An optimal control approach to robust tracking of linear systems

In our early work, we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem. If the system is linear, then the optimal control problem becomes a linear quadratic regulator (LQR) problem, which can be solved by solving an algebraic Riccati equation. In this article, we extend the optimal control approach to robust tracking of linear systems. We assume that the control objective is not simply to drive the state to zero but rather to track a non-zero reference signal. We assume that the reference signal to be tracked is a polynomial function of time. We first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic Riccati equation. Because the state feedback is not always available in practice, we also investigated the output feedback. We show that if we place the poles of the observer sufficiently left of the imaginary axis, the robust tracking problem can be solved. As in the case of the state feedback, the observer and feedback can be obtained by solving two algebraic Riccati equations.     

线性不确定系统的给定阶最优控制器设计

为避免间接法设计降阶控制器的模型近似引起的性能下降,本文在静态输出反馈控制器设计的基础上,直接设计了线性不确定系统的给定阶混合H2/H∞动态反馈控制器.利用系统内外分解方法,得到了最优降阶状态观测器.通过求解降维状态观测器的静态输出反馈,可得到降阶控制的最优反馈增益阵.给定阶控制器由两个Ric cati方程和一个Lyapunov方程参数化表示.最后,通过一个例子,说明了本文提出的给定阶控制器设计方法.     

New results in state estimation and regulation

An alternative technique to design linear state estimators and regulators is presented. This technique is based on transfer matrix considerations. The optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices.This approach provides further insight, displays the duality of estimation and control nicely, and bridges the state-space and frequency-domain techniques. The resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form.     

基于微分对策的最优状态观测器和最优状态反馈控制器的设计

研究了线性系统基于二次型指标的最优状态观测器和最优状态反馈控制器的设计问题．将观测状态的状态反馈和状态误差的输出反馈分别作为两个对局方,应用微分对策理论研究了系统的最优控制问题．给出了最优状态观测器和基于状态观测器的最优状态反馈控制的存在性条件．将系统的最优状态观测器和最优控制器的设计问题转化为一对Riccati方程的求解问题．研究表明最优状态观测器在一般情况下不存在．并进一步研究了基于状态观测器的次优控制问题,给出了基于LMI的优化算法．     

奇异线性系统的最优调节器问题

考虑奇异线性系统的最优调节器问题.所得调节器由线性反馈增益和积分器组成,反馈 增益矩阵由解一个稀疏的矩阵Riccati方程而算出.闭环稳定性也予以讨论.     

基于数据的网络化系统最优控制

针对网络介入导致的系统动态复杂性增加及建模困难等问题,提出了基于数据驱动控制的思想,独立于系统模型设计控制器。利用在线获取的系统输入/输出当前数据和历史数据,分别构造系统的输入数据矩阵和输出数据矩阵,建立两者之间的线性关系,获得当前采样时刻系统的Markov参数,将该参数代入到Markov参数形式表示的Riccati方程解中,获取最优控制增益。同时利用输入与输出之间的线性关系,构造一个控制器状态观测器,利用估计的控制器状态参与计算最优控制律。最后,在Truetime1.5和Matlab仿真平台上验证了提出的控制器是有效的。     

Numerical solution of the discrete-time periodic Riccati equation

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with     

Optimization of formation for multi-agent systems based on LQR

In this paper, three optimal linear formation control algorithms are proposed for first-order linear multiagent systems from a linear quadratic regulator (LQR) perspective with cost functions consisting of both interaction energy cost and individual energy cost, because both the collective object (such as formation or consensus) and the individual goal of each agent are very important for the overall system. First, we propose the optimal formation algorithm for first-order multi-agent systems without initial physical couplings. The optimal control parameter matrix of the algorithm is the solution to an algebraic Riccati equation (ARE). It is shown that the matrix is the sum of a Laplacian matrix and a positive definite diagonal matrix. Next, for physically interconnected multi-agent systems, the optimal formation algorithm is presented, and the corresponding parameter matrix is given from the solution to a group of quadratic equations with one unknown. Finally, if the communication topology between agents is fixed, the local feedback gain is obtained from the solution to a quadratic equation with one unknown. The equation is derived from the derivative of the cost function with respect to the local feedback gain. Numerical examples are provided to validate the effectiveness of the proposed approaches and to illustrate the geometrical performances of multi-agent systems.     

具有指定闭环特征值的离散时间最优调节器的设计

本文研究了LQ最优调节器的逆问题。在控制变量加权矩阵R给定的条件下,通过引入一组自由变量,给出了满足闭环系统特征值要求的状态加权矩阵Q的一种参数化表示结果。基于这一结果,文中研究了一类系统的LQ最优调节器之逆问题的解析解法。通过所求得的自由变量解,就可以直接确定系统的最优状态反馈控制器。     

LMI optimization for nonstandard Riccati equations arising instochastic control

We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations that arise, e.g., in the optimal control of linear systems subject to state-dependent multiplicative noise. Some nonstandard Riccati equations (such as those connected to linear systems subject to both state- and control-dependent multiplicative noise) are not amenable to the method. We show that we can still use LMI optimization to compute the optimal control law for the underlying control problem without solving the Riccati equation     

线性系统的动态输出反馈最优扰动抑制

首先通过求解Riccati方程和Sylvester方程,推导出前馈反馈最优扰动抑制控制律．然后构建了能同时预估状态和扰动的降维观测器,解决了前馈控制的物理不可实现问题．进而结合降维观测器和前馈反馈最优控制律,提出了一种动态输出反馈扰动抑制控制器的设计算法．最后通过仿真实例表明本文提出的控制算法的可行性．     

Cooperative output regulation of heterogeneous multi-agent systems based on passivity

This paper investigates the problem of cooperative output regulation of heterogeneous linear multi-agent systems. A passive framework is presented for the stabilisation analysis of cooperative output regulation, which can overcome the difficulty caused by the fact that the global dynamics of heterogeneous multi-agent systems depends on the global communication structure. An adaptive distributed observer is proposed to estimate the state of the exosystem, and the proposed distributed observer is independent of any global information of the communication graph. Based on passivity design and adaptive distributed observer, both a distributed state feedback and a distributed output feedback protocol are designed for output synchronisation of heterogeneous multi-agent systems. The gain matrices of the distributed protocols and observers are obtained by a Riccati equation design approach. Furthermore, sufficient local conditions for solving the problem of cooperative output regulation of heterogeneous multi-agent systems are presented. Finally, numerical simulation results are given to illustrate the effectiveness of the proposed distributed control schemes.     

Observer-based controller for robust pole clustering in a vertical strip and disturbance rejection in structured uncertain systems

This paper presents an observer-based multi-objective robust feedback controller to achieve robust pole clustering within a vertical strip and disturbance rejection with an H_∞-norm constraint for the uncertain linear systems. The systems of interest include both matched and mismatched uncertain linear systems with structured uncertainties existing in both the system and input matrices. The controller is obtained by solving two Riccati equations (one for the controller and the other for the observer) and checking three conditions (two for robust pole clustering). A set of tuning parameters is incorporated to enhance flexibility in finding the controller. © 1998 John Wiley & Sons, Ltd.     

A convergent algorithm for computing stabilizing static output feedback gains

We revisit the approach by Cao et al. that uses a fixed-structure control law to find stabilizing static output feedback gains for linear time-invariant systems. By performing singular value decomposition on the output matrix, together with similarity transformations, we present a new stabilization method. Unlike their results that involve a difficult modified Riccati equation whose solution is coupled with other two intermediate matrices that are difficult to find, we obtain Lyapunov equations. We present a convergent algorithm to solve the new design equations for the gains. We will show that our new approach, like theirs, is a dual optimal output feedback linear quadratic regulator theory. Numerical examples are given to illustrate the effectiveness of the algorithm and validate the new method.     

Robust observer design for nonlinear uncertain systems with LQ regulators

For a class of uncertain systems with linear nominal dynamics and nonlinear uncertainties, it has been shown (Katayama and Sasaki 1987) that linear quadratic (LQ) state feedback regulators can be used to provide robust asymptotic stability. In this paper, we study the combined observer-controller design problem, based on the linear state feedback regulator proposed by Katayama and Sasaki (1987), so that only output feedback is needed. Both full-order and reduced-order observers are considered. For the full-order observer, we propose an algorithm to synthesize the robust observer gain matrix. It is shown that with the observer it is still possible to achieve robust asymptotic stability. For the reduced-order observer, some conditions are derived to guarantee the robust asymptotic stabilizability of the uncertain systems. The trade-off between the magnitudes of controller and observer gains is clear in our approach. An example is used to illustrate the design process of the robust controller with full-order as well as reduced-order observers.     

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 observers for systems with colored noises

The problem of optimal full-order observers for continuous-time linear systems with colored process and measurement noises is considered. In such cases, optimal estimation of the state involves augmenting the system, thus a higher-order observer is required. The structure of a full-order observer is assumed and necessary conditions for the optimal observer are derived. The conditions are given for the general case where the intensity of the white-noise component of the measurement noise may be singular. The solution consists of a modified Riccati equation and a Lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case     

线性离散不确定系统区域极点配置的状态反馈实现方法

研究利用状态反馈对线性离散不确定系统进行区域极点配置的问题。导出了一个由离散代数Ｒｉｃａｔｉ方程表示的对线性离散不确定系统进行区域极点配置的充分条件,并给出了一个通过求解该方程来确定状态反馈增益矩阵的算法。     

船舶航向非线性H∞逆优化输出反馈控制

为抑制船舶航向非线性优化控制中模型参数摄动和由状态观测器引入的不确定观测误差,提出了一种非线性H∞逆优化控制算法.首先,基于无源理论设计观测器以实现海浪滤波,该观测器无需海浪扰动的方差信息从而减少了观测器参数数量.然后,考虑模型参数摄动对观测误差的影响,给出了描述系统局部(全局)性态的局部(全局)H∞优化性能指标.在以广义黎卡提方程(GARE)对局部优化问题的求解的基础上,应用逆优化方法将全局H∞优化问题转化为构造闭环系统的Lyapunov函数问题,得到同时满足两种指标的优化控制器,并证明了稳定性.仿真结果证明了该算法的有效性.     

基于统计线性化的随机非线性微分对策逼近最优策略

针对二人零和随机非线性微分对策问题, 利用统计线性化技术并提出一种新的逼近策略控制方法. 通过求解具有统计线性化参数的Riccati微分方程得到逼近控制策略, 该Riccati微分方程与一般线性系统的Riccati微分方程具有明显的区别; 同时对控制量受约束情形的控制策略也进行了求解; 最后通过仿真验证了所得结论的正确性.