Lagrangian manifolds and asymptotically optimal stabilizing feedback control

Approximations to nonlinear optimal control based on solving a Riccati equation which varies with the state have been put forward in the literature. It is known that such algorithms are asymptotically optimal given large scale asymptotic stability. This paper presents an analysis for estimating the size of the region on which large scale asymptotic stability holds. This analysis is based on a geometrical construction of a viscosity-type Lyapunov function from a stable Lagrangian manifold. This produces a less conservative estimate than existing approaches in the literature by considering regions of state space over which the stable manifold is multi-sheeted rather than just single sheeted.     

Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations

In this paper, we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic Riccati equation (CCARE) and the discrete coupled algebraic Riccati equation (DCARE), which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each Riccati equation. By utilizing the upper bounds for the maximal eigenvalues of each equation, we then derive upper matrix bounds for the solutions of the CCARE and DCARE. Following the development of each bound, an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds. Finally, we give numerical examples to demonstrate the effectiveness of the proposed results, making comparisons with existing results.     

Data-driven policy iteration algorithm for continuous-time stochastic linear-quadratic optimal control problems

This paper studies a continuous-time stochastic linear-quadratic (SLQ) optimal control problem on infinite-horizon. Combining the Kronecker product theory with an existing policy iteration algorithm, a data-driven policy iteration algorithm is proposed to solve the problem. In contrast to most existing methods that need all information of system coefficients, the proposed algorithm eliminates the requirement of three system matrices by utilizing data of a stochastic system. More specifically, this algorithm uses the collected data to iteratively approximate the optimal control and a solution of the stochastic algebraic Riccati equation (SARE) corresponding to the SLQ optimal control problem. The convergence analysis of the obtained algorithm is given rigorously, and a simulation example is provided to illustrate the effectiveness and applicability of the algorithm.     

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.

     

Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems

In this paper we consider the stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. For the case in which the quadratic-term matrices are non-negative, this necessary and sufficient condition can be written in a more explicit way. The results are applied to a problem of portfolio optimization.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

Robust control of nonlinear feedback passive systems

In this paper we consider a class of nonlinear systems with uncertain parameters which enter the system nonlinearly. We assume that the uncertain nonlinear system is minimum phase and the uncertain parameters are from a bounded compact set. The problem under consideration is the design of a nonlinear static state feedback controller such that the closed-loop system is passive for all admissible uncertainties.     

State and output feedback boundary control for a coupled PDE-ODE system

This note is devoted to stabilizing a coupled PDE-ODE system with interaction at the interface. First, a state feedback boundary controller is designed, and the system is transformed into an exponentially stable PDE-ODE cascade with an invertible integral transformation, where PDE backstepping is employed. Moreover, the solution to the resulting closed-loop system is derived explicitly. Second, an observer is proposed, which is proved to exhibit good performance in estimating the original coupled system, and then an output feedback boundary controller is obtained. For both the state and output feedback boundary controllers, exponential stability analyses in the sense of the corresponding norms for the resulting closed-loop systems are provided. The boundary controller and observer for a scalar coupled PDE-ODE system as well as the solutions to the closed-loop systems are given explicitly.     

Optimization of dependently controlled jump rates of JLQG problem

The Jump Linear Quadratic Gaussian (JLQG) model is well studied due to its wide applications. However, JLQG with controlled jump rates are rarely researched, while the existing studies usually impose an assumption that jump rates are independent and separately controlled. In practical systems, their jump rates may not be independent of each other. In this paper, we consider a continuous‐time JLQG model with dependently controlled jump rates and formulate it as a two‐level control problem. The low‐level problem is a standard JLQG problem, thus we focus on solution of high‐level problem. We propose a Markov decision process‐based approach to calculate performance gradient with respect to jump rates control variable and develop a gradient‐based optimization algorithm. We present an application of manufacturing system to illustrate the main results of this paper. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust stabilization and guaranteed cost control for discrete-time linear systems by static output feedback

This paper proposes a systematic approach for the static output feedback control design for discrete-time uncertain linear systems. It is shown that if the open-loop system satisfies some particular structural conditions and the uncertainty has a specific structure, a static output feedback gain can be calculated easily, using a formula only involving the original system matrices. Among the conditions the system has to satisfy, the strongest one relies on a minimum phase argument. Square and nonsquare systems are considered. The performance problem through a quadratic criterion is also discussed (guaranteed cost control).     

Dissipative control systems synthesis with full state feedback

In this paper we present a general framework for synthesizing state feedback controllers to achieve any desired closed loop dissipative behavior. Special cases include positive real andH _∞ controller synthesis. We show that the solution to this dissipative controller synthesis problem is equivalent to the existence of a solution to a partial differential inequality (nonlinear systems case) or an algebraic Riccati inequality (linear systems case). Stability results can be obtained under appropriate detectability assumptions, and a generalization of the strict bounded real lemma is given. We also present an application of the results to a robust stabilization problem. This work was supported by the Cooperative Research Centre for Robert and Adaptive Systems (CRASys), Australia.     

Nonlinear model-state feedback control for nonminimum-phase processes

A general nonlinear controller design methodology for continuous-time nonminimum-phase systems is presented, which utilizes synthetic outputs that are statically equivalent to the original process outputs and make the system minimum-phase. A systematic procedure is proposed for the construction of statically equivalent outputs with prescribed transmission zeros. The calculated outputs are used to construct a model-state feedback controller. The proposed method is applied to a nonminimum-phase chemical reactor control problem where a series/parallel reaction is taking place.     

About the Newton iteration for spectral factorization

It is shown, that Theorem 1.2 of Willems (Operators, Systems and Linear Algebra (Kaiserslautern, 1997), European Consort. Math. Indust., Jeubner, Stuggart, 1997, pp. 214–223), is analog of the outcomes of the statement 5 (Aliev and Lorin, Systems Contr. Lett. 21 (1993) 485–491).     

Monotonicity of algebraic Lyapunov iterations for optimal control of jump parameter linear systems

In this paper we show that the sequences of the solutions of the decoupled algebraic Lyapunov equations are monotonic under proper initialization. These sequences converge from above to the positive-semidefinite stabilizing solutions of the system of coupled algebraic Riccati equations of the optimal control problem of jump parameter linear systems.     

Analysis of SDC matrices for successfully implementing the SDRE scheme

The state-dependent Riccati equation (SDRE) approach for stabilization of nonlinear affine systems was recently reported to be effective in many practical applications; however, there is no guideline on the construction of state-dependent coefficient (SDC) matrix when the SDRE solvability condition is violated, which may result in the SDRE scheme being terminated. In this study, we present several easy checking conditions so that the SDRE scheme can be successfully implemented. Additionally, when the presented checking conditions are satisfied, the sets of all feasible SDC matrices and their structures are explicitly depicted for the planar system.     

基于LMI不确定离散随机系统输出反馈保性能控制

针对一类范数有界不确定离散随机系统,研究了具有输出反馈控制器的保性能控制,由此得到闭环系统的二次型性能指标上界的一般形式;然后应用LMI凸优化技术,推导出使闭环系统渐近稳定的最优保性能控制器的设计方法;最后通过仿真算例验证了所提出算法的可行性和有效性.     

线性反馈实现Liu系统的混沌同步

讨论了新混沌系统Liu系统的混沌同步问题.基于Lyapunov函数分别提出了单变量以及多变量的线性状态反馈控制方案,采用这两种线性控制方案均可实现Liu系统的混沌同步.线性反馈控制比起非线性控制具有结构简单、易于实现的特点.数值模拟结果验证了两种方案的可行性.     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.