In this chapter, a Closed Loop Reaction (CRC) curvemethod for the identification of stable TITO system is discussed. The system under consideration is controlled by decentralized PI or PID controllers. The responses and interactions are modelled by the extension of Yuwana and Seborg (1982) method and the transfer function matrix for the closed loop system is obtained. Using the relation between the open loop and the closed loop transfer function matrices, the open loop transfer function matrix is obtained in Laplace (s) domain. The responses of the obtained and the actual transfer functions matrix with the original controller settings are compared. Better results are obtained if the parameters of the identified transfer function models are used as initial guess values for any optimization method and the transfer function model is identified. The higher order models are approximated to a FOPTD model. Since the method is proposed for stable systems,many of the higher order stable systems can be approximated to a FOPTD system.

Identification Method

Identification of Individual Responses

Consider a stable 2 × 2 transfer function matrix (GP ) as in Eq. (4.1a).

Let the controller settings be defined by the matrix given in Eq. (4.2).

The PI/PID controllers are selected so as to get a closed loop under-damped response. Initial controller settings are taken, based on the knowledge of the gain alone, using Davison method (Section 3.13).

Consider Fig. 4.1a. A step change of known magnitude is given to the set point yr1 and the other set point is kept unchanged. The main response obtained is y11 and the interaction response is y21. Similarly for Fig. 4.1b. A step change of the same magnitude is given as yr2 and the other set point is kept unchanged. Let the set point matrix be given as in Eq. (4.3).

The main response is denoted as y22, the interaction as y21, and the output matrix is given in Eq. (4.4).

The transfer function of the closed loop responses is assumed to be an under-damped SOPTD system. From the closed loop response curve for each case, the values yp1, yp2, ym1 and T are noted (Fig. 2.6). Using the formulas given by Yuwana and Seborg (1982), the value of τe and ζ are noted for each case. The closed loop time delay is noted from the responses.