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Stability of Dynamic Systems

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Static Equilibrium and Stability for Conservative Systems

n degree of freedom system Static equilibrium implies

  A system in a state of rest stays at rest

An equilibrium state can be  Stable  Unstable  Critically stable (or neutrally stable)

Let qe be state of equilibrium.

q1

V(q1) stable

unstable neutrally stable

Potential energy

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Static Equilibrium and Stability for Conservative Systems

n degree of freedom system Static equilibrium implies

  potential energy is stationary

An equilibrium state can be   Stable   Unstable   Critically stable (or neutrally stable)

Why?

Set

in Lagrange’s equations

[Toricelli’s least total energy principle (17th century)]

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Kinetic Energy and the Generalized Inertia Matrix

T = T0 + T1 + T2   T0 terms independent of speeds   T1 terms that are linear functions of the speeds   T2 terms that are quadratic functions of the speeds

For Natural Systems

  H = [Hij] is the generalized inertia matrix   H is a positive definite matrix that is a function of the generalized

coordinates

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Holonomic Conservative Systems

Equations of Motion

If

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Stability Analysis

Analyze the dynamics of the system “near” the equilibrium   Does the system return to the equilibrium point after small perturbations?

€

f ˜ x ( ) = f 0( ) +∂f ˜ x ( )∂˜ x ˜ x = 0

˜ x ( ) + O ˜ x 2( )

≈ A ˜ x

Change of variables

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Example 1 Equation of motion

State space representation

Equilibrium points q

m

Viscous friction, c

Change of variables

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Example 1 (continued) Equilibrium point number 1

Equilibrium point number 2

q

m

Viscous friction, c

3

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Example 1 (continued) Equilibrium point number 1

Linearization q

m

Viscous friction, c

A

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Stability of a Single Degree-of-Freedom Conservative System

q1

V(q1) stable

unstable

marginally stable

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Example MEAM 535

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Example (continued)

Using the second derivative it can be verified that two equilibrium points are stable and one of them is unstable. (The second derivatives at the first, second and third equilibrium points are respectively 150.2463, -1.0148e3, and 2.3405e3.)

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Conservative Multiple Degree of Freedom Holonomic Systems

State Equations of Motion

1. Equilibrium 2. Perturbations about equilibrium

3. Equations of Motion for Small Perturbations

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Stability of a Multi Degree-of-Freedom Conservative System

Hessian matrix, D   D is positive definite, q is a local minimum in the n-dimensional space, and the

equilibrium is stable;

  D negative definite, q is a local maximum in the n-dimensional space, and the equilibrium is unstable;

  D is indefinite, q is a saddle point in the n-dimensional space. (Equilibrium is neutrally stable?) Unstable!

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Positive Definite Hessian For any non-zero vector, x, if xTDx is positive, D is said to be positive definite.   D is positive (semi-) definite if all of its eigenvalues are strictly positive (non

negative).   D is negative (semi-) definite for all non-zero , if all of its eigenvalues are

strictly negative (non positive).   D is indefinite, if some (at least one) of its eigenvalues are positive and some

(at least one) are negative.

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Beyond Conservative Systems

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Dynamical Systems State (mechanical systems)

  q describes the configuration (position) of the system

  x describes the state of the system

Phase Portrait Trajectory

(assuming holonomic)

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  Vector field   Trajectories   State space + trajectories in state space = phase portrait

Vector Field

Dynamics: The Geometry of Behavior, Abraham and Shaw, 1982

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Integral Curve

Dynamics: The Geometry of Behavior, Abraham and Shaw, 1982

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Equilibrium Points are called Critical Points

Dynamics: The Geometry of Behavior, Abraham and Shaw, 1982

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Critical Point

  Critical points are equilibrium points.   The critical points of the vector field, f(x), are found by solving f(x)=0.

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Stability in the sense of Lyapunov If a disturbance at time t0 of a trajectory produces changes that remain permanently bounded, the motion is said to be stable in the sense of Lyapunov and possesses L-stability.   Beyond static stability   Not limited to conservative systems

x1 x2

t

We will limit our discussion to autonomous (time-independent) systems

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Dynamical Systems State (mechanical systems)

  q describes the configuration (position) of the system

  x describes the state of the system

Phase Portrait Trajectory

(assuming holonomic)

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Lyapunov Stability

Equilibrium point   A point where f(x) vanishes

Translate the origin to the equilibrium point of interest   Without loss of generality, let 0 be an equilibrium point

f(0) = 0   The equilibrium solution or the null solution is

x(t) = 0, t > t0

Local and global stability   Recall that stability refers to small perturbations and therefore is intrinsically a local property   However, can establish global stability for special cases

Note: We have replaced 2n by n for convenience