| This work considers the problem of stabilizing and designing fixed order controllers for linear time-invariant plants. Specific controller structures considered are the constant gain (P), proportional-integral (PI) and proportional-integral-derivative (PID). First, a generalization of the Hermite-Biehler Theorem is derived and shown to be useful in providing a solution to this stabilization problem. Based on this, a complete analytical characterization of all stabilizing feedback gains is provided as a closed form solution. This is in stark contrast to the highly nonlinear inequalities that one may have to deal with in trying to solve this problem using the Routh-Hurwitz criterion or the graphical procedures which must be followed while using the Nyquist criterion or the Root Locus technique. Using the solution to the constant gain stabilization problem as a stepping stone, a linear programming characterization of all stabilizing PI and PID controllers for a given plant is obtained. This characterization besides being computationally efficient reveals important structural properties of PI and PID controllers. For example, it is shown that for a fixed proportional gain, the set of stabilizing integral and derivative gains lie in a convex set. The characterization is then shown to be useful for carrying out constant gain, PI, and PID designs which optimize given performance indices. The results given here should have a widespread impact on control applications because of the prevalence of PID controllers in industry.; Then using a similar approach developed for the continuous-time case, the analogous results on constant gain stabilization for the discrete-time case are also provided. It is also shown that by using appropriately generalized versions of the Hermite-Biehler Theorem derived for solving the stabilization problems, it is possible to provide a simple derivation of the Routh-Hurwitz criterion. |
