disturbance rejection. It extends the results of [11–13] for linear periodic controllers to general LTV control of LPTV plants. It is shown that LPTV control provides strictly better control performance than linear strictly time varying control for LPTV plants. The analysis is carried out in frequency domain. This approach provides not only new results on disturbance rejection of LTV control but also some new insight into properties of general LTV systems.

Relationship between an LPTV system and the equivalent LTI MIMO structure

It is understood how to transform a linear periodically time-varying (LPTV) filter/ difference equation into an equivalent multiple-input/multiple-output (MIMO) structure, or transfer matrix, with linear time-invariant (LTI) elements, but no published method exists for the reverse operation. The paper presents a technique to transform from the LTI MIMO structure to the original single-input/single-output LPTV difference equation and discusses its implications. In addition, it is shown how this new result is then used to represent parallel and cascade connections of LPTV systems as single LPTV filters, implement order reduction of an LPTV difference equation, and finally obtain an LPTV difference equation representation that is equivalent to an LPTV state-space structure.

Optimal Estimation of the Parameters of Linear Periodically Time‐varying Systems

This paper discusses the linear periodically time-varying (LPTV) system parameter estimation using a block approach. An block algorithm is proposed for optimal estimation of the parameters of LPTV system from the input sequence and the output sequence corrupted by additive Gaussianly distributed noise. In the proposed method, the least squares error criterion has been used.The algorithm provides a useful computational tool based on an appropriate theoretical foundation for parameter estimation of linear time-invariant (LTI) systems from input and output data. Simulation results are presented that demonstrate the performance of the approach.

Representations of linear periodically time-varying and multirate systems

The linear switched time-varying (LSTV), polyphase (blocked), and alias-component representations of linear periodically time-varying (LPTV) systems are studied. In particular, alias-components are related to the time-shifted versions of the system. It is shown that in general, a filterbank is equivalent to the cascade connection of two LPTV systems. By generalizing some of the results of the LSTV representation of LPTV systems, it is shown that for relatively coprime integers p and m, a p-channel filterbank that has LPTV filters with period m is equivalent to an mp-channel filterbank with LTI filters. The representation of multirate systems that have LPTV kernels is discussed next. Due to the presence of the upsampler and downsampler, there are some degrees of freedom in the choice of the kernel. This redundancy is dealt with by choosing from various subclasses so that there is a one-to-one relationship between a multirate system and its kernel. Then, we find the LPTV kernel that has the least period.

Symbolic modeling of periodically time-varying systems using harmonic transfer matrices

This paper presents an algorithm for generating symbolic expressions for the harmonic transfer functions of linear periodically time-varying (LPTV) systems, like mixers and PLLs. The harmonic transfer functions characterize the up- and downconversion behavior of the wanted and unwanted signal components. The algorithm, which the authors call Symbolic HTM, is based on the organization of the harmonic transfer functions into a harmonic transfer matrix. This representation allows one to manipulate LPTV systems in a way that is similar to linear time-invariant systems, making it possible to generate symbolic expressions relating the overall harmonic transfer functions to the building block parameters. These expressions can be used as design equations or as parameterized models for use in simulations or synthesis. Comparison of the symbolic models with numerical data shows them to be accurate, even for small numbers of modeling terms.

A Study on the Optimal Cost for the $H_infty$ Problem of Discrete Linear Periodically Time-Varying Systems

In this paper, we propose a new method to obtain the optimal cost (the infimum of the achievable $H_\infty$ norm of a closed-loop system) for discrete linear periodically time-varying (LPTV) systems. This method reduces via the lifting technique the original problem to the $H_\infty$ problems of linear time-invariant (LTI) systems without the causality constraint, and therefore the optimal cost can be easily calculated. It is one of the primary advantages over other existing methods. We also show, by applying our method to LTI systems, that the optimal cost for the $H_\infty$ problem of discrete LTI systems cannot be improved even if we use a class of noncausal controllers. An intriguing implication of this fact is further shown in the context of causal controllers.

On the Periodic Realisation of Transfer Matrices

It is understood how to transform a linear periodically time-varying (LPTV) filter/difference equation into an equivalent multiple-inputlmultiple-output (MIMO) structure, or transfer matrix, but no published method exists for the reverse operation. This paper now presents a technique to return from the MIMO structure to the original LPTV difference equation. This new result is then used to 'approximate' a MIMO structure with a single LPTV filter, represent parallel and cascade connections of LPTV systems as single LPTV filters, implement order reduction of a LPTV difference equation, and finally obtain a LPTV difference equation representation that is equivalent to a LPTV state-space structure.

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.

Robustness of discrete periodically time-varying control under LTI unstructured perturbations

Investigates robustness of linear periodically time varying (LPTV) control of discrete linear time invariant (LTI) plants subject to LTI unstructured perturbations. The note first derives a necessary and sufficient condition for robust stability of an LPTV system subject to LTI perturbations, which is less conservative than the well known small gain condition. It then presents a quantitative analysis on the robustness of LPTV control under LTI unstructured perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect.

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period p̂? The optimality is measured using H 2 norms. The study is extended to general multirate periodic systems.

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.

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.

One-dimensional linear periodically time-varying structures: derivations, interrelationships and properties

The author presents all the derivations of, and the interrelationships for, four principle models of a one-dimensional, linear periodically time-varying (LPTV) system. These results are then used to both analyse cascade and parallel structures and solve the LPTV inverse filter problem.

Oversampling PCM techniques and optimum noise shapers for quantizing a class of nonbandlimited signals

We consider the efficient quantization of a class of nonbandlimited signals, namely, the class of discrete-time signals that can be recovered from their decimated version. The signals are modeled as the output of a single FIR interpolation filter (single band model) or, more generally, as the sum of the outputs of L FIR interpolation filters (multiband model). These nonbandlimited signals are oversampled, and it is therefore reasonable to expect that we can reap the same benefits of well-known efficient A/D techniques that apply only to bandlimited signals. We first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. We can achieve a substantial decrease in bit rate by appropriately decimating the signals and then quantizing them. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing prefilters and postfilters around the quantizer. We start with a scalar time-invariant quantizer and study two important cases of linear time invariant (LTI) filters, namely, the case where the postfilter is the inverse of the prefilter and the more general case where the postfilter is independent from the prefilter. Closed form expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models. The class of noise shaping filters and quantizers is then enlarged to include linear periodically time varying (LPTV)/sub M/ filters and periodically time-varying quantizers of period M. We study two special cases in great detail.

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.