Minimizing control volatility for nonlinear systems with smooth piecewise-quadratic input signals

Abstract

We consider a class of nonlinear optimal control problems in which the aim is to minimize control variation subject to an upper bound on the system cost. This idea of sacrificing some cost in exchange for less control volatility—thereby making the control signal easier and safer to implement—is explored in only a handful of papers in the literature, and then mainly for piecewise-constant (discontinuous) controls. Here we consider the case of smooth continuously differentiable controls, which are more suitable in some applications, including robotics and motion control. In general, the control signal's total variation—the objective to be minimized in the optimal control problem—cannot be expressed in closed form. Thus, we introduce a smooth piecewise-quadratic discretization scheme and derive an analytical expression, which turns out to be rational and non-smooth, for computing the total variation of the approximate piecewise-quadratic control. This leads to a non-smooth dynamic optimization problem in which the decision variables are the knot points and shape parameters for the approximate control. We then prove that this non-smooth problem can be transformed into an equivalent smooth problem, which is readily solvable using gradient-based numerical optimization techniques. The paper includes a numerical example to verify the proposed approach.