Michael Malisoff

Constructions of Strict Lyapunov Functions

Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, but the functions it provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved through standard feedback designs or robustness arguments. Non-strict Lyapunov functions are often readily constructed. This book contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Their explicitness and simplicity make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples.

Brief paper: Further results on input-to-state stability for nonlinear systems with delayed feedbacks

We analyze a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions are both known. We prove that the closed loop systems are input-to-state stable (ISS) relative to actuator errors when small time delays are introduced in the feedbacks. We explicitly construct ISS Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the known Lyapunov functions for the original undelayed closed-loop dynamics. We also provide a general result on ISS for cascade systems with delays. We demonstrate the efficacy of our results using a generalized pendulum dynamics and other examples.

Robustness of nonlinear systems with respect to delay and sampling of the controls

We consider continuous time nonlinear time varying systems that are globally asymptotically stabilizable by state feedbacks. We study the stability of these systems in closed loop with controls that are corrupted by both delay and sampling. We establish robustness results through a Lyapunov approach of a new type.

Universal formulas for feedback stabilization with respect to Minkowski balls

This note provides explicit algebraic stabilizing formulas for clfs when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artsteins theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.

Further Results on Stabilization of Periodic Trajectories for a Chemostat With Two Species

We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation.

Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying non-linear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.

Global Asymptotic Controllability Implies Input-to-State Stabilization

The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed-loop systems input-to-state stable (ISS) with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well.

Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements

The stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem, and is of significant ongoing interest in bioengineering and population dynamics. In this paper, we solve an output feedback stabilization problem for chemostat models having two species, one limiting substrate, and either Haldane or Monod growth functions. Our feedback stabilizers depend on a given linear combination of the species concentrations, which are both measured with a constant time delay. The values of the delays are unknown. Instead, one only knows an upper bound on the delays, and we allow the upper bound to be arbitrarily large. The stabilizing feedback depends on the known upper bound for the delays as well. Our work is based on a Lyapunov-Krasovskii argument.

Reduction Model Approach for Linear Time-Varying Systems With Delays

We study stabilization problems for time-varying linear systems with constant input delays. Our reduction method ensures input-to-state stability with respect to additive uncertainties, under arbitrarily long delays. It applies to rapidly time-varying systems, and gives a lower bound on the admissible rapidness parameters. We also cover slowly time-varying systems, including upper bounds on the allowable slowness parameters. We illustrate our work using a pendulum model.

Optimal control, stabilization and nonsmooth analysis

Part I: Optimal Control, Optimization, and Hamilton-Jacobi-Bellman Equations.- Part II: Stabilization and Lyapunov Functions.- Part III: Nonsmooth Analysis and Applications.