november 17, 2011 - mech.pku.edu.cnaerocontrol/wangjinzhi/stability-09/lecture... ·...
TRANSCRIPT
CXê�5XÚ
CXê�5XÚ'å~Xê�5XÚ`5,�¹�E,�õ.
1�Ü©¥cn!?ØÙ�~Xê�5XÚ�aq�5�,=�55�, ±ÏXêXÚÚ�êìC½.
14!0�'u)�O�Gronwall-Bellman Ø�ª.
1�Ü©¹5-8!.Ì�l��,*ÿ(�þ?1?Ø, Ñ\Ñѽ�XÚ")ìC½�'X.
() November 17, 2011 2 / 32
CXê�5XÚ�A�
CXê�5XÚx = A(t)x (4.1.1)
Ù¥A(t)´©ãëY¼êÝA(t) : J→ Rn×n, x ∈ Rn.
x = 0´Ù�²ï �.
XÚ(4.1.1)��Ü)|¤�n��5�mS,
S = {x(t)∣x(t) = A(t)x(t)} (4.1.2)
C�x(t0) = x0 → x(t) (4.1.3)
ïá��dx0 ∈ Rn�S��_�5C�.
() November 17, 2011 3 / 32
CXê�5XÚ�A�
∀� ∈ J. Pfj(t, �)�÷vЩ^�
xj(�) = ej
�)§Ù¥e1, e2, ⋅ ⋅ ⋅ , en�g,Ä."Ä�)Ý:
F (t, �) = (f1(t, �), ⋅ ⋅ ⋅ , fn(t, �)) (4.1.4)
(1) dF (t,�)dt = A(t)F (t, �),=F (t, �)´(4.1.1)�Ý);
(2)Ä�)ÝF (t, �)äkeã+(�:
(∀�1, �2 ∈ J) : F (t, �2) = F (t, �1)F (�1, �2)
(∀� ∈ J) : F (�, �) = I
(∀t, � ∈ J) : F (t, �) = F (�, t)−1
�NF (t, �)|¤��C�+,�´Ø����.
() November 17, 2011 4 / 32
CXê�5XÚ�A�
(3)duF (t, �)F (�, t) = I
KkdF (t, �)
dtF (�, t) + F (t, �)
dF (�, t)
dt= 0
½�d/�¤
A(t) + F (t, �)dF (�, t)
dt= 0
dd��F (�, t)÷v�§
dF (�, t)
dt= −F (�, t)A(t)
ePF ∗(t, �)´(4.1.1)��ÝXÚ
x = −AT (t)x
�Ä�)Ý,KF ∗(t, �) = F (�, t)T
() November 17, 2011 5 / 32
CXê�5XÚ�A�
~Xê�5XÚx = Ax (4.1.5)
Ä�)Ý�eA(t−�),ù�C�+´���+.
CXê�5XÚ�(J,Ì�3uvk���{,U3�½ÝA(t)�ÒUá=�äÙéAXÚ�½5.
XÚ(4.1.5)�")ìC½⇔
�(A) ⊂oC− (4.1.6)
éuCXê�5XÚ(4.1.1),=¦k
(∀t ∈ J) : �(A(t)) ⊂oC− (4.1.7)
�Ã{��(4.1.1)�")´ÄìC½.
() November 17, 2011 6 / 32
CXê�5XÚ�A�
~~~4.1.1 �ÄCXê�5XÚ
x =
[−� M [1− '(t)]
M'(t) −�
]x = A(t)x
Ù¥
'(t) =1
2[1 + (−1)i], i ≤ t < i+ 1,
A(t) =
[−� 0M −�
]= A1, 2k ≤ t < 2k + 1,
A(t) =
[−� M0 −�
]= A2, 2k + 1 ≤ t < 2k + 2.
eA1(t−�) = e−�(t−�)[
1 0M(t− �) 1
]eA2(t−�) = e−�(t−�)
[1 M(t− �)0 1
]() November 17, 2011 7 / 32
CXê�5XÚ�A�
�ÄÐ�X(0) = I2�Ý),��X(t).K
X(1) = e−�[
1 0M 1
]
X(2) = e−2�[1 M0 1
] [1 0M 1
]= e−2�
[M2 + 1 MM 1
]dde�´y
X(2n) = e−2n�[M2n + ∗ ∗∗ ∗
]y�� > 0®�½.eÀM¦Me−� > 1,KokX(2n)Ã..u´XÚ")�½Ø½.�é?¿� > 0§k
(∀t ∈ J) : �(A(t)) = {−�,−�} ⊂oC−
() November 17, 2011 8 / 32
CXê�5XÚ�A�
~~~4.1.2 ��XÚ
� = �(t)�, �(t) =
{−4 2k ≤ t < 2k + 12 2k + 1 ≤ t < 2k + 2
XÚ(´ì?½�§��(t) ⊈oC−.
Lyapunov1��{�ä~Xê�5XÚìC½: 3�½W =W T�½e, Ý�§
V A+ATV = −W
´Ä�3�½)V = V T .
() November 17, 2011 9 / 32
CXê�5XÚ�A�
CXê�g.V (t, x) = xTV (t)x
��XÚ�Lyapunov¼ê,K
V ∣(4.1.1) = xT [V (t) + V (t)A(t) +AT (t)V (t)]x
éA�Lyapunov�§C�
V (t) + V (t)A(t) +AT (t)V (t) = −W (t) (4.1.8)
=3�½�½ÝW (t)�, �©�§(4.1.8)´Ä�3�½k.Ý(ÄKV (t, x)�7äá�þ.)V (t).
�±���~XêXÚ3nØþ�éA. ��ý¦)(4.1.8)Ù(J´����.
() November 17, 2011 10 / 32
CXê�5XÚ�A�
�ÄXÚx = A(t)x+B(t)u
F (t, �)´éAgdXÚ(1.1)�Ä�)Ý,K
x(t; t0, x0) = F (t, t0)x0 +
∫ t
t0
F (t, �)B(�)u(�)d� (4.1.9)
3"Ð^�eXÚ�Ñ\�G��m´���5C�
x(t) =
∫ t
t0
F (t, �)B(�)u(�)d�
ù�'X^5ïáCXê�5XÚBIBO½�gdXÚ��ìC½�m�'X. ��5!����5!�*ÿ5����*ÿ5,,,§¤å���^�~Xê�5XÚ¥eA(t−�)��^�aq.
() November 17, 2011 11 / 32
CXê�5XÚ�A�
éuXÚ(4.1.1),ÏÙ?Û)�~ê�þE�).Ï XÚ(4.1.1)��
Ü)�A5þdÐ�u)3�:�����oBr= {x ∣∥ x ∥< r}S)
�5�LyÑ5,Ï éuCXê�5XÚ`5,eã(Ø3ïÄ?Ø�k¿Â.
10XÚ�")´½��duXÚ��Ü)þk..
20XÚ")��ÛáÚ�du")´ÛÜáÚ�.
30XÚ")��Û��ìC½�du")��ìC½.
() November 17, 2011 12 / 32
LyapunovC��±Ï�5XÚ
½½½ÂÂÂ4.2.1 CXê�5C�
x = T (t)z (4.2.1)
¡�´�LyapunovC�,X�10 (∀t ∈ J) : T (t) ∈ C1(J);20 (∃M1,M2 > 0)(∀t ∈ J) : ∥T (t)∥ ≤M1, ∥T (t)∥ ≤M2;30 (∃N > 0)(∀t ∈ J) : ∥T−1(t)∥ ≤ N.e
(∃� > 0)(∀t ∈ J) : T (t) = T (t+ �)
KT (t)´±��±Ï�±Ï�5C�;�T (t)ét ∈ [0, � ]þ�_q��,KT (t)´�LyapunovC�.
�ÛÉ~Xê�5C�Ñ´LyapunovC�.
() November 17, 2011 13 / 32
LyapunovC��±Ï�5XÚ
½½½ÂÂÂ4.2.2 CXê�5XÚ
x = A(t)x (4.2.2)
¡�´Lyapunov¿Âe�z,½�zXÚ,X��3LyapunovC�(4.2.1)ò(4.2.2)z¤~Xê�5XÚ
z = Az, A ∈ Rn×n (4.2.3)
LyapunovC�¿Ø��, Ïd^ØÓLyapunovC�¤��~Xê�5XÚÒ�U�Oé�, ØK�½5.
LyapunovC�9Ù_þ���k.�5C�, ÏdC�c�XÚ�½5�ØC"
() November 17, 2011 14 / 32
LyapunovC��±Ï�5XÚ
~~~4.2.1 ïÄXÚ
x =
[0 1−1 0
]x (4.2.4)
C�
x =
[cos t sin t− sin t cos t
]z
´LyapunovC�. TC�ò(4.2.4)C�
z = 0
ØÓLyapunovC��ò�XÚC�*d¿Ø�q�~Xê�5XÚ.�üXÚ�:þ½�Øì?½.
() November 17, 2011 15 / 32
LyapunovC��±Ï�5XÚ
ÚÚÚnnn4.2.1 �(4.2.1)��LyapunovC�, KÙ_½�LyapunovC�.yyy²²² x = T (t)z�LyapunovC�,KT−1�3�k..-S(t) = T−1(t),duT (t)S(t) = I,K
d
dtT (t)S(t) = T (t)S(t) + T (t)S(t) = 0
u´S(t) = −T−1(t)T (t)S(t)7k.. =z = S(t)x´LyapunovC�.
½½½nnn4.2.1 eXÚ(4.2.2)´�z5XÚ,K
(I) ")½=⇒")��½.
(II) ")ìC½=⇒")��ìC½.
() November 17, 2011 16 / 32
LyapunovC��±Ï�5XÚ
x = A(t)x, V = xTV (t)x (4.2.5)
LyapunovC�x = T (t)z,
z = A(t)z, A = (S + SA)T, S = T−1 (4.2.6)
V = xTV (t)x = zTT TV Tz = zTWz
KW +WA+ ATW = T T (V + V A+ATV )T
=d
dt(zTW (t)z)∣(4.2.6) =
d
dt(xTV (t)x)∣(4.2.5)
() November 17, 2011 17 / 32
LyapunovC��±Ï�5XÚ
�ıÏXê�5XÚµ
x = A(t)x (4.2.7)
�F (t, �)´(4.2.7)�Ä�)Ý,KF (t+ T, �)E,´(4.2.7)�),u´�3���ÝG1(�),¦
F (t+ T, �) = F (t, �)G1(�)
⇒F (t+ T, t) = G1(t)
d��5�F (t, �) = F (t+ T, � + T )
⇒ G1(t+ T ) = G1(t), ¿�´�_�.
() November 17, 2011 18 / 32
LyapunovC��±Ï�5XÚ
�ÝXÚz = −AT (t)z
�´±ÏXê�"F T (�, t)�ÙÄ�)Ý. Ó��3ÝG2(t),¦
F (t+ T, �) = G2(t)F (t, �)
¿�G2(t+ T ) = G2(t). 2-� = t,K
G2(t) = F (t+ T, t) = G1(t)
½½½nnn4.2.2 eF (t, �)´±ÏXê�5XÚ(4.2.7)�Ä�)Ý, K�3����_±Ï¼êÝG(t),¦k
F (t+ T, �) = F (t, �)G(�) = G(t)F (t, �)
�é?Ût��þkG(t)�G(�)�q.
() November 17, 2011 19 / 32
LyapunovC��±Ï�5XÚ
½½½nnn4.2.3 ±ÏXê�5XÚ(4.2.7)´�zXÚ, ��3±Ï�5C�òÙC�¤~Xê�5XÚ,éØÓëê� ó, éAXÚ�m´�q�.
y²µduG(t)´�ÛÉ�,K�3B(t)¦k
G(t) = eB(t)T (4.2.8)
-J(t, �) = F (t, �)e−B(�)t
KJ(t+ T, �) = F (t+ T, �)e−B(�)(t+T )
= F (t, �)G(�)G−1(�)e−B(�)t = J(t, �)
() November 17, 2011 20 / 32
LyapunovC��±Ï�5XÚ
��5C�x = J(t, �)z (4.2.9)
duJ(t, �)´t����ä±ÏT��ÛÉÝ,Ï (4.2.9)��LyapunovC�.
þªü>ét¦�, K
z = J−1(t, �)[A(t)J(t, �)− J(t, �)]z = J−1(t, �)[A(t)J(t, �)
−A(t)F (t, �)e−B(�)t + F (t, �)e−B(�)tB(�)]z = B(�)z
duG(�)3ØÓ��*d�q,K��B(�)3ØÓ��½*d�q££4.2.8¤¤.
() November 17, 2011 21 / 32
LyapunovC��±Ï�5XÚ
5551µ~4.2.1¥
F (t, �) = e
⎛⎝ 0 1−1 0
⎞⎠(t−�)
G(t) = F (t+ T, t) = e
⎛⎝ 0 1−1 0
⎞⎠T
B(�) =
(0 1−1 0
)
J(t, �) = e
⎛⎝ 0 1−1 0
⎞⎠(t−�)e−B(�)t = e
−
⎛⎝ 0 1−1 0
⎞⎠�òXÚz¤~Xê�5XÚ�C��±´õ«/ª�,½n4.2.3�´Ù¥��«, 3ù�a¥éAØÓëê�¤C�¤�XÚ�mâ�½´�q�.
() November 17, 2011 22 / 32
LyapunovC��±Ï�5XÚ
(4.2.8)⇒Λ(G(�)) ⊂
oS1⇌ Λ(B(�)) ⊂
oC−
½½½nnn4.2.4 ±ÏXê�5XÚ(4.2.7)�")ìC½��=�
Λ(G(�)) ⊂oS1= {z∣ ∣z∣ < 1} (4.2.10)
")½��=�
Λ(G(�)) ⊂ S1 = {z∣ ∣z∣ ≤ 1} (4.2.11)
�éA∂S1 = {z∣ ∣z∣ = 1}þG(�)�A���éA�gÐ�Ïf.
() November 17, 2011 23 / 32
LyapunovC��±Ï�5XÚ
XÚ")ؽ��=�
Λ(G(�)) ∩o
Sc1 ∕= � (4.2.12)
½�k(4.2.11),�éA∂S1þG(�)�A����k��éA��g
Ð�Ïf.±þSc1 = {z∣ ∣z∣ ≥ 1},o
Sc1´ÙmØ.
éXÚ(4.2.7),ek(4.2.12),K¡éAXÚ")ؽ���ؽ.
() November 17, 2011 24 / 32
LyapunovC��±Ï�5XÚ
ü�¯¢:
±ÏXê�5XÚ�²LyapunovC�C�¤~Xê�5XÚ.
���5XÚ�gCq�~Xê�5XÚ�,�gCqXÚ�ìC½�y��5XÚìC½. �gCqXÚ��ؽKí���5XÚؽ.
() November 17, 2011 25 / 32
LyapunovC��±Ï�5XÚ
½½½nnn4.2.5 é±ÏXêXÚ
x = f(t, x), f(t, x) ≡ f(t+ T, x) (4.2.13)
Ù�gCqXÚ�
x = A(t)x, A(t) ≡ A(t+ T ) (4.2.14)
Ù¥A(t) = ∂f∂x ∣x=0. K
(I) (4.2.14)")ìC½�y(4.2.13)")ÛÜ�êìC½.(II) (4.2.14)")��ؽ�í�(4.2.13)")ؽ.
() November 17, 2011 26 / 32
LyapunovC��±Ï�5XÚ
5552µµµXJæ^~Xê½±ÏXê�g.��Lyapunov¼ê5ïÄ�5±ÏXÚ�ìC½, éuV (x) = xTV x,
V (x)∣(4.2.7) = −xTW (t)x (A(t)TV + V A(t) = −W (t))
�K½^�£J±��¤�±~f.
duV��½Ý,K2ÂA��¯K£V −1W�A��¤
�V x =Wx
k�����2ÂA���1(t)��2(t),¿k
(∀(t, x) ∈ J×Rn, x ∕= 0) : �1(t) ≥xTW (t)x
xTV x≥ �2(t)
u´dV (x)
dt∣(4.2.7) ≤ −�2(t)V (x)
() November 17, 2011 27 / 32
LyapunovC��±Ï�5XÚ
duA(t) = A(t+ T ),Ï �2(t) = �2(t+ T ). -∫ V (T )
V (0)
dV (x)
V (x)= lnV (x(T ))− lnV (x(0))
∫ T
0−�2(�)d� = −�
K
lnV (x(0))
V (x(T ))≥ �
�� > 0,KkV (x(T ))
V (x(0))< e−� < 1
dd U�äXÚ")�ìC½, ÃI�¦
(∀t ∈ J) : �2(t) ≥ � > 0
ù�¦xTW (t)x�½�^�.
() November 17, 2011 28 / 32
LyapunovC��±Ï�5XÚ
~~~4.2.2 �ÄXÚ{�1 = (−1
2 + "� cos 2t)�1 + (1− "� sin 2t)�2�2 = (−1− "� sin 2t)�1 + (−1
2 − "� cos 2t)�2(4.2.15)
éA" = 0�XÚÝ
C(1) =
[−1
2 1−1 −1
2
]-
V = �21 + �22
9V ∣(4.2.15) = −[�21 + �22 − 2"�[(�21 − �22) cos 2t− 2�1�2 sin 2t]]
() November 17, 2011 29 / 32
LyapunovC��±Ï�5XÚ
u´
W (t) =
[1− 2 cos 2t −2 sin 2t−2 sin 2t 1 + 2 cos 2t
], = "�
l �2(t) = 1− 2 §� < 1/2½" < 1/2��§éAXÚ")´ìC½�.
() November 17, 2011 30 / 32
LyapunovC��±Ï�5XÚ
½½½nnn4.2.6 éuXÚ(4.2.7),Ù")ìC½=I,�3V = V T ∈ Rn×n�½,¦
W (t) = −(V P (t) + P T (t)V )
�V|¤��KÝå(V,W (t))���2ÂA��
� = minx ∕=0
xTW (t)x
xTV x
k ∫ T
0�(t)dt = � > 0
() November 17, 2011 31 / 32