多项式非线性系统局部镇定控制

针对多项式非线性系统,提出一种用于验证二次型候选Lyapunov函数的数值计算方法.在该方法中,多项式系数被分解成带自由变量的系数矩阵,将正定性验证问题转化为矩阵不等式问题求解.对于局部稳定性分析,采用多个Lyapunov函数来趋近吸引域.每个Lyapunov 函数均在各指定方向上进行最大半径优化.在稳定性分析基础上,提出保收敛率的局部镇定控制器设计方法以扩大吸引域.

     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

一类多项式非线性系统鲁棒H∞控制

针对一类具有多项式向量场的仿射型不确定非线性系统,给出一种基于多项式平方和(sum of squares,SOS)技术的鲁棒H∞状态反馈控制器设计方法.该方法的优点在于控制器的设计避开了直接求解复杂的哈密尔顿-雅可比不等式(Hamilton Jacobi inequality,HJI)和构造Lyapunov函数带来的困难.将鲁棒稳定性分析和控制器设计问题转化为求解以Lyapunov函数为参数的矩阵不等式,该类不等式可利用SOS技术直接求解.此外,在前文基础上研究了基于SOS规划理论与S-procedure技术的局部稳定鲁棒H∞控制器设计方法.最后以非线性质量弹簧阻尼系统作为仿真算例验证该方法的有效性.     

执行器饱和的线性连续系统的镇定

针对控制系统中广泛存在饱和问题,主要研究执行器饱和线性连续系统的镇定问题并进行吸引域估计。首先根据Finsler’s引理和Lyapunov函数方法研究系统稳定的充分条件,得到执行器饱和控制系统稳定的新判据。其次,在稳定条件下,应用凸组合方法和新引入的自由权矩阵使得系统吸引域估计具有更小的保守性,将所得非线性矩阵不等式转化为线性矩阵不等式,给出求解最大吸引域的优化方法和状态反馈控制器的设计方案。最后通过仿真算例验证结果的有效性和可行性。     

转移概率部分未知的随机Markov 饱和切换系统的非脆弱镇定

研究一类转移概率部分未知的随机Markov饱和切换系统的非脆弱镇定问题. 基于参数依赖型Lyapunov函数, 设计非脆弱状态反馈控制器以保证闭环饱和系统的随机稳定性, 在此基础之上, 通过求解线性矩阵不等式, 得到均方意义下的最大不变吸引域. 数值仿真验证了所提出方法的有效性.

     

奇异线性系统在执行器饱和受限下不变集条件的改进

本文考虑饱和线性反馈下奇异线性系统扩大吸引域估计的问题.根据每个输入是否饱和,将输入空间分成若干子区域.在每个子区域内部,系统模型中没有显示的部分状态的时间导数可被显式表达.利用含有全部系统状态的二次Lyapunov函数,建立一组双线性矩阵不等式形式的改进的不变集条件.该组条件下,二次Lyapunov函数的水平集可诱导出一个吸引域估计.为得到最大的吸引域估计,构建了以这些双线性矩阵不等式为约束条件的优化问题,并为其求解给出了迭代算法.仿真结果表明本文得到的吸引域估计明显大于现有结果.     

切换系统的二类共同Lyapunov 函数

研究切换系统的共同Lyapunov函数存在问题.对于一类正切换系统,给出了共同Lyapunov函数存在的充分条件.当系统矩阵集为二阶矩阵紧集时,给出了判断共同Lyapunov函数存在的方法,并给出了计算共同Lyapunov函数的算法.最后通过算例验证了所提出算法的有效性.     

一类含多面体不确定性多项式系统鲁棒镇定

针对一类含多面体不确定性的多项式系统,研究其局部稳定鲁棒镇定问题。基于多项式平方和(SOS)技术,将该类非线性控制问题转换为凸的SOS规划问题,并通过引入S-procedure技术,保证了所得结论在局部范围内是有效的。同时,结合参数依赖Lyapunov函数方法,给出了该类系统鲁棒性分析与鲁棒镇定控制问题的充分条件,并将其描述为可由SOS规划技术直接求解的状态依赖线性矩阵不等式约束集。最后,通过数值仿真验证了该方法的有效性。     

跳变双线性随机系统饱和执行器的镇定

对于一类具有Markov跳变参数的双线性离散随机系统,研究其饱和执行器问题.分别采用一般二次型Lyapunov函数、饱和关联Lyapunov函数进行系统随机稳定性分析,以椭圆不变集构造随机稳定域,提出两种依赖于模态跳变率的饱和状态控制器设计方法,两种方法均以线性矩阵不等式的形式给出.     

Regional stability of two‐dimensional nonlinear polynomial Fornasini‐Marchesini systems

This paper addresses the problem of regional stability analysis of 2‐dimensional nonlinear polynomial systems represented by the Fornasini‐Marchesini second state‐space model. A method based on a polynomial Lyapunov function is proposed to ensure local asymptotic stability and provide an estimate of the domain of attraction of the system zero equilibrium point. The proposed results that build on recursive algebraic representations of the polynomial vector function of the system dynamics and Lyapunov function are tailored via linear matrix inequalities that are required to be satisfied at the vertices of a given bounded convex polyhedral region of the state space. Numerical examples demonstrate the effectiveness of the proposed method.     

Robust H∞ performance using lifted polynomial parameter-dependent Lyapunov functions

This paper investigates the robust H _∞ performance of time-invariant linear uncertain systems where the uncertainty is in polytopic domains. Robust H _∞ is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving κ powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler's Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of linear matrix inequalities. As κ increases, less conservative H _∞ evaluations are obtained. Both continuous and discrete-time systems are investigated. Numerical examples illustrate the method and compare the present results with similar works in the literature.     

An improved sum‐of‐squares based approach to fuzzy tracking control design of nonlinear systems

In this paper, using a more general Lyapunov function, less conservative sum‐of‐squares (SOS) stability conditions for polynomial‐fuzzy‐model‐based tracking control systems are derived. In tracking control problems the objective is to drive the system states of a nonlinear plant to follow the system states of a given reference model. A state feedback polynomial fuzzy controller is employed to achieve this goal. The tracking control design is formulated as an SOS optimization problem. Here, unlike previous SOS‐based tracking control approaches, a full‐state‐dependent Lyapunov matrix is used, which reduces the conservatism of the stability criteria. Furthermore, the SOS conditions are derived to guarantee the system stability subject to a given H_∞ performance. The proposed method is applied to the pitch‐axis autopilot design problem of a high‐agile tail‐controlled pursuit and another numerical example to demonstrate the effectiveness and benefits of the proposed method.     

Sum‐of‐Squares‐Based Finite‐Time Adaptive Sliding Mode Control of Uncertain Polynomial Systems With Input Nonlinearities

This paper proposes a novel adaptive sliding mode control (ASMC) for a class of polynomial systems comprising uncertain terms and input nonlinearities. In this approach, a new polynomial sliding surface is proposed and designed based on the sum‐of‐squares (SOS) decomposition. In the proposed method, an adaptive control law is derived such that the finite‐time reachability of the state trajectories in the presence of input nonlinearity and uncertainties is guaranteed. To do this, it is assumed that the uncertain terms are bounded and the input nonlinearities belong to sectors with positive slope parameters. However, the bound of the uncertain terms is unknown and adaptation law is proposed to effectively estimate the uncertainty bounds. Furthermore, based on a novel polynomial Lyapunov function, the finite‐time convergence of the sliding surface to a pre‐chosen small neighborhood of the origin is guaranteed. To eliminate the time derivatives of the polynomial terms in the stability analysis conditions, the SOS variables of the Lyapunov matrix are optimally selected. In order to show the merits and the robust performance of the proposed controller, chaotic Chen system is provided. Numerical simulation results demonstrate chattering reduction in the proposed approach and the high accuracy in estimating the unknown parameters.     

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.     

Homogeneous Lyapunov functions for systems with structured uncertainties

The problem addressed in this paper is the construction of homogeneous polynomial Lyapunov functions (HPLFs) for linear systems with time-varying structured uncertainties. A sufficient condition for the existence of an HPLF of given degree is formulated in terms of a linear matrix inequalities (LMI) feasibility problem. This condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the Lyapunov function. The maximum ?_∞ norm of the parametric uncertainty for which there exists a homogeneous polynomial Lyapunov function is computed by solving a generalized eigenvalue problem. The construction of such Lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in LMI form. Comparisons with other classes of Lyapunov functions through numerical examples taken from the literature show that HPLFs are a powerful tool for robustness analysis.     

Secure Synchronization Control for a Class of Cyber-Physical Systems With Unknown Dynamics

This paper investigates the secure synchronization control problem for a class of cyber-physical systems (CPSs) with unknown system matrices and intermittent denial-of-service (DoS) attacks. For the attack free case, an optimal control law consisting of a feedback control and a compensated feedforward control is proposed to achieve the synchronization, and the feedback control gain matrix is learned by iteratively solving an algebraic Riccati equation (ARE). For considering the attack cases, it is difficult to perform the stability analysis of the synchronization errors by using the existing Lyapunov function method due to the presence of unknown system matrices. In order to overcome this difficulty, a matrix polynomial replacement method is given and it is shown that, the proposed optimal control law can still guarantee the asymptotical convergence of synchronization errors if two inequality conditions related with the DoS attacks hold. Finally, two examples are given to illustrate the effectiveness of the proposed approaches.