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?