微分几何方法与非线性控制系统(5)

DIFFERENTIAL GEOMETRIC METHODS AND NONLINEAR CONTROL SYSTEMS

8 控制系统中的对称结构及相似结构 8.1 对称性及相似性 当前,对系统科学的发展,国内外都给予极大的关注,并致力于从不同的侧面、观点探索其规律。 今考虑自然形成的控制系统,这种系统在其发展过程中,需要适应外界环境,并力求以最佳状态运行,故应是“自寻最优”地逐渐演化而形成其结构。

This paper introduces a new kind of methods—differential geometric methods (DGMs), which are effective in dealing with nonlinear control system. The relations between DGMs and nonlinear control system are discussed in section 2. The next section gives the definitions of (differentisble) manifold and differentiable map. In section 4, the concepts of vector field are given, and the relations between vector fields and dynamic systems are introduced. Lie algebra and Lie derivatives are briefly diacussed in section 5. Section 6 introduces the concepts of distribution and integral manifoid, and gives a very important theorem—Frobenius Theorem. In section 7, the linearization and input-output decoupling problems are investigated respectively. The structure of nonlinear control systems with symmetries are studied, and the properties of large-acale compsite systems with similarity are investigated in section 8.