Applications of Singular Perturbation Techniques to Control Problems

Abstract

This paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years. The first three sections are devoted to the standard model and its time-scale, stability and controllability properties. The next two sections deal with linear-quadratic optimal control and one with cheap (near-singular) control. Then the composite control and trajectory optimization are considered in two sections, and stochastic control in one section. The last section returns to the problem of modeling, this time in the context of large scale systems. The bibliography contains more then 250 titles. D IS T R I B U T I O N / A V A I L A B I L I T Y O F A B S T R A C T U N C L A S S I F I E D / U N L I M I T E D S S A M E A S R PT. □ O T I C U S E R S □ 21. A B S T R A C T S E C U R I T Y C L A S S I F I C A T I O NUnclassified S E C U R I T Y C L A S S I F I C A T I O N O F T H IS P A G E S E C U R IT Y C L A S S IF IC A T IO N O F TH IS P A G E S E C U R I T Y C L A S S I F I C A T I O N O F T H I S P A G E APPLICATIONS OF SINGULAR PERTURBATION TECHNIQUES

Forest Economics, Natural Disturbances and the New Ecology

“…Models designed to describe the evolution of a forest ecosystem over time in the presence of wildfire would have to simultaneously integrate the equations of motion for the slow and fast variables which, for practical purposes, is not possible. However, mathematicians have developed special methods for solving this type of problem, known as singular perturbation theory (Kokotovic 1984). In this section, we provide a simple example that demonstrates how singular perturbation methods can be used to characterize the temporal dynamics of an important forest pest, the spruce budworm, which causes severe mortality in boreal forests in eastern Canada and the northeastern United States on roughly 40 year cycles (Boulanger and Arseneault 2(04).9 Simply stated, the singular perturbation method separates the dynamic variables into slow and fast categories which allow the fast and slow dynamics to be studied sequentially rather than simultaneously (Simon and Ando 1961, May 1977, Rinaldi and Muratori 1992, Rinaldi and Scheffer 2000.…”

Quality control of binary distillation columns via nonlinear aggregated models

“…It can be reduced by time-scale considerations. The standard form of a two-scale system is [see Kokotovic (1984) or Marino and Kokotovic (1988) for example]…”

“…Otherwise stated, the proposed synthesis by output feedback does not remix the time scales of the original system [for an extended discussion see Kokotovic (1984) or Marino and Kokotovic (1988)]. …”

“…More precisely, we show that the classical distillation model studied by Rosenbrock (1962) can be approximated, via singular perturbation techniques [see Kokotovic (1984) for example], by a reduced-order model including only the slow transients but having the global asymptotic properties of the original model. This aggregated model proves very useful for control: we apply nonlinear perturbation rejection techniques [see Isidori (1989)] and obtain, around every slowly varying reference trajectory, a feedback law without singularity, producing asymptotically stable closed-loop dynamics, that can be synthesized via output feedback when temperature measurements are available.…”

An Energy-Based Nonlinear Control for a Two-Link Flexible Manipulator

“…The exception is when we consider the dynamic characteristics of a special system; for example, the two-time-scale characteristics of a flexible manipulator with larger stiffness and use approximate methods such as the singular perturbation approach. 7,8) Therefore, energy shaping must include both potential energy and kinetic energy. Since there are traditionally mainly two kinds of methods to describe the behavior of dynamic systems (i.e., the dynamics of the systems can be either represented by Hamilton equation or expressed by Euler-Lagrange equation), the former systems are named the Hamiltonian systems and the latter systems are generally called the Euler-Lagrange systems.…”