Robust Input Shaper Design using Linear Matrix Inequalities

Abstract

This paper proposes an Linear Matrix Inequality based problem formulation to determine input shaped profiles. The cost function is the residual energy, a quadratic function of the amplitude of the shaped profile, for each sampling interval. The Schur complement permits representing the quadratic function as a Linear Matrix Inequality. Augmenting the state space model with the sensitivity of the states to uncertain parameters, input shaped profiles which are robust to model uncertainties can be derived. Finally, a minimax input shaped profile which minimizes the maximum magnitude of the residual energy over the domain of uncertainties is determined using the LMI problem. The proposed technique is illustrated on the single spring-mass-dashpot example. The solutions derived are shown to coincide with the solutions presented in the literature, without the requirement of solving a nonlinear programming problem.

Input shaping design using sequential linear programming (SLP) for non-linear systems

“…(4)- (7). Like any nonlinear optimal control method, we desire to compute the optimal control iteratively by assuming an initial control profile u 0 (t) and determining the corresponding evolution of the states.…”

“…Kim and Singh [6] designed robust controllers for rest-to-rest motion of a vibratory system subject to friction using linear programming. Conord and Singh [7] solved the minimax input shaper for linear systems using LMI which ensures that the globally optimal solution is achieved. Design of Input Shapers for nonlinear systems have required nonlinear programming [8], [9] which are sensitive to initial guesses.…”

Optimal input shaping filters for non-zero initial states

“…In the convolution based input-shaping technique, the reference command is computed for pure rigid-body and is later convolved with a single filter. In the closed-form method, the input is modified for each on-off switch in the rigidbody command using different filters designed for different transitions [6], [21]- [24], [26]. The rigid-body switch times are required to be corrected in order to satisfy the rest-torest motion conditions [6].…”