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Lyapunov stability theory

Consider the nonlinear system

)( x f x = (1)

Let us assume that xeq = 0 is an equilibrium point of (1).

Q.:

What if the equilibrium point xeq ≠ 0?

A.:

There is no loss of generality by this assumption. We can always

choose a shifted coordinates system in the form

y = x – xeq. The derivative of y is given by0)()()( ==+=== y g x y f x f x y eq

The system described by the means of the new variable has the

equilibrium at the origin.

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Definition

The equilibrium point x = 0 of (1) is

• stable, if for each ε > 0, there is δ = δ(ε) > 0 such that

0,)()0( ≥∀ε<⇒δ< t t x x

• unstable, if not stable

• asymptotically stable, if it is stable and δ can be chosen such

that

0)(lim)0( =⇒δ<

∞→

t x xt

• marginally stable if it is stable but not asymptotically stable

δ

ε

xeq

Marginally stable

unstable

Asymptotically stable

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Lyapunov First Method

(The indirect method )

According to the basic definitions, stability properties depend

only on the nature of the system near the equilibrium point.

⇓Let us linearize the system description!

⇓For small deviations from the equilibrium point, the performance

of the system is approximately governed by the linear terms.

These terms dominate and thus determine stability – provided

that the linear terms do not vanish.

The idea of checking stability by examination of a linearized

version of the system is referred to as Liapunov’s first method or

Liapunov’s indirect method .

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Theorem

Let x = 0 be an equilibrium point of a nonlinear system

)( x f x =

where f : D → Rn is continuously differentiable and D is the

neighborhood of the equilibrium point.

Let λi denote the eigenvalues of the matrix0=∂

∂=

x x

f A

1. If Re λi < 0 for all i then x = 0 is asymptotically stable for the

nonlinear system.

2. If Re λi > 0 for one or more i then x = 0 is unstable for the

nonlinear system.

3. If Re λi ≤ 0 for all i and at least one Re λ j = 0 then x = 0 may

be either stable, asymptotically stable or unstable for the

nonlinear system

Conclusion:

Except for the boundary situation, the eigenvalues of the

linearized system completely reveal the stability properties of an

equilibrium point of a nonlinear system.

If there are boundary eigenvalues, a separate analysis is required

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An example

)()()()(

2

2

t f t Kydt

t dy B

dt

t yd M =++

Moreover, since we are interested in stability properties, f (t ) = 0.

0=++ Ky y B y M , equilibrium point : y = 0

State variables

=

=

)()(

)()(

2

1

t yt x

t yt x

−−==

==

212

21

x M B x

M K y x

x y x

f M

K

B y

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The total stored energy is given by

22

21

2

1

2

1)( Mx Kxt V +=

which have the following properties:

• positive for all nonzero values of x1(t ) and x1(t )

• equals zero when x1(t ) = x1(t ) = 0

The time derivative of V (t ) is given by:

2

2

1

1

)()()( x xt V x

xt V

dt t dV

∂∂+

∂∂=

22

)( Bx

dt

t dV −=

dV /dt is negative ⇒ the state must move from its initial state in

the direction of smaller values of V (t )

x1

x2V = C

3

V = C 2

V = C 1

C 1

< C 2

< C 3

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Lyapunov Second Method

(The direct method )

Theorem


)( x f x =

Let V : D → R be a continuously differentiable function on a

neighborhood D of x = 0 , such that

V (0) = 0 and V ( x) > 0 in D – 0,

0)( ≤ xV in D

Then, x = 0 is stable.

Moreover, if 0)( < xV in D – 0 then x = 0 is asymptotically stable

The task:

To find V ( x), called a Lyapunov function, which must satisfy the

following requirements:

• V is continuous

• V ( x) has a unique minimum at xeq with respect to all other

points in D

• Along any trajectory of the system contained in D the value of

V never increases

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What if the stability of x = 0 has been established?

The first Lyapunov method determines stability in the immediate

vicinity of the equilibrium point.

The second Lyapunov method allows to determine how far from

the equilibrium point the trajectory can be and still converge to it

as t approaches ∞ ⇓region of asymptotic stability (region of attraction, basin)

Let φ(t ; x) be the solution of the system equation that starts at

initial state x at time t = 0.

Then the region of attraction is defined as the set of all points x

such that

limt→∞ φ(t ; x) = 0

If Ωc = x ∈ Rn | V ( x) ≤ c is bounded and contained in D, then

every trajectory starting in Ωc remains in Ωc and approaches the

equilibrium point as t → ∞.

Thus, Ωc is an estimate of the region of attraction.

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Types of stability with reference to the region of attraction:

• local stability ( stability in the small ) – when a system remains

within an infinitesimal region around the equilibrium when

subjected to small perturbation

• finite stability – when a system returns to the equilibrium point

from any point within a region R of finite dimensions

surrounding it• global stability ( stability in the large) – if the region R includes

the entire state space

Theorem


)( x f x =

Let V : Rn → R be a continuously differentiable function such that

V (0) = 0 and V ( x) > 0 ∀ x ≠ 0,

|| x|| → ∞ ⇒ V ( x) → ∞

0)( ≤ xV ∀ x ≠ 0

then x = 0 is globally asymptotically stable

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Another example (a pursuit problem)

Suppose a hound is chasing a rabbit (in such a way that his

velocity vector always points directly toward the rabbit).

The velocities of the rabbit and the hound are constant and

denoted by R and H , respectively (see the picture)

Let xr , yr , and xh, yh denote the x and y coordinates of the rabbit

and hound, respectively. Then

0==

=

r r

r

y y

R x

222

H y x hh =+

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The fact that velocity vector of the hound always points toward

the rabbit means that

)(

)(

r hh

r hh

y yk y

x xk x

−−=

−−=

k – a positive constant

So

22

22

)(

)(

)(

hr h

hh

hr h

r hh

y x x

Hy y

y x x

x x H x

+−

−=

+−

−−=

Let us introduce the relative coordinates – the coordinates of thedifference in position of the hound and the rabbit:

h

r h

y y

x x x

=

−=

22

22

y x

Hy y

R y x

Hx x

h

+

−=

−

+

−=

(*)

Will the hound always catch the rabbit?

⇓

Will a trajectory with an arbitrary initial condition eventually get

to the point where the relative coordinates are zero?

⇓

We can consider the origin as an equilibrium point

⇓

What are conditions for global stability of the system?

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⇓

We have to find a suitable Lyapunov function for the system

given by (*)

Let us choose as a Lyapunov function

V ( x, y) = x2 + y2

Then

Rx y x H y xV 22),( 22 −+−=

If H > R:

• if x = 0 and y ≠ 0, it is clear that 0),( < y xV

• if x ≠ 0 then

0)(22<−−<−+− x R H Rx y x H

Thus, 0),( < y xV for all x, y except the origin.

⇓

If the hound runs faster than the rabbit, he always catches the

rabbit

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Comments on the second Lyapunov’s method:

• determines stability without actually having to solve the

differential equation

• can be applied even if the system model cannot be linearized

• allows to estimate the stability region

• in some cases there are natural Lyapunov function candidates,

like energy functions in electrical or mechanical systems

• the stability conditions are sufficient, but not necessary

• there is no systematic method for finding Lyapunov functions

– sometimes a matter of trial and error

• a Lyapunov function for any particular system is not unique

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Nonlinear phenomena:

• Finite escape time: The state of an unstable linear system goes

to infinity as time approaches infinity; a nonlinear system’s

state, however, can go to infinity in finite time.

• Multiple isolated equilibria: a linear system can have only one

isolated equilibrium point; hence it can have only one steady-

state operating point which attracts the state of the system

irrespective of the initial state. A nonlinear system can have

more than one isolated equilibrium point. The state may

converge to one of the several steady-state operating points,

depending on the initial state of the system.

• Limit cycles: For a linear time-invariant system to oscilate, it

must have a pair of eigenvalues on the imaginary axis, which is

a nonrobust condition that is almost impossible to maintain in

the presence of perturbations. Even if we do, the amplitude of

the oscillation will be dependent on the initial state. In real life

stable oscillation must be produced by nonlinear systems.

There are nonlinear systemswhich can go into an oscillation of

fixed amplitude and frequency, irrespective of the initial state

(so called limit cycle)

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• Subharmonic, harmonic or almost–periodic oscillations: A

stable linear system under a periodic input produces an output

of the same frequency. A nonlinear system under periodic

excitation can oscillate with frequencies which are

submultiples or multiples of the input frequency. It may even

generate an almost–periodic oscillation, an example of which

is the sum of periodic oscillations with frequencies which are

not multiples of each other

• Chaos: A nonlinear system can have more complicated steady-

state behavior that is not equilibrium, periodic oscillation or

almost–periodic oscillation. Such behavior is usually referred

to as chaos. Some of these chaotic motions exhibit

randomness, despite the deterministic nature of the system

• Multiple modes of behavior : It is not unusual for two or more

modes of behavior to be exhibited by the same nonlinear

system. An unforced system may have more than one limit

cycle. A forced system with periodic excitation may exhibit

harmonic, subharmonic or more complicated steady-state

behavior, depending upon the amplitude and frequency of the

input. It may even exhibit a discontinuous jump in the mode of

behavior as the amplitude or frequency of the excitation is

smoothly changed

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