线性离散奇异系统的不定二次最优控制问题: Krein空间方法

The finite time horizon indefinite linear quadratic(LQ) optimal control problem for singular linear discrete time-varying systems is discussed. Indefinite LQ optimal control problem for singular systems can be transformed to that for standard state-space systems under a reasonable assumption. It is shown that the indefinite LQ optimal control problem is dual to that of projection for backward stochastic systems. Thus, the optimal LQ controller can be obtained by computing the gain matrices of Kalman filter. Necessary and sufficient conditions guaranteeing a unique solution for the indefinite LQ problem are given. An explicit solution for the problem is obtained in terms of the solution of Riccati difference equations.     

Optimal LQ tracking control for continuous-time systems with pointwise time-varying input delay

This paper investigates the linear quadratic (LQ) tracking problem for linear continuous-time systems with pointwise time-varying input delays. It is assumed that the time-varying input delay takes only one value from a finite set at a given time instant. By defining an indicator, the investigated LQ tracking problem is firstly transformed into a special optimal control problem for continuous-time systems with multiple delays in a single input channel. Then the duality between the LQ tracking for the system with multiple input delay and the smoothing for an associated backward delay free system is established. Finally, an explicit analytical solution to the proposed LQ tracking problem is presented by solving the smoothing problem.

     

Optimal control for linear systems with multiple time delays in control input

This note presents the optimal linear-quadratic (LQ) regulator for a linear system with multiple time delays in the control input. Optimality of the solution is proved in two steps. First, a necessary optimality condition is derived from the maximum principle. Then, the sufficiency of this condition is established by verifying that it satisfies the Hamilton-Jacobi-Bellman equation. Using an illustrative example, the performance of the obtained optimal regulator is compared against the performance of the optimal LQ regulator for linear systems without delays and some other feasible feedback regulators that are linear in the state variables. Finally, the note establishes a duality between the solutions of the optimal filtering problem for linear systems with multiple time delays in the observations and the optimal LQ control problem for linear systems with multiple time delays in the control input.     

一类奇异时滞系统的奇异二次指标最优控制问题

利用基本的代数等价变换，将一类奇异滞后系统的奇异二次指标最优控制问题转化为正常状态滞后系统的非奇异二次指标最优控制问题，并讨论了二的等价性，在一些常规条件下，给出了问题的解，并把最优控制综合为最优状态反馈。     

乘性随机离散系统的最优控制

基于对系统随机不确定因素的分析,文中定义了一种新型随机离散系统--乘性随机离散系统,并研究该类系统的线性二次型(LQ)最优控制问题.首先给出了该类系统的有限时间和无限时间LQ最优控制律,并着重分析、证明了无限时间LQ最优控制问题的Riccati方程的正定矩阵解的存在性及相应数值求解算法与收敛性,以及闭环系统的稳定性等问题.仿真结果表明了该方法的有效性.     

线性调节器问题的一种次优控制律和最有性能的一个估计不等式

本文通过把一个线性定常系统看成为对称系统的一个虚参数摄动系统,给出了线性二次型调节器问题的次优控制律和次优性能值的一种简单计算方法。这种方法的一个优点是无需对通常的Riccati方程进行数值求解。文中同时给出了估计最优性能的一个不等式,提供了估计其上、下界的一个方便的途径。     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

带有饱和执行器的T-S离散模糊系统的LQ模糊控制

研究了带有饱和执行器的Takagi-SugenoT-S离散模糊系统的LQ模糊控制问题,利用Lyapunov稳定理论、PDC(平行分配补偿)技术以及线性矩阵不等式方法,得到了闭环模糊系统的渐近稳定的充分条件,给出了闭环系统的LQ模糊控制律的设计方法和吸引域的一个估计,并建立了闭环系统的LQ性能函数上界的计算公式.进一步,针对两类优化问题,即:LQ性能最小化问题和吸引域最大化问题,给出了相应的带有线性矩阵不等式约束的计算方法.最后,一个仿真例子说明了所给方法的有效性.     

Multiple models,multiplicative noise and linear quadratic control—algorithmic aspects

Numerical solution of certain linear quadratic (LQ) control problems for robust design is addressed. An iterative method is suggested and analysed for solving a minimax multiple model LQ control problem. A convergent iterative method is studied for finding the constant feedback gains of a given linear controller so as to solve a spectral radius functional minimization problem for multiple plants. Furthermore, a convergent iterative method is proposed for solving the maximum entropy parametric LQ control problem with multiplicative noise.     

方块脉冲函数应用于线性时变二次型最优控制的动态规划解法

本文通过应用方块脉冲函数的一些基本性质，将线性时变二次型最优控制问题化成多段动态规划问题，通过求解得到的动态规划问题，可得原问题的分段常值解。文中给出了形式简单且易于计算机求解的递推算法，与参考文献中的方法比较，本文得到的递推算法简单，明了，且计算时间及存储空间极大地减少，文中给出了算法的具体算例，具有明显的优越性。     

Improved control using PFC and modified linear quadratic regulator for a kind of nonlinear systems

This paper describes the combination design of predictive functional control (PFC) and optimal linear quadratic (LQ) method for a kind of nonlinear process with output feedback coupling. In many existing control methods for this kind of nonlinear systems, the nonlinear part is either ignored or represented as a rough linear one when corresponding predictive control methods are designed. However, by assuming that the nonlinearity can be ignored or simplified to a linear time-varying part may not lead to the good control performance of subsequent linear control designs. The paper is a further investigation on this kind of systems, in which a procedure of PFC plus a modified optimal LQ control is developed. With respect to the proposed control strategy and the corresponding processes, the closed-loop performance is improved concerning tracking ability and disturbance rejection compared with previous predictive control methods. In addition, the proposed control is easy to implement as it selects a simple structure and a modification of the classical control scheme.     

基于策略迭代的连续时间系统的随机线性二次最优控制

针对模型参数部分未知的随机线性连续时间系统, 通过策略迭代算法求解无限时间随机线性二次(LQ) 最优控制问题. 求解随机LQ最优控制问题等价于求随机代数Riccati 方程(SARE) 的解. 首先利用伊藤公式将随机微分方程转化为确定性方程, 通过策略迭代算法给出SARE 的解序列; 然后证明SARE 的解序列收敛到SARE 的解, 而且在迭代过程中系统是均方可镇定的; 最后通过仿真例子表明策略迭代算法的可行性.

     

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     

Saturated linear quadratic regulation of uncertain linear systems: Stability region estimation and controller design

This paper considers the problems of estimating the stability region (domain of attraction) and controller design for uncertain linear continuous-time systems with input saturation when linear quadratic (LQ) optimal controller is used. By exploiting the structure of the LQ controller and the property of saturation functions, it is established that the estimation of stability region can be obtained by solving linear matrix inequality (LMI) problems. Moreover, an iterative LMI (ILMI) algorithm is presented to design an LQ controller such that the largest estimated stability region can be obtained. Two examples are given to compare our results with existing ones.     

LQ control for constrained continuous-time systems

A design method of LQ optimal control law is considered for constrained continuous-time systems. By introducing singular value decomposition for finite-time horizon linear systems, the sequence of LQ sub-optimal control laws, which converges to the exact solution, is obtained based on quadratic programming. It is also shown that the nonlinear sub-optimal feedback gain is found by the union of affine state functions.     

一种新的离散系统最优闭环极点配置方法

分析了LQ逆问题解的存在条件,以便合理选择期望的闭环极点,使之成为一组最优极点．提 出了一种以离散系统LQ逆问题分析为基础的新的最优控制系统设计方法,得到了开环、闭环 特征多项式系数与加权矩阵之间的解析关系,只要给定一组期望闭环极点,即可确定与 之对应的加权矩阵Q和R,从而得到一个具有指定极点的最优控制系统．     

LQ终端控制的生成函数法与Riccati变换法的等价性

通过定义线性哈密顿系统新形式的第3类生成函数, 建立了求解线性二次终端控制问题的生成函数方法与Riccati变换方法的直接联系, 并证明两种方法导出的最优控制律是等价的. 生成函数方法的优势在于可以灵活地处理边界约束条件. 本文根据Riccati变换方法沿时间逆向求解问题的特点, 定义了适于逆向正则变换的第3类生成函数, 用于求解哈密顿两点边值问题, 可得到用生成函数表示的最优控制律. 并通过验证生成函数方法所得最优控制律的各部分与Riccati变换法所得的结果相对应, 证明了两种方法所得控制律的等价性.