Quadratic stability and block diagonal quadratic stability analysis for a special class of nonlinear systems

Yin-Lam Chow, Purdue University

Abstract

First of all, quadratic stability of a system is introduced where it directly implies global uniform exponential stability. By introducing the notions of multiplier matrices and set nonlinearity, a necessary and sufficient condition for checking quadratic stability for a special type of systems is given based on Linear Matrix Inequalities (LMIs). By using the Kalman-Yakubovich-Popov (KYP) lemma and the concepts in Strictly Positive Realness (SPR), quadratic stability for this type of systems can be verified using frequency domain methods. Furthermore, for fixed multiplier matrices, quadratic stability of the above system can be checked via computing the generalized eigenvalues of a particular Hamiltonian matrix plus satisfying some side conditions. Next, the concept of block diagonal quadratic stability is introduced. It is shown that if a nonlinear system is block diagonally quadratically stable, then both the interconnected system and every sub-systems are quadratically stable. There are several approaches for checking block diagonal quadratic stability. The first one is by structuring the Lyapunov matrix in the LMI result that checks quadratic stability. Therefore, a sufficient condition for checking block diagonal quadratic stability of a special class of nonlinear interconnected systems is obtained. Secondly, block diagonal quadratic stability of a special class of nonlinear interconnected systems is guaranteed if 1) every subsystem is quadratically stable and 2) a LMI condition is satisfied. Moreover, by considering a specific augmented system, checking block diagonal quadratic stability of the above nonlinear interconnected system is strongly related to checking quadratic stability of this augmented system with a structured multiplier matrix. By using analogous arguments, block diagonal quadratic stability can be verified using both LMIs and frequency domain methods. In addition, the problem of checking input-output L 2 stability and quadratic stability when the exogenous inputs of the system vanish is studied. By formulating an appropriate augmented system with a structured multiplier matrix, a nonlinear system is input-output L2 stable and quadratically stable when the exogenous inputs of the system vanish, if a corresponding augmented system is quadratically stable. As a result, this stability criterion check can be done using both LMIs and frequency domain methods. Similar analysis can be done for block diagonal quadratic stability and passivity analysis as well. In this thesis, some common nonlinearities/system uncertainties are introduced and the efficiencies of verifying quadratic stability and block diagonal quadratic stability with both LMIs and frequency domain methods are compared.

Degree

M.S.E.

Advisors

Corless, Purdue University.

Subject Area

Mathematics|Aerospace engineering|Electrical engineering

COinS