Periodic Time-varying Systems Research Articles

Observer-based output reachable set synthesis for periodic piecewise time-varying systems

In this paper, the output reachable set synthesis problem for periodic piecewise time-varying systems has been addressed for the first time. Generalized periodic time-varying conditions of the closed-loop system to guarantee the system output bounded by a collection of finite ellipsoids are developed. The decoupling of time-varying observer and controller gains is achieved by a technical lemma . Moreover, by using time-varying matrix functions in multiple linear-interpolative forms and the relaxed quadratic matrix polynomial definiteness results, sufficient condition is developed to determine the periodic time-varying observer and controller with reduced conservatism. A tractable iterative algorithm is proposed to obtain the periodic time-varying observer and controller gains. A numerical example is presented to illustrate the validity of proposed methods.

Energy-to-Peak Output Tracking Control of Actuator Saturated Periodic Piecewise Time-Varying Systems With Nonlinear Perturbations

This article is focused on the design of an output tracking control scheme for a class of continuous-time periodic piecewise time-varying systems (PPTVSs) with actuator saturation and nonlinear perturbations. The energy-to-peak tracking performance is studied based on an equivalent condition on the definiteness property of matrix polynomials. Considering the actuator saturation and nonlinear perturbation, matrix polynomial-based sufficient conditions are derived through the Lyapunov method using periodic matrix functions. From a perspective of subinterval segmentation aimed at PPTVSs, the proposed conditions can achieve less conservatism for tracking the output of a periodic time-varying reference system, while the controller gains can be computed using convex optimization. Moreover, a heuristic algorithm is constructed to simultaneously guarantee the closed-loop state convergence and the output tracking performance. The reduction in conservatism and the effectiveness of algorithm are demonstrated by illustrative case studies.

Generalized Actor-Critic Learning Optimal Control in Smart Home Energy Management

This article is concerned with a new generalized actor-critic learning (GACL) optimal control method. It aims at the optimal energy control and management for smart home systems, which is expected to minimize the consumption cost for home users. In the present GACL optimal control method, it is the first time that three iteration processes, which are global iteration, local iteration, and interior iteration, respectively, are established to obtain the optimal energy control law. The main contribution of the developed method is to establish a common iteration structure for both value and policy iterations in adaptive dynamic programming based on a control law sequence in each iteration for periodic time-varying systems, instead of a single control law, and simultaneously accelerates the convergence rate. The monotonicity, convergence, and optimality of the iterative value function for the GACL optimal control method are proven. Finally, numerical results and comparisons are displayed to show the superiority of the developed method.

A Novel Scheme of Nonfragile Controller Design for Periodic Piecewise LTV Systems

In this article, a novel nonfragile controller design scheme is developed for a class of periodic piecewise systems with linear time-varying subsystems. Two types of norm-bounded controller perturbations, including additive and multiplicative ones, are considered and partially characterized by periodic piecewise time-varying parameters. Using a new matrix polynomial lemma, the problems of nonfragile controller synthesis for periodic piecewise time-varying systems (PPTVSs) are made amenable to convex optimization based on the favorable property of a class of matrix polynomials. Depending on selectable divisions of subintervals, sufficient conditions of the stability and nonfragile controller design are proposed for PPTVSs. Case studies based on a multi-input multi-output PPTVS and a mass-spring-damper system show that the proposed control schemes can effectively guarantee the close-loop stability and accelerate the convergence under controller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existing methods.

Non-fragile H ∞ control of periodic piecewise time-varying systems based on matrix polynomial approach

This paper investigates the problem of non-fragile control for periodic piecewise time-varying systems. Based on a Lyapunov function with continuous time-varying Lyapunov matrix polynomial, and combining with the positiveness and negativeness properties of matrix polynomials, the performance analysis is first accomplished. Then consider two types of controller gain perturbations that are formulated by time-varying matrix parameters and norm-bounded uncertainties. The additive and multiplicative non-fragile controllers to guarantee the performance of the system are formed, of which the controller gain could be solved with linear matrix inequalities directly. The designed non-fragile controller is desirable in applications. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

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.

Efficient Calculation of Stabilization Parameters in RF Power Amplifiers

This article proposes an efficient method for the calculation of the stabilization parameters in RF power amplifiers operating in periodic large-signal regimes. Stabilization is achieved by applying the principles of linear control theory for periodic linear time-varying (PLTV) systems. A numerical method is proposed to obtain the harmonic transfer function that represents the system linearized around the large-signal steady state. Then, a feedback analysis is performed to calculate the closed-loop poles of the PLTV system. The proposed approach is demonstrated with two examples. First, a three-stage amplifier that exhibits a low-frequency oscillation for increasing values of input power is correctly stabilized. Next, the stabilization of an unstable design that exhibits an odd-mode parametric oscillation is presented. The results of the proposed technique are compared to those obtained with the conventional parametric stability simulation. These examples serve to illustrate the capability and efficiency of the proposed approach.

A Control Solution for a Tethered Kite Trajectory Tracking With Application to Ship Propulsion

Abstract In the field of clean and green energy production systems, airborne wind energy (AWE) systems are really promising technologies. Replacing lightweight tethers with expensive towers, is an innovative solution to get energy from higher altitudes than classical wind turbines. But AWE systems can also be used to produce a traction force through a fast-flying airfoil, or kite. In this paper, the traction force is supposed to be used for ship propulsion. Such an application is permitted if the kite is kept on an appropriate trajectory. This paper thus considers the design of an automatic pilot for trajectory tracking. Its design requires a path-planning and a control strategies that are described here. In terms of control strategy, a robust CRONE controller is designed based on a family of linear models that stands for the nonlinear model of the kite and obtained numerically around several trajectories and for various flight conditions. Efficient trajectories for ship traction being periodic, a method involving tools dedicated to time-varying periodic systems is then used to validate the approach used for CRONE controller design.

Finite-Time Non-Fragile Extended Dissipative Control of Periodic Piecewise Time-Varying Systems

This paper studies the problem of finite-time non-fragile extended dissipative control for the periodic piecewise time-varying systems. First, based on the time-varying Lyapunov matrix and the matrix polynomial condition, a sufficient condition of finite-time boundedness for periodic piecewise time-varying subsystem is derived, and the finite-time extended dissipative performance is then analyzed. Furthermore, considering two types of time-varying controller perturbations, the non-fragile controllers are developed, which can guarantee the finite-time boundedness of periodic piecewise time-varying systems and satisfy the extended dissipative performance at the same time. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

Stability analysis problems of periodic piecewise polynomial systems

This paper studies the stability analysis problems of periodic piecewise systems, in which subsystems are given in the time-varying polynomial forms. A Lyapunov function with continuous time-varying Lyapunov matrix is adopted, which relaxes constraints on the variation of Lyapunov function in each subsystem. Using the time interval information, the exponential stability and stabilizing controller synthesis are studied. The results provide a possible alternative method to study the general periodic time-varying systems, which may further support the analysis and synthesis of general periodic systems. The effectiveness of the proposed method is validated and illustrated via numerical examples.

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.

Non‐fragile control of periodic piecewise linear time‐varying systems with time delay

In this study, the stability and stabilisation issues of a class of periodic piecewise linear time-varying systems with time delay are studied. Based on a lemma with respects to the negative definite matrix polynomial, a sufficient delay-dependent exponential stability condition is obtained via constructing a Lyapunov–Krasovskii functional with periodic matrix functions. Then, for two types of norm-bounded uncertainties of time-varying controller gains, the non-fragile stabilising controllers under the multiplicative and additive gain perturbations are designed to stabilise the closed-loop system. The obtained controller gains could be solved with linear matrix inequalities directly instead of using the iterative algorithm. That is valued in engineering applications. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Stability and $L_2$ Synthesis of a Class of Periodic Piecewise Time-Varying Systems

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

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.

H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function

In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method.

A novel H∞ tracking control scheme for periodic piecewise time-varying systems

In this work, a novel H∞ tracking control scheme is designed for a class of periodic piecewise systems with time-varying subsystems. The reference model is a periodic piecewise linear system that allows the existence of unstable subsystems. Based on periodically time-varying Lyapunov functions, a sufficient condition is proposed to guarantee the exponential stability of augmented tracking system in the H∞ sense. Taking advantage of the negative definiteness property of a type of matrix polynomials, an H∞ state tracking controller with periodically time-varying gains is provided via a series of continuous matrix functions, which can be efficiently online computed by the solutions of straightforward convex optimization. Numerical simulations are presented to demonstrate the effectiveness of the proposed scheme.

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.

Model predictive control with discrete actuators: Theory and application

Despite the presence of discrete actuators in many industrial processes, model predictive control (MPC) theory typically considers only continuous actuators, which requires discrete decisions to be removed from the MPC layer. However, if discrete inputs are chosen optimally, process performance may be greatly improved, and thus, discrete decisions should be treated directly in MPC theory. In this paper, we develop the idea that discrete actuators can be added to MPC theory without major modification, i.e., results established with sufficient generality for standard MPC with continuous actuators hold also for MPC with discrete actuators. First, we show that standard exponential stability for suboptimal MPC can be extended without modification to cover discrete actuators by avoiding restrictive assumptions about the geometry of the control set. Then, we prove stability results for tracking MPC applied to a time-varying periodic system. Finally, we demonstrate these results with two example systems.

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.