Seminar Telematik systeme zur Fernwartung und Ferndiagnose Passivity and Stability Ph.D Candidate...

Seminar Telematik systeme zur Fernwartung und Ferndiagnose

Passivity and StabilityPassivity and Stability

Ph.D Candidate Yohko Aoki

3rd June, 2004


Ph.D Candidate Yohko Aoki Informatik VII: Robotik und Telematik

Contents

Introduction to Stability

The Stable System without Control Input

The Stable System with Control Input



The Definition of “Stability”

Stable state Quasi-stable state Unstable state

The tendency of the variables or components of a system to remain within defined and recognizable

limits despite the impact of disturbances.



The Definition of “Passivity”

Stored input 0E t E t E t

1. There is no Energy resource inside of the system

2. The criteria equation is shown below,



The Stable System without Control Input

Exponential Function : A<0

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Time [sec]

Expo

nent

ial

x Ax

0A

expx At

Exponential Function : A>0

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3

Time [sec]Exp

onen

tial

0A Stable Unstable

# A, x is a scalar



Example : Thermal Equilibrium

0

1dCdt R

C: Thermal Capacity of the ball

R: Thermal Resistance

0 : Constant0

Equation of State

0 Fig. 3 The ball is inserted to the box



Rigorous Proof

0̂

0 0

1d

dt CR

ˆ 1 ˆd

dt CR

x Ax0A

Describe the System in the following Equation

ControlControl



Higher-order Differential Equation

1

1 1 010

n n

n nn n

d x d x dxa a a adt dt dt

General Equation of State

No more x Ax

Stable ?Stable ?



n Differential Equations

1

12

22

32

11

1

nn

nn

nn

n

x x

dxdxx

dt dt

dxd xx

dt dt

dxd xx

dt dt

dxd x

dt dt

A First order Linear Differential Equation

1 1 2 0n

n n n

dxa a x a x adt

N linear differential equations N linear differential equations



Criteria of Stability

0 A I

x Ax

λ: Eigen value

Scalar Matrix

x Ax Re 0 0A

New CriteriaNew Criteria



Stable and Unstable System : Pendulum

2J mlJ : Inertial Moment

sin 0J mgl sin 0J mgl

Stable at 0

Unstable at 0



Stable System

0 1

0

dg

dtl

gil

Re 0

Limit of Stability

2 sin 0ml mgl

No convergence!



Viscous Damping

0 1d

gdt d

l

2

cd

ml

Motion stop in real world …Friction

convergence 1Re 02

d

2 sin 0ml c mgl



Energy Dissipation

2 sin 0ml c mgl

Due to this damping term (friction)energy is dissipated.

Energy Dissipation is essential

0E t 0dE

E tdt



Lyapunov Stability Theorem

0eV x

Lyapunov second theorem on stabilityConsider a function V(x) : Rn → R such that

0V x

ex x 0dV

Vdt

Then V(x) is called a Lyapunov function candidateLyapunov function candidateThe systemis asymptotically stable around ex



Application of Lyapunov Stability Theorem

0

0

0e

E

dEE

dtE

00E

00E

V EEnergy

of System



Passivity and Stability

If there is no energy input,

Stability

Passive system

without control

0 0dEdE

Edx dx

in 0E t E t E t Passivity =



Performance of Control

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.53

0 1 2 3 4

Time [sec]

x(t)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4

Time [sec]

x(t)

Without ControlWith Control

Low Performance

Asymptotically Stable… lim 0tx t



Control Scheme

x Ax + Bu

y = Cx

u : Control Inputy : Output

x Ax Re 0

How to verify the Stability?Stability?

Not Compatible

u



PID control

Define u(t) is a linear function of x(t)

t

0

dx tu t = Px t + I x t + D

dt

ddt

dt

t

0

x Ax + Bu

xAx + B Px + I x + D

P, I, D …User defined parameter

This scheme is so called

PID control



Problem … Sampling Rate

Increasing ??

Sampling rate must be enough high.

Max : Nyquist theorem

Sampling Rate

AnalogAnalog DigitalDigital

Sampled Data Control



Next Step…

Time Delay, Low Level of output, Aging…

1. The system is really Stable?2. How much error is allowable?3. How much time delay is allowable?

Stability CriterionStability Margin

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Seminar Telematik systeme zur Fernwartung und Ferndiagnose Passivity and Stability Ph.D Candidate...

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