阿尔贝托·伊西多里 nonlinear control systems 第三版：经典指南

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《阿尔贝托·伊西多里著：非线性控制系统（第三版）》是一本控制理论领域的经典之作，收录在Springer-Verlag出版社的"LectureNotesinControlandInformationSciences"系列中，由M.Thoma担任系列编辑。作者阿尔贝托·伊西多里是意大利罗马大学计算机与系统科学学院的教授，他的专长在于非线性控制系统的理论与应用。本书旨在为读者提供非线性控制系统的基础知识和深入理解，对那些寻求理解和掌握非线性动态系统分析、设计以及控制器开发的工程师和研究人员来说，具有极高的参考价值。伊西多里教授以其丰富的经验和深入研究，将复杂的非线性现象简化，使得非线性控制理论的艰深概念变得易于理解。在第三版中，内容可能包括但不限于以下知识点： 1. **非线性系统概述**：介绍了非线性系统的特性，如状态空间描述、稳定性分析（如Lyapunov稳定性、混沌理论等）、非线性动力学基础，如相图和分岔分析。 2. **Lyapunov方法**：作为非线性控制设计的核心工具，讲解了如何通过Lyapunov函数来证明系统稳定性并设计控制器。 3. **反馈控制理论**：探讨了反馈控制的设计原则，如全状态反馈、输出反馈、自适应控制等，以及它们在处理非线性系统中的应用。 4. **非线性控制器设计**：如滑模控制、自抗扰控制等特殊类型的控制器，以及针对特定非线性系统的优化设计策略。 5. **混沌与同步控制**：对于混沌系统的理解和控制技术，以及如何通过同步技术将其转化为可控制的行为。 6. **应用实例与案例**：书中可能包含实际工业系统中的非线性控制问题，如航空航天、电力系统、机器人技术等领域的应用案例，以展示理论在实践中的应用。 7. **最新进展和技术前沿**：考虑到第三版更新，可能包括了近年来非线性控制领域的最新研究成果，如模型预测控制、控制理论与人工智能的融合等。《NonlinearControlSystems: An Introduction》这本书不仅适合高级学生和研究者深入学习非线性控制，也适合专业人士查阅参考，帮助他们解决复杂工程中的控制问题。在阅读过程中，读者可以期待通过作者严谨的逻辑和丰富的实践经验，提升对非线性控制系统的理论与实践理解。

dimension of a codistribution at p, and construct codistributions by

taking the span of a given set of covector fields, or else by adding

or intersecting two given codistributions, etc. always looking at a

pointwise characterization of the objects we are dealing with.

Sometimes, one can construct codistributions starting from given

distributions and conversely. The natural way to do this is the fol-

lowing: given a distribution A on N, the

annihiZator

of A, denoted A i,

is the codistribution on N defined by the rule

A i :p

v* *

~ { ETpN : (v*,v)= 0 for all v 6 A(p)}

Conversely, the

annihilator

of ~, denoted ~-, is the distribution

defined by the rule

~i ,v )= 0 for all v E ~(p)}

p ~ {v E TpN: (v* *

Distributions and codistributions thus related possess a number

of interesting properties. In particular, the sum of the dimensions of

A and of A ± is equal to the dimension of N. The inclusion A I C A 2 is

± D ±

satisfied if and only if the inclusion A I A 2 is satisfied. The an-

nihilator (A I A &2 )± of an intersection of distributions is equal to

± I

the sum A 1 + A 2.

Like in the case of the distributions, some care is required when

dealing with the quality of being smooth for codistributions construc-

ted in some of the ways we described before. Thus it is easily seen

that the span of a given set of smooth covector fields, as well as sum

of two smooth codistributions is again smooth. But the intersection of

two such codistributions may not need to be smooth.

Moreover, the annihilator of a smooth distribution may fail to be

smooth, as it is shown in the following example.

(2.15)

Example.

Let N =

A =

sp{x ~}

Then

Ai(x) = {0} if x ~ 0

AA(x) = T N if x : 0

and we see that A i is not smooth because it is not possible to find a

smooth covector field on I~ which is zero everywhere but on the point

covector field dh° It is easy to verify that the two objects thus de-

fined are one the annihilator of the other, i.e. that

(sp(dh))± = ker(h#) .

3. Frobenius Theorem

In this section we shall establish a correspondence between the

notion of distribution on a manifold N and the existence of partitions

of N into lower dimensional submanifolds. As we have seen at the be-

ginning of this chapter, partitions of the state space into lower di-

mensional submanifolds are often encountered when dealing with reach-

ability and/or observability of control systems.

We begin our analysis with the following definition. A

nonsing~lar

d-dimensional distribution A

N is

completely integrable

if at each

p 6 N there exists a cubic coordinate chart (V,~) with coordinate func-

tions ~1,...,~n , such that

(3.1) A(q) = span{ (~I) q ..... (~d)q}

for all q C V.

There are two important consequences related to the notion of

completely integrable distribution. First of all, observe that if there

exists a cubic coordinate chart (V,~), with coordinate functions

~I .... '~n ' such that (3.1) is satisfied, then any

slice

of V passing

through any point p of V and defined by

(3.2)

Sp = {q E V:~i (q) = ~i (p) ; i = d+1 ..... n}

(which is a d-dimensional imbedded submanifold of N), has a tangent

space which, at any point q, coincides with the subspace &(q) of T N.

Since the set of all such slices is a

partition

of V, we may see

that a completely integrable distribution A induces, locally around

each point p E N, a partition into lower dimensional submanifolds, and

each of these submanifolds is such that its tangent space, at each

point, agrees with the distribution A at that point.

The second consequence is that a completely integrable distribu-

tion is

inuoiutive.

In order to see this we use the definition of in-

volutivity and compute the Lie bracket of any pair of vector fields

belonging to A. For, recall that in the ~ coordinates, any vector

field • defined on N is represented by a vector of the form

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