The Nash Equilibrium Point of Dynamic Games Using Evolutionary Algorithms in Linear Dynamics and Quadratic System

In this paper, examining some games, we show that classical techniques are not always effective for games with not many stages and players and it can’t be claimed that these techniques of solution always obtain the optimal and actual Nash equilibrium point. For solving these problems, two evolutionary algorithms are then presented based on the population to solve general dynamic games. The first algorithm is based on the genetic algorithm and we use genetic algorithms to model the players' learning process in several models and evaluate them in terms of their convergence to the Nash Equilibrium. in the second algorithm, a Particle Swarm Intelligence Optimization (PSO) technique is presented to accelerate solutions’ convergence. It is claimed that both techniques can find the actual Nash equilibrium point of the game keeping the problem’s generality and without imposing any limitation on it and without being caught by the local Nash equilibrium point. The results clearly show the benefits of the proposed approach in terms of both the quality of solutions and efficiency.     

Disturbance attenuation by dynamic output feedback for input-delay systems

This paper addresses the optimal disturbance attenuation problem by output feedback for multivariable linear systems with delayed inputs. The attenuation is optimal in the sense that the controller minimizes the maximal amplitude of the plant output in response to such a disturbance. The controller is a general feedback, involving an observer, a state predictor, and a predicted state feedback. The optimal disturbance attenuation problem is formulated in terms of an equivalent system without delay. The optimal bound of the disturbance attenuation is then characterized, and it is shown that the optimal controller tends to have high gains. A necessary and sufficient condition to guarantee the existence of an optimal solution is provided using the geometric approach.     

Lorenz系统的线性反馈控制

设计了状态变量的线性反馈控制器对Lorenz系统的平衡点和周期轨道进行控制.首先,利用Routh-Hurwitz准则对受控系统进行了稳定性分析,证明了达到控制目标反馈系数的选择原则.然后,通过数值计算证明了该方法能够有效地控制混沌系统达到稳定的平衡点同时也能使系统控制到1p周期轨道,并且得到了相应的稳定的1p周期轨道的控制参数和系统幅值的关系曲线.最后给出了控制到1p周期轨道的控制参数的选取范围.     

Disturbance attenuation by output feedback for linear systems subject to actuator saturation

In this paper, we study the problem of disturbance attenuation by output feedback for linear systems subject to actuator saturation. A nonlinear output feedback, expressed in the form of a quasi‐linear parameter‐varying system with state‐dependent scheduling parameter, is constructed that leads to the attenuation of the effect of the disturbance on the output of the system. The level of disturbance attenuation is measured in terms of the restricted ℒ︁₂ gain and the restricted ℒ︁₂–ℒ︁_∞ gain over a class of bounded disturbances. Copyright © 2008 John Wiley & Sons, Ltd.     

Linear state feedback regulator for rigid link manipulators

A new generic representation of the gravity vector in the rigid link robot dynamic model is proposed. We use this representation to design a linear state feedback regulator and show that the closed loop nonlinear system is globally asymptotically stable and exponentially stable in any closed ball. We exploit the fact that the gravity vector is the gradient of the potential function. We also consider robustness of the linear state feedback regulator to parameter uncertainty.     

Quadratic stabilization of bilinear systems: Linear dynamical output feedback

We consider stabilization of bilinear control systems by means of linear output dynamical controllers. Using the linear matrix inequality technique, quadratic Lyapunov functions, and a special iterative method, we propose a regular approach to the construction of the stabilizability ellipsoid having the property that the trajectories of the system emanating from the points of this ellipsoid asymptotically tend to zero. The developed approach enables for an efficient construction of nonconvex inner approximations of domains of stabilizability of bilinear control systems.     

线性二次闭环微分对策次优对策的双线性矩阵不等式方法

The suboptimal control program via memoryless state feedback strategies for LQ differential games with multiple players is studied in this paper. Sufficient conditions for the existence of the suboptimal strategies for LQ differential games are presented. It is shown that the suboptimal strategies of LQ differential games are associated with a coupled algebraic Riccati inequality. Furthermore, the problem of designing suboptimal strategies is considered. A non-convex optimization problem with BMI constrains is formulated to design the suboptimal strategies which minimizes the performance indices of the closed-loop LQ differential games and can be solved by using LMI Toolbox of MATLAB. An example is given to illustrate the proposed results.     

Suboptimal Strategies of Linear Quadratic Closed-loop Differential Games: An BMI Approach

The suboptimal control program via memoryless state feedback strategies for LQ differential games with multiple players is studied in this paper. Sufficient conditions for the existence of the suboptimal strategies for LQ differential games are presented. It is shown that the suboptimal strategies of LQ differential games are associated with a coupled algebraic Riccati inequality. Furthermore, the problem of designing suboptimal strategies is considered. A non-convex optimization problem with BMI constrains is formulated to design the suboptimal strategies which minimizes the performance indices of the closed-loop LQ differential games and can be solved by using LMI Toolbox of MATLAB, An example is given to illustrate the proposed results.     

Disturbance attenuation of a class of non‐linear systems via output feedback

This paper studies the problem of disturbance attenuation with internal stability via output feedback for a family of non‐linear systems. Using a feedback domination design which substantially differs from the separation principle, we explicitly construct a dynamic output compensator attenuating the disturbance's effect on the output to an arbitrary degree of accuracy in the L₂‐gain sense, and achieving global asymptotic stability in the absence of disturbance. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical analysis of the Nash strategy for weakly coupled large-scale systems

This note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm; it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation.     

线性时变二次微分对策Nash策略的小波分析法(Ⅰ)——小波逼近解

基于小波多尺度逼近特性,提出了一种求解线性时变系统中微分对策Nash策略的新方法.该法避免求解耦合Riccati微分方程,而只需求解代数方程,适合于计算机求解.     

An integrated perturbation analysis and Sequential Quadratic Programming approach for Model Predictive Control

Computationally efficient algorithms are critical in making Model Predictive Control (MPC) applicable to broader classes of systems with fast dynamics and limited computational resources. In this paper, we propose an integrated formulation of Perturbation Analysis and Sequential Quadratic Programming (InPA-SQP) to address the constrained optimal control problems. The proposed algorithm combines the complementary features of perturbation analysis and SQP in a single unified framework, thereby leading to improved computational efficiency and convergence property. A numerical example is reported to illustrate the proposed method and its computational effectiveness.     

Disturbance attenuation for a class of distributed parameter systems

This paper deals with the problem of robust stabilization and disturbance attenuation via measured feedback, for a class of dissipative collocated distributed systems with disturbances affecting both the input and the measured output. The proposed solution is based on a direct ℒ︁₂-gain characterization which avoids the usual Riccati equation argument. For this purpose only strong stabilizability of the semigroup is required. A flexible slewing link is chosen as an application example. © 1997 John Wiley & Sons, Ltd.     

线性时变二次微分对策Nash策略的小波分析法(Ⅱ)——小波逼近解的收敛性

研究小波逼近分析方法的收敛性问题, 对线性时变二次微分对策Nash策略情形, 证明了Nash策略的小波逼近解收敛于精确解, 基于小波逼近的多尺度多分辨特性, 给出了误差估计的阶数.     

超混沌Bao系统线性状态反馈控制及自适应控制

研究超混沌Bao 系统的控制问题. 首先, 给出超混沌Bao 系统的Lyapunov 指数图, 用以验证系统存在超混沌现象; 然后, 将线性反馈控制方法和自适应反推控制方法应用于超混沌Bao 系统, 并设计了将系统混沌状态稳定到平衡点的混沌控制器; 最后, 通过数值仿真验证了所设计的控制器的有效性.

     

一个用于解决需求不一致的纳什均衡策略选取方法

即使在需求说明形成之后,项目涉众要求变更需求的情形一样屡见不鲜,对新需求的管理以及需求不一致的处理显得十分重要。将博弈论中纳什均衡的相关知识应用到需求协商,定义了协商时博弈双方的策略意愿值,给出了纯策略与混合策略纳什均衡策略的选取方法,并实现了该策略选取工具,为协商函数提供了所需的策略组合方案。     

N人合作博弈的Nash及演化均衡稳定策略分析

在有限理性的基础上,对N人合作博弈的对称Nash均衡进行了分析,并引入演化博弈理论分析了参与人的演化均衡稳定策略,得到了不同策略选择下的均衡点。进而应用生物复制动态理论对离散时间及连续时间下的复制动态稳定集进行了研究。最后通过实例说明了该方法在博弈均衡选择上的有效性。     

Sequential LMI approach for the design of a BMI‐based robust observer state feedback controller with nonlinear uncertainties

This paper aims at developing a robust observer–based estimated state feedback control design method for an uncertain dynamical system that can be represented as a linear time‐invariant system connected with an integral quadratic constraint–type nonlinear uncertainty. Traditionally, in existing design methodologies, a convex semidefinite constraint is obtained at the cost of conservatism and unrealistic assumptions. This paper avoids such assumptions and formulates, the design of the robust observer state feedback controller as the feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP‐hard problem in general. The applicability of a linearization method, such as the variable change method and the congruence transformation, depends on the specific structure of the problem at hand and cannot be generalized. This paper transforms the feasibility analysis of the BMI constraint into an eigenvalue problem and applies the convex‐concave–based sequential linear matrix inequality optimization method to search for a feasible solution. Furthermore, an augmentation of the sequential linear matrix inequality algorithm to improve its numerical stability is presented. In the application part, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real‐world estimated state feedback control design problem and the necessity of the augmentation for numerical stability.     

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.