Modern PID Control: Stabilizing Sets and Multiple Performance Specifications
In this chapter, we introduce some recent results on PID controller design. The new approach first computes the entire set of PID controllers that stabilize a given plant. Within this set, controllers that simultaneously satisfy some performance requirements such as gain, phase, and H ∞ margins are computed. The characterization for PI controllers is in a quasi-closed form while that for PID controllers involves the solution of a linear programming problem. This chapter also shows that the complete set of stabilizing PID controllers for a finite dimensional LTI plant, possibly cascaded with a delay, can be calculated directly from the frequency response (Nyquist/Bode) data P(jω) for ω∈[0,∞) without the need of producing an identified analytical model. The solutions have important new features. For example, the solution identifies, in the case of PID controllers, an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate. These constitute important new guidelines for identification when the latter is to be used for control design.
This research was supported in part by NSF Grants CMMI-0927664 and CMMI-0927652, and DOD Grant W911NF-08-0514.
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