10 - Identification of Unstable TITO Systems by Optimization Technique

Summary

In this chapter, a generalized technique is discussed to obtain the initial guess values for individual transfer function processes of the unstable Two Input Two Output (TITO) multivariable systems. To determine the lower and the upper bounds to be used in the optimization technique, a simple method is explained. Section 10.1 discusses the proposed design method to identify the model parameters of a TITO system under decentralized controller along with the analytical expressions to determine the initial guess values. The applicability of the method is demonstrated with two simulated unstable systems. The method is also extended to unstable TITO system under centralized controllers. Simulation examples show that the proposed method gives a quick convergence with less computational time. For solving the optimization problem, lsqnonlin routine in Matlab is used.

Identification of Systems with Decentralized PI Controllers

The decentralized multivariable system shown in Fig. 3.2 (Chapter 3) is considered. The main loop diagonal transfer function models are unstable FOPTD. The process transfer function matrix G_p(s) and the decentralized controller matrix G_c(s) are given as in Eq. (10.1).

The unstable system can be stabilized by the decentralized control scheme. The controller settings can be selected to obtain a reasonable stable process response. The present example focuses on the identification of the systems having the main loop diagonal transfer functions (g₁₁ and g₂₂) as unstable and the off diagonal transfer function (g₁₂ and g₂₁) as stable.

In general, the transfer function models used are expressed as in Eq. (10.2).

The set point yr₁ is perturbed with the other loops closed and other set points unchanged. From this set point change, the main response y₁₁ and the interaction y₂₁ is obtained. Similarly, yr₂ is perturbed to obtain the main response y₂₂ and the interaction y₁₂. The initial guess value plays an important role in the optimization technique. For the identification of the model, these response values are used to find the initial guess values for which a straightforward method is suggested. The initial guess values for time delay are considered to be the same as the corresponding closed loop time delay values and the time constant is considered as t_s/8 where t_s are the settling time of closed loop responses. The initial guess values for k_p11 and k_p22 are obtained from the relation given in Eq. (10.3) (Chidambaram, 1998).