Stability of controllers with on-line computations

The dynamical systems theory developed by Zufifia [1], Zufiria and Guttalu [2, 3], and Guttalu and Zufiria [4] is applied to the stability analysis of control systems in which the feedback control law requires in real time the solution of a set of nonlinear algebraic equations. Since a small sampling period is assumed, the stability and performance of the controlled process can be studied with a continuous-time formulation. A singularly perturbed system is used to model both the dynamics of the system being controlled and a numerical iterative algorithm required to compute the control law. An updating control procedure has been proposed based on the iterative nature of the control algorithm. The results obtained by Zufiria [1] regarding the behavior of a dynamical system that models the numerical algorithms lead to a considerable simplification in the analysis. For the case of a control problem involving inverse kinematics, the numerical algorithm that solves for inverse kinematics can be considered as an observer (or an estimator) of the state-space variables. The study provides an estimate of the required speed of computations to preserve the stability of the controller