Optimal control of coupled motors using PAM signals in presence of stochastic noises

In this paper the optimal stochastic control of coupled motors is discussed. The unperturbed currents of the motors are designed to follow the prescribed trajectory in the absence of noise. However owing to noise in the current sources, the motors do not follow the prescribed trajectory. In order to correct these errors, we introduce non random perturbation in the current which are controlled by Pulse Amplitude Modulation(PAM) signals. Based on perturbation theory we set up a sequence of equation of coupled second order differential equation in the angle perturbation terms(ε). These equations are solved sequentially using linear state variable theory. These solutions are expressed up to fourth order perturbation terms using first to fourth degree Volterra terms in the noise processes & first and second degree Volterra terms in the PAM current sources. These expressions are used to calculate mean square error(MSE) between the actual trajectories of the motors & desired trajectories up to fourth order perturbation terms(ε⁴). This expression for the MSE containing constant terms, linear terms & quadratic terms in the current perturbation. The Kernel coefficient appearing in this expression involve noise auto correlation function setting the variational derivatives of these MSE with respect to the PAM current sequences results in a set of linear matrix equation for the PAM current sequences which we solved using MATLAB.