Seminar Telematik systeme zur Fernwartung und Ferndiagnose Passivity and Stability Ph.D Candidate Yohko Aoki 3rd June, 2004

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Contents Introduction to Stability The Stable System without Control Input The Stable System with Control Input

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik 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.

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik The Definition of Passivity 1. There is no Energy resource inside of the system 2. The criteria equation is shown below,

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik The Stable System without Control Input StableUnstable # A, x is a scalar

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Example : Thermal Equilibrium C: Thermal Capacity of the ball R: Thermal Resistance : Constant Equation of State Fig. 3 The ball is inserted to the box

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Rigorous Proof Describe the System in the following Equation Control

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Higher-order Differential Equation General Equation of State No more Stable ?

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik n Differential Equations A First order Linear Differential Equation N linear differential equations

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Criteria of Stability λ: Eigen value Scalar Matrix New Criteria

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Stable and Unstable System : Pendulum J : Inertial Moment Stable atUnstable at

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Stable System Limit of Stability No convergence!

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Viscous Damping Motion stop in real world …Friction convergence

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Energy Dissipation Due to this damping term (friction) energy is dissipated. Energy Dissipation is essential

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Lyapunov Stability Theorem Lyapunov second theorem on stability Consider a function V(x) : R n R such that Lyapunov function candidate Then V(x) is called a Lyapunov function candidate The systemis asymptotically stable around

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Application of Lyapunov Stability Theorem Energy of System

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Passivity and Stability If there is no energy input, Stability Passive system without control Passivity =

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Performance of Control Without Control With Control Low Performance Asymptotically Stable…

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Control Scheme u : Control Input y : Output Stability? How to verify the Stability? Not Compatible u

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik PID control Define u(t) is a linear function of x(t) P, I, D … User defined parameter This scheme is so called PID control

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik Problem … Sampling Rate Increasing ?? Sampling rate must be enough high. Max : Nyquist theorem Sampling RateAnalogDigital Sampled Data Control

Seminar Telematik systeme zur Fernwartung und Ferndiagnose Ph.D Candidate Yohko AokiInformatik VII: Robotik und Telematik 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 Criterion Stability Margin