Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions

The determination of a stability domain of a control system, the motion of which is described by nonlinear differential equations, is often the object of intensive experimental and theoretical attack. This paper, partly tutorial and partly a presentation of new results, describes a method for obtaining a solution to this problem proposed recently by the Russian mathematician, V. I. Zubov. The tutorial part outlines the fundamental principles of V. I. Zubov's procedure for constructing Lyapunov functions for non-linear systems. If the construction problem can be solved, it leads to a Lyapunov function which uniquely defines the exact boundary of the stability region. For the application of the method, several simple examples are treated in which the exact stability region is found in analytic closed form. Since the construction procedure requires the solution of a linear partial differential equation, there are many cases for which an exact analytic solution is not possible. In some of these cases, however, it is possible to construct an approximate series solution which is always at least as good as the usual quadratic form Lyapunov function. The series construction procedure has been programmed (in IBM 7070 FORTRAN language) for a broad class of differential equations of the second order. A simple example solved by the digital computer program is described.     

Application of generalized Zubov's method to power system stability†

The generalized form of Zubov's partial differential equation (Szego 1962) is written using the dynamical equations of a power system. Then a Lyapunov function which satisfies the partial differential equation is obtained by using a transformation of state variables. The V function obtained is used to describe the region of stability. The application of the method to the power system stability problem is illustrated by considering a synchronous generator connected to an infinite bus, Three examples, using three different models for the synchronous machine, are given.     

具分布参数的随机Hopfield神经网络的指数稳定

基于随机Fubini定理,将随机偏微分方程描述的Hopfield神经网络系统转化为用相应的随机常微分方程来描述.利用关于空间变量平均的Lyapunov函数与Ito^公式,通过对所构造的Lyapunov函数在Ito^微分规则下对相应系统求导的方法,获得了系统指数稳定的代数判据及其Lyapunov指数估计.实现了运用Lyapunov直接法对分布参数系统稳定性的研究.     

动态区间系统的鲁棒稳定性

研究了动态连续间系统和离散区间系统的鲁棒稳定性问题,在给出了区间系统的一种等价描述之后,利用Lyapunov方法和RIccati方程方法,分别得到了连续间我系统和离散区间系统鲁棒稳定的充分条件,最后的数值例子说明本文的结果不仅保守性小,而且计算简单。     

Robust state observer and control design using command-to-state mapping

In this paper, by introducing the concept of command-to-state/output mapping, it is shown that the state of an uncertain nonlinear system can robustly be estimated if command-to-state mapping of the system and that of an uncertainty-free observer converge to each other. Then, a global Jacobian system is defined to capture this convergence property for the dynamics of estimation error, and a set of general stability and convergence conditions are derived using Lyapunov direct method. It is also shown that the conditions are constructive and can be reduced to an algebraic Lyapunov matrix equation by which nonlinear feedback in the observer and its corresponding Lyapunov function can be searched in a way parallel to those of nonlinear control design. Case studies and examples are used to illustrate the proposed observer design method. Finally, observer-based control is designed for systems whose uncertainties are generated by unknown exogenous dynamics.     

Stabilization of synchronous generators with the Hamiltonian function approach

Using the Hamiltonian function approach, this paper proposes an energy-based stabilizing method for a fifth-order model of synchronous generators to keep the terminal machine voltage (output) remaining at a given expected value. By constructing a Hamiltonian function as the total energy function for the fifth-order model and changing the system into a forced Hamiltonian system with dissipation, an energy-based Lyapunov function is obtained. As the result, a suitable stabilizing controller is constructed for the system. Simulation shows the effectiveness of the stabilizing method proposed in this paper.     

时滞切换系统指数稳定性分析: 微分不等式方法

考虑由多个时滞系统组成的切换系统,并研究在什么条件下,可以把无时滞切换系统的稳定性分析及结论推广至上述的时滞系统.方法是将时滞项作为线性常微分方程扰动项,利用常数变易公式与Halanay微分不等式,分析时滞项对于切换系统稳定性的影响.结论表明,在时滞项满足某些前提时,切换系统稳定性分析的Lyapunov方法仍然适用.仿真算例验证了方法的有效性.     

Stabilisation of time-varying linear systems via Lyapunov differential equations

This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.     

Analysis and control of the jump modes behavior of 2-D singular systems—Part I: Structural stability

This paper considers the problem of structural stability of 2-D singular systems. Firstly, some properties of structural stability of 2-D general singular systems are presented. Sufficient and necessary conditions for the structural stability of the 2-D singular systems are given. Then, by extending the Lyapunov approach for the structural stability of 1-D continuous singular systems to the discrete case, a generalized Lyapunov equation approach to the analysis of the structural stability of 2-D singular Roesser models (2-D SRM) is proposed. The existence of a solution to the generalized Lyapunov equation gives a sufficient condition for the structural stability of the 2-D SRM.     

Sequential circle criterion approach for robustly stabilizing individual generators with SVC

Absolute stability with the spatially defined linear time‐invariant (LTI) state‐space modelings is scrutinized by means of what we call the sequential Lyapunov approach, which possesses independent significance in stabilization when gain‐scheduling control laws are adopted. Then, this theoretical result is exploiting for stabilization of individual generators via SVC actions. More precisely, by remodeling the perturbed swing equations of synchronous generators in multimachine networks through spatially defined LTI state‐space expressions subjected to uncertainties and power disturbance, which are viewed as sector nonlinearities, we introduce frequency responses for coping with nonlinear power swing dynamics of individual generators. By sequentially relating the frequency responses to the circle criterion (substantially, the KYP theorem or the positive real lemma) claimed for LTI systems subject to sector disturbances, output feedback control laws for static VAR compensators are worked out to stabilize individual generators. The frequency‐domain approach is also useful in steady‐state specification besides stabilization in individual generators. Examples show efficacy of the suggested stabilization and steady‐state specification technique. Copyright © 2014 John Wiley & Sons, Ltd.     

A novel stability analysis of linear systems under asynchronous samplings

This article proposes a novel approach to assess the stability of continuous linear systems with sampled-data inputs. The method, which is based on the discrete-time Lyapunov theorem, provides easy tractable stability conditions for the continuous-time model. Sufficient conditions for asymptotic and exponential stability are provided dealing with synchronous and asynchronous samplings and uncertain systems. An additional stability analysis is provided for the cases of multiple sampling periods and packet losses. Several examples show the efficiency of the method.     

Quadratic-type Lyapunov functions for singularly perturbed systems

Asymptotic and exponential stability of nonlinear singularly perturbed systems are investigated via Lyapunov stability techniques. A quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratic-type Lyapunov functions of two lower order systems. Estimates of domain of attraction, of upper bound on perturbation parameter, and of degree of exponential stability are obtained. The method is illustrated by studying the stability of a synchronous generator connected to an infinite bus.     

具有端点质量的一维波动方程半离散格式的一致指数稳定性

无穷维系统主要由偏微分方程描述, 可是大部分用偏微分方程描述的控制系统, 无论是单纯的数值实验还是需要应用到实际的问题中去, 都需要对方程进行有限数值离散. 本文考虑了端点带有质量的波动方程在边界反馈控制下半离散格式的一致指数稳定性. 首先, 原闭环系统通过降阶法变成低阶的等价系统, 通过一种间接Lyapunov函数方法证明了降阶等价的连续系统是一致指数稳定的. 其次, 对等价系统空间变量离散得到半离散的差分格式.平行于连续系统, 间接Lyapunov函数方法证明了半离散系统的一致指数稳定性. 数值实验证明了基于降阶法的一致指数稳定性和经典半离散格式的非一致指数稳定性.     

Power-shaping control of reaction systems: The CSTR case

Power-shaping control is a recent approach for the control of nonlinear systems based on the physics of the dynamical system. It rests on the formulation of the dynamics in the Brayton-Moser form. One of the main obstacles for using the power-shaping approach is to write the dynamics in the required form, since a partial differential equation system submitted to sign constraints has to be solved. This work comes within the framework of control design approaches that could possibly generate a closer link between the notions of energy that are specific to reaction systems as derived from thermodynamics concepts, and the dynamic system stability theory. The objective of this paper is to address the design of power-shaping control to reaction systems, and more particularly the step of solving the partial differential equation system. In order to illustrate the approach, we have selected the classical yet complex continuous stirred tank reactor (CSTR) as a case study. We show how using the power-shaping approach leads to a global Lyapunov function for the unforced exothermic CSTR. This Lyapunov function is then reshaped by means of a controller in order to stabilize the process at a desired temperature.     

An approach to stability criteria of neural-network control systems

This paper discusses stability of neural network (NN)-based control systems using Lyapunov approach. First, it is pointed out that the dynamics of NN systems can be represented by a class of nonlinear systems treated as linear differential inclusions (LDI). Next, stability conditions for the class of nonlinear systems are derived and applied to the stability analysis of single NN systems and feedback NN control systems. Furthermore, a method of parameter region (PR) representation, which graphically shows the location of parameters of nonlinear systems, is proposed by introducing new concepts of vertex point and minimum representation. From these concepts, an important theorem, which is useful for effectively finding a Lyapunov function, is derived. Stability criteria of single NN systems are illustrated in terms of PR representation. Finally, stability of feedback NN control systems, which consist of a plant represented by an NN and an NN controller, is analyzed.     

On application of the implemented semigroup to a problem arising in optimal control

In this paper we show that the concept of an implemented semigroup provides a natural mathematical framework for analysis of the infinite-dimensional differential Lyapunov equation. Lyapunov equations of this form arise in various system-theoretic and control problems with a finite time horizon, infinite-dimensional state space and unbounded operators in the mathematical model of the system. The implemented semigroup approach allows us to derive a necessary and sufficient condition for the differential Lyapunov equation with an unbounded forcing term to admit a bounded solution in a suitable space. Whilst our focus is on the differential Lyapunov equation, we show that the same framework is also appropriate for the algebraic version of this equation. As an application we show that the approach can be used to solve a simple decoupling problem arising in optimal control. The problem of infinite time admissibility of the control operator and an infinite-dimensional version of the Lyapunov theorem serve as additional illustrations.     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

通信带宽有限的重置控制系统

针对经由具有通信带宽限制的网络通道构成反馈的重置控制系统,讨论了时变网络化控制系统的建模与镇定问题,利用Lyapunov方法获得了这类系统的指数稳定性判据,并将该结论推广到有界摄动条件下的鲁棒稳定性问题．随后又利用Riccati矩阵微分方程方法,得到了一种更加实用的指数稳定性条件,并证明该结论对于渐进跟踪问题和扰动抑制问题仍然成立．     

网络环境下固定拓扑的多电机领航跟随控制

基于多智能体一致性理论,从领航跟随控制的角度解决网络环境下多电机的同步控制问题,其中考虑了网络时延对同步控制的影响。每个电机均被视为一个智能体,且每个智能体能且只能得到其邻域的输出测量信息,在此条件下,研究了在有向固定网络拓扑情况下多电机系统的同步控制问题。为了解决网络时延对电机同步控制的影响,提出了一种带有分布式观测器的时延一致性控制协议,应用 Lyapunov稳定性理论证明了若单个电机是能观能控的,只要网络连接满足简单的拓扑结构,则多电机系统能够达到领航跟随一致性。最后通过仿真实验验证了该方法的正确性和有效性。     

Observers design for linear time-delay systems

In this contribution we propose a simple and useful approach to design observers for discrete-time systems with delays in the state and output variables. The main feature is that the necessary and sufficient conditions for the existence of such observer are derived. The stability analysis is performed by the Lyapunov approach, where the obtained conditions are expressed in terms of a modified Riccati equation. Numerical examples are provided to show efficiency of the proposed observer.