Physically unrealizable controllers

The presence of pronounced dead times in the process, leading to a physically unrealizable controller (the reader can show this easily)... 

If i > 0, we have physically unrealizable controllers because they require future output values, as depicted by the term y .t, which are not available. In this case we can develop suboptimal controllers by omitting the future output values. Thus if t = 1, we can neglect the term yn+t from eq. (30.22) and take the following suboptimal control action ... 

We must then select the type of controller and set its tuning constants so that, when coupled with the process, it gives the desired closedloop response. Naturally, the control specification must be physically attainable. We cannot make a Boeing 747 jumbo jet behave like an F-IS lighter. We cannot violate constraints on the manipulated variable (the control valve can only go wide open or completely shut), and we cannot require a physically unrealizable controller (more about the mathematics ol this in Chapter 8). 

The next step is to make sure that no physically unrealizable control schemes have been proposed—for example, to attempt to control all of the component concentrations in a stream plus its total flow rate. Even with controllers in place, there must be some way for each species to leave the plant. For example, when the purge line is closed, there is no way for component D to leave the recycle path thus, constraint handling methods associated with the control system cannot close for a significant period of time. 

Again, this controller is physically unrealizable because the order of the numerator is greater than the order of the denominator. We would have to modify our specified so as to make this controller realizable. 

The Bode plots for a PD controller show that as co - oo, the AR - oo. This is, again, physically unrealizable. How should we modify the transfer function... 

The exponential term implies that we need future values of the disturbance in order to compute the current value of the manipulated variable. But, such future values of the disturbance cannot be available. In such case the feedforward controller, described by the last equation above, is characterized as physically unrealizable and cannot be applied in real situations. 

What do we mean when we say that a control algorithm is physically unrealizable What are the necessary conditions for designing physically realizable deadbeat and Dahiin algorithms ... 

The control algorithm given by eq. (30.9) is physically unrealizable because it requires a future value of the error (i.e., +i) to compute the current control action (i.e., c ). 

It is not always possible to achieve perfect feedforward control. If the Gm(s) transfer function has a deadtime that is larger than the deadtime in the Gl(s) transfer function, the feedforward controller will be physically unrealizable because it requires predictive action. Also, if the Gm(s) transfer function is of higher order than the Gi( s) transfer function, the feedforward controller will be physically unrealizable [see Eq. (9.28)]. 

Unfortunately, the ideal proportional-derivative control algorithm in Eq. 8-11 is physically unrealizable because it cannot be implemented exactly using either analog or digital controllers. For analog controllers, the transfer function in (8-11) can be approximated by... 

This controller transfer function can be interpreted as the product of the transfer functions for PI and PD controllers. Because the transfer function in (14-50) is physically unrealizable and amplifies high-frequency... 

Because the term is a negative time delay, implying a predictive element, the ideal feedforward controller in (15-25) is physically unrealizable. However, we can approximate the term by increasing the value of the lead time constant from to Xp -I- 0. 

Note that Gy(s) is physically unrealizable. The static (or steady-state) version of the controller is simply a gain, Gf(s) = -4.17. In order to derive a physically realizable dynamic controller, we approximate the unrealizable controller in (15-33) by a lead-lag unit ... 

This equation requires a future input u k + 1) to influence the present value of the output y k), which is physically impossible (unrealizable). Physical realizability of models (process or controller) should be checked prior to their use for simulation or control. 

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