Optimal PD control design via dominant pole assignment for a class of TORA systems

It is known that dominant pole of closed-loop system gives rise to the longest lasting terms in the transient response of systems. In this paper, we address how to design the PD control gains via dominant pole assignment for stabilizing a class of TORA (Translational Oscillator with a Rotational Actuator) systems which has been proposed as a benchmark problem for nonlinear system design. We solve analytically an optimization problem, which achieves a fast control response of the TORA. We present two steps to solve the optimization problem analytically. First, we perform a coordinate transformation to the TORA´s characteristic equation, and prove its unstability. Second, we design the optimal control gains which assign the unstable poles to the imaginary-axis by using the Routh-Hurwitz stability criterion. By using these two steps, we provide the optimal control gains and show that the closed-loop system consisting of the TORA and the PD controller has two pairs of complex-conjugate poles with the same real part. We verify the validity of the analytical result by comparing with numerical computation and numerical simulation.