On the Synthesis of a Stabilizing Feedback Control via Lie Algebraic Methods

Abstract

Let the n-dimensional system of differential equations ${{dx} / {dt}} = X(x(t))$ have $p \in \mathbb{R}^n $ as a rest solution, i.e., $X(p) = 0$. Even in cases when this rest solution is unstable, one can often induce a strong stability (asymptotic stability) by the inclusion of one or more controls, e.g., via a controlled system (a) ${{dx} / {dt}} = X(x) + uY(x)$, where, say, $| u | \leqq 1$. Lie theory gives a computable, sufficient, condition to determine when, by use of the control u, one can steer a full n-dimensional neighborhood of p to p by solutions of (a). This condition is assumed to hold. One prefers a feedback control, i.e., that $u = u(x)$. The main result in this paper is an algorithm which determines a “modified” stabilizing feedback control. Specifically, for given $\varepsilon > 0$, one measures the current state q and the algorithm determines $u(t;q)$, $0 \leqq t \leqq \varepsilon $, such that the solution $x( \cdot ;u)$ of (a) initiating from q and corresponding to this control u, satisfies distance $| {x(\varepsilon ;u) - p} | < | {q - p} |$. In fact, iterates are theoretically shown to converge to p. Numerical examples computed via a simple FORTRAN program are included. These substantiate the strong stability achieved via such a modified feedback control.