Specification for integrating processes

There are also some other characteristics that the closed-loop system must have when the open-loop transfer function contains a double integrator. Suppose that the process transfer function G(s) has the following form, with the stable part represented by H s) 

Suppose we choose the desired closed-loop transfer function relating the setpoint to the control signal to be of the form 

We refer to this type of integrating process as being lag dominant, which represents the majority of integrating processes encountered in the process industries. The time constant of the desired control signal is chosen as 

We define s = 7i s as a scaled Laplace transform variable which allows us 

The scaling with 7i in the Laplace domain naturally leads to a scaling in the time domain with t =, where t represents the normalized time. The desired control signal response for a given step setpoint change of magnitude f has an initial change of and then exponentially decays to 

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The control signal trajectories presented in the previous section require relatively little information about the process to be controlled. For instance, the specification for stable processes contains only the steady state process gain K and the specification for integrating processes contains only K and 71. However, the design of the PID controller itself, with its limited degrees of freedom, will ultimately have to be based on some further process information. The information to be used here is given by the frequency response of the process, G jw). From the desired transfer function of the control signal... 

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