Lect_4 Stability of Equilibrium Points

8/9/2019 Lect_4 Stability of Equilibrium Points

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Nonlinear Control

Lecture # 4

Stability of Equilibrium Points

Nonlinear Control Lecture # 4 Stability of Equilibrium Points


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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



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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



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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



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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



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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



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Two-dimensional Systems(n = 2)

Type of equilibrium point Stability PropertyCenter

Stable NodeStable Focus

Unstable NodeUnstable Focus

Saddle



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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



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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



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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



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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



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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



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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]


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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



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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)=



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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



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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



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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



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Example 3.3

x=ax3

A= fx

x=0

= 3ax2x=0

= 0

Stable ifa= 0; Asymp stable ifa 0

When a

Lect_4 Stability of Equilibrium Points

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