PID PARAMETERS LEAST SQUARES APPROACH

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 

Oiur objective is to find the PID controller parameters such that the actual closed-loop frequency response is in some sense close to the desired closed-loop fi quency response Gr- y jw). However, the direct approach to this problem leads to a nonlinear optimization problem. Instead, we choose to work with the equivalent open-loop transfer function because, in this case, the problem becomes linear in the controller parameters, enabling us to consider a linear least squares approach to solving this problem. 

FVom the desired closed-loop frequency response, the desired open-loop frequency response can be obtained firom 

The actual open-loop transfer function, G jw), for the process under PID control is given by 

Our task is to find the coefficients C2, ci and cq such that the sum of squared errors between the desired and actual open-loop frequency response is minimized over a set of frequencies u . We choose the loss function to be 

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