LQ最优控制之逆问题的研究

本文通过适当地选取LQ性能指标函数中的加权矩阵R,给出了该二次型性能指标函数中的另一个加权矩阵Q与系统的开环特征多项式、闭环特征多项式的系数以及系数的系数矩阵A、B之间的对应关系。如果给定一个系统以及该系统的一组最优闭环极点,就可以求得矩阵Q。同时,用本文的研究结果,还可以直接确定系统的最优状态反馈系数矩阵。     

具有给定闭环极点的最优控制系统的设计

文中给出了线性二次型性能指标函数中的加权系数矩阵 Q 和 R 与控制系统的闭环极点之间的关系.只要给定系统的最优闭环极点,就能方便地求得矩阵 Q 和 R.     

确定线性调节器加权矩阵的简易方法

本文给出了线性二次型性能指标中加权矩阵与开环、最优闭环系统特征多项式系数之间的解析关系。只要给定一组期望的闭环极点,即可直接确定与之对应的加权矩阵。     

连续系统线性二次型期望极点配置问题的研究

本文以线性二次型性能指标中的加权矩阵和最 优闭环系统在频域内的解析关系为基础,提出了一种新的期望极点配置方法．该方法的主要 优点是不必求解复杂的矩阵Riccati方程也可很容易地确定满足指定闭环极点配置要求的状 态反馈矩阵．本文还讨论了指定闭环极点的选择方法,并用例子说明这种极点配置方法的有 效性和简便性．     

执行器有时滞的馈能式主动悬架的鲁棒控制

车辆用的馈能式主动悬架系统具有不确定参数,而且执行器有时滞.为了保证其稳定性、减振性和能量回收性能,我们提出了一种保成本/H∞鲁棒控制器设计方法.对车辆悬架系统性能方面的要求,用二次型加权性能指标和H∞性能指标反映.定义了一个Lyapunov函数代表这两个性能指标;根据这个Lyapunov函数,把闭环系统设计问题转化为一组线性矩阵不等式以求解控制器.根据直流电机工作原理,分析了参数摄动和执行器时滞对系统能量平衡的影响,推导出了能量平衡方程.最后对二自由度1/4车悬架模型进行了仿真;结果表明:对一定范围内的参数摄动和有界时滞,悬架系统在有效减振的同时,实现了能量的回收.     

一类关联不确定离散系统的分散保性能控制

结合一个给定的二次型性能指标, 研究一类关联不确定离散大系统的分散保性能状态反馈控制器设计问题. 采用线性矩阵不等式处理方法, 导出了分散控制器存在的条件, 并给出了一组分散保性能控制器的参数化表示, 据此, 通过建立和求解一个凸优化问题给出了使得闭环性能指标的上界最小化的最优分散保性能控制器设计方法.     

离散系统最优调节器的逆问题

本文证明了在闭环特征多项式的系数,以及二次型指标函数中的矩阵Q和系统的系数矩阵A,B之间所存在的一组关系,并对系统(A,B)的形式无特别要求。对于SiSO系统来说,如果给定开环系统以及所希望的闭环特征,通过这组关系,就能方便地求出矩阵Q。     

一种新的最优控制系统设计方法

本文采用LQ逆问题方法得到了一种新的最优控制系统设计方法，推导了线性二次型性能指标中的加权矩功Q与开环特征多项式，最优闭环特征多项式之间的关系。并研究LQ逆问题解的存在性和唯一性问题。只要给定期望的闭环极点，即可确定与之对应的加权矩阵Q，从而获得一个具有指定闭环极点的最优控制系统。     

不确定离散切换系统具有极点约束的保性能控制

对一类含有范数有界不确定性的离散切换系统和一个二次型性能指标，研究其具有闭环极点约束的鲁棒状态反馈保性能控制问题．利用二次Lyapunov函数方法和线性矩阵不等式技术，给出了鲁棒保性能控制器存在的一个充分条件，在所构造切换规则下，闭环系统二次D稳定，且满足给定的性能指标．在此基础上，将次优保性能控制器设计问题转化为一组线性矩阵不等式约束下的凸优化问题．数值例子说明了所提方法的有效性．     

多变量双线性广义预测极点配置自校正解耦控制器（续）

对一类多变量双线性系统提出了一种基于预测状态空间实现的GPP自校正控制算法,建立了预测状态与模型结构参数和输入输出信息之间的直接关系,给出了含有多个加权矩阵的多变量二次型性能指标。增加了系统设计的自由度。由于加权因子可以根据闭环系统稳定性要求以及系统动态特性、前馈零点增补、输出滤波和跟踪要求分别加以选取,可以保证闭环系统稳定并改善了系统动态特性增强了鲁棒性,仿真结果表明了该算法具有GPP的诸多优点。     

Robust control for linear uncertain systems via linear quadratic state feedback

By the Lyapunov stability criterion and the algebraic Riccati equation, conditions of selecting the weighting matrices in the quadratic cost function are derived so that linear quadratic state feedback can exponentially stabilize a linear uncertain system, provided the uncertainties satisfy the so-called matching conditions and within a given bounding set. Furthermore, two simple but effective algorithms are proposed for systematically selecting the weighting matrices. The main features of this approach are that the uncertain system can be exponentially stabilized with prescribed exponential rate and no precompensator is needed. Two examples are given to illustrate the results.     

Receding Horizon Controls for Input-Delayed Systems

This paper presents a receding horizon control (RHC) for an unconstrained input-delayed system. To begin with, we derive a finite horizon optimal control for a quadratic cost function including two final weighting terms. The RHC is easily obtained by changing the initial and final times of the finite horizon optimal control. A linear matrix inequality (LMI) condition on two final weighting matrices is proposed to meet the cost monotonicity, under which the optimal cost on the horizon is monotonically nonincreasing with time and hence the asymptotical stability is guaranteed only if an observability condition is met. It is shown through simulation that the proposed RHC stabilizes the input-delayed system effectively and its performance can be tuned by adjusting weighting matrices with respect to the state and the input.      

Equivalence of quadratic performance criteria†

Two quadratic cost functionals are defined to be equivalent, if they generate the same optimal control law. Necessary and sufficient conditions are derived in terms of the system parameters and the quadratic weighting matrices for two cost functionals to be equivalent.     

H2 guaranteed cost control for singularly perturbeduncertain systems

The H₂ guaranteed cost control problem for a singularly perturbed norm-bounded uncertain system is addressed using the quadratic stabilizability framework. After defining the corresponding slow and fast uncertain subsystems, the set of quadratic stabilizing composite controls is characterized. Two Riccati equations have to be solved, one for the slow subsystem and the other for the fast subsystem. Choosing appropriately the weighting matrices, it is shown how to pick up in the set of quadratic stabilizing composite controls, a control minimizing an upper bound on the H₂ norm of a certain transfer matrix. The case of the guaranteed cost control problem for the reduced system is also investigated     

Constrained optimal control

The explicit form of the optimal control law of a given linear, discrete-time, time-invariant process subject to a quadratic cost criterion is well known. In some applications it is desirable that the state of a controlled dynamic process be nonnegative, given a certain class of initial disturbances. Using the controllable block companion transformation, sufficient conditions on the weighting matrices of the cost criterion are derived to ensure that the closed-loop response of the original process with the standard, unconstrained optimal feedback law will be nonnegative. It is shown that the nondiagonal elements of the transformed weighting matrices can be chosen to ensure nonnegativity     

Efficient optimal controller for nuclear power plants

A new scheme for the optimal control of nuclear power plants is proposed. It differs from previous applications of optimal control theory to nuclear reactors in that here it is possible to select the weighting matrices in the quadratic cost functional so that desired pole placement, and subsequent transient response, is achieved. The desired weighting matrix and corresponding optimal control law are found sequentially. From the computational point of view the proposed design algorithm is very efficient since the dynamical system is aggregated at each stage of the sequential process to a first- or a second-order system. Thus, almost the entire computation involves second- or fourth-order matrices.     

Lower bounds on the quadratic cost of optimal regulators

Lower bounds for linear optimal regulators with quadratic cost functional are derived, explicitly containing information pertaining to some parameters of the system, e.g. the weighting matrices of the quadratic cost. If the optimal solution is not available, but rather a sub-optimal design is known, these bounds provide a means for estimating how close a sub-optimal design is to the optimal one, without having to actually calculate the latter. Being explicit in terms of the system parameters, these bounds enable the ranking of the various sub-optimal designs in terms of these parameters.     

Linear quadratic networked control of uncertain polytopic systems

This paper investigates the problem of designing robust linear quadratic regulators for uncertain polytopic continuous‐time systems over networks subject to delays. The main contribution is to provide a procedure to determine a discrete‐time representation of the weighting matrices associated to the quadratic criterion and an accurate discretized model, in such a way that a robust state feedback gain computed in the discrete‐time domain assures a guaranteed quadratic cost to the closed‐loop continuous‐time system. The obtained discretized model has matrices with polynomial dependence on the uncertain parameters and an additive norm‐bounded term representing the approximation residual error. A strategy based on linear matrix inequality relaxations is proposed to synthesize, in the discrete‐time domain, a digital robust state feedback control law that stabilizes the original continuous‐time system assuring an upper bound to the quadratic cost of the closed‐loop system. The applicability of the proposed design method is illustrated through a numerical experiment. Copyright © 2015 John Wiley & Sons, Ltd.     

Sub-optimal control for multivariable systems based on an orthogonal decomposition technique†

This paper considers the optimization of time-varying multivariable control systems with quadratic cost function which may have time-varying weighting matrices. The problem is first formulated in the context of functional analysis which leads to the optimal control being denned by a functional equation. An approximate solution of this equation is then obtained by replacing the kernel of the equation by a double Fourier series expansion. It is shown that the resulting sub-optimal control may be made to approach the absolute optimum as closely as desired by extending the double Fourier series, and bounds on the norm of the error in the control and the cost function are derived for any finite expansion. Application of the method to a particular system shows that Chebyshev basis functions lead to a better approximation than the classic sinusoidal functions. Finally, the use of the method as an adjunct to iterative optimization based on the contraction mapping algorithm is discussed briefly.