【作者】 张端；

【导师】 孙优贤；

【作者基本信息】 浙江大学 ， 控制理论与控制工程， 2005， 博士

【摘要】 本文研究三角形式非线性系统的控制策略。本文的主要思路是:对非线性系统的控制可以这样实现,通过系统等价把应用中的非线性控制系统等价转换为具有三角形式的系统,然后用后推法(及其他方法如前推法)完成控制器的设计。 第一章,回顾了控制理论的发展历程,特别是非线性系统规范型和后推法的进展。 第二章是关于微分流形的简单介绍。 第三章利用向量场的李导数研究非线性系统等价性问题,目的是将较复杂的非线性系统转换成为较为简单的系统,主要工作有: (1)讨论了反馈线性化与方程的标准化之间的关系,给出了方程可以局部标准化的充要条件。对于一个常微分方程能否转化为标准型的问题,给出了两种判断方法。其一是,证明了当且仅当存在以常微分方程为漂移项的仿射单输入受控系统能被状态反馈线性化,该常微分方程可转化为标准型;其二,从局部看,当表示常微分方程的向量场在某点不是奇点时,常微分方程必能在该点附近转化为标准型,否则能否在该点附近转化为标准型取决向量场的雅可比矩阵的特征多项式是否是其最小多项式。 (2)考察了两个仿射非线性系统通过坐标变换相互等价的问题。当系统的漂移向量场与输入向量场有正则闭包时我们解决了这一问题。 (3)给出在线性坐标变换和反馈下等价到上三角系统,下三角系统及同时是上三角系统和下三角系统的条件。 (4)利用奇异分布的有关理论,给出了非自治非线性系统通过反馈和坐标变换等价于非自治下三角形式系统的充要条件。 (5)给出了非自治的非线性系统通过反馈和坐标变换等价于非自治p-标准型的两种充要条件。 第四章研究离散时间非线性系统等价性,目的是将较复杂的系统转换成为较为简单的系统。在离散时间非线性系统中,关于具有三角形式的系统研究较为为丰富。本章的主要工作是: (1)给出单输入离散时间系统反馈等价于具有下三角形式的非线性系统的充要条件;并给出离散时间系统反馈等价于离散p-标准型的充要条件。 (2)给出单输入离散时间系统反馈等价于具有严格上三角形式的非线性系统的充要条件。 第五章研究具有下三角形式的非线性控制系统的后推法,主要工作有: (1)对反馈形三角非仿射系统,推进了有界反馈后推法的成果,使我们可以在更广泛的情况中应用有界反馈后推法。我们的方法的进步在于,第一,限制条件更少,尤其是我们可以在不假设上一步的虚控制有界的情更多还原

【Abstract】 This paper is concerned with control strategies of the triangular nonlinear control systems. It is the fundamental idea of this paper that, for a class of nonlinear system, one may design controller by equivalently transforming this system to a triangular nonlinear systems and then using the technique of backstepping or forwarding to design the controllers.In chapter 1, we reviewed the history of control theory, especially the progress of the canonical form of nonlinear systems and the backstepping approach.Chapter 2 includes a brief introduction of differentiable manifold.In chapter 3, using differential-geometric control theory, we solved some problems about the equivalence of nonlinear systems which concern whether a complicated system may be, in some sense, equivalent to a simpler system. The major contributions of this chapter are as follows:(1) Two criteria for converting an ordinary differential equation to normal form via coordinate transformation were presented. Firstly, it was proved that ordinary differential equations can be converted to normal forms if and only if there exists a single-input control system which treats the vector field of the ordinary differential equation as drift vector field and can be fully linearized by state feedback. Secondly, near the non-singular point, there always exists some coordinate transformation to perform the converting;and near the singular point, the converting can be found if and only if, at this point, the eigenpolynomial of the Jacobian matrix of the vector field is equal to the minimal polynomial of the same matrix.(2) The problem of two affine nonlinear systems equivalent to each other via coordinate transformation was considered. The solution of the problem was given when the distribute spaned by the drift vector field and input vector field of either of the two systems have regular involutive closure.(3) We showed the conditions of nonlinear systems equivalent to, via linear coordinate transformation and feedback, upper-triangular systems, low-triangular systems and the systems which are both upper-triangular systems and low-triangular systems.(4) Using the theories of singular distributions, a necessary and sufficient condition under which an affine non-autonomous system is locally feedback equivalent to, via a change of coordinates and state feedback, a non-autonomous low-triangular system.(5) We presented two necessary and sufficient conditions under which a non-autonomous system is locally feedback equivalent to, via coordinate transformation and state feedback, a non-autonomous p-normal form.In chapter 4, we solved two problems about the equivalence of discrete-time nonlinear systems which concern whether a complicated problem may be, in some sense, equivalent to a simpler system. The major contributions of this chapter are as follows:(1) A necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to a low-triangular nonlinear system via feeback and coordinate transformation. Moreover, a necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to a low-triangular nonlinear system via feeback and coordinate transformation.(2) A necessary and sufficient condition was provided for transforming a discrete-time nonlinear system to an upper-triangular nonlinear system via feeback and coordinate transformation.In chapter 5, we study the backstepping approach of the low-triangular nonlinear systems. The major contributions of this chapter are as follows:(1) For non-affine systems with feedback form, we improve on the former result of bounded backstepping, so the technique can be used in more general cases. The advances of our new approach are as the follows. Firstly, there are fewer restrictions in the new design, especially we reject the assume that the virtual control is bounded. Secondly, the bound of feedback is compressed successfully, in some sense, this result is stronger with the less restriction. Thirdly, the proposed Lyapunov function is not including the saturation functions, which is technically different from all of the former results on bounded backstepping. Finally, the proposed method allows applying to a more general class of non-affine control systems.(2) V° bounded robust control, the purpose of which is to restrict the trajectory of system in a given region, is really practical in engineering. Using the nonsmooth analysis theory, the applications of L°° bounded robust backstepping were extended to nonsmooth control systems. The right sides of considered system equations are locally IT bounded, and the designed Lyapunov functions are regular and local Lipschitz.(3) Consider a hybrid system which is combined by a nonlinear control system and a continuous Petri net, and the latter can represented as a set of min-plus linear algebraic equations. We applied the backstepping approach to make the nonlinear system asymptotically stable and make the Petri net limited buffer capacity.In the chapter 6, we considered the observablity. First, we reviewed the definition of observablity and the necessary and sufficient conditions under which a nonlinear system is completely uniformly locally weakly observable, respectively, a single-output nonlinear system is completely uniformly locally weakly observable. Then, a sufficient condition under which a multi-output nonlinear system is completely uniformly locally weakly observable was given. We also gave a canonical form of these multi-output systems. The sufficient conditions become necessary and sufficient conditions when the system is single-output.In the chapter 7, we applied the nonlinear control theory to model and observe the avermectin fermentative process. According to the mechanism of fermentation, we constructed a mathematic model of the fermentative process. The parameter of the model is identified by genetic algorithm. Based on thetheory of nonlinear observer and the main theorem of the former chapter, discrete and continuous observers are designed for observing the concentration of mycelia respectively.更多还原