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On averaging and input optimization of high-frequency mechanical control systems

This paper presents the optimization of input amplitudes for mechanical control-affine systems with high-frequency, high-amplitude inputs. The problem consists of determining the input waveform shapes and the relative phases between inputs to minimize the input amplitudes while accomplishing some control objective. The effects of the input waveforms and relative phases on the dynamics are investigated using averaging. It is shown that of all zero-mean, periodic functions, square waves require the smallest amplitudes to accomplish a control objective. Using the averaging theorem the problem of input optimization is transformed into a constrained optimization problem. The constraints are algebraic nonlinear equalities in terms of the amplitudes of the inputs and their relative phases. The constrained optimization problem may be solved using analytical or numerical methods. A second approach uses finite Fourier series to solve the input optimization problem. This second approach confirms the earlier results concerning minimum amplitude inputs and is then applied to the problem of minimizing control energy.