The Real Stabilizability Radius of the The Real Stabilizability Radius of the Multi-Link Inverted PendulumMulti-Link Inverted Pendulum

Connections 2006ECE Graduate Symposium

Presenter: Simon LamSupervisor: Professor E. J. DavisonSystems Control Group, ECEUniversity of TorontoDate: June 9, 2006

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

Introduction

classic problem in control theory

widely used as a benchmark for testing control algorithms

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

u

θ1

θ2

θv

M1

M2

Mv

L1

Lv

L2

Properties of a v-link Inverted Pendulum

Linearized Model:

System is stabilizable for any number of links

Can design a controller to (locally) stabilize a pendulum with any number of links!

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

Gap

Of course, in reality, we can’t stabilize an inverted pendulum with too many links.

Possible factors?

i) Nonlinear effects (e.g. friction)

ii) Initial conditions

iii) Sensitivity to physical disturbances

Conjecture:

real stabilizability radius is too small

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

Real Stabilizability Radius

Given a stabilizable LTI system:

the real stabilizability radius measures the smallest

such that the perturbed system:

is no longer stabilizable.

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

Real Stabilizability Radius of Pendulum

v rc,norm Sig. Fig.

1 1.00E+00 1-2

2 1.11E-01 2-3

3 4.69E-02 3-4

4 2.46E-02 3-4

5 1.47E-02 3-4

6 9.56E-03 3-4

7 6.64E-03 4-5

Department of Electrical and Computer EngineeringSystems Control Group, University of Toronto

Thank you!