Lyapunov Stability of Periodic Control on a Continuous Stirred Tank Reactor

Abstract

Abstract Periodic control uses zero mean parametric excitation as a tool to influence the transient behavior of a dynamical system. Unlike conventional methods, based on feedback or feedforward principles, the method of periodic control may not require any measurement of the deviations or disturbances to stabilize an unstable system. The choice of amplitude and frequency in periodic control provides two additional degrees of freedom to stabilize an open-loop system. Stability analysis of periodically forced systems is often limited to linearization methods. This is often not sufficient to assess the global properties of a non-linear system like finding the region of convergence. In this paper, forced oscillations are introduced in the input flow rates of a continuous stirred-tank reactor (CSTR) to operate the process around an unstable point. We use Lyapunov analysis to demonstrate the exponential stability of a CSTR system under the operation of periodic control. For exponentially convergent systems, we derive a theorem to estimate the region of convergence and show that the rate of convergence for a system under weak oscillations is almost the same as that of its averaged system. Numerical simulations of a propylene glycol process are carried out to illustrate the exponential stability of periodic control and verify the results of our analysis.