Closedloop regulator transfer function

The first closedloop transfer function in Eq. (10.5) relates the controlled variable to the load variable. It is called the closedloop regulator transfer function. The second closedloop transfer function in Eq. (10.5) relates the controlled variable to the setpoint. It is called the closedloop servo transfer function. 

It is useful to consider the ideal situation. If we could design an ideal controller without any regard for physical realizability, what would the ideal elosed-loop regular and servo transfer functions be Clearly, we would wish a load disturbance to have no effect on the controlled variable. So the ideal closedloop regulator transfer function is zero. For setpoint changes, we would like the controlled variable to track the setpoint perfectly at all times. So the ideal servo transfer function is unity. 

Equation (15.64) gives the effects of setpoint and load changes on tbe controlled variables in the closedloop multivariable environment. The matrix (of order N X N) multiplying the vector of setpoints is the closedloop servo transfer function matrix. The matrix (N x 1) multiplying the load disturbance is the closed-loop regulator transfer function vector. 

In most chemical processes the principal control problem is load rejection. We want a control system that can keep the controlled variables at or near their setpoints in the face of load disturbances. Thus the closedloop regulator transfer function is the most important. 

For a 3 X 3 system there are three elements in the jt vector, so three curves are plotted for each of the three elements in the vector [[I + G (tt.>)Qc(w)] Gl(uo)] for the specified load variable L. Table 13.3 gives a MATLAB program that calculates these curves for the 3 X 3 Ogunnaike and Ray column. Figure 13.4 plots the log modulus of the three closedloop regulator transfer functions. 

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