representation and stability - purdue university · 3. representation and stability concepts 3.1...

Representation and Stability

Concepts

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ME 680- Spring 2014

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3. Representation and stability concepts

3.1 Continuous time systems:

Consider systems of the form

nx F(x), x (1)

The family of solutions generated by the collection of

solutions through every point 𝑥 0 ∈ 𝑈 is called a flow.

where F • : 𝑈 → 𝑉is a mapping U,V⊂ ℜ 𝑛

The function F • is also called a vector field.

Given an initial condition 𝑥 0 ∈ 𝑈 in the domain of the

definition of the function F • , one gets a solution

𝑥 (𝑥 0, 𝑡), 𝑡 ∈ ℜ+ which is called an orbit through 𝑥 0 ∈ 𝑈.

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3.1 Continuous time systems…..

Thus, the flow is a mapping:

n

t t 0 0: U with (x ) x(x ,t) (2)

t

d(x,t) F( (x, )) (3)

dt

Note that the mapping

satisfies the differential

equation (1). Thus,

One of the key properties

is the semi-group property:

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Examples of flow include:

Ex 1.

Ex 2. Consider the flow defined by:

tA

t

n n

t

ktA

k 0

(x) e x (4)

Here : and

(At)e

k!

n n

n n

0

nn n

n

n

0 0

n

n 0

n 0

1cos t sin t

c sinn x(u ,t) (5)

d sinn x1sin t cos t

c sinn xso that (u ,0) u

d sinn x

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Ex 2... A careful thought will suggest that the flow in

(5) represents solution of a partial differential

equation where 𝒖 (𝒙, 𝒕) represents the state of the

system. In fact, the flow represents solution of a

time-dependent boundary value problem:

This represents equation of motion for linear (small)

motions for an axially loaded elastic rod:

4 2 2

4 2 2

u u up 0, t 0, 0 x 1. (6)

x x t

x P

x=1 u(x,t)

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Ex 2... The boundary conditions for the pinned-pinned rod are: 2

2

uu(x,t) 0, t 0, x 0,1. (7)

x

n

n

n 0

n 0

u(x,0) c sinn x,

u(x,0) d sinn x, 0 x 1. (8)

t

nn n n

n

2 2 2 2 2

n

n 0

du(x,t) (c sinn xcos t sinn xsin t) (9)

with n (n p)

x

p x=1 u(x,t)

The solution of (6) with the given boundary and initial

conditions is

The initial conditions are:

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3.2 Discrete time systems:

Consider systems of the form

n

p 1 px F(x ), x (10)

where 𝐹 (•) : 𝑈 → 𝑉 is a mapping with , 𝑈, 𝑉 ⊂ ℜ𝑛

The function 𝐹 (•) is also called a Poincare mapping.

Given an initial condition 𝑥 1 ∈ 𝑈, in the domain of definition

of the function 𝐹 (•), one gets a solution 𝑥 (𝑥 1, 𝑝), 𝑝 ∈ ℜ+

which is called an orbit through 𝑥 1 ∈ 𝑈.

The sequence of points generated as p is increased is also

called the forward iterates.

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3.2 Discrete time systems……

Such discrete time systems can arise in two different

ways: through the natural definition of discrete-time events

for a system where states are only defined at discrete times;

and in systems where discrete events are defined for

convenience.

ground

table

Y(t)

ball

g mg

Ex 3. Consider the system of the

ball bouncing on an oscillating

table:

As was shown, its dynamic

response can be studied by two

simultaneous finite-time

equations; they relate relative

velocities.

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Thus, the dynamics is governed by:

i 1 i i 1

2

i i i 1 i i 1 i

W ( ) [sin(2 ) sin(2 )]2

[W cos(2 )]( ) ( ) 0 (11)

i 1 i i 1 i i 1 iW e[ {cos(2 ) cos(2 )} W 2( )] (12)

ground

table

Y(t)

ball

g mg where W() is relative displacement, W’ is relative velocity, i is time instant of ith impact, W’(i) is velocity just after ith imact, W’(i+1) is velocity just before (i+1)th impact.

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Ex 4. Consider the impulsively excited rigid rod: The force P is applied at discrete times 𝑡 = 𝑚𝜏, 𝑚 ⊂ (−∞,∞) or 𝑚 ⊂𝑍(set of integers).

2

2

m

d dI b c

dt dt

[PL (t m )]sin 0 (13)

The equation between impulsive actions of the force is linear and can be solved exactly. At an impulse, the angular velocity undergoes a jump with no change in position. So, develop a mapping relating position and velocity just after an impulse to time instant just after the next impulse is applied.

The equation of motion of the rod is:

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3.3 Equations of motion and dynamics: We have already

considered equations of motion for some systems. In finite-

dimensional continuous time systems, the equations of

motion are of the form

n

p 1 px F(x ), x U , p (15)

nx F(x), x U , t (14)

For systems defined by infinite-dimensional state variables,

the formulation has to consider some function space.

where as for finite-dimensional discrete time systems, the

dynamics is defined by iterates of mappings

L L

2

0 0

1V EI ( ) ds P[L cos ds] (16)

2 s

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3.3 Equations of motion and…..

As an example of a system defined in a function space,

consider the dynamics of a axially loaded beam:

x P

s=L

u(s,t)

s (s)

where the horizontal component of displacement of

a material point, 𝑢(𝑠, 𝑡) is: 𝑢(𝑠, 𝑡) = 𝑠 − 𝑥

Then, 𝑢(𝑠, 𝑡) = 𝑠 − cos𝜓𝑑𝑠𝐿

0 (17)

Assuming an inextensible rod, the strain energy of the system

can be shown to be

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Accounting for changes in kinetic energy of the rod, and using

Hamilton’s principle, one can show that the equation of

motion of the rod (neglecting inertia terms) has the form

2

2EI Psin 0 (17)

s

0, s 0,L (18)

s

x P

s=L

u(s,t)

s (s)

along with the appropriate boundary conditions (for pinned

ends):

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Clearly, any solution of the system must belong to the set of

functions that satisfy the boundary condition. So, we define a

function space:

2H {u C u(0) u(L) 0} (18)

This is he space of all twice differentiable functions that

satisfy the boundary condition. It is a linear vector space, an

example of a Banach space. The operation in (17) takes

such a function and gives a function that is only a continuous

function of s. So, the resulting function is not a member of H,

even if it may form a linear vector space. It belongs to the

space 𝐾 = {𝑢 ∈ 𝐶0

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Equation (17) then can be written as

G(u, ) 0, u H (19)

G:H K (20)

Such mappings arise naturally in infinite-dimensional systems

– physical systems which are dependent on space variables,

in addition to time.

Here, G is a mapping from H into K, i.e.,

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3.4 Definitions of stability: First, consider the concept of a metric – distance in the space of variables one is using to define the system under consideration. As an example, consider the beam system (equation (6)):

v(x,t) u t

4 2 2

4 2 2

u u up 0, t 0, 0 x 1. (6)

x x t

L 22 2 2 2 1/ 2

2

0

u u(w,0) [ {v ( ) ( ) u }ds] (21)

ss

Then, a couple of examples of norms are:

Let us define velocity of a point on the beam:

and the state vector: 𝑤 (𝑥, 𝑡) = 𝑢, 𝑣 𝑇

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3.4 Definitions of stability…..

or L 2

2 2 1/ 2

2

0x [0,L]

u(w,0) [u] [ {v ( ) }ds] (22)

ssup

There are quite a few definitions of stability. The most significant and important one is called Lyapunov stability.

Lyapunov stability: Consider the geometric picture in the Fig. The flow is defined by 𝝋 𝒕: 𝑼 → 𝑲. For a specific

trajectory, we consider the solution started at an initial condition 𝒖𝟎, 𝒕𝟎); 𝝋 𝒕(𝒖𝟎, 𝒕𝟎 .

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Consider also the solution started at a neighboring initial condition 𝑣0, 𝑡0); 𝜑 𝑡(𝑣0, 𝑡0

At a time ′𝑡′, the two solutions are at 𝜑 𝑡 𝑢0, 𝑡0 and𝜑 𝑡(𝑣0, 𝑡0).

The stability of the trajectory𝜑 𝑡(𝑢0, 𝑡0) is related to the

deviations of all other solutions from 𝜑 𝑡(𝑢0, 𝑡0).

0 00 0 0 0 0 0

0 0t tt t( (u ,t ), (v ,t )) (u ,v )

0 0 0 0t t( (u ,t ), (v ,t ))

Thus, the definition: The solution 𝜑 𝑡 𝑢0, 𝑡0 is stable (in the

sense of Lyapunov) if and only if, for any initial time t0 and any prescribed positive number ε > 0, there exists a number > 0 such that for all t > t0

implies that

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Asymptotic stability: If the solution 𝜑 𝑡(𝑢0, 𝑡0) is stable (in the

sense of Lyapunov) and in addition, 𝜌(𝜑 𝑡(𝑢0, 𝑡0), 𝜑 𝑡 𝑣0, 𝑡0 for

𝑡 → ∞, then the solution 𝜑 𝑡(𝑢0, 𝑡0)is called asymptotically stable.

In some cases, this definition is too restrictive. For example,

consider motion of a nonlinear pendulum.

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For the nonlinear pendulum, periodic oscillations around the

bottom equilibrium are common, and the period of oscillations

depends on the amplitude, or initial conditions. Shown here

are the phase plane trajectories 𝜑𝑡(𝜓0) and 𝜑𝑡(𝜓0 + 𝛿)

corresponding to two different initial conditions

𝜓0 𝑎𝑛𝑑 𝜓0 + 𝛿 started on the

displacement axis. The orbit

𝜑𝑡(𝜓0) has period T, so that

𝜑0(𝜓0) = 𝜑𝑇(𝜓0) =𝜑2𝑇(𝜓0). . . .

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The orbit 𝜑𝑡(𝜓0 + 𝛿), even though periodic, will have period

> 𝑇 since energy is larger. Thus, at time 𝑇, the solution on this

orbit is at 𝜑𝑇(𝜓0 + 𝛿) 𝑤𝑖𝑡ℎ 𝜑𝑇(𝜓0 + 𝛿) ≠ 𝜑0(𝜓0 + 𝛿)

As time marches, the solution𝜑𝑡(𝜓0 + 𝛿) deviates more from

the solution𝜑𝑡(𝜓0)at each instant of time though the two orbits

remain close to each other for all time.

Thus, need a different way to

characterize stability of

periodic solutions – Poincare

(orbital) stability.

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An attractor is a set of states (points in the phase

space), invariant under the dynamics, towards which

neighboring states in a given basin or domain of attraction

asymptotically approach in the course of dynamic evolution.

An attractor is defined as: the smallest unit which cannot be

itself decomposed into two or more attractors with distinct

basins of attraction. This restriction is necessary since a

dynamic system may have multiple attractors.

So, if 𝜑𝑡(.) is the flow, by invariant we mean a set 𝐴

such that 𝜑𝑡(𝐴) ⊆ 𝐴) Furthermore, A must be asymptotically

stable, i.e., there is a neighborhood 𝑉.

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With 𝐴 ⊆ 𝑉 such that for every

t

tx V, lim ( (x),A) 0

The region V – domain of attraction of A.

2(1 ) 0y y y y

representation and stability - purdue university · 3. representation and stability concepts 3.1...

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