Video 8.1 Vijay Kumar 1 Property of University of Pennsylvania, - - PowerPoint PPT Presentation

1 Property of University of Pennsylvania, Vijay Kumar

Video 8.1 Vijay Kumar

2 Property of University of Pennsylvania, Vijay Kumar

Definitions

State State equations Equilibrium

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Stability

• Stable
• Unstable
• Neutrally

(Critically) Stable

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Stability

Translate the origin to xe

e d

x1 x2

x(t) =0 is stable (Lyapunov stable) if and only if for any e > 0, there exists a d(e) > 0 such that x(t) =0 is asymptotically stable if and only if it is stable and there exists a d > 0 such that

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Asymptotic Stability

x(t) =0 is asymptotically stable if and only if it is stable and there exists a d > 0 such that

x1 x2

d

x(t) =0 is globally asymptotically stable if and

• nly if it is asymptotically stable and it is

independent of x(t0)

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Example

q m Viscous friction, c x1 x2

Suppose you want

6

1

• x

E(t) cannot increase

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Global Asymptotic Stability of Linear Systems

• Global Asymptotic Stability
• Lyapunov Stability, not Global Asymptotic Stability
• Unstable

if and only if the real parts of all eigenvalues of A are negative if and only if the real parts of all eigenvalues are non positive, and zero eigenvalue is not repeated if and only if there is one eigenvalue of A whose real part is positive

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Linear Autonomous Systems

Eigenvalues and eigenvectors for non defective X

for non defective A

but similar story for defective A

Solution

Exponential of a matrix, X

eigenvectors eigenvalues

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Video 8.2 Vijay Kumar

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Stability of “Almost Linear” Systems

• Global Asymptotic Stability
• Lyapunov Stability, not Global Asymptotic Stability
• Unstable

if and only if the real parts of all eigenvalues of A are negative if and only if the real parts of all eigenvalues are non positive, and zero eigenvalue is not repeated if and only if there is one eigenvalue of A whose real part is positive Significant dynamics

Not Significant

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Lyapunov’s theorem

• Nonlinear, autonomous systems
• Near equilibrium points

If the linearized system exhibits significant behavior, then the stability characteristics of the nonlinear system near the equilibrium point are the same as that of the linear system.

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Example

• Equation of motion
• State space representation
• Equilibrium points
• Change of variables

Viscous friction, c

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Example

• Equilibrium point number 1
• Equilibrium point number 2

q m Viscous friction, c

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Example

• Equilibrium point number 1
• Linearization

q m Viscous friction, c

The system is locally asymptotically stable If c>0 and g>0, real parts of both eigenvalues are always negative

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• Equilibrium point number 2
• Linearization

Example

q m Viscous friction, c

The system is unstable If c>0 and g>0, both eigenvalues are real,

• ne is positive.

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Example (c=0)

• Equilibrium point number 1
• Linearization

q m No friction

No conclusive results Real parts of both eigenvalues are non negative

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Summary for Nonlinear Autonomous Systems

• Write equations of motion in state space notation
• Solve f(x)=0
• Identify equilibrium point(s), xe
• Linearize equations of motion to get the coefficient matrix A
• Compute eigenvalues of A. Use Lyapunov’s theorem. If the linearized system have

significant dynamics, we can make an inference about stability.

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Lyapunov’s Direct Method

• Avoids linearization (hence direct)

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Example

q m Viscous friction, c x1 x2

E(t) cannot increase

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Lyapunov’s Direct Method

• V(x) is a continuous function with continuous first

partial derivatives

• V(x) is positive definite

V acts like a norm What if you can show that V never increases?

Such a function V is called a Lyapunov Function Candidate

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Theorem

• 1. The (above) system is stable if there exists a Lyapunov

function candidate such that the time derivative of V is negative semi-definite along all solution trajectories of the system.

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Theorem

• 2. The (above) system is asymptotically stable if there

exists a Lyapunov function candidate such that the time derivative of V is negative definite along all solution trajectories of the system.

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Example 1

• Equation of motion
• State space representation
• Equilibrium point

q m Viscous friction, c

What is a candidate Lyapunov function?

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Example 1

q m Viscous friction, c

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Example 2

• One-dimensional spring-mass-dashpot with a nonlinear spring

k x O M

What is a candidate Lyapunov function?

Linearized system does not have significant dynamics

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Video 8.3 Vijay Kumar

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Fully-actuated robot arm (n joints, n actuators)

Equations of Motion

symmetr ic, positive definite inertia matrix

n- dimensiona l vector of Coriolis and centripetal forces n- dimensi

• nal

vector of gravitati

• nal

forces n- dimensi

• nal

vector

• f

actuator forces and torques

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Fully-actuated robot arm (continued)

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PD Control of Robot Arms

Reference trajectory Error Proportional + Derivative Control

assume

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Assume no gravitational forces

PD Control achieves Global Asymptotic Stability Proof Lyapunov function candidate Identity

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Assume no gravitational forces

PD Control achieves Global Asymptotic Stability Proof Lyapunov function candidate

decreasing as long as velocity is non zero can it reach a state where ?

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Assume no gravitational forces

PD Control achieves Global Asymptotic Stability Proof

decreasing as long as velocity is non zero can it reach a state where La Salle’s theorem guarantees Global Asymptotic Stability

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With gravitational forces

PD Control achieves Global Asymptotic Stability but with a new equilibrium point

q g

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PD control with gravity compensation

Global Asymptotic Stability with the correct equilibrium configuration Use the same Lyapunov function candidate:

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Computed Torque Control

Reference trajectory Compensate for gravity and inertial forces

Global Asymptotic Stability

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Computed Torque Control and Feedback Linearization

y

• riginal

system nonline ar feedbac k new system

Nonlinear feedback transforms the original nonlinear system to a new linear system Linearization is exact (distinct from linear approximations to nonlinear systems)

v u

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Joint Space versus Task Space Control

Task coordinates Reference trajectory Kinematics Task space control

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Task Space Control

Task coordinates Kinematics Task space control Commanded joint accelerations Computed torque control