Absolute Stability

Work Supported by NSF under Grant CMS-0324329    

Absolute Stability
Constrained Control
Frequency Response
Magnetic Suspension
Nonquadratic Lyapunov Functions
Switching/Switched Systems


Absolute stability is about the stability of a system with a uncertain nonlinear component
(possibly with time-varying uncertainties):

Text Box: y(u,t)
In classic absolute stability theory, the function y(u,t) describing the uncertain nonlinear
component (dashed curves) is bounded by a conic sector (formed by two straight lines
v =k1u and v=k2u):

Text Box: u
Text Box: v
Text Box: k2u
Text Box: k1u
Text Box:  




The system is said to be absolutely stable if the origin is globally stable for any y(u,t) 
within  the conic sector. Sufficient conditions for absolute stability have been obtained via
linear systems corresponding to the linear functions bounding the conic sector. Popular
conditions are the Circle Criterion and the Popov Criterion.

Motivation for a more flexible sector: In many situations, we know some properties
about the nonlinear component. With a particular type of nonlinearity, e.g., saturation or
dead-zone, a conic sector might be too conservative.  It is clear that a pair of piecewise
linear functions can form a tighter envelope:





The question is:
1. Are envelopes with piecewise linear boundaries numerically tractable?
2. How to use a tighter envelope to improve stability analysis results?
These problems are addressed in the following papers:

In these papers, numerically tractable methods are derived for stability analysis of systems
whose uncertain nonlinearities are bounded by a pair of piecewise linear functions. If
the piecewise linear functions are convex/concave, necessary and sufficient conditions
are obtained for quadratic stability. These conditions are interpreted with Linear Matrix
Inequalities (LMIs) which are easily solvable with numerical tools. These tools will help the
design of more efficient dynamic control systems. The main idea and key steps can be found
in the power point file:





This site was last updated 01/11/07