非连续连接下级联非线性系统的状态反馈稳定研究

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本文档标题"State feedback stabilization of cascaded nonlinear systems with discontinuous connection (2008年)"聚焦于实际工程中的一个关键问题，即非连续连接下的串联非线性系统稳定性控制。在许多工程现象中，如机械系统的设定点控制问题，状态变量的不连续性常常导致控制性能下降，例如静态摩擦力，它作为接触面速度的sgn函数，会引入定位误差。作者Maiko HIRANO、Tielong SHEN、Kazuhisa ITO和Katutoshi TAMURA针对此类问题展开研究。他们首先从一个特殊案例出发，探讨了如何通过状态反馈稳定这种系统结构。状态反馈是一种常见的控制策略，通过实时测量系统的状态并据此调整输入来保持系统稳定。对于存在不连续性的系统，这需要特殊的方法来处理，因为传统的控制理论可能无法直接适用。论文的核心贡献在于提出了一种方法，旨在解决这类非线性系统的稳定问题，并且能够扩展到更一般的情况。他们可能采用了分段连续化技术或者滑模控制等手段来克服不连续性带来的挑战，确保在控制过程中系统的稳定性得到保障。此外，文中还给出了若干数值例子，这些实例展示了他们的方法在实际应用中的有效性，通过对比实验结果，证实了所提理论的实际可行性和优越性。对于工程师来说，这篇论文的重要性在于它提供了一种处理工程中常见非线性问题的新思路，特别是在那些动态特性复杂，受不连续因素影响的系统设计中。通过阅读和理解此篇论文，工程师可以更好地理解和应对系统不稳定或性能退化的挑战，从而提高控制系统的可靠性和精度。同时，这也为控制理论的发展和实际应用之间的桥梁铺设了基础，促进了未来在类似领域的进一步研究和创新。

J Control Theory Appl 2008 6 (1) 45–52

DOI 10.1007/s11768-008-7188-6

State feedback stabilization of cascaded nonlinear

systems with discontinuous connection

Maiko HIRANO

, Tielong SHEN

, Kazuhisa ITO

, Katutoshi TAMURA

(1.Department of Mechanical Engineering, Sophia University, Tokyo 102-8554 Japan;

2.Department of Mechanical Engineering, Tottori University, Tottori 680-8552, Japan)

Abstract: In practical engineering, many phenomena are described as a discontinuous function of a state variable,

and the discontinuity is usually the main reason for the degradation of the control performance. For example, in the set-

point control problem of mechanical systems, the static friction (described by a sgn function of velocity of the contacting

faces) causes undesired positioning error. In this paper, we will investigate the stabilization problem for a class of nonlinear

systems that consist of two subsystems with cascaded connection. We will show the basic idea with a special case ﬁrst, and

then the result will be extended to more general cases. Some interesting numerical examples will be given to demonstrate

the effectiveness of the proposed design approach.

Keywords: Cascaded connection; Discontinuous systems; Filippov solution

1 Introduction

In practical engineering, the cascaded connection of two

subsystems is a typical structure in control systems. The

normal form of a smooth nonlinear afﬁne system is de-

scribed by a nonlinear subsystem connected with an inte-

grator chain in cascaded connection [1]. For example, in

the application of motion control, the controlled mechanical

plant is driven by active forces which are usually provided

by electric or hydraulic actuators, and the latter is forced by

the control signal, which is the design control input.

Indeed, the control design for cascaded nonlinear systems

is an old topic in control theory, and Lyapunov framework-

based design approaches to the feedback stabilizing con-

troller have been well established by many researchers (for

example, [2∼6] and their references). It has been shown

by [2] that if the driving subsystem is linear and stabiliz-

able, and the unforced driven subsystem is asymptotically

stable, the cascaded system may be stabilized by state feed-

back control under the condition that the connecting func-

tion is spanned by the output of the driving subsystem. This

result is established based on the fact that if we can ﬁnd a

Lyapunov function for the unforced subsystem and a stor-

age function for the linear driving subsystem, then the sum

of the functions will play the role of Lyapunov function for

the cascaded system. As shown in [6, 7], this result can be

extended to the case in which the driving subsystem is non-

linear but strictly passive. On the other hand, under the con-

dition that the driven system is input-to-state stable with re-

spect to the driving signal, an alternative design approach

has been provided where additional cross term is required

in the sum. An approach has been presented by [8] for con-

structing the cross term such that the sum plays the role of

Lyapunov function. However, all previous works targeted

the systems that are described by differential equations with

smooth functions.

In this paper, we focus our attention on a class of cas-

caded nonlinear systems with discontinuous connection. As

is well-known, the dynamics of the nonlinear systems with

discontinuity is described by differential equations with dis-

continuous right-hand side, and a feasible way to analyze

the behavior of the solution is the Filippov-framework. We

will target a class of nonlinear systems that consist of two

smooth subsystems with discontinuous connection. We will

show that if the driven system is in the weakly minimum

phase with relative degree one and the driving system is pas-

sive, then we can ﬁnd a state feedback control such that the

closed loop system is globally stable and the state converges

ultimately to any given small closed set. Finally, three nu-

merical examples will be shown to demonstrate the effects

of the proposed controllers.

2 Motivation and problem formulation

Consider the nonlinear systems described as follows:

⎧

⎪

⎨

⎪

⎩

˙x = f

(x)+c(x, sgn(φ(x)),y),

ξ = f

(ξ)+g(ξ)u,

y = h(ξ),

(1)

where f

(·)(i =1, 2), g(·) and c(·, ·, ·) are nonlinear smooth

functions with appropriate dimensions according to the di-

Received 10 September 2007.

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