脉宽调制控制系统的适定性

不连续动态系统的适定性(即解的唯一存在性)问题是混合系统研究的基本问题之一.脉 宽调制控制系统可以表示成具有时变仿射不等式约束的分段仿射系统.该文首先在Carathéodory 的解定义下导出了脉宽调制控制系统无滑模和跳跃的解的适定性的充分必要条件.然后利用该 结果对PWM型DC-DC降压变换器的适定性的研究表明当载波信号h(t)=O时闭环PWM DC-DC 变换器总是不适定的,当载波信号h(t)的参数和比例控制器的参数满足一定的条件时，比例 控制的闭环PWM DC-DC变换器是适定的.     

Characterization of well-posedness of piecewise-linear systems

One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multimodal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed     

On solution concepts and well-posedness of linear relay systems

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two different solution concepts, Filippov solutions and forward solutions. We derive necessary and sufficient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. To be precise, uniqueness of Filippov (and also forward) solutions follows in this case if the first non-zero Markov parameter is positive. By means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two. The influence of the Zeno phenomenon (an infinite number of relay switching times in a finite length time interval) on well-posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems, the consequences of the presence of the Zeno behaviour is typical for many other classes of non-smooth and hybrid systems.     

Time delay systems with distribution dependent dynamics

General delay dynamical systems in which uncertainty is present in the form of probability measure dependent dynamics are considered. Several motivating examples arising in biology are discussed. A functional analytic framework for investigating well-posedness (existence, uniqueness and continuous dependence of solutions), inverse problems, sensitivity analysis and approximations of the measures for computational purposes is surveyed.     

Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations

Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.     

Well-posedness analysis of switch-driven piecewise affine systems

The well-posedness problem (existence and uniqueness of solutions) of a class of multi-modal piecewise affine systems is addressed, where binary-switches individually act under autonomous switching. First, a new transition rule on the discrete state, called the switch-based transition rule, is introduced and some relations with the mode-based transition rule are discussed. Next, a sufficient condition for such a multi-modal system to be well-posed for all external inputs is derived in terms of well-posedness of its subsystems of lower complexity "bimodal systems". Finally, an easily checkable condition for the bimodal system to be well-posed for all external inputs is given, which consequently allows us to algebraically determine well-posedness of the multi-modal systems in question.     

Analysis of the Caratheodory's Theorem on Dynamical System Trajectories Under Numerical Uncertainty

The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discrepancy between mathematical proofs, algorithms, and their implementations, which may affect the correct functioning of a control system. Due to growing demands on security and compliance with specifications, correctness of the control system functioning is becoming ever more important. Since in both dynamical systems and many control design approaches, one of the central notions is the system trajectory, it is important to address existence and uniqueness of system trajectories in a way which incorporates numerical uncertainty. Constructive analysis is a particular approach to formalizing numerical uncertainty and is used as the basis of the current work. The major difficulties of guaranteeing existence and uniqueness of system trajectories arise in the case of systems and controllers which possess discontinuities in time, since classical solutions to initial value problems do not exist. This issue is addressed in Caratheodory's theorem. A particular constructive variant of the theorem is proven which covers a large class of problems found in practice.      

多模式网络化控制系统稳定性分析

网络传输引起的时滞在网络化控制系统中经常出现．针对存在多个运动模式的网络化控制系统,建立了时滞混杂自动机模型,分析了系统平衡点的稳定性,给出了平衡点保持稳定的充分条件在于构造的与离散状态相关的Lyapunov泛函在离散状态的切换时刻是非增的．当时滞混杂自动机中连续性子系统为线性时不变时滞系统时,可以利用线性矩阵不等式方法来寻找系统的公共Lyapunov泛函,并求解使系统平衡点保持稳定的时滞上界．最后,给出了一个例子说明了分析结果．     

The complementary-slackness class of hybrid systems

In this paper we understand a “hybrid system” to be one that combines features of continuous dynamical systems with characteristics of finite automata. We study a special class of such systems which we call the complementary-slackness class. We study existence and uniqueness of solutions in the special'cases oflinear andHamiltonian complementary-slackness systems. For the latter class we also prove an energy inequality.     

Robust stability of reset control systems with uncertain output matrix

The main difficulty in stability analysis of reset control systems comes from their state-dependent reset mechanism. It becomes more challenging when output matrix suffers from uncertainties which make the reset time instants uncertain. In this paper, existing quadratic stability results for uncertainty-free reset systems are extended to the case with uncertainty in the output matrix. By constructing special parameter-dependent full rank annihilators of the output matrix, it is proved that the dissipativeness of reset action is equivalent to the feasibility of an LMI plus some structural constraints on the Lyapunov matrix. For the time-varying uncertainty case, necessary and sufficient conditions for quadratic stability are obtained. For the constant uncertainty case, a sufficient condition for affine quadratic stability is also obtained. All the results are given in terms of LMIs which can be efficiently verified.     

Stability analysis and design of reset systems: Theory and an application

This paper investigates stability analysis and design of reset control systems where the reset time instants are pre-specified. Firstly, in order to quantify stability of reset control systems, we propose an approach which does not rely on the stability of base linear systems. Based on this, some Lie-algebraic conditions for stability of reset control systems are obtained. Then, we develop a method for reset law design which aims at improving transient responses of the base linear systems. In addition, observer-based design is also considered. We prove that under some mild conditions, reset controller and observer can be designed separately. The obtained results are applied to short-span-seeking control of a kind of single stage hard disk drive servo systems. Simulations show that the proposed design is much more capable of improving transient response than traditional control design techniques.     

Stability of simultaneously block triangularisable switched systems with partial state reset

We study the stability of a certain class of switched systems where discontinuous jumps (resets) on some of the state components are allowed, at the switching instants. It is known that, if all components of the state are available for reset, the system can be stabilisable by an adequate choice of resets. However, this question may have negative answer if there are forbidden state components for reset. We give a sufficient condition for the stabilisability of a switched system, under arbitrary switching, by partial state reset in terms of a block simultaneous triangularisability condition. Based on this sufficient condition, we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset. We also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems.     

On controllability and observability for a class of impulsive systems

Many dynamic systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. These complex dynamic behaviors can be modeled by impulsive differential systems. This paper studies the controllability and observability for a class of time-varying impulsive control systems. Several sufficient and necessary conditions for state controllability and state observability of such systems are established and the corresponding criteria for time-invariant impulsive control systems are also obtained. Meanwhile, several new results associated with variation of parameters for time-varying impulsive control systems are derived.     

Event‐triggered adaptive backstepping control for parametric strict‐feedback nonlinear systems

This paper proposes a novel adaptive backstepping control method for parametric strict‐feedback nonlinear systems with event‐sampled state and input vectors via impulsive dynamical systems tools. In the design procedure, both the parameter estimator and the controller are aperiodically updated only at the event‐sampled instants. An adaptive event sampling condition is designed to determine the event sampling instants. A positive lower bound on the minimal intersample time is provided to avoid Zeno behavior. The closed‐loop stability of the adaptive event‐triggered control system is rigorously proved via Lyapunov analysis for both the continuous and jump dynamics. Compared with the periodic updates in the traditional adaptive backstepping design, the proposed method can reduce the computation and the transmission cost. The effectiveness of the proposed method is illustrated using 2 simulation examples.     

Dynamical properties of hybrid automata

Hybrid automata provide a language for modeling and analyzing digital and analogue computations in real-time systems. Hybrid automata are studied here from a dynamical systems perspective. Necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized. The results on existence, uniqueness, and continuity serve as a starting point for stability analysis. Lyapunov's theorem on stability via linearization and LaSalle's invariance principle are generalized to hybrid automata.     

Linearization of discrete-time systems by exogenous dynamic feedback

It is shown that a discrete-time system may be linearizable by exogenous dynamic feedback, even if it cannot be linearized by endogenous feedback. This property is completely unexpected and constitutes a fundamental difference with respect to the continuous-time case. The notion of exogenous linearizing output is introduced. It is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability. Necessary and sufficient conditions for the existence of an exogenous linearizing output are provided. The results of the paper are obtained using transformal operator matrices. The properties of such operators are studied. The theory is applied to the exact discrete-time model of a mobile robot, showing that the above-mentioned property concerns not only academic examples, but also physical systems.     

Positivity and stability of mixed fractional‐order systems with unbounded delays: Necessary and sufficient conditions

This article provides a comprehensive study on quantitative properties of linear mixed fractional‐order systems with multiple time‐varying delays. The delays can be bounded or unbounded. We first obtain a result on existence and uniqueness of solutions to these systems. Then, we prove a necessary and sufficient condition for their positivity. Finally, we provide a necessary and sufficient criterion to characterize asymptotic stability of positive linear mixed fractional‐order systems with multiple time‐varying delays.     

Lyapunov and converse Lyapunov theorems for stochastic semistability

This paper develops Lyapunov and converse Lyapunov theorems for stochastic semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a stochastic dynamical system almost surely converge to (not necessarily isolated) Lyapunov stable in probability equilibrium points determined by the system initial conditions. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose infinitesimal generator decreases along the dynamical system trajectories and is such that the Lyapunov function satisfies inequalities involving the average distance to the set of equilibria.