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第 四 章. 控制系统的稳定性分析 ----Lyapunov 稳定性理论. 稳定性是系统正常工作的必要条件,它描述初始条件下系统方程解是否收敛,而与输入作用无关。. 给定一个静止的系统, ( 1 )施加一个初始扰动,它 是否会恢复到静止状态; ( 2 )在持续扰动下,系统的输出是否有界;. 不同的稳定性概念 : ( 1 )李雅普诺夫意义下的稳定性(内部稳定性); ( 2 )输入输出稳定性 ( 外部稳定性, BIBO 稳定性 ) 。. 输入输出稳定性. 经典控制理论:

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第 四 章

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----Lyapunov


1

2

1

2(BIBO)



1892Lyapunov,

18571918

  • 1892

  • 190715

  • 1992100


Lyapunov

__,

__Lyapunov



Lyapunov

Lyapunov

LyapunovLTI

MATLAB


SISO

X0=0|u|m1|y|m2

BIBOP133~134


MIMO

t0=0g(t) G(s) BIBO

kg(t)

G(s) Gij(s)


  • 4


t xe

1

Ax=0

A

2

3

4



1()

x0xeS()x(tx0t0)txeS()

xe t0


x2

S()

S()

x0

xe

x1


x2

x0

xe

x1

S()

S()

xe

2

xet

xexe


3

x0

xet0


4

S()x0S()xe

x2

x0

S()

  • ? S()

xe

x1

S()


t0

*


Lyapunov()

:


S

s

As


[4-6]

[]

1

ss=2

2


Taylor

xe=0

f(x)--

h(x)--


Jacobian


  • 1)

    • 2)

    • 3)


  • v(x)

Lyapunov

  • Lyapunov

  • v(x)v(x)


v(x)

1()

2

3 P


V(x)V(x)|x=0=0


1.

Pij=Pji


PSylvester

  • 2. Sylvester

  • P

PV(x)


  • P

  • PV(x)

  • V(x)

  • V(x)


1

1

2

1V(x)


2

1

2

3


3


[1]

[]1

2

1


3

()


[2]

[]1

2

2


[3]

[]:


x1(x10

x2=0 )

[4]

[]:


Lyapunov

1

4

QP


1QQ

2

3 , 2Q Q


[1]

[]


P


[2]K

[]

1


2u=0

1Q

2Pk

4


P

P12-2k>0k>0

k0<k<6


5

Xe=0QP


1

QPP

2 Q


[3]

k

[]


P k<2k0<k<2


MATLAB

1.

eig(A) %0

2.

P=lyap(A,Q) %

P=dlyap(A,Q) %

deltapi=det(p(1:i;1:i)) %i=1~n


  • 4 - 1

  • 4 - 2

  • 4 - 3

  • 4 - 6

  • 4 - 8


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