滑模与非线性控制：变结构系统讲座笔记

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《变结构系统、滑模控制与非线性控制》是Springer出版于1999年的LectureNotes in Control and Information Sciences系列中的第247卷，由M. Thoma编辑，由K.D. Young和U. Ozguner共同担任合著者。这本书专为控制领域的专业人士设计，深入探讨了变结构控制系统的理论与应用，滑模控制（SMC）以及非线性控制技术。在本书中，作者们首先介绍了变结构控制的基本概念，这是一种动态调整系统结构来应对复杂环境或避免混沌现象的方法。它强调了结构的变化对系统性能的影响，特别是在面对不确定性和快速变化的系统需求时，这种灵活性显得尤为重要。书中可能涵盖了变结构控制器的设计原则，如切换律、状态反馈和适应性控制策略。滑模控制部分则是讨论的核心内容之一，它是一种通过引入斜面（或滑模面）来实现系统快速跟踪和鲁棒性的控制方法。滑模控制器能够有效地处理参数不确定性、外部扰动和模型简化带来的挑战，其优点包括快速响应、无需要精确模型以及较强的抗噪声能力。书中可能会涉及滑模控制的稳定性分析、控制器设计方法、以及如何在实际系统中实施和优化滑模算法。非线性控制理论部分则深入探讨了诸如混沌理论、分段光滑系统、非线性动力学等高级主题。读者可以在这里学习到如何分析和设计非线性系统的控制器，如何利用Lyapunov稳定性理论、反馈线性化技术、以及基于神经网络或其他智能算法的非线性控制方法。此外，本书还可能包含了案例研究和实践应用，帮助读者将理论知识转化为实际工程解决方案。每一章末尾通常会有习题和思考题，鼓励读者进行深入理解和独立研究。书后的参考文献也提供了进一步探索相关领域的广阔途径。《变结构系统、滑模控制与非线性控制》是一本实用的工具书，对于寻求理解复杂系统动态行为、提升控制性能以及解决现代工程问题的工程师和技术人员来说，具有很高的价值。无论是初次接触这些领域的研究人员，还是经验丰富的专业人士，都能从中受益匪浅。

[1511"br any controllable plant (9), the PR-sliding sector defined

Theorem 2.2

in (14) can be rewritten as

S = {x] ]s(x)] < 5(x) } (21)

which is determined by linear function s(x) and quadratic function 52(x)

s(x) = Sx(t), S = BT p (22)

~(x) = ~/xT(t)Ax(t), A e R"• >_ 0 (23)

where P is the solution of the following Riccati equation:

PA + AT p -- PBBT p = -Q, (24)

R E R '~• is a symmetric positwe semi-definite matrix which is chosen so that

A = Q - R > O, Q E R '~xn is a symmetric positive definite matrix.

In conclusion of the section, steps to design a PR-sliding sector (21) using

Riccati equation are given as follows.

1. Choose a n • n positive definition symmetric matrix as Q,

2. Solve the Riccati equation (24) for the positive definition symmetric solution

3. Choose a positive constant r ( 0 < r < 1) and let A = rQ and R = (1 - r)Q.

3 VSC for Continuous-time Systems

The VS controller with PR-sliding sector for continuous-time systems is designed

such that

1. the system state is contro]led to move from the outside toward the inside of

the PR-sliding sector if the initial state is outside the PR-sliding sector,

2. the P-norm decreases in the state space.

To design a chattering free VS controller based on the PR-sliding sector, an

inner sector S~ and an outer sector So are defined as subsets of the PR-sliding

sector (21), i.e.

s~ = {zl Js(x)l < a~(x) } (25)

So = {x[ aS(x) < Is(x)l <_ 5(x) } (26)

where positive constant a satisfies that 0 < a < 1. It is obvious that S = Si U So

and 8~ A So = 9 where ~ is an empty set in R'.

With the inner sector Si (25) and the outer sector So (26) of the PR-sliding

sector (21), the VS control law designed in the paper will be active to move the

system state from the outside of the PR-sliding sector to the inside of the inner

sector, then will become zero as soon as the state enters the inner sector and

will not be active until the system state moves out of the PR-sliding sector.

4.1 Robust VSC

The robust VS control law is designed to move the system state into the inner

sector of the PR-sliding sector (21) and to let the P-norm decrease in the state

space even if there exist parameter uncertainty and external disturbance.

Theorem 4.1 The robust VS control law

u(t) = a(x(t), 5(x) )uo(t) + (1 - a(x(t), 5(x) ) )u~ (t) (31)

quadratically stabilizes the uncertainty plant (30) i/the pair (A, B) in (9) is

controllable, where the hysteresis dead-zone/unction a(s(x),5(x)) on s(x) and

5(x) is defined in (29), the PR-sliding sector ,S (21) is designed in Theorem 2.2

with A = rQ and R -- (1 - r)Q for some positive definite symmetric matrix

Q E R n•200 and some positive constant r (0 < r < 1), the inner sector $~ and

the outer sector $~ of the PR-sliding sector (21) are defined in (25) and (26),

respectively, and u~(t) and uo(t) are defined as

u~(t) = -(r + kl (x, t))sgn(s(x)) (32)

Uo(t) = -(SB)-I(SAx(t) + ks(x)) - (r + k2(x, t))sgn(s(x)) (33)

k, kl (x, t) and k2(x, t) satisfy the/ollowing inequalities, respectively.

> max{ ~B_, ko }, (34)

kl(X,t) > H(x,t) Vx G S (35)

k2(x,t) > H(x,t) + (SB)-IISAxl Vx~8, (36)

and the positive constant ko enables the following inequality hold.

2kor~Q + (ST SA -4- AT sT s) > O.

Proof. Because it is assumed that the pair (A, B) is controllable, the PR-sliding

sector can be designed for the nominal system (9) by using Riccati equation as

shown in (21).

Consider the square of the P-norm as a Lyapunov function candidate, i.e.,

L(t) = xT(t)Px(t) > O, Vx E Rn,x # O. (37)

It is assumed that the initial state is outside the PR-sliding sector. In this

case, the control law u(t) should be uo(t) (33), i.e.,

u(t) = uo(t) = -(SB)-I(SAx(t) + ks(x)) - (r + k2(x, t))sgn(s(x))

which will be active before the system state is moved into the inner sector. With

the control law uo(t) (33), tile derivate of s2(x) satisfies that

d 2

~s (~) = 2s(x)~(~)

= 2s(x)(SAx(t) + SBh(x, t) + SBd(t) + SBuo(t))

= -2ks2(x) + BTpBs(x)(h(x, t) + d(t) + (r + k2(x, t))sgn(s(x)))

<_ -2ks2(x) < 0

which means that the absolute value of the linear function

s(x)

will decrease

so that the system state moves inside the inner sector in a finite time if the

decreasing rate of (f(x) is slower than the one of

Is(x)l

for some large enough k.

While the system state is being moved from the outside of the PR-sliding

sector to the inside of the inner sector by the VS control law (33), the P-norm is

decreasing because according to the definition of the PR-sliding sector (21) with

A = rQ

and R = (1 -

r)Q,

it follows from [s(x)l >

aS(x)

and the inequality

(34) that

L(t) = (Ax(t) + Bu(t))Tpx(t) + xT(t)P(Ax(t) + Bf(x, t) + Bd(t) + Bu(t))

= xT(t)(ATp + PA)x(t) + 2xT(t)PB(f(x, t) + d(t) + u(t))

= s2(x) - ~2(x) - xr(t)Rx(t) + 2s(x)(y(z, t) + d(t) + ~(t))

< -(2(SB)-lk -

1)s2(x) -

2s(x)(SB)-lSAx(t) -

52(x) -

xT(t)Rx(t)

<_ -(2(SB)-l k -

1)a52(x)

2xT(t)sT (sB)-I SAx(t)

(f2(x) -

xT(t)Rx(t)

< -2(SB)-lka~2(x) - 2xT(t)sT(SB)-ISAx(t) - xr(t)Rx(t)

= -(SB)-lxT(t)(2kraQ + STSA + ATsTs)x(t) - xT(t)Rx(t)

< --xT(t)Rx(t)

where

SB = BT pB > O.

After the state is moved into the inner sector and before goes out of tile

PR-sliding sector, the control input

u(t)

should be

ui(t)

(32), i.e.,

u(t) = u~(t) =

-(r

+ kl (x, t))sgn(s(x))

In this case, Is(x)] < 5(x) and the derivate of the Lyapunov function

L(t)

L(t) = s2(x) -

52(x) - xr (t)Rx(t) + 2s(x)(f(x, t) + d(t) + ui(t))

< --xT(t)Rx(t).

If the system state goes out of the PR-sliding sector with

ui(t)

(32), the

control input

Uo(t)

(33) will let it back to the inside of the inner sector again

while the P-norm keeps decreasing.

Thus the system state will be moved from the outside of the PR-sliding

sector to the inside of the inner sector and the Lyapunov function

L(t)

(37)

keeps decreasing in the state space with the VS control law (27), i.e.

L(t) < -xr(t)Rx(t) < O, Vx 9 R"

which means that the VS control law (27) quadratically stabilizes the plant (9).

Remark 4. 1

The control law ui(t) inside the PR-sliding sector includes the

sign function

sgn(s(x)) ,

which may result in the chattering for some systems.

If the external disturbance is not taken into consideration, i.e., letting d(t) = 0

in (30), then u~(t)

does

not included r If the unknown parameter

matrix t9 is estimated by some adaptive rule, then a chattering free adaptive VS

controller is available for the uncertainty plant (30) without external disturbance.

Such adaptive VS control is proposed in the next subsection.

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