Symbolic aids for modelling, analysis and synthesis of nonlinear control systems

To help along the analysis and design of nonlinear control systems the NONLINCON package, an acronym for nonlinear control, was developed. This note addresses the usefulness of symbolic computation, and of the NONLINCON package in particular, for the symbolic analysis and design of nonlinear control systems. The symbolic computation program MAPLE is used as computing substratum. Textbook problems show that the NONLINCON package can be used successfully. A larger scale problem is too complex, however, to be solved with the current versions of NONLINCON and MAPLE. The conclusion is that symbolic computation is a viable approach for textbook problems, but not yet for more complex ones. >

In this work we study the design and analysis for nonlinear discrete time active control system with boring charter. It is shown that most analysis result for continuous time nonlinear system can be extended to the discrete time case. In previous studies, a method of nonlinear Model Following Control System (MFCS) was proposed by Okubo (1985). In this study, the method of nonlinear MFCS will be extended to nonlinear discrete time system with boring charter. Nonlinear systems which are dealt in this study have the property of norm constraints ║ƒ (v (k))║≤α + β║v (k)║^γ , where α≥0, β≥0, 0≤γ≤1. When 0≤γ<1. It is easy to extend the method to discrete time systems. But in the case γ = 1 discrete time systems, the proof becomes difficult. In this case, a new criterion is proposed to ensure that internal states are stable. We expect that this method will provide a useful tool in areas related to stability analysis and design for nonlinear discrete time systems as well.

The concept of a sampled-data nonlinearity matrix is defined. It is shown that, by means of this matrix, a direct correlation can be establihed between the nonlinearity and its equivalent gain. Thus, symmetrical single-valued and/or double-valued nonlinearities, however complicated, can be transformed directly into the respective equivalent gains. This is obtained by means of linear transformations, using numerically known matrices, i.e. matrices which are independent of the nonlinearity. Similarly, in synthesis procedures, the nonlinerities are obtained from the characteristics of the desired equivalent gain. A theorem concerning these transformations is stated and proved, and some properties of the sampled-data nonlinearity matrix are emphasised. The matrices permitting direct and reciprocal transformations are given, and two non-linearities are calculated. A stability criterion is stated, and the stability of a nonlinear system is examined. Thus, it is shown that, in the design of the system, the matrix correlations and the stability criterion allow one to obtain the edesired behaviour by a very large class of nonlinear cntrol system by acting directly on the nonlinearity.

Analysis of Nonlinear Systems with Stochastic Input and Stochastic Parameters—II

Surprisingly, computer algebra systems have not been used for the analysis of discrete-time nonlinear control systems. The only exceptions seem to be papers based on Reduce and on Maple. This paper presents certain modelling problems of discrete-time nonlinear system as an example of what can be done with symbolic computation in nonlinear discrete-time control with Mathematica.

Theory of nonlinear control

Some new concepts in nonlinear systems

摘要: 确定学习运用自适应控制和动力学系统的概念与方法, 研究未知动态环境下的知识获取、表达、存储和利用等问题. 针对产生周期或回归轨迹的连续 非线性动态系统, 确定学习可以对其未知系统动态进行局部准确建模, 其基本要 素包括: 1)使用径向基函数(Radial basis function, RBF)神经网络; 2)对于周期(或回归)状态轨迹 满足部分持续激励条件; 3)在周期(或回归)轨迹的邻域内实现对非线性系统动态的局部准确神经网络逼近(局部准确建模); 4)所学的知识以时不变且空间分布的方式表达、以常值神经网络权值的方式存储, 并可在动态环境下用于动态模式的快速识别或者闭环神经网络控制. 本文针对离散动态系统, 扩展了确定学习理论, 提出一个根据时态数据序列对离散动态系统进行建模与控制的框架. 首先, 运用确定学习原理和离散系统的自适应辨识方法, 实现对产生时态数据的离散非线性系统的未知动态进行局部准确的神经网络建模, 并利用此建模结果对时态数据序列进行时不变表达. 其次, 提出时态数据序列的基于动力学的相似性定义, 以及对离散动态系统产生的时态数据序列(亦可称为动态模式)进行快速识别方法. 最后, 针对离散非线性控制系统, 实现了基于时态数据序列对控制系统动态的闭环辨识(局部准确建模). 所学关于闭环动态的知识可用于基于模式的智能控制. 本文表明确定学习可以为时态数据挖掘的研究提供新的途径, 并为基于数据的建模与控制等问题提供新的研究思路. 关键词: 确定学习 / 时态数据序列 / 离散动态系统 / 基于数据的建模 / 部分持续激励条件 / 时态数据挖掘 / 动态模式识别 / 基于模式的控制

Modal series is a new approach for modeling and analysis of nonlinear systems. This paper provides application of modal series to state estimation of nonlinear systems and introduces a new state estimation approach for nonlinear systems which uses a modal series model of nonlinear systems for Kalman filtering. The method implies a sequential use of Kalman filters which each one tries to decrease estimation errors of states. To validate the proposed approach, results of simulation of LQG control of a cart and pole using proposed approach has been compared with classical LQG control.

Theory of nonlinear systems

Transfer Function Approach to the Model Matching Problem of Nonlinear Systems

Automatic modeling of mechatronic systems via symbolic approach

The vast majority of the mathematically oriented literature in the areas of robotics and control has been heavily influenced by a differential geometric point of view. For nonlinear systems in particular, most of the research has concentrated on the analysis of the Lie algebras associated with controllability, reachability and observability. In recent years, however, a small but influential trend has begun in the literature on the use of other methods, such as differential algebra [9, 8, 10] and exterior differential systems [13, 11] for the analysis of nonlinear control systems and nonlinear implicit systems. In this paper we survey some key results from the theory of exterior differential systems and their application to current and challenging problems in robotics and control.

EXTENSION OF SOME of the techniques of sinusoidal analysis familiar in linear-system theory to systems containing nonlinear elements provides much useful information concerning their design and performance. Most important is the ability to predict possible periodic behavior, both stable and unstable, of the unexcited systems. This ability allows nonlinearities to be accounted for in system synthesis so that the usually undesirable behavior implied by the periodic solutions can be prevented or at least controlled. Either linear or nonlinear correcting devices can be employed effectively for this purpose. Steady-state responses to external sinusoidal excitation, including the hysteretic frequency responses common in nonlinear systems, are determined readily when the response is of excitation frequency. The method is helpful also in obtaining a qualitative understanding of simple transient behavior.

Analysis of fast-sampled non-linear systems: generalised frequency response functions for δ-operator models