Linear Periodic Time-varying System Research Articles

Controllability and control for discrete periodic systems based on fully-actuated system approach

The problem of controllability and control of linear discrete-periodic systems is investigated in this paper. The high-order fully-actuated models for linear discrete periodic time-varying systems are constructed, and a controllability criterion based on fully-actuated system models is proposed. On this basis, stabilization as the fundamental issue is studied and periodic state-feedback control laws are designed via fully-actuated systems approach and parametric design, which converted the original problem into the pole assignment problem for linear constant systems. Finally, a numerical example is given to verify the validity and feasibility of the proposed method.

Model order reduction based on discrete-time Laguerre functions for discrete linear periodic time-varying systems

Many engineering problems can be modelled as linear periodic time-varying (LPTV) systems, which naturally leads to the need for model order reduction of LPTV systems. This paper investigates a new model order reduction method for discrete LPTV systems. First, the state-space realization in the Fourier-lifted form of discrete LPTV system is constructed by representing periodic matrices in exponentially modulated periodic (EMP) Fourier series. By using Laguerre functions to expand the transfer function of the resulting Fourier-lifted system, the corresponding model order reduction algorithm is developed. Furthermore, the proposed algorithm is used to reduce the discrete LPTV system in the standard-lifted form. Theoretical analysis indicates that the transfer functions of both reduced order systems can match a certain number of moments. Finally, two numerical examples are given to verify the effectiveness of the proposed method.

Multirate Harmonic Compensation Control for Low Switching Frequency Converters: Scheme, Modeling, and Analysis

Using the grid-interfacing voltage-source converters (VSCs) to perform harmonic control as a smart ancillary function is increasingly studied in recent years. Different from the traditional dedicated harmonic compensation converters, the main function of the high-power smart converters is still delivering real power at a low switching frequency (e.g., 2 kHz). It is challenging to compensate harmonics with low switching frequency VSCs due to the low sampling rate and large system delay. This article proposes a multirate control scheme and derives the accurate multirate model to improve harmonic control. The multirate control scheme consists of high-rate harmonic control and synchronous sampled fundamental control. With the proposed control scheme, the advantages of the conventional synchronous sampling method and the benefits of the high sampling rate are achieved simultaneously. To accurately model the low switching VSC with two different sampling rates, the lifting method is introduced to achieve the discrete-time system model for this linear periodic time-varying system. The frequency response and stability analysis are carried out based on the lifted model. The modeling method, as well as the enhanced harmonic control performance of the multirate control structure, is validated in the experiments.

A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix $A(t).$ Finally we analyze the robustness of the stability property under external disturbance.

Adaptive identification and control of linear periodic systems using second‐level adaptation

SummaryAt the present time, there is a great deal of interest in the adaptive control of systems with rapidly varying parameters. Such problems arise in different forms and in many disciplines including finance, sociology, biology, and engineering. In general, these problems are intractable mathematically and the time variations have to be classified in some form to obtain rigorous results. In this paper, we consider the adaptive identification and control of linear systems with periodically varying parameters (referred to as linear time‐varying periodic (LTP) systems). The class of systems with known periodic parameters has been investigated widely since the pioneering work of Floquet and lends itself to rigorous mathematical analysis. However, very little was known until the early years of the last decade concerning the adaptive control of such systems. The problem was introduced by Xu and explored in detail by Narendra and Zhiling in 2006 and 2009, respectively. In 2012, a new method was introduced in adaptive control by Han and Narendra and is referred to as second‐level adaptation. In this paper, the adaptive identification and control of LTP systems using second‐level adaptation is discussed.

Analysis and Design of a Ripple Reduction Chopper Bandpass Amplifier

A low-power low-noise chopper amplifier for biosensor applications is proposed. To tackle the inherent ripple artifacts, it employs a simple ripple reduction method using a bandpass amplifier. The chopper amplifier is a linear periodic time-varying system. The method of harmonic transfer matrix is used to derive the signal and the noise harmonic transfer functions, and the theoretical results are confirmed by simulation and measurement results. Fabricated in a 130-nm standard CMOS process, the proof-of-concept prototype occupies a chip area of 0.28 mm2 and consumes $3.3~\mu {\text{A}}$ from a 1.2-V supply. The input referred noise is only 43 nV/ $({\mathrm {Hz}})^{1/2}$ .

Stabilizing unstable periodic orbits with delayed feedback control in act-and-wait fashion

A delayed feedback control framework for stabilizing unstable periodic orbits of linear periodic time-varying systems is proposed. In this framework, act-and-wait approach is utilized for switching a delayed feedback controller on and off alternately at every integer multiples of the period of the system. By analyzing the monodromy matrix of the closed-loop system, we obtain conditions under which the closed-loop system’s state converges towards a periodic solution under our proposed control law. We discuss the application of our results in stabilization of unstable periodic orbits of nonlinear systems and present numerical examples to illustrate the efficacy of our approach.

An efficient algorithm for periodic Riccati equation with periodically time-varying input matrix

This paper proposes an efficient algorithm for the periodic discrete-time Riccati equation arising from a linear periodic time-varying system (Ak,Bk). Although the algorithm can be used for general problem, it is particularly efficient when Ak is time-invariant and Bk is periodic, a special property associated with the problem of spacecraft attitude control using magnetic torques. We will compare the proposed algorithm, by using analysis and simulation, to some well-established algorithm to confirm our claim.

A novel extended precise integration method based on Fourier series expansion for the H2-norm of linear time-varying periodic systems

ABSTRACTA new reliable algorithm for computing the H2-norm of linear time-varying periodic (LTP) systems via the periodic Lyapunov differential equation (PLDE) is proposed. By taking full advantage of the periodicity, the transition matrix of the underlying LTP system associated with the PLDE is effectively computed by developing a novel extended precise integration method based on Fourier series expansion, where the time-consuming work for the computation of the matrix exponential and its related integrals in every sub-interval is avoided. Then, a highly accurate and efficient algorithm for the PLDE is derived using the block form of the transition matrix. Thus, the H2-norm is evaluated by solving a simple first-order ordinary differential equation. Finally, two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm.

Identification of pseudo-natural frequencies in a beam-moving mass system with periodic passages

The response of a linear time-invariant (LTI) system to harmonic input generates a harmonic output with constant frequency but varying magnitude and phase. Many structural dynamic systems have been modeled as linear time-varying periodic (LTP) systems. Previous studies have reported that the response of an LTP system to an exponential input establishes an infinite number of frequencies. These studies have presented a new, exponentially modulated periodic signal space and a corresponding harmonic transfer function as useful tools in the operational modal analysis of LTP systems. In consideration of this new approach, this study mainly identifies the frequencies of a typical LTP system, such as a beam that is subject to the intermittent passage of moving masses. Upon obtaining the harmonic transfer function for the beam-moving mass system, conventional frequency domain methods for LTI systems are used to derive the frequency characteristics of the LTP system from the system response. These methods include the peak-picking method. As expected in an LTP system, an infinite number of pseudo-natural frequencies resonate in the beam-moving mass system.

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.

Delayed Feedback Control for Linear Time-Varying Periodic Systems in Act-and-Wait Fashion

A delayed feedback control framework for stabilizing periodic orbits of a linear periodic time-varying system is proposed. Act-and-wait approach is utilized for switching the proposed delayed feedback controller on and off alternately at every integer multiples of the period of the system. We obtain conditions for the convergence of the state trajectories towards a periodic solution of the closed-loop system under our proposed control law. We present a numerical example to illustrate the efficacy of our approach.

Nonparametric Tracking of the Time-Varying Dynamics of Weakly Nonlinear Periodically Time-Varying Systems Using Periodic Inputs

In this paper, a nonparametric estimation procedure is presented in order to track the evolution of the dynamics of continuous (discrete)-time (non)-linear periodically time-varying (PTV) systems. Multisine excitations are applied to a PTV system since this kind of excitation signals allows us to discriminate between the noise and the nonlinear distortion from a single experiment. The key idea is that a linear PTV system can be decomposed into an (in)finite series of transfer functions, the so-called harmonic transfer functions (HTFs). Moreover, a systematic methodology to determine the number of significant branches is provided in this paper as well. Making use of the local polynomial approximation, a method that was recently developed for multivariable (non)-linear time invariant systems, the HTFs, together with their uncertainties embedded in an output-error framework, are then obtained from only one single experiment. From these nonparametric estimates, the evolution of the dynamics, described by the instantaneous transfer function (ITF), can then be achieved in a simple way. The effectiveness of the identification scheme will be first illustrated through simulations before a real system will be identified. Eventually, the methodology is applied to a weakly nonlinear PTV electronic circuit.

A numerical procedure to compute stabilizing state feedback gains for linear time-varying periodic systems

A procedure to synthesize stabilizing controllers for linear time-varying periodic continuous-time systems is proposed in this paper. The controller is a periodic state-feedback gain whose construction is based on the utilization of the transition matrix of the open-loop system, and the stability of the closed-loop system is guaranteed if the system is controllable and if a observability-based condition is satisfied. The periodic state feedback gain is obtained through the numerical integration of two differential matrix equations over two periods, being the resolution of such equations considerably simpler and computationally more viable than the resolution of Ricatti differential equations considered in the standard LQR approach. Some examples illustrates the validity of the technique.

H 2-norm computation of linear time-varying periodic systems via the periodic Lyapunov differential equation

A new reliable algorithm for computing -norm of linear time-varying periodic (LTP) systems in continuous-time domain is proposed in this article. The solving of -norm is firstly transformed into the solving of the periodic Lyapunov differential equation (PLDE). Then, the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure-preserving Magnus series method is proposed for solving the PLDE. With the solutions of PLDE, the -norm of LTP systems is evaluated by a simple first-order ordinary differential equation. Finally, the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples.

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.

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.

The steady-state impedance operator of a linear periodically time-varying one-port network and its determination

The steady-state impedance operator of a linear periodically time-varying one-port network and its determinationThe main subject of the paper is the description and determination of the impedance operator of a linear periodically timevarying (LPTV) one-port network in the steady-state. If the one-port network parameters and the supply vary periodically with the same period, the network reaches a periodic steady state. However, the sinusoidal supply may induce a non-sinusoidal voltage or current. It is impossible to describe such a phenomenon by means of one complex number. A periodically time-varying one-port network working in a steady-state regime can be described with a circular parametric operator. Within the domain of discrete time, such an operator takes the form of a matrix with real-valued entries. The circular parametric operator can be transformed into the frequency domain using a two-dimensional DFT. This description makes it possible to quantitatively assess LPTV system input and output harmonics aliasing. The paper also presents the derivation and the proof of convergence of an iteration scheme for the identification of circular parametric operators. The scheme may be used to determine the impedance of an LPTV one-port network. Some results of computer simulations are shown.

ALGORITHMS FOR LPTV SYSTEM PARAMETERS ESTIMATION

A nonrecursive and recursive algorithms are proposed for estimation of the linear periodically time- varying (LPTV) system parameters. Using a block approach, the parameter estimation problem is transformed into one-dimensional minimization problem for L set parameters of the LPTV system. The algorithm can be performed in parallel at L time moments for each set of parameters to reduce the computation time.

Robustness analysis of linear periodic time-varying systems subject to structured uncertainty

In this paper, we show how Floquet theory may be combined with a technique known as Lifting to cast a linear periodically time-varying system subject to structured linear time invariant uncertainty in the form of a linear fractional transformation. The stability and performance robustness of the resulting system may then be analysed using standard μ -analysis methods. A significant advantage of the proposed approach is that it allows the computation of a worst-case destabilising uncertainty combination which may be used to estimate the conservatism of the computed robustness margin. An example is given to illustrate the application of the proposed approach.