Tuning PID Controllers for Minimizing ISE and Satisfying Specified Gain and Phase Margins

A standard PID controller has three adjustable parameters, i.e., the proportional gain, and the integral and derivative time constants. Usually, conventional frequency domain techniques of single-loop feedback control design use only gain and phase margin specifications to determine these parameters. As a result, there remains one degree of freedom in selecting controller parameter to meet the specification of gain and phase margins. Motivated by the availability of analytical and/or efficient numerical methods of computing integral of square error (ISE) for systems with/without a time delay, we suggest in this paper to adjust the remaining degree of freedom in PID parameters to minimize a quadratic cost functional. Instead of solving a constrained nonlinear optimization problem, we formulate the design problem in a form such that it can be easily solved by a path-following algorithm while giving the global optimum.