Type 3 integral desaturation

The form of integral desaturation which we shall call lype r specifies that the desaturated integral term should be held constant as soon as the controller output reached either its lower limit, Cmin. or its upper limit, Cmix- It can be shown that a Type 1 integral desaturation system leads to a controller that will come back out of saturation into its normal range when both of the following conditions are satisfied (see Appendix 9)  

the errm must have returned to the value present at entry to saturation. 

We may model Type 1 integral desaturation by replacing the pure integral term, /, in our simulation by the desaturated integral term. Ip, given at a general time, t, by  

Here C is the partially desaturated controller output, given by 

The (fully) desaturated controller output is then given by 

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Chapter 22 provides equations for typical process controllers and control valve dynamics. The controllers considered are the proportional controller, the proportional plus integral (PI) controller and the proportional plus integral plus derivative (PID) controller. Integral desaturation is an important feature of PI controllers, and mathematical mc els are produced for three different types in industrial use. The control valve is almost always the final actuator in process plan. A simple model for the transient response of the control valve is given, which makes allowance for limitations on the maximum velocity of movement. In addition, backlash and velocity deadband methods are presented to model the nonlinear effect of static friction on the valve. 

The essential features of integral desaturation can be explained by reference to a P +1 controller. Three types will be considered and their describing equations given. Types 2 and 3 are implemented using digital techniques and so it has been found helpful to provide a subscripted label for time, t , which signifies a point in time n controller timesteps after the simulation start time, to. (The controller timestep need not be the same as the integration timestep of the simulation.)... 

The equations presented above are sufficient to describe the action of a controller with Type I integral desaturation. They are also in a suitable form for direct implementation within a continuous system simulation language. 

The Type 2 integral desaturation system checks the controller output continually. If its value is found to lie outside the output limits Cmin and Cm, then the integral term is adjusted to ensure that the controller output lies on the nearest output limit. 

The above equations constitute a full and sufficient description of the Type 2 integral desaturation system. However, equation (22.20) shows that the controller resets the integrator state, and this is not an option usually included within a continuous simulation language. Accordingly we need to find a different but equivalent formulation if we are to model Type 2 integral desatu-ration successfully. Let us reconsider equation (22.20), but move on in time to see the value of the desaturated... 

Type 3 integral desaturation is a variant of Type 2. The arrangement is shown in Figure 22.S. 

Hgura 22.5 P +1 controller with Type 3 integral desaturation. 

To implement this form of integral desaturation within a standard continuous simulation language, we introduce once again the concept of the remainder, R, just as we did in the case of Type 2 integral desaturation. The analysis then mirrors that of Section 22.5.2, and this gives the following equation set for R ... 

With Type I integral desaturation, the integral term is limited when the partially desaturated output, C (r), takes a value outside the controller s output limits (see equations (22.16) and (22.17)). 

The condition of equation (A9.7) is a necessary but not sufficient condition for the emergence from saturation for a controller with a Type 1 integral desaturation algorithm. 

A9.2.3 Summary of conditions for a controller with Type 1 integral desaturation emerging from controller saturation... 

With the Type 2 algorithm for integral desaturation the controller output, c(t), is made up of the proportional term, P, and foe adjusted integral term, /a ... 

A similar analysis reveals that the controller with Type 3 integral desaturation cannot come out of low saturation until... 

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