Stabilization by three term controllers
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This paper presents recent techniques for determining the complete set of stabilizing, three term controllers for a given single-input single-output, linear, time invariant, continuous or discrete time plant of arbitrary order. Important special cases are first order and PID controllers. For digital PID controllers, a Generalized Hermite-Biehler theorem is utilized to characterize the stabilizing region in the controller parameter space in term of sets of linear programming problems. For the case of general three term (and first order) controllers, the method involves mapping the stability boundary in the frequency domain into the parameter space. The resulting surface(s) divides the parameter space into regions. Every point (i.e. controller) in a region will result in a corresponding closed loop characteristic polynomial with a fixed number of unstable closed loop poles. The stability region will consist of such regions corresponding to no unstable closed loop poles.