Robust input shaper design using linear matrix inequalities

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.