Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.     

On singular perturbations due to fast actuators in hybrid control systems

A stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators. If a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted, or equivalently, are infinitely fast, then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics. This result, which generalizes classical results for differential equations, justifies using a simplified plant model that ignores fast but continuous actuator dynamics, even when using a hybrid feedback control algorithm.     

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

It is known that if a system can be (robustly) globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable (UCO), then this system can be practically semiglobally stabilized by means of (possibly dynamic) output feedback. This papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by UCO functions is guaranteed. The class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback. In particular, the hypothesis does not require the system to be minimum phase.     

A theorem for UGAS and ULES of (passive) nonautonomous systems: robust control of mechanical systems and ships

The main contribution of this paper is a theorem to guarantee uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) for a class of nonlinear non‐autonomous systems which includes passive systems. These properties (and a uniform local Lipschitz condition) guarantee robustness of stability while weaker properties, like uniform global stability plus global convergence, do not. Our main result is then used in the tracking control problem of mechanical systems and ships. We use an adaptive backstepping design and prove UGAS of the closed‐loop tracking error system, in particular, we obtain that both the tracking and parameter estimation errors converge uniformly globally to zero. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis for a class of singularly perturbed hybrid systems via averaging

A class of singularly perturbed hybrid dynamical systems is analyzed. The fast states are restricted to a compact set a priori. The continuous-time boundary layer dynamics produce solutions that are assumed to generate a well-defined average vector field for the slow dynamics. This average, the projection of the jump map in the direction of the slow states, and flow and jump sets from the original dynamics define the reduced, or average, hybrid dynamical system. Assumptions about the average system lead to conclusions about the original, higher-dimensional system. For example, forward pre-completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains. In addition, global asymptotic stability for the average system implies semiglobal, practical asymptotic stability for the original system. We give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems. We also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous-time plants implemented by fast but continuous actuators.     

Matched disturbance suppression for nonlinear systems stabilizable by logic-based feedback

We focus on a class of nonlinear systems that can be stabilized at a desired compact set by means of logic-based continuous feedback and we show how to design regulators that are robust to matched exosystem-generated disturbances. The proposed design methodology, of interest also for purely continuous-time systems, combines recent advances in the field of hybrid control systems and output regulation for nonlinear systems. As an illustrative example, the problem of global stabilization of a 6-DOF rigid body affected by a force periodic disturbance of uncertain amplitude, phase and frequency is presented.     

Gronwall inequality for hybrid systems

The Gronwall inequality, a well-known and useful result both for continuous-time and discrete-time signals, is extended to hybrid signals, namely those that combine continuous time and discrete time. An application of the result to establishing a bounded energy bounded state property for hybrid systems with inputs is provided.     

Mineralization of a sorbed polycyclic aromatic hydrocarbon in two soils using catalyzed hydrogen peroxide

Hydrogen peroxide (H2O2) catalyzed by soluble iron or naturally occurring soil minerals, (i.e., modified Fenton's reagent) was investigated as a basis for mineralizing sorbed and NAPL-phase benzo[a]pyrene (BaP), a hydrophobic and toxic polycyclic aromatic hydrocarbon, in two soils of different complexity. 14C-Benzo[a]pyrene was added to silica sand and a silt loam soil, and mineralization was investigated using three-level central composite rotatable experimental designs. The effects of H2O2 concentration, slurry volume, and iron(II) amendment were investigated in the silica sand systems. In a Palouse loess silt loam soil, the variables included H2O2 concentration, slurry volume, and pH, with H2O2 catalyzed by naturally occurring iron oxyhydroxides. Regression equations generated from the data were used to develop three-dimensional response surfaces describing BaP mineralization. Based on the recovery of 14C-CO2, 70% BaP mineralization was achieved in the sand within 24 h using 15 M H2O2 and an iron(II) concentration of 6.6 mM with a slurry volume of 0.3 x the field capacity of the sand. For the silt loam soil, 85% mineralization of BaP was observed using 15 M H2O2, no iron amendment, and a slurry volume of 20 x the soil field capacity. The balance of the radiolabeled carbon remained as unreacted BaP in the soil fraction. Gas-purge measurements over 5 d confirmed negligible desorption under nontreatment conditions. However, oxidation reactions were complete within 24 h and promoted up to 85% BaP mineralization, documenting that the natural rate of desorption/dissolution did not control the rate of oxidation and mineralization of the BaP. The results show that catalyzed H2O2 has the ability to rapidly mineralize sorbed/NAPL-phase BaP and that partitioning, which is often the rate-limiting factor in soil remediation, does not appear to limit the rate of vigorous Fenton-like treatment.     

Discrete-time certainty equivalence output feedback: allowing discontinuous control laws including those from model predictive control

We present certainty equivalence output feedback results for discrete-time nonlinear systems that employ possibly discontinuous control laws in the feedback loop. Coupling assumptions of nominal robustness with uniform observability or detectability assumptions, we assert nominally robust stability for output feedback closed loops. We further show that model predictive control (MPC) can be used to generate a feedback control law that is robustly globally asymptotically stabilizing when used in a certainty equivalence output feedback closed loop. Allowing for discontinuous feedback control laws is important for systems employing MPC, since the method can, and sometimes necessarily does, result in discontinuous control laws.     

Constructive nonlinear anti-windup design for exponentially unstable linear plants

In this paper we give a constructive method for anti-windup design for general linear saturated plants with exponentially unstable modes. The constructive solution is independent of the controller dynamics so that the size of the (necessarily bounded) operating region in the exponentially unstable directions of the plant state space is large. Desirable properties of the closed-loop are formally proved and shown to induce a very desirable behavior on a MIMO example with two exponentially unstable modes.