The relay (on–off) controller can stabilize wide ranges of processes including open-loop stable, integrating, and unstable processes, producing sustained oscillations. For improved proportional-integral-derivative controller tunings, methods to find process models with mixed closed-loop tests of relay feedback and proportional-derivative (PD) controllers are proposed. For unknown processes with arbitrary initial states, relay feedback tests are first applied and, after cyclic steady states are obtained, PD controllers or other relay feedback tests with set point changes are followed. This full closed-loop operation is desirable for integrating and unstable processes and will be useful even for stable processes when processes are far from their desirable operating points. Refined methods to find exact frequency responses of processes from initial and final cyclic steady states are derived. Whole relay feedback responses need not be saved. Several integrals at the relay switching times are used without iterative tests or computations.

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Step and pulse responses have long been used to identify process dynamics and design control systems such as PID (proportional-integral-derivative) and model predictive control. They are simple and can represent different types of process dynamics intuitively. However, for some processes, long test times in the open-loop environment are required to obtain them. Here, utilizing the time-optimal control techniques, methods to reduce test times significantly are investigated. From initial short-term responses, exact step and pulse responses are obtained without extrapolations or predictions. Since short-term responses are used, the proposed methods can also be used to restore step responses from fragments of failed step tests. Simulations and experiments are given to illustrate applicability of the proposed methods.

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