Fundamental performance limitations in tracking and sinusoidal signals

This paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies. The tracking performance is measured by an integral square error between the output of the plant and the reference signal. Our purpose is to find the fundamental limitation for the attainable tracking performance, under any control structure and parameters, in terms of the characteristics and structural parameters of the given plant, as well as those of the reference signal under consideration. It is shown that this fundamental limitation depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency-dependent directional information. The main results of this paper are based on the assumption that the controller accesses the full information of the reference. However, when the full information of the reference is unavailable, we also obtain a simple performance limit, for an important special case, which clearly shows the extra cost one has to pay due to the information restriction.