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8/9/2019 Lect_4 Stability of Equilibrium Points
1/20
Nonlinear Control
Lecture # 4
Stability of Equilibrium Points
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
2/20
Basic Concepts
x=f(x)fis locally Lipschitz over a domain D Rn
Suppose x D is an equilibrium point; that is, f(x) = 0
Characterize and study the stability ofx
For convenience, we state all definitions and theorems for thecase when the equilibrium point is at the origin ofRn; that is,x= 0. No loss of generality
y=x x
y= x =f(x) =f(y+ x) def
= g(y), where g(0) = 0
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
3/20
Definition 3.1
The equilibrium point x= 0ofx= f(x) isstable if for each >0 there is >0 (dependent on )such that
x(0) < x(t) < , t 0
unstable if it is not stable
asymptotically stable if it is stable and can be chosensuch that
x(0) < limt
x(t) = 0
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
4/20
Scalar Systems (n = 1)
The behavior ofx(t) in the neighborhood of the origin can bedetermined by examining the sign off(x)
The requirement for stability is violated ifxf(x)>0 oneither side of the origin
f(x)
x
f(x)
x
f(x)
x
Unstable Unstable Unstable
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
5/20
The origin is stable if and only ifxf(
x) 0
in someneighborhood of the origin
f(x)
x
f(x)
x
f(x)
x
Stable Stable Stable
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
6/20
8/9/2019 Lect_4 Stability of Equilibrium Points
7/20
Definition 3.2
Let the origin be an asymptotically stable equilibrium point ofthe system x=f(x), where f is a locally Lipschitz functiondefined over a domain D Rn ( 0 D)
The region of attraction (also called region of asymptoticstability, domain of attraction, or basin) is the set of all
points x0 in D such that the solution of
x= f(x), x(0) =x0
is defined for all t 0and converges to the origin as ttends to infinity
The origin is globally asymptotically stable if the region ofattraction is the whole space Rn
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
8/20
Two-dimensional Systems(n = 2)
Type of equilibrium point Stability PropertyCenter
Stable NodeStable Focus
Unstable NodeUnstable Focus
Saddle
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
9/20
Example: Tunnel Diode Circuit
0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x1
x 2
Q2
Q3
Q1
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
10/20
Example: Pendulum Without Friction
x = y
y = sin(x)
4 3 2 1 0 1 2 3 4
3
2
1
0
1
2
3
x
y
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
11/20
Example: Pendulum With Friction
8 6 4 2 0 2 4 6 84
3
2
1
0
1
2
3
4
x2
B
A
x1
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
12/20
Linear Time-Invariant Systems
x=Ax
x(t) = exp(At)x(0)
P1AP =J= block diag[J1, J2, . . . , J r]
Ji=
i 1 0 . . . . . . 00 i 1 0 . . . 0...
. . . ...
..
.
. .. 0...
. . . 10 . . . . . . . . . 0 i
mm
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
13/20
exp(At) =Pexp(Jt)P1 =r
i=1
mi
k=1
tk1 exp(it)Rik
mi is the order of the Jordan block Ji
Re[i]0 for some i Unstable
Re[i] 0 i& mi >1for Re[i] = 0 Unstable
Re[i] 0 i & mi = 1for Re[i] = 0 StableIf an n n matrix A has a repeated eigenvalue i of algebraicmultiplicity qi, then the Jordan blocks ofi have order one ifand only ifrank(A iI) =n qi
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
14/20
Theorem 3.1
The equilibrium point x= 0ofx= Axis stable if and only if
all eigenvalues ofAsatisfy Re[i] 0and for every eigenvaluewith Re[i] = 0and algebraic multiplicity qi 2,rank(A iI) =n qi, where nis the dimension ofx. Theequilibrium point x= 0 is globally asymptotically stable if and
only if all eigenvalues ofAsatisfy Re[i]
8/9/2019 Lect_4 Stability of Equilibrium Points
15/20
Exponential Stability
Definition 3.3
The equilibrium point x= 0ofx= f(x) is exponentiallystable if
x(t) kx(0)et
, t 0k 1, >0, for all x(0) < c
It is globally exponentially stable if the inequality is satisfiedfor any initial state x(0)
Exponential Stability Asymptotic Stability
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
16/20
Example 3.2
x= x3
The origin is asymptotically stable
x(t) = x(0)
1 + 2tx
2
(0)
x(t) does not satisfy|x(t)| ket|x(0)| because
|x(t)| ket|x(0)| e2t
1 + 2tx2
(0)
k2
Impossible because limt
e2t
1 + 2tx2(0)=
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
17/20
Linearization
x=f(x), f(0) = 0f is continuously differentiable over D= {x < r}
J(x) = f
x
(x)
h() =f(x) for 0 1, h() =J(x)x
h(1) h(0) =
1
0 h
() d, h(0) =f(0) = 0
f(x) =
1
0
J(x) d x
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
18/20
f(x) = 10 J(x) d x
Set A= J(0) and add and subtract Ax
f(x) = [A+G(x)]x, where G(x) =
1
0 [J(x) J(0)] d
G(x) 0 as x 0
This suggests that in a small neighborhood of the origin wecan approximate the nonlinear system x=f(x)by itslinearization about the origin x= Ax
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
19/20
Theorem 3.2
The origin is exponentially stableif and only ifRe[i]0 for some i
Linearization fails when Re[i] 0for all i, with Re[i] = 0for some i
Nonlinear Control Lecture # 4 Stability of Equilibrium Points
8/9/2019 Lect_4 Stability of Equilibrium Points
20/20
Example 3.3
x=ax3
A= fx
x=0
= 3ax2x=0
= 0
Stable ifa= 0; Asymp stable ifa 0
When a
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