STABILITY ANALYSIS OF DYNAMIC SYSTEMS - LAR · PDF file · 2013-10-16The given definition of stability refers to a local stability property (for small perturbations). x=0, u( .) is

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C. Melchiorri (DEI) Automatic Control & System Theory 1

AUTOMATIC CONTROL AND SYSTEM THEORY

STABILITY ANALYSIS OF DYNAMIC SYSTEMS

Claudio Melchiorri

Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione (DEI) Università di Bologna

Email: [email protected]

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Definition of stability

Assume that U, Uf, X, Y are normed vector spaces (equipped with a norm).

Given the initial time instant t0, the initial state x(t0) and the input function u(.), let consider the reference motion

Let us consider a generic dynamic systems (linear/non-linear, time varying, stationary) described by

We are interested to study two types of “perturbations”:

1) Variation of motion due to a perturbation of the initial state:

2) Variation of motion due to a perturbation of the input:

Difference between motions

Difference between motions

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Local stability

The motion x(.) corresponding to the input function u(.) is stable with respect to perturbations of the initial state if:

Initial state perturbation

The motion x(.) is asimptotically stable with respect to perturbations of the initial state if:

if

if

R3

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The motion x(.) corresponding to the input function u(.) is stable with respect to perturbations of the input function if:

Input perturbation

Notes 1.  These definitions of stability refer to a particular motion, defined by the

three elements t0, x(t0), u(.).

2.  The stability definitions, besides to ϕ(t,t0,x(t0),u(.)), can also be applied to the response function γ(t,t0,x(t0),u(.)).

3.  Stability can be referred to an equilibrum state x(t)=ϕ(t,t0,x(t0),u(.)), that is a particular motion of the system.

Local stability

if

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Local stability

6.  The given definition of stability refers to a local stability property (for small perturbations).

x=0, u(.) is an equilibrium state: u > Mg stable u < Mg unstable

4.  If the system is stationary, stability does not depend on t0 but on x(t0), u(.) only

5.  A particular motion may correspond to different input functions, and its stability may depend on the applied input

Example:

Notes

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Stability in large Stability in large: initial state perturbations

Let consider the asymptotic stability with respect to perturbations of the initial state; if

∀ δx1(t0) such that x(t0)+δ x1(t0) ∈ X0(t0, x(t0), u(·)), then X0 is called

asymptotic stability domain with respect to t0, x(t0), u(·).

•  If X0(t0, x(t0), u(·)) = X (i.e. the whole state space) then the motion x(·) corresponding to the input function u(·) is globally asymptotically stable.

•  If this condiiton holds for all t0 and all u(·) the system ∑ is globally asymptotically stable.

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Stability in large

Example:

x=0 is an equilibrium point whose stability domain is –X < x < X.

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Stability in large

Stability in large: input perturbations

Let consider again the definition of stability with respect to variations of the input function. The system is bounded input - bounded state stable (b.i.b.s.) with respect to the motion t0, x(t0), u(·) if two positive real numbers Mu and Mx exist, in general function of t0, x(t0), u(·), such that

for all δu(·) such that ||δu(t)|| < Mu , t ≥ t0. Similarly, by considering the response function γ(t,t0,x(t0),u(·)), it is possible to define the bounded input – bounded output stability (b.i.b.o.).

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Stability of the equilibrium

Stability of an equilibrium state

Let consider an autonomous (homogeneous) system

Def. 1: xe is an equilibrium point (state) if

Let consider now system (1) and assume, without loss of generality, that the zero state is an equilibrium state (xe = 0), i.e.

0 = f(xe, t), 8t � t0

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The assumption of considering the zero state as an equilibrium state is not restrictive. As a matter of fact, if the equilibrium is for xe ≠ 0 (i.e. f(xe,t)=0), it is possible to consider a new state z and a new function g(z, t) such that

In which, obviously, we have


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Let consider an autonomous system

Def. 2: The zero state of system (1) is stable in the sense of Lyapunov at time instant t0 if for all ε > 0 there exists a parameter η > 0 such that

Def. 3: The zero state of system (1) is asymptotically stable in the sense of Lyapunov at time instant t0 if it is stable and if

Note: These definitions apply to the equilibrium state, not to “the system.” A dynamic system can have several equilibrium points.

Note: A system is autonomous if it is without input and does not depend explicitly on time.

x(t) = f(x(t)), (1)

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Def. 4: The zero state of system (1) is said to be globally asymptotically stable or asymptotically stable at large in the sense of Lyapunov at time instant t0 if it is stable and if

where X is the whole state space. In this case, since no other equilibrium state exists, the system (1) is said to be globally asymptotically stable.

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In summary:

1.  Stable 2.  Asymptotically stable 3.  Unstable

1

2

3

ε

η x1

x2

t

t

If the previous definitions hold for all initial time instants t1 ∈ [t0, ∞], the stability property is said to be uniform in [t0, ∞].

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Def. 5: The zero state of system (1) is said to be (locally) exponentially stable if there exist two real constants α, λ > 0 such that

|| x(t) − xe || ≤ α ||x(0) − xe|| e−λt ∀t > 0

whenever ||x(0) − xe|| < δ. It is said to be globally exponentially stable if this property holds for any x ∈ Rn. Note: Clearly, exponential stability implies asymptotic stability. The converse is, however, not true.

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Example: the integrator

We want to study the time evolution of the state x(t) with initial condition x0 and null input. 1) Time evolution of the state x(t) due to perturbations of x0. Given x0 x(t) = x0 Given a perturbation of the initial state δx x(t) = x0 + δx The perturbations on the time evolution of the state and of the output are bounded

(globally stable in the sense of Lyapunov – non asymptotically stable) 2) Time evolution of the state x(t) due to perturbations of the input. Given x0 and u(t)=0 x(t) = x0 In case of a (constant) perturbation of the input δu x(t) = x0 + δu t The perturbations on the time evolution of the state and of the output are not bounded,

x(t) → ∞ for t → ∞ (non b.i.b.s. stable and non b.i.b.o. stable)

u(s) y(s)

Integrator

s 1

The integrator as a dynamic system

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Analysis of stability for dynamic systems

The study of the stability properties is of fundamental importance for the analysis of dynamic systems and the design of control laws. In the following, we discuss two methods applicable to both linear and non linear systems expressed in state form 1)  First Lyapunov method (local linearization)

2)  Second Lyapunov method (direct method)

x(t) = f(x, u, t), x(t0) = x0

Aleksandr Mikhailovich Lyapunov (1857 – 1918) was a Russian mathematician, mechanician and physicist. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory.

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Let us consider the autonoumous system

where x = 0 is an equilibrium point. If it is possible to compute the derivative of the functions f(·), then in a proper neighbourhood of the origin it is possible to write, by using the Taylor series expansion:

where f0(x(t)) are the residuals of the Taylor series expansion, assumed negligible. Since f(0) = 0, by defining

then the homogeneous linear system

represents the local linearization or local approximation of the non linear system (2).

First Lyapunov method (local linearization)

x(t) = f(x(t)) = f(0) +�f

�x

��x=0

x(t) + f0(x(t))

x(t) = A x(t) (3)

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Theorem: (Lyapunov linearization method)

1) If the linearized system (3) is strictly stable (all the eigenvalues

of A are with negative real part), the equilibrium point is asymptotically stable for system (2).

2) If the linearized system (3) is unstable (at least an eigenvalue

of A has real positive part), the equilibrium point is unstable for system (2).

3)  If A has eigenvalues with null real part, the linearization does not give information on the stability of the considered equilibrium point.

First Lyapunov method (local linearization)

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Asimptotycally stable system

First Lyapunov method (local linearization) – Example 1

�f(x)

�x=

2

4�f1(x)�x1

�f1(x)�x2

�f2(x)�x1

�f2(x)�x2

3

5=

"� sinx2 � 2x1 � cosx1 �x1 cosx2

2x1 �2 cosx2

#

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Original system

0 1 2 3 4 5 0

0.2

0.4

0.6

0.8

1

1.2

1.4 Evolution of the original and of the linearized system (dash)

Linearized system

First Lyapunov method (local linearization) – Example 1

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First Lyapunov method – Example 2 - Pendulum Gravity force

Viscous friction Input

If u(t) = 0, the system has 2 equilibrium states:

m g

xe,2 =

⇡

0

�xe,1 =

00

�

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Then:

UNSTABLE STABLE An eigenvalue is positive

First Lyapunov method – Example 2 - Pendulum

�f(x)

�x=

2

4�f1(x)�x1

�f1(x)�x2

�f2(x)�x1

�f2(x)�x2

3

5=

"0 1

� gL cosx1 � d

L2m

#

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By considering d = 0, then:

??????? UNSTABLE

First Lyapunov method – Example 2 - Pendulum

�f(x)

�x=

2

4�f1(x)�x1

�f1(x)�x2

�f2(x)�x1

�f2(x)�x2

3

5=

"0 1

� gL cosx1 � d

L2m

#

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Lyapunov method

A function V(x): Rn -> R is positive definite in ϑ(0,ρ), a spherical neighbourhood of the origin of the state space, if 1.  V(0) = 0; 2.  V(x) > 0 ∀ x ≠ 0

Positive definite functions:

Lyapunov functions: A function V(x): Rn à R is a Lyapunov function in ϑ(0,ρ) for system (2) if

1.  V(x) is positive definite in ϑ(0,ρ);

2.  V(x) has continuous first-order partial derivatives with respect to x;

3. 

Definitions

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Lyapunov method

-5

0

5

-5

0

50

5

10

15

20

25

-5

0

5

-5

0

50

5

10

15

20

25

Positive definite functions

-4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4

-4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4

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Lyapunov method

-4-2

02

4

-4-2

02

4-6

-4

-2

0

2

4

6

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

-4-2

02

4

-4-2

02

4-300

-200

-100

0

100

200

300

X

Positive definite function Undefinite function

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1.  (simple stability): The zero state is stable in the sense of Lyapunov if there exists a Lyapunov function V in a neighbourhood of the origin ϑ(0,ρ) .

2.  (asymptotic stability): The zero state is asymptotically stable (AS) in the sense of Lyapunov if there exists a Lyapunov function V in a neighbourhood of the origin ϑ(0,ρ) such that dV(x)/dt < 0 for all x ∈ ϑ(0,ρ), x ≠ 0.

3.  (global asymptotic stability): System (2) is globally asymptotically stable (GAS) in x = 0 if there exists a Lyapunov function V defined in all the state space such that:

(radially unlimited Lyapunov function).

Lyapunov theorems

Lyapunov method

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4.  (asymptotic stability domain): Let V(x) be a Lyapunov function and h a positive real number such that the (open) set

is bounded and let dV(x)/dt < 0 for all x ∈ D, x ≠ 0. Then, all trajectories starting from a point in D converge asymptotically to zero.

x1

x2

V(x)

t

x(0)

Time evolution of the Lyapunov function

Lyapunov method

Time evolution of the state of an autonomous system

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NOTE: The function describes the variation in time of function V(x) when this is computed along a solution x(t) of the differential equation.

NOTE: The previous theorems can be applied to non autonomous systems This can be made by defining a Lyapunov function V(x,t) (with time) such that and by properly modifying the theorems in order to consider the factor ∂ V/∂ t.

Lyapunov method

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If there exists a function V(x) positive definite, with positive definite derivative in an arbitrary small neighbourhood ϑ(0, ρ) of the origin, that is

then the zero state x = 0 of the system is unstable.

Chetaev theorem (instability theorem)

Lyapunov method

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Consider an autonomous dynamical systems and assume that x = 0 is an equilibrium point. Let V : D → R have the following properties:

(i) V(0) = 0

(ii) ∃ x0 ∈ Rn, arbitrarily close to x = 0, such that V(x0) > 0

(iii) dV/dt > 0 ∀ x ∈ U, where the set U is defined as follows:

U = {x ∈ D : ||x|| ≤ ρ, and V (x) > 0}.

Under these conditions, x = 0 is unstable.

Lyapunov method

Chetaev theorem (instability theorem -- another formulation)

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Lyapunov method Example: mass with spring and dumper

k

m

x(t) b

Function V(x) decreases since a dissipative element (“b”) is present, otherwise V(x) would be constant and therefore the energy would not be dissipated. The system evolves until velocity reaches the zero value, and then stops (in a stable equilibrium configuration). In this configuration, from the diff. equation of the system we obtain: 0 = k x + 0, x = 0 Therefore, indeed the system stops in the zero state x = 0.

mx = �k x� b x

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Lyapunov method

-1 -0.5 0 0.5 1

-3 -2 -1 0 1 2 3 0 10 20 30 40 50

m = 5 K = 50 b = 2

0 5 10 15 20 -3 -2.5

-2 -1.5

-1 -0.5

0 0.5

1 1.5

2

V (x)

V (x)

x = [1, 0]T

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Lyapunov method

-1 -0.5 0 0.5 1

-4 -2

0 2

4 0 10 20 30 40 50

0 5 10 15 20 -4 -3 -2 -1 0 1 2 3 4

m = 5 K = 50 b = 0

V (x)

V (x)

x = [1, 0]T

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Lyapunov method – The pendulum Gravity force


By considering u(t) = 0, the system has 2 equilibrium states:

m g

xe,2 =

⇡

0

�xe,1 =

00

�

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Let define the function V(x) as the sum of the potential and kinetic energy

Clearly, V(x) is positive definite

Lyapunov method – The pendulum Gravity force


m g

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-4-2

02

4

-4-2

02

40

5

10

15

20

25

x2: velocità angolare ω(t)x1: posizione angolare θ(t)

Lyapunov method – The pendulum

x1: angular position θ(t) x2: angular velocity ω(t)

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x 1 An

gula

r pos

ition

θ (t

)

x 2 : angular velocity ω (t) -3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3


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The derivative of V(x) is:

If d = 0:

The system is conservative: V(x) = c, a constant depending on the initial condition

V(x) = c is a closed curve about x = 0

If c is “sufficiently small”, then x(t) is constrained to remain “close” to zero

x = 0 is then a stable equilibrium.

If d ≠ 0:

The system dissipates energy: V(x) decreases and dV(x)/dt < 0

V(x) decreases until the system is “in motion” (velocity non zero)

x goes to 0

x = 0 is then an asymptotically stable equilibrium.


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First order systems Let consider the following first order system:

with f(x) continuous, and the condition:

Let consider the Lyapunov function V = x2, then

The origin x = 0 is a global asymptotically stable (GAS) point. In fact V is unlimited for x ➝ ∞

Applications of the Lyapunov method

x f(x) > 0, x 6= 0 (2)

x+ f(x) = 0 (1)

for

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Given the system

the function f(x)= -sin2x + x verifies condition (2) since:

-10 -5 0 5 10 -10 -8 -6 -4 -2 0 2 4 6 8

10


First order systems - Example

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C. Melchiorri (DEI) Automatic Control & System Theory 42 0 0.5 1 1.5 2 2.5 3 0

1 2 3 4 5 6 7 8 9

Phase plane

Plot of V(x) = x2

x0 = 3

Position Velocity

0 5 10 15 20 -3

-2

-1

0

1

2

3

-1 -0.5 0 0.5 1 1.5 2 2.5 3 -3 -2.5

-2 -1.5

-1 -0.5

0 0.5

1 First order systems - Example


x

x

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Second order systems Let consider the following second order system:

where f and g are continuous functions, and the conditions:


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Consider the Lyapunov function

If the integral function

is unlimited for x ➝ ∞, V is unlimited and the origin is GAS.

Then:

Applications of the Lyapunov method Second order systems

for for

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Equations such as (3) and (4) are common when mechanical or electrical systems are considered with elements able to accumulate energy (potential or kinetic) (mass-spring-damper or LRC).

Let consider the circuit shown in the figure. The equation

The system is GAS

is of type (3) and verifies (4). If x(t)=i(t), then:

The integral is radially unlimited

Applications of the Lyapunov method Second order systems - Example

The Lyapunov function can be considered as the sum of the kinetic and potential energy, the fact that it decreases is related to the dissipativity of the system.

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Applications of the Lyapunov method Second order systems - Example

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Limit cycles and invariant sets

Non-linear systems may have: 1.  Equilibrium points

Ø  stable Ø  unstable

2.  Limit cycles Ø  stable Ø  unstable (closed trajectory)

Definition: Limit set •  A positive limit set S+ is a set in the state space that “attracts” sufficiently close

trajectories when t ➝ ∞ •  A negative limit set S+ is a set in the state space that “attracts” sufficiently

close trajectories when τ ➝ ∞ where –τ, = t

An invariant J is a set defined in the state space with the property that for each initial state belonging to J, the whole trajectory belongs to J for both positive and negative values of t.

-3 -2 -1 0 1 2 3 4

-5 0

5 0 0.2 0.4 0.6 0.8

1

Limit cycles

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Let consider a non-linear autonomous system:

Assume that 1) a positive definite function V1 exists such that in the limited domain

D1={ x: V1(x) < h1} the following condition holds:

2) a positive definite function V2 exists such that in the domain D2={ x: h2< V2(x) < h3} the following condition holds:

D2 can be unlimited: in this case the term h3 is not considered. It is possible to show that the closed domain contains a positive limit set.

Limit cycles Limit cycles and invariant sets

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x1

x2

D1

D1={ x: V1(x) < h1}

D2

D2={ x: h2< V2(x) < h3}

D3

h2 h1

D2 D1 D2

V2

V1

D3 D3

Limit cycles

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Limit cycles Let consider the system

It is simple to verify that a limit cycle exists for r = 1 and that such limit cycle is globally asymptotically stable for the system, that is for any initial conditions the trajectories tend to this limit cycle.

-2 -1 0 1 2 -2 -1.5

-1 -0.5

0 0.5

1 1.5

2

The limit cycle r = 1 is asymptotically stable N.B.: r > 0 in any case

-2-1

01

2

-2-1

01

20

0.2

0.4

0.6

0.8

X1X2

V(z)

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Limit cycles Let consider the system

x1 = x2

x2 = µ (1� x21) x2 � x1

x3 = � x3

Van der Pool oscillator (λ < 0)

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Limit cycles

•  Evolution of the system defined by

In the state space the system has an evolution, with non zero initial conditions, as shown in the figure. NB: this system is linear, then this is NOT a limit cycle!

-4 -3 -2 -1 0 1 2 3 4

-4 -2

0 2

4 0 0.2 0.4 0.6 0.8

1

0 5 10 15 20 -3 -2 -1 0 1 2 3

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Non stationary linear systems

Let us consider the non stationary homogeneous system:

The system is asymptotically stable in the sense of Lyapunov if it is stable and

As known, this system is stable in the sense of Lyapunov if, for all t0 and for all ε > 0, a parameter η > 0 exists such that:

Lyapunov method

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The following theorem relates the stability properties of system (1) to the state transition matrix φ(t,t0).

Theorem: System (1) is stable in the sense of Lyapunov if and only if for all t0 a positive real number M exists such that: System (1) is asymptoticaly stable in the sense of Lyapunov if it is stable and if:


Lyapunov method

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Let us consider the non stationary linear system :

Theorem: System (3) is B.I.B.S. stable if and only if:

The system is B.I.B.S. if for all t0 and for all ε > 0 a parameter η > 0 exists such that if ||u(t)|| < η , t ≥ t0 then

Lyapunov method


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Theorem: System (3) is B.I.B.O. stable if and only if:

Similarly, system (3) is B.IB.O. if for all t0 and for all ε > 0 a parameter η > 0 exists such that if ||u(t)|| < η , t ≥ t0 then

Lyapunov method


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By considering V(x) = xT P x, then:

where:

Let us consider the following linear time invariant system:

If matrix M is positive definite, system (1) is globally asymptotically stable. The following theorem holds: Theorem: The Lyapunov matrix equation (2) admits a unique solution P (symmetric positive definite matrix) for each symmetric positive definite matrix M if and only if all the eigenvalues of A are with negative real part.

Lyapunov method

Continuous-time, time invariant linear systems

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where:

If matrix M is positive definite, system (3) is globally asymptotically stable. The following theorem holds: Theorem: The Lyapunov matrix equation (4) admits a unique solution P (symmetric positive definite matrix) for each symmetric positive definite matrix M if and only if all the eigenvalues of A have magnitude less than 1.

Lyapunov method Discrete time, time invariant linear systems Let us consider the following linear time invariant system:

By considering V(x) = xT P x, then:

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AUTOMATIC CONTROL AND SYSTEM THEORY

STABILITY ANALYSIS OF DYNAMIC SYSTEMS

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STABILITY ANALYSIS OF DYNAMIC SYSTEMS - LAR · PDF file · 2013-10-16The given definition of stability refers to a local stability property (for small perturbations). x=0, u( .) is

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