This document discusses control of nonholonomic systems. It presents a globally stabilizing nonlinear controller for a 3D nonholonomic integrator with drift terms. The controller provides optimal steering with sinusoids that enables steady state analysis and minimal input control. It also addresses the singularity issue at zero initial states using a time-optimal control scheme for internal states. The control approach was successfully implemented for induction machine torque control.

1. Automatica 48 (2012) 2888-2893 The 5th (Q1) in the subject area of Engineering, and
Subject category: Control and Systems Engineering
Speaker: Ittidej Moonmangmee
November 17, 2012

2. 2/19
Key references for this presentation
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.

3. 3/19
Holonomic vs Nonholonomic
Mechanical systems ex. Mobile robotics
Holonomic
(or integrable)
Systems:
Nonholonomic
(or nonintegrable)
Systems:
Electro-magnetics and Electromechanical systems
Ex.

4. 4/19
1. Introduction
A Squirrel Cage Induction Motor System: A Nonholonomic Integrator System:
The Heisenberg System:
Geometric Nonlinear Control
The Analytic Affine Control System:
Drift-Free Control System:

5. 5/19
1. Introduction (cont)
A Three-dimensional non-holonomic integrator system with drift terms:

6. 6/19
1. Introduction (cont)
An Aside:

7. 7/19
1. Introduction (cont)
Analysis:
For any constant output x3, the following
condition must be fulfilled
Proof:
Hence,
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.

8. 8/19
2. Optimal steering with sinusoids
Proof (extended):
Closed-loop system:
For x1:

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

10. 10/19
3. Feedback control and stability
The nonlinear state controller:
[Grcar, Cafuta, Štumberger, Stankovic, and Hofer (2011)]
where , , and are design parameters.
The closed-loop system:
Initial condition
Must be satisfied

11. 11/19
3. Feedback control and stability (cont)
Sketch of Proof:
Change of variables
Closed
Equilibrium points Lyapunov function candidate
orbit
negative semidefinite
Globally Asymptotically Stable (by LaSelle’s theorem)
(or (locally) exponentially stable)

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

13. 13/19
4. Singularity & time-optimal control of internal state
The Time Optimal
Controller:
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.

14. 15/19
4. Singularity & time-optimal control of internal state

15. 16/19
5. Experimental results
Implementation example : Induction machine torque control
A Squirrel Cage Induction Motor System:
and estimator
where

16. 17/19
5. Experimental results (cont)
Time optimal control from the singular point

17. 19/19
6. Conclusion
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