Compound matrices for stability analysis of nonlinear systems

Although Lyapunov exponents have been widely used to characterize the dynamics of nonlinear systems, few methods are available so far to obtain a-priori bounds on their magnitudes. In this thesis, sufficient conditions to rule out the existence of attractors with positive Lyapunov exponents and to ensure 2-contractivity of the system are derived via a Lyapunov approach based on the second additive compound matrices of the system Jacobian. Moreover, insights into this approach are provided by showing how several available techniques for computing Lyapunov functions can be fruitfully applied to Lorenz and Thomas systems to derive explicit conditions on their system parameters, which ensure that there are no attractors with positive Lyapunov exponents. Then, the approach is extended to the case of nonlinear systems with a first integral of motion and its application to the memristor Chua's circuit is discussed. Furthermore, sufficient small-gain like conditions for 2-contraction of feedback interconnected systems, on the basis of individual gains of suitable subsystems arising from a modular decomposition of the second additive compound equations, are introduced. The condition applies even to cases when individual subsystems might fail to be contractive (due to the extra margin of contraction afforded by the second additive compound matrix). Some examples are provided to illustrate the theory and show its degree of conservatism and scope of applicability. Finally, the second additive compound approach is also used to derive conditions, formulated in terms of 2-contractivity of the closed loop system, for designing a feedback control law to remove the dense set of Unstable Periodic Orbits (UPOs) and chaotic attractors, while preserving the system equilibrium points. Matrix inequalities for computing the control gain matrix are derived and applied to the Lorenz and Thomas systems.