Defense Announcement

Here’s the BlueJeans link for the meeting:

https://bluejeans.com/4832389748

Title: Methods of Analysis and Design of Dynamical Systems Using homogeneous Polynomial Lyapunov Functions

Advisor: Prof. Eric Feron

Date: Monday July 26 at 11am

Abstract:

Lyapunov functions are the mainstay for systems analysis and control, and in particular quadratic Lyapunov functions have been used successfully for many classes of problems. When using quadratic Lyapunov functions for analysis and design of nonlinear systems, there is a measure of conservatism due to the inherent limitations of the associated ellipsoid as a covering for the stability region of the system, and of course, there are classes of systems where quadratic Lyapunov forms yield no results at all. This is the case for switched linear systems, where there may not exist a common quadratic Lyapunov function for each of the switched modes, even though they system is stable. However, Mason et al. have shown that there exists homogeneous polynomial Lyapunov functions that certify stability of the system even though the degree of the homogeneous polynomial may not be known a priori. Quadratic Lyapunov functions are amenable to energy-based problems because energy type bounds and constraints are captured efficiently by ellipsoids. On the other hand, analysis of peak-input bounded types of problems lack closed-form solutions, often utilize approximations and relaxations, in addition to being computationally expensive to compute due to the norm expressing the bounds.

This research investigates generalizations of quadratic Lyapunov functions, specifically homogeneous Lyapunov functions that are constructed through a process called lifting. The state vector x is lifted via a recursive Kronecker product to a higher degree, homogeneous form resident in a higher-dimensional space. Linear dynamical systems can similarly be constructed in the lifted space with this process. This research demonstrates a method of constructing homogeneous Lyapunov functions that provide good estimates of system characteristics such as domain of attraction and inescapable sets. This method is demonstrated for stability of switched linear systems and implicit systems, as well as for the analysis of the L1 problem. We show that using higher-order homogeneous Lyapunov functions improves estimates of the domain of attraction and inescapable sets. The main contribution of this research is applying this methodology to the L1 problem and improving upper-bounds to the 1-norm of a linear time-invariant system. Moreover, this method is accessible through linear matrix inequalities (LMI) constructions and computationally solvable with standard semidefinite programming.

Committee:

Prof. Eric Feron - School of Aerospace Engineering

Prof. Tsiotras Panagiotis - School of Aerospace Engineering

Prof. Kyriakos Vamvoudakis - School of Aerospace Engineering

Prof. Yongxin Chen - School of Aerospace Engineering

Prof. Jeff S. Shamma - Chair of Industrial & Ent. Systems Engineering, University of Illinois Urbana-Champaign

Prof. Eric N. Johnson - School of Aerospace Engineering, Pennsylvania State University