Numerical Solution of the Optimal Periodic Control Problem Using Differential Flatness

Abstract

Optimal periodic control (OPC) is of interest in many engineering applications. In practice, the numerical solution of the OPC problem has been found to be quite challenging. In this note, we present a method which uses differential flatness for the solution of OPC problems. The OPC problem is reformulated using the flatness of the underlying dynamical system to eliminate the differential equations and the periodicity constraints, resulting in simpler and generally more efficient computation. The effect of point-wise constraints and the analytical computation of the Jacobian matrix are also discussed. The approach is demonstrated using two examples.Index Terms-Differential flatness, dynamic optimization, optimal periodic control (OPC).

Optimal periodic control of a drug delivery system

“…First, we employ a method based on differential flatness described in [27] to obtain the optimal periodic drug infusion strategy. Differential flatness (or simply, flatness), is a property of a dynamical system that is related to the concepts of absolute equivalence and dynamic feedback linearizability [26,13].…”

“…The need for integration of differential equations is removed by restating the problem in terms of the so-called flat outputs. A computational method for OPC problems using flatness has been presented in [27], and this approach is used here for the current drug delivery problem to compute the OPC solution.…”

“…In this paper, a simple pharmacokinetic/pharmacodynamic (PKPD) model of a drug is considered and the question whether periodic drug infusion is more effective in a time averaged sense is investigated. In section 2 the problem is formulated as an optimal periodic control (OPC) problem so that powerful analytical and computational techniques developed for OPC problems can be employed [4,2,14,27]. In section 3, the so-called π-test is used to assess whether small periodic variations around the optimal steady state (OSS) can improve performance, a situation known as local properness.…”

“…The solution to the OPC problem is obtained using two computational methods, both of which produce a consistent solution. The first method is based on flatness as described in [27] and the second is a shooting method implemented within a commercially available process modeling software tool, gPROMS TM [22].…”

On‐Line Feedback Control for Optimal Periodic Control Problems

“…Fradkov and Pogromsky (1998) proposed a gradient-based adaptive control schemes to steer a given system to periodic orbits for passive non-linear systems. In the case of fl at nonlinear systems, the study by Varigonda et al (2004a) suggested a method to construct off-line control laws that yield to the optimal periodic orbit. Using extremum-seeking techniques, online construction of optimal periodic orbits solving the OPC problem was proposed recently by Guay et al (2005) for differentially fl at systems where the periodic behaviour is enforced through a periodic input that is tracked by a fl atness based controller.…”