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!