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The controller can also be implemented as
3.3
This structure has the advantage that we can develop the design methods for an ideal PID controller and use an iterative design procedure. The controller is first designed for the process P(s). The design gives the controller parameter Td. An ideal controller for the process P(s)/(1+sTd/N)2 is then designed giving a new value of Td etc. Such a procedure will also give a clear picture of the tradeoff between performance and filtering.
Set Point Weighting
When using the control law given by （3.1） it follows that a step change in the reference signal will result in an impulse in the control signal. This is often highly undesirable there for derivative action is frequently not applied to the reference signal. This problem can be avoided by filtering the reference value before feeding it to the controller. Another possibility is to let proportional action act only on part of the reference signal. This is called set point weighting. A PID controller given by （3.1） then becomes
where b and c are additional parameter. The integral term must be based on error feedback to ensure the desired steady state. The controller given by DE3.4 has a structure with two degrees of freedom because the signal path from y to u is different from that from r to u. The transfer function from r to u is
Figure 3.4 Time t
Figure 3.4 Response to a step in the reference for systems with different set point weights b= 0 dashed, b = 05 full and b=10 dash dotted. The process has the transfer function P（s）=1/（s+1）3 and the controller parameters are k = 3, ki = 15 and kd = 15.
and the transfer function from y to u is
3.6
Set point weighting is thus a special case of controllers having two degrees of freedom.
The system obtained with the controller （3.4） respond to load disturbances and measurement noise in the same way as the controller （3.1） . The response to reference values can be modified by the parameters b and c. This is illustrated in Figure 3.4, which shows the response of a PID controller to set-point changes, load disturbances, and measurement errors for different values of b. The figure shows clearly the effect of changing b. The overshoot for set-point changes is smallest for b = 0, which is the case where the reference is only introduced in the integral term, and increases with increasing b.
The parameter c is normally zero to avoid large transients in the control signal due to sudden changes in the set-point.
IV. Different Parameterizations
The PID algorithm given by Equation（2.1）can be represented by the transfer function
An interacting controller of the form Equation DE4.2 that corresponds to a non-interacting controller can be found only if
The parameters are then given by
The non-interacting controller given by Equation （4.1） is more general, and we will use that in the future. It is, however, sometimes claimed that the interacting controller is easier to tune manually.
It is important to keep in mind that different controllers may have different structures when working with PID controllers. If a controller is replaced by another type of controller, the controller parameters may have to be changed. The interacting and the non-interacting forms differ only when both I and the D parts of the controller are used. If we only use the controller as a P, PI, or PD controller, the two forms are equivalent. Yet another representation of the PID algorithm is given by
4.5
The parameters are related to the parameters of standard form through