Design of switched controllers for discrete singular bilinear systems

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems.The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems that are not open-loop stable.     

Continuous stabilization controllers for singular bilinear systems: The state feedback case

Global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper. New approaches are developed by means of the LaSalle invariant principle for nonlinear systems. A new set of sufficient condition is first derived via the continuous static state feedback, the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system.     

On singular hybrid switched and impulsive systems

In this work, the problem of stability analysis for a class of singular hybrid switched and impulsive system (HSIS) is addressed. Corresponding to each subsystem, a hybrid switched and impulsive controller is designed and then the exponential stability property of the proposed singular HSIS is discussed for linear and nonlinear cases. Because switched systems without impulses are a special case of HSISs, the results are also given to switched system with synchronous and asynchronous controllers. The obtained results apply to control singular systems, and the introduced theorems allow knowing how the control must be designed. Two numerical examples are given to show the effectiveness of the proposed approaches. At first, by using MATLAB^® software, the proposed method is applied to a class of physiological processes of endocrine disruptor diethylstilbestrol models to illustrate the effectiveness of the results obtained here for the linear case. Thereafter, another numerical example is provided to support the presented theoretical results for the nonlinear case.     

Stabilisation of a class of positive switched nonlinear systems under asynchronous switching

This paper addresses the stabilisation problem for a class of positive switched nonlinear systems under asynchronous switching, which means that the switches between the candidate controllers and the system modes are not synchronous. The continuous and discrete cases are considered respectively. Sufficient conditions are firstly provided for the existence of the asynchronous switching controllers to guarantee the closed-loop system to be positive and exponentially stable, and the corresponding admissible switching signals are presented. As a special case, the stabilisation results for positive switched linear systems under asynchronous switching are provided accordingly. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

一类离散时间切换系统鲁棒控制器设计

考虑一类非线性离散时间切换系统的鲁棒二次镇定和渐近镇定问题．利用公共李亚普诺夫函数方法和多李亚普诺夫函数方法，分别设计了切换系统鲁棒状态反馈控制器和输出反馈控制器，保证了切换系统的二次稳定性和渐近稳定性．仿真结果验证了所提出算法的有效性．     

Stability and static output feedback stabilization for a class of nonlinear discrete-time singular switched systems

In this paper, the stability and static output feedback stabilization problems for a class of nonlinear discrete-time singular switched systems are discussed. First, based on Lyapunov theory and the implicit function theorem, linear matrix inequalities sufficient conditions are developed which guarantee that the nonlinear discrete-time singular switched systems are regular, causal, have unique solution in a neighborhood of the origin, and are uniformly asymptotically stable. Then, with these conditions, and based on singular value decomposition approach, the design method of static output feedback controllers is given. Last, numerical examples are provided to illustrate the effectiveness of the proposed methods.     

基于反步法的一类非线性切换系统控制器设计

研究一类子系统是纯反馈非线性系统的切换系统控制器设计问题．基于多Lyapunov函数和反步法，提出一种状态反馈控制律及切换规则设计方法。通过递推设计可得到使非线性切换系统稳定的反馈控制律和切换规则．将所提出的方法应用于该类非线性切换系统含有不确定性的情形，仿真算例结果表明了所提出方法的有效性．     

A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control

In this paper we consider the asymptotic stability of a class of discrete-time switching linear systems, where each of the constituent subsystems is Schur stable. We first present an example to motivate our study, which illustrates that the bilinear transform does not preserve the stability of a class of switched linear systems. Consequently, continuous time stability results cannot be transformed to discrete time analogs using this transformation. We then present a subclass of discrete-time switching systems that arise frequently in practical applications. We prove that global attractivity for this subclass can be obtained without requiring the existence of a common quadratic Lyapunov function (CQLF). Using this result, we present a synthesis procedure to construct switching stabilizing controllers for an automotive control problem, which is related to the stabilization of a vehicle’s roll dynamics subject to switches in the center of gravitys (CG) height.     

Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings

This paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings. Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping. The first class uses the common state feedback controller which is independent of switching signals; the other class utilizes individual state feedback controllers for the subsystems. As an extension of the designed method, the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied. An example is given to show the effectiveness of the proposed method.     

Global finite-time control for a class of switched nonlinear systems with different powers via output feedback

This paper is concerned with the problem of global finite-time control for switched nonlinear systems whose nonlinear terms satisfy homogenous growth conditions. At first, we design homogenous output feedback controllers for nominal switched nonlinear systems by adding a power integrator method. Then, we employ the homogeneous domination approach to scale homogeneous observers and controllers to render switched nonlinear systems with lower-triangular homogenous growth condition globally finite-time stable. Finally, the proposed control method can be extended to switched nonlinear systems satisfying with upper-triangular homogeneous growth condition. Two examples are provided to demonstrate the effectiveness of the proposed control scheme.     

Robust Stability and Stabilization Criteria for Discrete Singular Time‐Delay LPV Systems

This paper is concerned with establishing robust stability and stabilization criteria for discrete singular time‐delay linear parameter varying (LPV) systems. Firstly, a robust stability criterion is obtained for this class of systems by a delay‐partition approach, and thereby a less conservative sufficient condition which guarantees discrete singular time‐delay LPV systems to be admissible is given. Secondly, a class of state feedback controllers for stabilizing discrete singular time‐delay LPV systems is designed. Finally, compared with existing results, the numerical results of several examples illustrate the effectiveness of the approach proposed in this paper. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stabilization of Discrete‐time Switched Systems with State Constraints Based on Mode‐Dependent Average Dwell Time

This paper considers the stabilization problem for a class of switched systems with state constraints based on mode‐dependent average dwell time (MDADT) in discrete‐time context. An improved average dwell time method is proposed, which is less conservative than the common average dwell time method. The sufficient conditions and stabilizing state feedback controllers for stabilization of discrete‐time switched systems with state constraints under MDADT switching are derived. Finally, the simulation results show that the approach designed by this paper is effective.     

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered.

In the deterministic problem, for the initial state in a certain set ∑₀, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path.

The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.     

一类不确定离散切换模糊时滞系统的鲁棒非脆弱控制

针对一类不确定离散切换模糊时滞系统,研究了在控制器增益存在摄动情况下的稳定性问题.利用切换技术和多Lyapunov函数方法,并以矩阵不等式形式给出了非脆弱状态反馈控制器存在的充分条件和切换律设计.最后通过数值仿真算例验证了设计方法的有效性和可行性.     

一类离散时间切换混杂系统鲁棒控制

由于切换规则的存在使得切换混杂控制系统的稳定性研究变得极为复杂,如何针对给定的系统设计适当的控制器和切换规则没有统一的方法.本文考虑一类线性不确定离散时间切换混杂系统的鲁棒二次镇定和渐近镇定问题.利用公共李雅普诺夫函数方法和多李雅普诺夫函数方法,分别设计了切换混杂系统鲁棒状态反馈控制器和鲁棒输出反馈控制器,保证了切换混杂系统的二次稳定性和渐近稳定性.仿真结果验证了所提算法的正确有效性.     

Model‐free adaptive formation control for unknown multiinput‐multioutput nonlinear heterogeneous discrete‐time multiagent systems with bounded disturbance

Based on the model‐free adaptive control, the distributed formation control problem is investigated for a class of unknown heterogeneous nonlinear discrete‐time multiagent systems with bounded disturbance. Two equivalent data models to the unknown multiagent systems are established through the dynamic linearization technique considering the circumstances with measurable and unmeasurable disturbances. Based on the obtained data models, two distributed controllers are designed with only using the input/output and disturbance data of the neighbor agents system. The tracking error of the closed‐loop system driven by the proposed controllers is shown to be bounded by the contraction mapping principle and inductive methods. An example illustrates the effectiveness of the proposed two distributed controllers.     

Stability analysis of hybrid switched nonlinear singular time-delay systems with stable and unstable subsystems

The issue of exponential stability of a class of continuous-time switched nonlinear singular systems consisting of a family of stable and unstable subsystems with time-varying delay is considered in this paper. Based on the free-weighting matrix approach, the average dwell-time approach and by constructing a Lyapunov-like Krasovskii functional, delay-dependent sufficient conditions are derived and formulated to check the exponential stability of such systems in terms of linear matrix inequalities (LMIs). By checking the corresponding LMI conditions, the average dwell-time and switching signal conditions are obtained. This paper also highlights the relationship between the average dwell-time of the switched nonlinear singular time-delay system, its stability and the exponential convergence rate of differential and algebraic states. A numerical example shows the effectiveness of the proposed method.     

非线性奇异系统的能控性子分布

研究非线性奇异控制系统的能控性子分布问题.提出了非线性奇异系统的能控性子分 布的概念;研究了非线性奇异系统能控性子分布的反馈不变性质;给出了非线性奇异系统能控 性子分布的算法,并讨论这个算法的一些性质;证明了在一定条件下,该算法给出的分布确为包 含在某给定分布中的非线性奇异系统的最大能控性子分布.     

Incremental passivity and output regulation for switched nonlinear systems

This paper studies incremental passivity and global output regulation for switched nonlinear systems, whose subsystems are not required to be incrementally passive. A concept of incremental passivity for switched systems is put forward. First, a switched system is rendered incrementally passive by the design of a state-dependent switching law. Second, the feedback incremental passification is achieved by the design of a state-dependent switching law and a set of state feedback controllers. Finally, we show that once the incremental passivity for switched nonlinear systems is assured, the output regulation problem is solved by the design of global nonlinear regulator controllers comprising two components: the steady-state control and the linear output feedback stabilising controllers, even though the problem for none of subsystems is solvable. Two examples are presented to illustrate the effectiveness of the proposed approach.     

Switched seesaw control for the stabilization of underactuated vehicles

This paper addresses the stabilization of a class of nonlinear systems in the presence of disturbances, using switching controllers. To this effect we introduce two new classes of switched systems and provide conditions under which they are input-to-state practically stable (ISpS). By exploiting these results, a methodology for control systems design—called switched seesaw control—is obtained that allows for the development of nonlinear control laws yielding input-to-state stability. The range of applicability and the efficacy of the methodology proposed are illustrated via two nontrivial design examples. Namely, stabilization of the extended nonholonomic double integrator (ENDI) and stabilization of an underactuated autonomous underwater vehicle (AUV) in the presence of input disturbances and measurement noise.