Automatica 48 (2012) 2888-2893 The 5th (Q1) in the subject area of Engineering, andSubject category: Control and Systems Engineering

Speaker: Ittidej MoonmangmeeNovember 17, 2012

Key references for this presentation2/19

Textbook:

[1] A.M. Bloch, Nonholonomic Mechanics and Control, Springer,

Springer, New York, 2003.

[2] S. Sastry, Nonlinear Systems; Analysis, Stability, and Control,

Springer, New York, 1999.

Holonomic vs Nonholonomic

Holonomic (or

integrable) Systems:

Nonholonomic (or

nonintegrable) Systems:

Ex.

Mechanical systems ex. Mobile robotics

Electro-magnetics and Electromechanical systems

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A Nonholonomic Integrator System:

The Analytic Affine Control System:

Drift-Free Control System:

Geometric Nonlinear Control

1. Introduction

A Squirrel Cage Induction Motor System:

The Heisenberg System:

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1. Introduction (cont)

A Three-dimensional non-holonomic integrator system with drift terms:

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1. Introduction (cont)

An Aside:

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Analysis:

For any constant output x3, the followingcondition must be fulfilled

1. Introduction (cont)

Meaning: The trajectory of vector are closed orbits, which, under consideration of linearity of the first two equations of (*), implies that input vector u must generate closed orbits as well. This important property is satisfied by amplitude and frequency modulated harmonic functions.

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Proof:

Hence,

2. Optimal steering with sinusoids

Proof (extended):

Closed-loop system:

For x1:

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2. Optimal steering with sinusoids (cont)9/19

Initial condition

Must be satisfied

3. Feedback control and stability

The nonlinear state controller:[Grcar, Cafuta, Štumberger, Stankovic, and Hofer (2011)]

The closed-loop system:

where , , and are design parameters.

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Sketch of Proof:

3. Feedback control and stability (cont)

Equilibrium points

Change of variables

Lyapunov function candidateClose

d orbit

Globally Asymptotically Stable(or (locally) exponentially stable)

negative semidefinite

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(by LaSelle’s theorem)

3. Feedback control and stability (cont)12/19

This relation is valid not only in steady state but also during the transients, i.e. the nonlinear state controller always keeps the input norm minimal for the required output.

4. Singularity & time-optimal control of internal state13/19

Given a desired value and any initial internal state determine the control inputs u1 and u2 such that the desired internal state magnitude according to

is reached in minimum time.

The Time Optimal Controller:

4. Singularity & time-optimal control of internal state15/19

5. Experimental results

Implementation example : Induction machine torque control

A Squirrel Cage Induction Motor System:

where

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and estimator

5. Experimental results (cont)17/19

Time optimal control from the singular point

6. Conclusion19/19

A globally stabilizing nonlinear controller is proposed for a three-dimensional nonholonomic integrator with drift terms

The optimal steering with sinusoids enabling steady state analysis, system inversion and calculation of the minimal input norm is introduced. The structure of the proposed feedback control provides amplitude and frequency modulation of the input vector that implicitly imposes periodic orbits in the internal state vector.

The problem of singularity at zero initial states was solved by a

time-optimal control scheme for the internal states

The control proposition, successfully implemented for an induction machine torque control offers new possibilities and is conceptually different from existing IM control solutions