Subharmonic Oscillations, Relay On-Line Tuning and Process Identification

In this paper a multiple relay arrangement for on-line process identification and controller tuning is proposed. The standard relay feedback method proposed by Astrom and Hagglund (1984) is very popular, because it is a simple and time efficient technique; moreover it is a closed-loop method and allows a tight control of the magnitude of the oscillations. Unfortunately only the critical point of the process (i.e. the process frequency response at the phase lag of -π) is detected and the acquired information may be insufficient for a correct identification of a large class of processes. Since in chemical processes the duration of each experiment may be critical, multiple identification sessions are undesirable and impracticable; therefore several alternative techniques have been proposed. However the information on the critical point of the process must be retained, often with the need of including some information at frequencies near the crossover point or in the low frequency range. Friman and Waller (1997) introduced a two-channel relay structure in order to extract more information about the process, e.g. to get a point in the third quadrant of Nyquist plane, Wang and Yang (2000) have provided the cascade relay for multipoint identification of the frequency response, Balestrino et al. (2006) have proposed a technique based on a standard relay with a variable hysteresis width, in order to originate relay transients. Several variants of the standard method have been presented in literature (see Wang et al., 2003): most of them consider various relay connections and give rise to very complex waveforms, usually difficult to analyse. In this paper a master relay is configured as in the standard method of Astrom and Hagglund. Assume that the input of the master relay is periodic. The output of the master relay, um, is converted from real to binary format and, through a shift register or a chain of flip-flops, it gives rise to a sub-harmonic signal which is again converted from binary to real as the input to an auxiliary relay. Any sub-harmonic signal can be generated, but we restrict our attention to the simplest arrangement, where the subharmonic signal us (the output of the auxiliary relay) shows a frequency 2-m of the master relay frequency. The input signal is the sum of the master and auxiliary relay outputs. The conditions assuring a limit cycle can be derived as in the classical approach of Tsypkin (1984). Eventually hysteresis can be added to the master relay, so that a strong robustness is assured with respect to output measurement noises. The implementation of this technique is straightforward by using low cost electronics; examples of simulation tests using Matlab implementation illustrate the technique for some typical plants.