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TTK4150 Nonlinear Control Systems
Lecture 5
Stability analysis for autonomous systems
Direct Lyapunov method continued
Invariance principle
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Previous lectures:
stability of autonomous systems
Stability of equilibrium points (autonomous systems)
Denitions (stability, asymptotic/exponential stability, global
asymptotic/exponential stability, region of attraction)
Stability analysis using phase plane analysis
Stability analysis using indirect (linearization) Lyapunov method
Stability analysis using energy functions
Direct Lyapunov method for stability analysis
Theorems for stability, asymptotic stability, global asymptotic stability, exponential
stability, global exponential stability
Application examples
Lyapunov functions: quadratic Lyapunov functions, examples, tricks
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Today:
Direct Lyapunov method for stability analysis
Lyapunov functions: examples, tricks, continued
Invariance principle
Denitions: compact sets, invariant sets, convergence to sets
Invariance principle + corollaries
Application example
Estimation of the region of attraction
Denition, examples
An algorithm
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Literature
Khalil
Chapter 4 (Lyapunov functions tricks and examples)
Chapter 4, Section 4.2, (Invariance principle)
Chapter 8, Secton 8.2 (estimation of the region of attraction)
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Stability notions for autonomous systems
Stability = start close (to equilibrium) => stay close
Asymptotic stability = Stability + Local Convergence
Exponential stability = Stability + Local Exp. Convergence
Global asymptotic stability = Stability + Global Convergence
Global exponential stability = Stability + Global Exp. Conv.
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Terminology
Direct Lyapunov method
stability
asymptotic stability
!
D= R
n
global asymptotic stability
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Lyapunov functions: tips and tricks
8
Invariance principle
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Example: pendulum
Lyapunov function:
1 2
2 1 2
sin
x x
x a x bx
=
= ! !
&
&
!
1
2
x
x
!
!
=
=
&
0, 0 a b > >
!
V(x) =
x
2
2
2
+ a(1"cos x
1
) > 0
Can we conclude asymptotic stability of x=0?
V(x(t)) decreases everywhere where
(stability)
!
"V
"x
(x) f (x) = #bx
2
2
$ 0
!
D={x : x
1
<
"
2
, x
2
# R}
!
x
2
" 0
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Estimates of the region of attraction
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Region of attraction
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Example: Asymptotic stability
Pendulum with friction (l=1, k=1, m=1)
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An estimate of the region of attraction
Starting point: You have proved asymptotic stability of the origin by
either finding a strict Lyapunov function or by using LaSalles theorem
1) Choose the largest set
that is contained in D (where and )
or in which (invariance principle)
and which is bounded
2) Choose the connected component in this set that
contains 0
Then it is a subset of the region of attraction of the
origin, and can hence be used as an estimate
} ) ( : { c x V x
n
c
! " # = $
0 V >
0 V <
&
0 V !
&
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Example: An estimate of the region of attraction
(Do not always trust your intuition)
Equilibrium point (0,0)
Lyapunov linearization method: Locally asymptotically stable
Corollary 4.3: Locally exponentially stable
Is it globally asymptotically/exponentially stable?
Intuition may suggest yes...
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For this particular system it is possible to find an analytic
solution:
The equilibrium point is clearly not globally asymptotically
stable. It is locally exponentially stable and the region of
attraction is given by
Finite escape time!
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-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
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x1-x2 plot
(1,4)
(1,3)
(1,2)
x
1
x
2
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-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x1-x2 plot
(1,4)
(1,3)
(1,2)
x
1
x
2
An estimate of the region of attraction
Note: Conservative
