仿射非线性系统的一种观测器设计方法

对仿射非线性系统全维观测器设计方法进行了讨论，提出了一种设计方法．该方法基于微分Riccati方程的正定解，观测器的增益矩阵通过Riccati方程的正定解给出，因而观测器增益矩阵为时变的．通过坐标变换，用李亚普诺夫方法对观测器的存在性进行了论证．对一个实际模型的仿真分析表明了该方法的实用性和正确性．     

The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise

The optimal projection equations obtained in [2,3] for reduced-order, discrete-time state estimation are generalized to include the effects of state- and measurement-dependent noise to provide a model of parameter uncertainty. In contrast to the single matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order discrete-time estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by both an oblique projection and stochastic effects.     

The set of positive semidefinite solutions of the algebraic Riccatiequation of discrete-time optimal control

In this paper the algebraic Riccati equation (ARE) of the discrete-time linear-quadratic (LQ) optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability. With respect to an appropriate basis, the discrete-time algebraic Riccati equation (DARE) decomposes into a Lyapunov equation and an irreducible Riccati equation. The focus is on the Riccati part which amounts to studying a DARE where all unimodular modes are controllable. A bijection between positive semidefinite solutions and certain well-defined sets of F-invariant subspaces is established which, together with its inverse, is order reversing. As an application, issues concerning positive definite or strong solutions are clarified. Analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace     

摄动离散LYAPUNOV方程解的上下界估计

给出了离散Lyapunov方程在参数摄动下正定解上、下界的一种新估计,它允许的 摄动结构更为一般,且只需解两个代数Riccati方程,从而避免了高阶代数方程的求解.从而 为基于李雅普诺夫方程的系统分析及控制问题提供必要的理论分析基础.数值算例说明了本 文结果的优越性.     

Continuation methods for solving modified discrete-time algebraicRiccati equations

It is well known that the continuation methods have been successfully applied to solve polynomial systems and fixed point problems, etc. In this paper, we consider a discrete-time algebraic Riccati equation with an admissible, low rank, and symmetric perturbation. Our attention is directed primarily to this modified discrete-time algebraic Riccati equation and the numerical method for its solution based on proceeding along the continuation path     

Robust consensus tracking of linear multiagent systems with input saturation and input‐additive uncertainties

The problem of robust global consensus tracking of linear multiagent systems with input saturation and input‐additive uncertainties is investigated in this paper. By using the parametric Lyapunov equation approach and an existing dynamic gain scheduling technique, a new distributed nonlinear‐gain scheduling consensus‐trackining algorithm is developed to solve this problem. Under the assumption that each agent is asymptotically null controllable with bounded control, it is shown that the robust global consensus tracking can be achieved under the undirected graph provided that its generated graph contains a directed spanning tree. Compared with the existing algebraic Riccati equation approach, which requires the online solution of a parameterized algebraic Riccati equation, all the parameters in the proposed nonlinear algorithm are offline determined a priori. Finally, numerical examples are provided to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Control of slowly-varying linear systems

State feedback control of slowly varying linear continuous-time and discrete-time systems with bounded coefficient matrices is studied in terms of the frozen-time approach. This study centers on pointwise stabilizable systems. These are systems for which there exists a state feedback gain matrix placing the frozen-time closed-loop eigenvalues to the left of a line Re s=-γ<0 in the complex plane (or within a disk of radius ρ<1 in the discrete-time case). It is shown that if the entries of a pointwise stabilizing feedback gain matrix ar continuously differentiable functions of the entries of the system coefficient matrices, then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small. It is also shown that for pointwise stabilizable systems with a sufficiently slow rate of time variation in the system coefficients, a stabilizing feedback gain matrix can be computed from the positive definite solution of a frozen-time algebraic Riccati equation     

Solving the Matrix Differential Riccati Equation: A Lyapunov Equation Approach

In this technical note, we investigate a solution of the matrix differential Riccati equation that plays an important role in the linear quadratic optimal control problem. Unlike many methods in the literature, the approach that we propose employs the negative definite anti-stabilizing solution of the matrix algebraic Riccati equation and the solution of the matrix differential Lyapunov equation. An illustrative numerical example is provided to show the efficiency of our approach.      

Reduced-Order Observer-Based Leader-Following Formation Control for Discrete-Time Linear Multi-Agent Systems

Formation control of discrete-time linear multi-agent systems using directed switching topology is considered in this work via a reduced-order observer, in which a formation control protocol is proposed under the assumption that each directed communication topology has a directed spanning tree. By utilizing the relative outputs of neighboring agents, a reduced-order observer is designed for each following agent. A multi-step control algorithm is established based on the Lyapunov method and the modified discrete-time algebraic Riccati equation. A sufficient condition is given to ensure that the discrete-time linear multi-agent system can achieve the expected leader-following formation. Finally, numerical examples are provided so as to demonstrate the effectiveness of the obtained results.      

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

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.     

Guaranteed cost stabilization for a class of uncertain discrete-time systems

A technique to design stabilizing state feedback controllers is presented for a class of linear discrete-time systems subject to possibly time-varying uncertainties that affect the system and input matrices. The controller design procedure is based on the solution of one of three modified algebraic Riccati matrix equations. Stability considerations rely on the second method of Lyapunov. To evaluate the system's performance degradation due to uncertainly, an upper bound (guaranteed cost) is derived for the cost of the controlled uncertain system with respect to a designer-chosen quadratic cost function. An example adapted from the literature is used to illustrate the method.     

Exponential synchronization of a class of neural networks with time-varying delays.

This paper aims to present a synchronization scheme for a class of delayed neural networks, which covers the Hopfield neural networks and cellular neural networks with time-varying delays. A feedback control gain matrix is derived to achieve the exponential synchronization of the drive-response structure of neural networks by using the Lyapunov stability theory, and its exponential synchronization condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. This condition can avoid solving an algebraic Riccati equation. Both the cellular neural networks and Hopfield neural networks with time-varying delays are given as examples for illustration.     

Stability margin evaluation for uncertain linear systems

A sufficient stability condition for parameter variation in a perturbed, continuous time, multivariable linear system, represented by a state space model is presented. Starting with the existence of an algebraic Riccati equation, a stability bound is derived from the polar decomposition of the nominal system matrix. Unlike previous work, the results are not dependent on the solution of the Lyapunov equation and, consequently, not a function of an arbitrarily selected positive definite matrix. In addition, the bound would appear to be the tightest possible, in that violation of the presented inequality can be shown to lead to instability     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Solvability conditions and design for synchronization of discrete‐time multiagent systems

This paper provides solvability conditions for state synchronization with homogeneous discrete‐time multiagent systems with a directed and weighted communication network under partial‐ or full‐state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H₂ discrete‐time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H₂ discrete‐time algebraic Riccati equation.     

LQ逆问题解的一种有效算法

本文研究了LQ最优控制逆问题解的参数化表示结果和基于这一参数化表示结果的矩阵变换解法。研究的对象是线性时不变离散时间系统。此外,文中还给出了不求解代数矩阵Riccati方程确定系统的最优状态反馈系数矩阵K的方法。     

不确定离散时间系统鲁棒稳定控制

根据Lyapunov稳定性定理,针对不确定离散时间系统鲁棒稳定状态反馈控制问题,提出 一种方法,即通过不确定矩阵秩1分解,计算加权矩阵,沿用线性最优调节器问题的Riccati代 数方程,设计鲁棒稳定调节器,并讨论了控制矩阵的不确定程度与鲁棒稳定调节器的存在性关 系问题.     

Optimal reduced-order state estimation for unstable plants

The problem of optimal reduced-order steady-state state estimation is considered for the case in which the plant has unstable poles. In contrast to the standard full-order estimation problem involving a single algebraic Riccati equation, the solution to the reduced-order problem involves one modified Riccati equation and one Lyapunov equation coupled by a projection matrix. This projection is completely distinct from the projection obtained by Bernstein and Hyland (1985) for stable plants.