#### Work Supported by NSF under Grant CMS-0324329

 Absolute stability is about the stability of a system with a uncertain nonlinear component (possibly with time-varying uncertainties): 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):                                                   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:  T. Hu, B. Huang and Z. Lin, ``Absolute stability with a generalized sector  condition," IEEE Transactions on Automatic Control, Vol. 49, pp.535-548, 2004.  T. Hu and Z. Lin, ``Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions," IEEE Transactions on Automatic Control, Vol.50, pp.781-797, 2005. 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:

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