Linear Periodically Time-varying Systems Research Articles

A generalized analysis of high‐order N‐path filters

AbstractIn N‐path filter analysis, assumptions are used to simplify their analysis to avoid the complexity of the calculations. These assumptions limit the number of examinable circuits. This paper analyzes a general system including an arbitrary number of N‐path filters to generalize the analysis. In the proposed method, due to the calculation of harmonic transfer functions for each independent capacitor in each time interval, it is possible to examine N‐path filters in the presence of common capacitors between multiple branches, leakage resistors, and branches with asymmetric switching. The absence of discontinuous jumps during the switching moments and switching operation in a non‐overlapping fashion with a specified periodicity are the only assumptions of the proposed method. As a result, more LPTV (linear periodically time‐varying) systems can be analyzed using the calculations provided. After presenting the method, we show the proposed method's ability by analyzing a recently published impedance‐transforming N‐path filter and examining its details.

Cyclic Signals and Systems in Power Line Communications

This paper provides a comprehensive discussion about the characteristics of power line communication (PLC) channels related to its synchronization with the mains period. This fact constitutes an interesting and distinguishing feature of these kind of channels. Under the mains voltage presence, the behavior of some loads connected to the power network allows representing the transmission medium as a linear periodically time varying (LPTV) system. Some components in the received noise exhibit periodicity as well, in terms of impulsive waveforms synchronized with the mains period and other cyclostationary (CS) signals. This fact has been taken into account in the design of PLC systems, for instance by defining medium access control (MAC) frames synchronized with the mains period and by adapting the modulation to the cyclic signal-to-noise ratio (SNR) to improve the system performance. Hence, in PLC, it is of particular interest to describe the general relations of LPTV systems and CS signals when both have a common period, which is the purpose of this work. It is not aimed at providing new mathematical tools for the analysis of these cyclic signals and systems, which have been extensively developed over the last five decades, but to serve the readers of the PLC community more accessible mathematical relations that can be applied to the practical problems they have to face. A review of recent works dealing with the cyclic properties of PLC channels and transmission techniques aware of these strategies is included as well.

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

ABSTRACTMany control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.

Robust Output Estimator Synthesis Based on Cycling-Based LPTV Scaling

This paper is concerned with robust output estimator synthesis for discrete-time linear periodically time-varying (LPTV) systems with uncertainties. To evaluate robustness of estimation, we exploit the separator-type robust stability theorem under the cycling-based treatment of LPTV systems; we call such an approach for robustness analysis cycling-based LPTV scaling. Then, we derive a linear matrix inequality condition for robust estimator synthesis. A noteworthy property of our cycling-based synthesis approach is that we can predetermine the estimator period in the synthesis regardless of the system period, which is difficult to achieve in the case with the lifting-based treatment of LPTV systems.

An Extended Modal Identification for Linear Periodically Time-Varying Systems Based on Stability-Preserving State Transformation

This paper proposes an extended modal identification method to extract the time-varying modes and poles of linear periodically time -varying (LPTV) systems. A set of poles defines a stabilitypreserving transformation relat ing the original LPTV state equation to a constant upper triangular state equation and the time-varying modes contain the informat ion regarding the stability of the original LPTV system. The relation between the time-varying modes and poles is presented. The time-varying modes and poles of LPTV systems are shown to be the extension of the linear time-invariant (LTI) case. The proposed identification method is finally illustrated by a numerical example.

Optimal $H_{\infty}$ Control for Linear Periodically Time-Varying Systems in Hard Disk Drives

Periodicity frequently occurs in hard disk drives (HDDs) whose servo systems with periodic phenomena can be usually modeled as linear periodically time-varying (LPTV) systems. This paper discusses optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_{\infty}$</tex></formula> control synthesis for discrete-time LPTV systems via discrete Riccati equations. First, an explicit minimum entropy <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_{\infty}$</tex> </formula> controller for general time-varying systems is obtained. Subsequently, the developed control synthesis algorithm is applied to LPTV systems and it is shown that the resulting controllers are periodic. The proposed control synthesis technique is evaluated through both single and multirate optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $H_{\infty}$</tex></formula> track-following control designs. The single-rate servo design shows that our proposed control synthesis technique is more numerically robust in calculating optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $H_{\infty}$</tex></formula> controllers for discrete-time linear time-invariant systems than the MATLAB function of “hinfsyn,” while the multirate servo design validates its ability of synthesizing multirate controllers to achieve the robust performance of a desired error-rejection function. Moreover, an experimental study—in which the developed control synthesis algorithm on a real HDD with missing position error signal sampling data is implemented—further demonstrates its effectiveness in handling LPTV systems with a large period and attaining desirable disturbance attenuation.

Subspace Instability Monitoring for Linear Periodically Time-Varying Systems

Most subspace-based methods enabling instability monitoring are restricted to the linear time-invariant (LTI) systems. In this paper, a new subspace method of instability monitoring is proposed for the linear periodically time-varying (LPTV) case. For some LPTV systems, the system transition matrices may depend on some parameter and are also periodic in time. A certain range of values for the parameter leads to an unstable transition matrix. Early warning should be given when the system gets close to that region, taking into account the time variation of the system. Using the theory of Floquet, some symptom parameter of stability- or residual- is defined. Then, the parameter variation is tracked by performing a set of parallel cumulative sum (CUSUM) tests. Finally, the method is tested on a simulated model of a helicopter with hinged blades, for monitoring the ground resonance phenomenon.

Track-Following Control Design and Implementation for Hard Disk Drives with Missing Samples

AbstractMissing position-error-signal sampling data in hard disk drives (HDDs) results in their servo systems having irregular sampling rates. Due to the natural periodicity of HDDs, which is related to the disk rotation, HDD servo systems with missing samples can be modeled as linear periodically time-varying (LPTV) systems. Based on previous H∞ control synthesis results for general LPTV systems developed by some of the authors, an optimal H∞ track-following control algorithm for HDD servo systems with missing samples is proposed. A simulation and implementation study, which has been carried out on a set of hard disk drives, demonstrates that the proposed control synthesis algorithm is able to handle missing samples and can be used to achieve the robust performance of a desired error rejection function for disturbance attenuation. Implementation results on multiple real hard disk drives validate the predicted robust performance.

Identification of LPTV systems in the frequency domain

We derive a new method for the identification of discrete linear periodically time-varying (LPTV) systems. Take an LPTV system with period M, and assume that an input with period N is applied to this system, where N is a multiple of M. The output of this system will be periodic with period N. Using such periodic inputs, we show that we can formulate the problem of identification of LPTV systems in the frequency domain. With the help of the discrete Fourier transform (DFT), the identification method reduces to finding the least-squares (LS) solution of a set of linear equations. We show that the method achieves the Cramer-Rao lower bound (CRLB). A sufficient condition for the identifiability of LPTV systems is given, which can be used to find appropriate inputs for the purpose of identification. Simulation results illustrate the efficiency of the proposed algorithm.

Alias reduction: Generalised pseudocirculant conditions

Complete alias cancellation, in an arbitrary nonuniform and/or nonmaximally decimated filter bank structure, is not guaranteed. Also it is well known that the inversion of a linear periodically time varying (LPTV) system is not always guaranteed. In this paper we exploit the ability of uniform filter bank (UFB) framework to provide alternative models for these structures. We propose a simplification by generalising the well known pseudocirculant conditions [1] (Vaidyanathan and Mitra, 1988). UFB can be characterised by the two switching representations of the LPTV systems. The set of linear time invariant (LTI) systems in each representation is uniquely related to the rows and columns of the matrix associated with the UFB. Under the pseudocirculant conditions the UFB reduces to an LTI system. Here we obtain a set of generalised pseudocirculant conditions which reduce the UFB to an LPTV of lower order.

Least square identification of alias components of linear periodically time-varying systems and optimal training signal design

A least-squares (LS) method for identifying alias components of discrete linear periodically time-varying (LPTV) systems is proposed. The authors apply a periodic input signal to a finite impulse response (FIR) - LPTV system and measure the noise-contaminated output. The output of this LPTV system has the same period as the input when the period of the input signal is a multiple of the period of the LPTV system. The authors show that the input and the output can be related by using the discrete Fourier transform. In the frequency domain, an LS method can be used to identify the alias components. A lower bound on the mean square error (MSE) of the estimated alias components is given for FIR-LPTV systems. The optimal training signal achieving this lower MSE bound is designed subsequently. The algorithm is extended to the identification of infinite impulse response (IIR) - LPTV systems as well. Simulation results show the accuracy of the estimation and the efficiency of the optimal training signal design.

Least-squares Support Vector Regression for Identification of Periodic Systems

Least-squares support vector regression is applied to identify linear periodically time-varying (LPTV) systems. In this contribution the LPTV systems are modelled using a Fourier series representation of the transfer function and orthonormal filter expansions, such as Laguerre or Kautz filters. The model parameters are determined using least-squares support vector regression. In order to decrease complexity of the identified model, smoothness priors which penalize fluctuations of the frequency response can be introduced in the support vector procedure. The performance of the identified models are assessed by means of several simulation trials.

Analysis of Uncertain Discrete-Time Linear Periodic Systems based on System Lifting and LMIs

In this article, we propose novel linear matrix inequality (LMI) conditions for the stability and l 2 gain performance analysis of discrete-time linear periodically time-varying (LPTV) systems. These LMIs are convex with respect to all of the coefficient matrices of the LPTV systems and this property is promising when we deal with several control system analysis and synthesis problems. For example, we can apply those LMIs straightforwardly to robust performance analysis problems of LPTV systems that are affected by polytopic-type uncertainties. Even though our approach for robust performance analysis is conservative in general, we can reduce the conservatism gradually by artificially regarding the original N-periodic system as pN-periodic and increasing p. In addition, thanks to the simple structure of the LMI conditions, we can readily derive a viable test to verify the exactness of the computation results.

Noncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators

This article is concerned with robust stability analysis of discrete-time systems and introduces a novel and powerful technique that we call noncausal linear periodically time-varying (LPTV) scaling. Based on the discrete-time lifting together with the conventional but general scaling approach, we are led to the notion of noncausal LPTV scaling for LPTV systems, and its effectiveness is demonstrated with a numerical example. To separate the effect of noncausal and LPTV characteristics of noncausal LPTV scaling to see which is a more important source leading to the effectiveness, we then consider the case of LTI systems as a special case. Then, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling when viewed in the context of the conventional LTI scaling, while causal LPTV scaling fails to do so. It is further discussed that the effectiveness of noncausal characteristics leading to the frequency-domain interpretation can be exploited even for LPTV systems by considering the ν N -lifted transfer matrices of N -periodic systems.

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.

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.

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.

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.