## Example of Exact Trade-Offs in Linear Controller DesignC. Barratt and S. Boyd
The design of a linear time-invariant controller for a given linear-time-invariant plant, like any engineering design, involves tradeoff among many desirable qualities, such as fast response to commands without excessive overshoot low actuator authority, robustness low controller complexity, and so on. Only for a few very special cases are analytic methods known for finding the exact form of these tradeoffs. Two examples of such analytic methods are Linear Quadratic Gaussian (LQG) theory (where the plant actuator and output variance can be traded off), and Nevanlinna-Pick (NP) 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 tradeoff, the paper treats the design of a regulator for a very typical plant, a double integrator with some excess phase. The tradeoffs are presented for two different measures of robustness with noise sensitivity. The exact form of these tradeoffs is numerically determined using the techniques described in the appendix. |