A parameter-adaptive control technique

Control of linear stochastic systems with unknown parameters is accomplished by means of an approximate solution of the associated functional equation of dynamic programming. The approximation is based on repeated linearizations of a quadratic weighting matrix appearing in the optimal cost function for the control process. This procedure leads to an adaptive control system which is linear in an expanded vector of state estimates with feedback gains which are explicit functions of a posteriori parameter probabilities. The performance of this controller is illustrated with a simple example.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

A double integral quadratic cost and tolerance of feedback nonlinearities

A double integral quadratic cost with an associated integral constraint on state trajectories is shown to result in a stable feedback control law for linear time-varying differential systems. The gain matrix for this control is obtained by integrating a Riccati-type matrix differential equation over a finite time interval and is shown to allow for a large class of nonlinearities in the feedback loop without destroying its asymptotically stable property. The class of nonlinearities is larger than that which is generally permitted for the standard steady-state linear optimal regulator, and a phase margin of 90 ° can be approached by the closed-loop system in the case of time-invarlant systems.     

Distributed robust control of linear multi-agent systems with parameter uncertainties

This article considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighbouring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the H _∞ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust H _∞ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.     

Quadratic optimal controller to stabilise symmetrical systems

In this paper, we design a controller for stabilising a control system. The technique used for designing the controller includes a linear regulator and an asymptotical observer which form the controller. The linear regulator designed is a feedback of estimated states and also it must minimise a quadratic performance index. The gain matrix of optimal feedback is obtained by solving the Riccati equation, whilst the gain observer matrix is computed by making use of symmetrical systems properties. The properties of symmetrical systems allow us to find the optimal gain matrix of the observer without solving the dual Riccati equation, we only need to compute the matrices of controllability and observability. Having calculated the gain matrices of regulator and of observer, we proceeded to compute the transfer function of the observer-based controller.     

Output feedback decentralized stabilization: ILMI approach

In this paper, the static output feedback decentralized stabilization problem is addressed using a linear matrix inequality approach. A necessary and sufficient condition for static output feedback decentralized stabilizability is derived for linear time-invariant large-scale systems. It is proven that the existence of a stabilizing decentralized gain is equivalent to that of the solution of a quadratic matrix inequality. The extension of the result to control is studied. An iterative LMI algorithm based on the linear matrix inequality technique is proposed to obtain the decentralized feedback gain. Examples show the effectiveness of the algorithm.     

A gradient flow approach to decentralised output feedback optimal control

This paper visits the quadratic optimal control problem of decentralised control systems via static output feedback. A gradient flow approach is introduced as a tool to compute the optimal output feedback gain. Several nice properties are revealed concerning the convergence of the gain matrix along the trajectory of an ordinary differential equation obtained from the gradient of objective cost, i.e. the objective cost is decreasing along this trajectory. If the equilibrium points are isolated, the convergence can be guaranteed. A simulation example is given to illustrate the effectiveness of this approach.     

基于学习的线性多智能体系统弹性最优协同容错控制

针对一类线性多智能体系统,研究其在网络间歇性拒绝服务攻击下的最优同步控制问题.首先,在时变非对称通讯网络拓扑结构下,提出一种弹性最优协同容错控制策略,并优化多智能体的合作二次性能指标,然后证明全局跟踪误差在出现执行器故障和网络攻击时仍然渐进收敛.进一步,当考虑多智能体子系统模型参数未知,同时系统发生执行器故障的情况下,提出利用局部系统状态和输入信息的自学习迭代算法求解代数Riccati方程,计算子系统的反馈控制器增益,实现弹性协同容错控制目标.最后,通过Chua电路网络仿真算例验证所提出的控制方法的有效性.     

Quadratic control of stochastic hybrid systems with renewal transitions

We study the quadratic control of a class of stochastic hybrid systems with linear continuous dynamics for which the lengths of time that the system stays in each mode are independent random variables with given probability distribution functions. We derive a condition for finding the optimal feedback policy that minimizes a discounted infinite horizon cost. We show that the optimal cost is the solution to a set of differential equations with unknown boundary conditions. Furthermore, we provide a recursive algorithm for computing the optimal cost and the optimal feedback policy. The applicability of our result is illustrated through a numerical example, motivated by stochastic gene regulation in biology.     

Solution of the optimal constant output feedback problem by conjugate gradients

The problem of finding the optimal, constant output feedback matrix for linear time-invariant multivariable systems with quadratic cost is reconsidered. Simple formulae for the gradient matrix are developed and used in a Fletcher-Powell-Davidon algorithm. Computational results are presented.     

A new digital control strategy for a nuclear boiling water reactor

A method is proposed for the design of constant gain optimal feedback control laws for discrete-time systems which reduce trajectory sensitivity to small system parameter variations and ensure the closed-loop eigenvalues at the prescribed locations. The selected performance index is of standard linear regulator type, modified to include a quadratic sensitivity term. An efficient computational procedure based on direct cost oprimization using gradient type algorithm is also reported. A case study of a Boiling Water Reactor power plant has been presented to illustrate the application of the design procedure.     

Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.     

一类双线性随机系统M量测不定LQ控制

研究了带有乘性噪声和受扰动观测的离散时间随机系统不定线性二次(Linear quadratic, LQ) 最优输出反馈控制问题. 对此类问题而言,二次成本函数的加权矩阵不定号,并且最优控制具有对偶效果.为在最优性和计算复杂度间 进行折衷,本文采用了一种M量测反馈控制设计方法.基于动态规划方法,将未来的测量结合到当前控制 计算当中的M量测反馈控制可以通过倒向求解一类与原系统维数相同的广义差分Riccati方程(Generalized difference Riccati equation, GDRE)得到.仿真结果 表明本文提出的算法与目前普遍采用的确定等价性方法相比具有优越性.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

Simultaneous stabilization via static output feedback and statefeedback

In this paper, the simultaneous stabilization problem is considered using the matrix inequality approach. Some necessary and sufficient conditions for simultaneous stabilizability of strictly proper multi-input/multi-output (MIMO) plants via static output feedback and state feedback are obtained in the form of coupled algebraic Riccati inequalities. It is shown that any such stabilizing feedback gain is the solution of some coupled linear quadratic control problems where every cost functional has a suitable cross term. A heuristic iterative algorithm based on the linear matrix inequality technique is presented to solve the coupled matrix inequalities. The effectiveness of the approach is illustrated by numerical examples     

Output feedback optimal guaranteed cost control of uncertain piecewise linear systems

This paper proposes the output feedback optimal guaranteed cost controller design method for uncertain piecewise linear systems based on the piecewise quadratic Lyapunov functions technique. By constructing piecewise quadratic Lyapunov functions for the closed‐loop augmented systems, the existence of the guaranteed cost controller for closed‐loop uncertain piecewise linear systems is cast as the feasibility of a set of bilinear matrix inequalities (BMIs). Some of the variables in BMIs are set to be searched by genetic algorithm (GA), then for a given chromosome corresponding to the variables in BMIs, the BMIs turn to be linear matrix inequalities (LMIs), and the corresponding non‐convex optimization problem, which minimizes the upper bound on cost function, reduces to a semidefinite programming (SDP) which is convex and can be solved numerically efficiently with the available software. Thus, the output feedback optimal guaranteed cost controller can be obtained by solving the non‐convex optimization problem using the mixed algorithm that combines GA and SDP. Numerical examples show the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous‐time polytopic systems. This generalized problem includes the robust stability, H₂ ‐norm, and H_∞ ‐norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H₂ ‐norm and H_∞ ‐norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail.     

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.     

线性不确定系统的最优保性能控制--线性矩阵不等式处理方法

针对一类具有范数有界时变参数不确定性的线性系统的和一个给定的二次型性能指标,研究了其最优保性能控制问题,用一个线性矩阵不等式系统的可解性给出了存在保性能状态反馈控制律的条件,并且这个线性矩阵不等式系统的可行解提供了一组保性能控制律的一个参数化表示,进而保性能控制律的这一刻划被用来求解最优保成本控制问题。     

Feedback control of a class of linear systems with jump parameters

A class of linear systems are studied which are subject to sudden changes in parameter values. An algorithm similar in form to Kushner's stochastic maximum principle is derived and the relationship between these algorithms discussed. Systems in which the performance measure is quadratic are investigated in detail and a differential equation is derived which yields the optimal feedback gains.