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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3.1 Continuous time systems…..
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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3.1 Continuous time systems…..
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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3.1 Continuous time systems…..
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. Representation and stability concepts
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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3.2 Discrete time systems……
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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3.2 Discrete time systems……
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. Representation and stability concepts
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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3.3 Equations of motion and…..
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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3.3 Equations of motion and…..
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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3.3 Equations of motion and…..
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. Representation and stability concepts
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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3.4 Definitions of stability…..
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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3.4 Definitions of stability…..
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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3.4 Definitions of stability…..
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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3.4 Definitions of stability…..
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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3.4 Definitions of stability…..
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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3.4 Definitions of stability…..
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