Frequency Response Analysis of the Process

In the Laplace domain, this control equation can be written as  

It can easily be seen that for the proportional controller t should approach infinity and Ta approach zero. For a proportional-differential controller t should be infinity. 

Determination of the frequency response of the controller is relevant since it can be nsed for controller tuning. To calculate the frequency response, the following transformation is used (see chapter 9)  

Using this substitution, the amplitude ratio and phase shift of a transfer function can be determined for different values of the frequency o). 

The gain (amplitude ratio AR) is the ratio between the amplitude of the input signal and output signal of a transfer function G(fi ). This is equal to the norm of the transfer function  

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