Performance of Suboptimal Controllers for Affine-Quadratic Problems

Author(s): Ankita Sharma, A. J. Shaiju

ABSTRACT
In this article, affine-quadratic control problems are studied. Error bounds are derived for the difference between the performance indices corresponding to the optimal and a class of suboptimal controls. In particular, it is shown that the performance of these suboptimal controls is close to that of the optimal control whenever the error in estimating the costate initial condition is small.

1. Introduction
One of the most active areas in control theory is optimal control and methods to find them [1]-[3]. It has a wide range of practical applications in engineering (Aerospace, Chemical, Mechanical, Electrical), science (Physics, Biology), and economics (see e.g. [4]-[7]). Optimal control theory has been developed for linear systems ([1] [2] [8]) and explicit formulae for computing optimal control inputs are available. However, control of nonlinear systems is much more challenging and obtaining formulae for optimal controls seems in general not possible. This motivated researchers to study various classes of nonlinear control problems separately, and affine-qudratic problems is one such class. In a recent paper [9], the optimal control for affine-quadratic problems is obtained in terms of the associated costate. But, in practice, it is difficult to compute the costate (at each time t ) as the knowledge of its terminal condition is required.

In this article, we study the affine-quadratic control problem given by ((1), (2)). We note that a method for finding the initial condition for the costate is recently proposed [10]. This allows one to compute the initial costate (at t = 0 ) exactly or approximately. This approximation of the initial costate and the explicit formula for optimal control (as in [11]) are shown, in this article, which give rise to suboptimal controls of practical importance. More precisely, our main theorem (Theorem 2) provides an upper bound for the difference in performance between these suboptimal and optimal control.

The article is organized as follows. In Section 2, the affine-quadratic control problem is described. We also explain how to obtain the optimal control in terms of costate. The main (Theorem 2) is proved in Section 3. This theorem provides a method to obtain the costate (without the knowledge of its terminal value) which results in an explicit formula and performance bounds for a class of suboptimal controls.

Source:

Journal: International Journal of Modern Nonlinear Theory and Application
DOI: 10.4236/ijmnta.2014.35025 (PDF)
Paper Id: 52211 (metadata)

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