lyapunov functions for nonlinear discrete-time … functions for inclusions strong stability: if is...
TRANSCRIPT
Lyapunov Functions for Nonlinear Discrete-Time
Systems
Chris Kellett
Outline
• Converse Lyapunov Theorems - Function Construction
• Nonlinear Discrete-Time Systems
• Converse Lyapunov Theorem - Computable Function?
• Relation to Stability Estimates
• Future Ideas
Comparison Functions
• Class- Functions:
• Continuous, strictly increasing, zero at zero
• Class- Functions: unbounded
• Class- Functions:
• Continuous, nonincreasing, zero in the limit
• Class- Functions:
• Class- in first argument, Class- in second
Jose L. Massera, “Contributions to Stability Theory”, Annals of Mathematics, 64:182-206, 1956. (Erratum: Annals of Mathematics, 68:202, 1958).
Wolfgang Hahn, Theorie und Anwendung der direkten Methode von Ljapunov, Springer-Verlag, 1959.C. M. Kellett, “A Compendium of Comparison Function Lemmas”, submitted November 2012.
“The manuscript is a joy to read.”
K ↵ : R�0 ! R�0
↵(s) = tanh(s)↵(s) = sK1
' : R�0 ! R�0L '(t) = e�t
'(t) =1
1 + t
KL � : R�0 ⇥ R�0 ! R�0 �(s, t) = tanh(s)e�t
�(s, t) = se�tK L
Converse Theorems
Suppose there exist , , and positive definite so that, for all
V : Rn ! R�0 ↵1, ↵2 2 K1
ddtV (x) �⇢(|x|)
↵1(|x|) V (x) ↵2(|x|)x 2 Rn
then the origin is globally asymptotically stable for .x = f(x)
Converse theorems: Conditions under which global asymptotic stability (or another appropriate stability notion) implies existence of a Lyapunov function.
Lyapunov’s Second Method
⇢
System Model
System: x = f(x), x 2 Rn
� : R�0 ⇥ Rn ! RnSolutions: ddt�(t, x) = f(�(t, x))
Definition: A system is KL-stable if there exists so that� 2 KL
|�(t, x)| �(|x|, t), 8x 2 Rn, t 2 R�0.
Proposition: A system is KL-stable if and only if the origin is uniformly stable and uniformly globally attractive.
Teel and Praly, “A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions”, ESAIM Control Optim. Calc. Var., 2000.
Continuous-Time Constructions
Jose L. Massera, “On Liapunoff’s Conditions of Stability”, Annals of Mathematics, 1949.Taro Yoshizawa, “Stability Theory by Liapunov’s Second Method”, 1966.
x = f(x), x 2 Rn � : R�0 ⇥ Rn ! RnSystem
V (x) .=Z 1
0�(|�(⌧, x)|)d⌧
- Positive definite and decreasing along trajectories- Regularity
Massera-type constructions:
V (x) .= supt�0
↵(|�(t, x)|)(t)
Yoshizawa-type constructions:
- Positive definite and nonincreasing along trajectories- Regularity
System: Solution Set:
Differential Inclusions
x 2 F (x) S(x)
|�(t, x)| �(|x|, t), 8t 2 R�0
Strong stability:There exists � 2 KL so that for all x 2 Rn
and for all � 2 S(x)
There exists � 2 KL so that for all x 2 Rn,
there exists � 2 S(x) so that
|�(t, x)| �(|x|, t), 8t 2 R�0
Weak stability:
Lyapunov Functions for Inclusions
Strong stability: If is strongly KL-stable and if satisfies some technical conditions, then there exists a Lyapunov function
x 2 F (x) F (·)
↵1(|x|) V (x) ↵2(|x|)sup
w2F (x)hrV (x), wi �V (x)
V (x) .= sup�2S(x)
supt�0
↵(|�(t, x)|)(t)Candidate function:
Weak stability: Similar converse theorem (small caveat)
V (x) .= inf�2S(x)
supt�0
↵(|�(t, x)|)(t)Candidate function:
A.R. Teel and L. Praly, “A Smooth Lyapunov Function from a Class-KL Estimate Involving Two Positive Semidefinite Functions”, ESAIM Control Optim. Calc. Var., 2000.
C.M. Kellett and A.R. Teel, “Weak Converse Lyapunov Theorems and Control-Lyapunov Functions”, SIAM J. Control Optim., 2004.
Discrete-Time Systems
System:
There exists � 2 KL so that for all x 2 Rn
|�(k, x)| �(|x|, k), 8k 2 Z�0
KL-stability:
x
+= f(x), f(·) continuous
Converse Lyapunov Theorem:
↵1(|x|) V (x) ↵2(|x|)
V (f(x)) �V (x)
Kellett and Andrew R. Teel, “Smooth Lyapunov functions and robustness of stability for differenceinclusions”, Systems & Control Letters, 2004.
Kellett and Teel, “On the robustness of KL-stability for difference inclusions: Smooth discrete-timeLyapunov functions”, SIAM J. Contr. Opt., 2005.
If x
+= f(x) is KL�stable, then there exists a
continuous V : Rn ! R�0 so that, for all x 2 Rn,
Lyapunov Function Candidates
• Massera Construction
• Yoshizawa Construction
V (x) =1X
k=0
�(|�(k, x)|)
V (x) = supk�0
↵(|�(k, x)|)(k)
For every � 2 (0, 1), � 2 KL there exists ↵1, ↵2 2 K1so that
Sontag’s Lemma on KL-Estimates
↵1(�(s, k)) ↵2(s)�2k, 8s 2 R�0, k 2 Z�0
Yoshizawa Specifics
V (x) .= supk�0
↵1(|�(k, x)|)��kCandidate Lyapunov Function:
↵1(|x|) V (x) supk�0
↵1(�(|x|, k))��k
supk�0
↵2(|x|)�2k�
�k ↵2(|x|)
Upper and Lower Bounds:
V (�(1, x)) = supk�0
↵1(| (k,�(1, x))|)��k
= supk�1
↵1(|�(k, x)|)��k+1
supk�0
↵1(|�(k, x)|)��k+1 = �V (x)
Decrease Condition:
Proving Continuity
Note that �
�
K(x) V (x)
↵2(|x|)�
V (x) = max
(sup
k2{0,}↵1(|�(k, x)|)��k
, sup
k�↵1(|�(k, x)|)��k
)
max
(sup
k2{0,}↵1(|�(k, x)|)��k
, sup
k�↵2(|x|)�2k
�
�k
)
max
(sup
k2{0,}↵1(|�(k, x)|)��k
, V (x)�
)
Calculate
Let µ.= ��1.
Fix � K(x)
.
=
⇠logµ
✓↵2(|x|)V (x)
◆⇡+ 1, x 6= 0
Finite Time Optimization Problem
K(x) =
⇠logµ
✓↵2(|x|)V (x)
◆⇡+ 1
⇠logµ
✓↵2(|x|)↵1(|x|)
◆⇡+ 1
.
= K(x)
Recall � 2 (0, 1), µ = ��1and consider
V (x) = max
k2{0,...,K(x)}↵1(|�(k, x)|)��k
Then a Lyapunov function is given precisely by
Recall ↵1, ↵2 2 K1 such that ↵1(�(s, k)) ↵2(s)�2k
Stability Estimates
• How to generally find a stability estimate ?
• Proof of Sontag’s Lemma on KL-estimates is nonconstructive
� 2 KL
↵1(s) =
Z s
0⇡(⌧)d⌧, ⇡(s) =
12�(0)e
�2���1(s)
↵2(s) = max
n
p
↵1(�(s, 0)), ↵1(�(s, 0))e�↵�1(s)o
Exponential Stability
Fix � = e
�1. Then K(x) =
⇠ln
✓↵2(|x|)↵1(|x|)
◆⇡+ 1.
Consider |�(k, x)| ↵(|x|)e�⌘k, ⌘ > 0, ↵ 2 K1
↵1(�(s, k)) ↵2(s)e�2k
Then ↵1(s).= s2/⌘, and ↵2(s)
.= (↵(s))2/⌘ satisfy
and K(x) =
⇠2
⌘
log
✓↵(|x|)
|x|
◆⇡+ 1
If ↵(s) = Ms ) K(x) =l
2⌘ lnM
m+ 1.
⌘ = 0.1, M = 2 ) K(x) = 15
⌘ = 10, M = 10 ) K(x) = 2
Other Estimates
K(x) =⇠ln
✓M |x|
ln(1 + |x|)
◆⇡+ 1
M = 10 : |x| = 1 ) K(x) = 4
|x| = 100 ) K(x) = 7
�(s, t) exp(Mse�2t)� 1 ) ↵1(s) = ln(1 + s), ↵2(s) = Ms
K(x) =⇠ln
✓M
2s
2
(es � 1)2
◆⇡+ 1
M = 10 : |x| = 1 ) K(x) = 5
|x| = 2 ) K(x) = 4
�(s, t) ln(1 + Mse�t) ) ↵1(s) = (es � 1)2, ↵2(s) = M2s2
Benefits of Yoshizawa-Type
- Provides an exact (not approximate) Lyapunov function
- Finite-time optimization problem
- (Hopefully) short time horizons dependent on stability estimate
- Continuous Lyapunov function
V (x) = max
k2{0,...,K(x)}↵1(|�(k, x)|)��k
K(x) =
⇠logµ
✓↵2(|x|)↵1(|x|)
◆⇡+ 1
x
+ = f(x)
Similar Situations
Difference Inclusions x
+ 2 F (x)
V (x) .= supk�0
sup�2S(x)
↵1(|�(k, x)|)��k
x
+ 2 F (x) .=\
�>0
f(x + �Bn)
Discontinuous Discrete-Time Systems
Kellett and Andrew R. Teel, “Smooth Lyapunov functions and robustness of stability for differenceinclusions”, Systems & Control Letters, 2004.
Kellett and Teel, “On the robustness of KL-stability for difference inclusions: Smooth discrete-timeLyapunov functions”, SIAM J. Contr. Opt., 2005.
Conclusions
• Brief overview of converse Lyapunov theorem constructions
• Massera vs. Yoshizawa type constructions
• Yoshizawa-type leads to a finite-time optimization problem, which in discrete-time may be easily computable
• Difficulty: Need to find stability estimates AND “solve” Sontag’s Lemma
• Difficulty: No estimate on modulus of continuity
• Ideas for control-Lyapunov functions?