Gain/phase margin design with first order controllers

The problem of determining all first order controllers (C(s) = x₁s+x₂/s+x₃) that stabilize a fixed plant Po(s) has been recently solved in [R.N. Tantaris et.al., Proceeding of The 10th Mediterranean Conference on Control and Automation, Lisbon, Portugal, July 9-12, 2002]. In the present paper we extend this solution to the problems of simultaneously stabilizing the continuum of plants a) KPo(s) with K lying in a prescribed interval and b) e^jθ Po(s) with θ lying in a known interval. Our solution answers the question of when prescribed gain and phase margins can be attained, for the given plant, with first order controllers and when possible, gives the complete set attaining specifications. It is shown how such sets can then be used to obtain prescribed target closed loop transfer functions within specified tolerances. Examples are given for illustrations.