Example of exact trade-offs in linear controller design

Two examples of analytic methods for finding the exact form of trade-offs among such desirable qualities as fast response to commands without excessive overshoot, low actuator authority, robustness, low controller complexity, and so on are considered. They are linear quadratic Gaussian theory, where the plant actuator and output variance can be traded off, and Nevanlinna-Pick theory, where, for instance, the achievable disturbance rejection can be traded off in two different bandwidths. In many cases, the limit of performance achievable with a linear time-invariant controller, and thus the exact form of the tradeoffs, can be computed numerically. To demonstrate this trade-off, the design of a regulator for a very typical plant, a double integrator with some excess phase, is examined. The trade-offs are presented for two different measures of robustness with noise sensitivity. The exact form of the trade-offs is determined numerically by using the techniques described in the appendixes.<>