The Framework for Linear Periodic Time‐Delay Systems Based on Semi‐Discretization: Stability Analysis and Control

Stability analysis and control for linear periodic time‐delay systems are investigated in this paper. In this framework, a semi‐discretization method is used to develop a mapping of the system response in a finite‐dimensional state space. With the mapping, the stability region and stability boundary can be identified by comparing the maximum absolute value of its eigenvalues to 1. More importantly, an efficient stability criterion is presented for periodic neutral systems. In addition, minimization of the maximum absolute value of the mapping's eigenvalues leads to optimal control gains. The tracking control problem is also discussed. Two numerical examples are given to illustrate the effectiveness of the proposed method.     

State waypoint approach to continuous‐time nonlinear optimal control problems

In this paper, we propose an optimal control technique for a class of continuous‐time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit‐form suboptimal solution, by means of linear approximation. Thus the original continuous‐time nonlinear control problem reduces to a finite‐dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior‐reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new methodology to compute stabilizing control laws for continuous‐time LTV systems

This paper is devoted to the problem of computing control laws for the stabilization of continuous‐time linear time‐varying systems. First, a necessary and sufficient condition to assess the stability of a linear time‐varying system based on the norm of the transition matrix computed over a sequence of successive finite‐time intervals is proposed. A link with a stability condition for an equivalent discrete‐time model is also established. Then, 3 approaches for the computation of stabilizing state‐feedback gains are proposed: a continuous‐time technique, ie, directly derived from the stability condition, not suitable for numerical implementation; a method based on the stabilization of the discrete‐time equivalent model along with a transformation to generate the desired continuous‐time gain; and the computation of stabilizing gains for a set of periodic discrete‐time systems. Finally, by adapting one of the existing methods for the stabilization of periodic discrete‐time systems, an algorithm for the computation of a stabilizing state‐feedback continuous‐time gain is proposed. A numerical example illustrates the validity of the technique.     

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

This paper presents a connection between dissipation inequalities and integral quadratic constraints (IQCs) for robustness analysis of uncertain discrete‐time systems. Traditional IQC results derived from homotopy methods emphasize an operator‐theoretic input–output viewpoint. In contrast, the dissipativity‐based IQC approach explicitly incorporates the internal states of the uncertain system, thus providing a more direct procedure to analyze uniform stability with non‐zero initial states. The standard dissipation inequality requires a non‐negative definite storage function and ‘hard’ IQCs. The term ‘hard’ means that the IQCs must hold over all finite time horizons. This paper presents a modified dissipation inequality that requires neither non‐negative definite storage functions nor hard IQCs. This approach leads to linear matrix inequality conditions that can provide less conservative results in terms of robustness analysis. The proof relies on a key J‐spectral factorization lemma for IQC multipliers. A simple numerical example is provided to demonstrate the utility of the modified dissipation inequality. Copyright © 2016 John Wiley & Sons, Ltd.     

Observer‐based finite‐time stabilization for extended Markov jump systems

In this paper, we study the problem of observer‐based finite‐time stabilization for a class of extended Markov jump systems with norm‐bounded uncertainties and external disturbances. The stochastic character under consideration is governed by a finite‐state Markov process, but with only partial information on the transition jump rates. Based on the finite‐time stability analysis, sufficient conditions for the existence of the observer‐based controller are derived via a linear matrix inequality approach such that the closed‐loop system trajectory stays within a prescribed bound in a fixed time interval. When these conditions are satisfied, the designed observer‐based controller gain matrices can be obtained by solving a convex optimization problem. Simulation results demonstrate the effectiveness of the approaches proposed in this paper. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Event‐triggered finite‐time reliable control for nonhomogeneous Markovian jump systems

This article investigates the event‐triggered finite‐time reliable control problem for a class of Markovian jump systems with time‐varying transition probabilities, time‐varying actuator faults, and time‐varying delays. First, a Luenberger observer is constructed to estimate the unmeasured system state. Second, by applying an event‐triggered strategy from observer to controller, the frequency of transmission is reduced. Third, based on linear matrix inequality technique and stochastic finite‐time analysis, event‐triggered observer‐based controllers are designed and sufficient conditions are given, which ensure the finite‐time boundedness of the closed‐loop system in an H_∞ sense. Finally, an example is utilized to show the effectiveness of the proposed controller design approach.     

Global finite‐time stabilization of planar nonlinear systems with disturbance

This paper proposes a continuous global finite‐time controller for a class of planar systems with disturbance. The proposed controller consists of nominal and compensated parts. The nominal part, designed by an homogeneity‐based technique, takes care only of the nominal system. The closed‐loop nominal system is asymptotically stable and satisfies negative homogeneity of degree. However, using a finite‐time convergent second order sliding mode super‐twisting algorithm, the compensated part enables cancelling of the disturbance which is time‐varying or unbounded as long as its derivative is bounded. The combination of the nominal and compensated parts makes the planar system globally finite‐time stable. Simulation results show the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Global adaptive finite‐time stabilization for a class of p‐normal nonlinear systems via an event‐triggered strategy

This article addresses the problem of global adaptive finite‐time control for a class of p‐normal nonlinear systems via an event‐triggered strategy. A state feedback controller is first designed for the nominal system by adding a power integrator method. Then, by the skillful design of adaptive dynamic gain mechanism, a novel event‐triggered controller is constructed for uncertain nonlinear system without homogeneous growth condition. It is proved that the global finite‐time stabilization of p‐normal nonlinear systems is guaranteed and the Zeno phenomenon is excluded. Finally, two examples are presented to indicate the effectiveness of the proposed control scheme.     

线性时变系统有限时间稳定性分析

针对一般非线性时变系统的有限时间稳定性分析问题,考虑到系统初始时刻在有限时间区间内的变化,本文分别提出了一般非线性时变系统的一致有限时间稳定性,一致收缩稳定性和固定调节时间一致收缩稳定性定义.针对一类线性时变系统,基于计算系统的所有轨线的包线的思想,本文分别给出了判定该类系统的收缩稳定性、固定调节时间收缩稳定性、一致有限时间稳定性、一致收缩稳定性、固定调节时间一致收缩稳定性的充分必要条件,同时给出了判定该类系统一致收缩稳定性及固定调节时间一致收缩稳定性的3个充分条件.进一步,本文将所得定理结果推广到了周期线性时变系统,所得结论为判定周期线性时变系统关于任意初始时刻的一致有限时间稳定性,一致收缩稳定性及固定调节时间一致收缩稳定性提供了理论依据.最后,以4个数值算例和两航天器相对运动过程为例验证了本文结果的正确性.     

On the design of approximate non‐linear parametric controllers

This paper focuses on the design of non‐linear parametric controllers, around a nominal input/output trajectory of a discrete‐time non‐linear system. The main result provided herein is a relationship between the tracking performance of the closed‐loop control system in the neighbourhood of a nominal trajectory, and some local features (the first‐order linear approximations about the nominal trajectory) of the non‐linear mappings which characterize the plant and the feedback controller. Such a result can be used to predict the dynamic behaviour of the control system, and to reduce the computational complexity of the optimization task associated with the tuning of the parametric feedback controller. Copyright © 2000 John Wiley & Sons, Ltd.     

Event‐triggered practical finite‐time output feedback stabilization of a class of uncertain nonlinear systems

This paper studies the event‐triggered practical finite‐time output feedback stabilization problem for a class of uncertain nonlinear systems with unknown control gains. First, a reduced‐dimensional observer is employed to implement the reconstruction of the unavailable states. Furthermore, a novel event‐triggered output feedback control strategy is proposed based on the idea of backstepping design and sign function techniques. It is shown that the practical finite‐time stability of the closed‐loop systems is ensured by Lyapunov analysis and related stability criterion. Compared with the existing methods, the main advantage of this strategy is that the observer errors and event‐trigger errors can be processed simultaneously to achieve the practical finite‐time stability. Finally, an example is adopted to demonstrate the validity of the proposed scheme.     

Robust Finite‐Time Stability and Stabilization of Linear Uncertain Time‐Delay Systems

Robust finite‐time stability and stabilization problems for a class of linear uncertain time‐delay systems are studied. The concept of finite‐time stability is extended to linear uncertain time‐delay systems. Based on the Lyapunov method and properties of matrix inequalities, a sufficient condition that ensures finite‐time stability of linear uncertain time‐delay systems is given. By virtue of the results on finite‐time stability, a memoryless state feedback controller that guarantees that the closed‐loop system is finite time stable, is proposed. The controller design problem is solved by using the linear matrix inequalities and the cone complementarity linearization iterative algorithm. Numerical examples verify the efficiency of the proposed methods.     

Tube‐based robust nonlinear model predictive control

This paper extends tube‐based model predictive control of linear systems to achieve robust control of nonlinear systems subject to additive disturbances. A central or reference trajectory is determined by solving a nominal optimal control problem. The local linear controller, employed in tube‐based robust control of linear systems, is replaced by an ancillary model predictive controller that forces the trajectories of the disturbed system to lie in a tube whose center is the reference trajectory thereby enabling robust control of uncertain nonlinear systems to be achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite‐Time Sliding Mode Trajectory Tracking Control of Uncertain Mechanical Systems

The problem of finite‐time tracking control is studied for uncertain nonlinear mechanical systems. To achieve finite‐time convergence of tracking errors, a simple linear sliding surface based on polynomial reference trajectory is proposed to enable the trajectory tracking errors to converge to zero in a finite time, which is assigned arbitrarily in advance. The sliding mode control technique is employed in the development of the finite‐time controller to guarantee the excellent robustness of the closed‐loop system. The proposed sliding mode scheme eliminates the reaching phase problem, so that the closed‐loop system always holds the invariance property to parametric uncertainties and external disturbances. Lyapunov stability analysis is performed to show the global finite‐time convergence of the tracking errors. A numerical example of a rigid spacecraft attitude tracking problem demonstrates the effectiveness of the proposed controller.     

Control of discrete‐time periodic linear systems with input saturation via multi‐step periodic invariant sets

This paper studies local control of discrete‐time periodic linear systems subject to input saturation by using the multi‐step periodic invariant set approach. A multi‐step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. The problems of (robust) estimation of the domain of attraction, (robust) local stabilization (with bounded uncertainties), and disturbance rejection are considered. Compared with the conventional periodic invariant set approach, which has been used in the literature for local stability analysis and stabilization of discrete‐time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing additional auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller, in the presence of norm bounded uncertainties, whose period is the same as the open‐loop system and is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open‐loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control‐Dependent Noise

In this paper, finite‐time stability and stabilization problems for a class of linear stochastic systems are studied. First, a new concept of finite‐time stochastic stability is defined for linear stochastic systems. Then, based on matrix inequalities, some sufficient conditions under which the stochastic systems are finite‐time stochastically stable are given. Subsequently, the finite‐time stochastic stabilization is studied and some sufficient conditions for the existence of a state feedback controller and a dynamic output feedback controller are presented by using a matrix inequality approach. An algorithm is given for solving the matrix inequalities arising from finite‐time stochastic stability (stabilization). Finally, two examples are employed to illustrate the results.     

Finite‐time stability of linear fractional‐order time‐delay systems

In this paper, a finite‐time stability results of linear delay fractional‐order systems is investigated based on the generalized Gronwall inequality and the Caputo fractional derivative. Sufficient conditions are proposed to the finite‐time stability of the system with the fractional order. Numerical results are given and compared with other published data in the literature to demonstrate the validity of the proposed theoretical results.     

参数不确定时滞系统的鲁棒稳定性

研究一类具有参数不确定时滞系统的鲁棒稳定性问题．假定其线性部分的参数不确定性由区间摄动模武描述，非线性部分的动态不确定性由积分二次约束(IQC)描述，给出了时滞互联系统鲁棒稳定的充分条件，并根据IQC乘子的特性，将无穷维稳定性检验问题转化为一维检验或有限检验问题．     

Finite‐time control for LPV systems with parameter‐varying time delays and exogenous disturbances

In this paper, we consider the stability analysis and control synthesis of finite‐time boundedness problems for linear parameter‐varying (LPV) systems subject to parameter‐varying time delays and external disturbances. First, the concepts of uniform finite‐time stability and uniform finite‐time boundedness are introduced to LPV systems. Then, sufficient conditions, which guarantee LPV systems with parameter‐varying time delays finite‐time bounded, are presented by using parameter‐dependent Lyapunov–Krasovskii functionals and free‐weight matrix technologies. Moreover, on the basis of the results on the uniform finite‐time boundedness, the parameter‐dependent state feedback controllers are designed to finite‐time stabilize LPV systems. Both analysis and synthesis conditions are delay‐dependent, and they are formulated in terms of linear matrix inequalities by using efficient interior‐point algorithms. Finally, results obtained in simulation demonstrate the effectiveness of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite‐time stability of switched positive linear systems

This brief paper addresses the finite‐time stability problem of switched positive linear systems. First, the concept of finite‐time stability is extended to positive linear systems and switched positive linear systems. Then, by using the state transition matrix of the system and copositive Lyapunov function, we present a necessary and sufficient condition and a sufficient condition for finite‐time stability of positive linear systems. Furthermore, two sufficient conditions for finite‐time stability of switched positive linear systems are given by using the common copositive Lyapunov function and multiple copositive Lyapunov functions, a class of switching signals with average dwell time is designed to stabilize the system, and a computational method for vector functions used to construct the Lyapunov function of systems is proposed. Finally, a concrete application is provided to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.