Linear Feedforward Control

A block diagram of a simple openloop process is sketched in Fig. 11.4o. The load disturbance L, and the manipulated variable affect the controlled variable 

A conventional feedback control system is shown in Fig. 11.4b. The error signal ( is fed into a feedback controller (,) that changes the manipulated variable M, . 

Thus the transfer function of a feedforward controller is a relationship between a manipulated variable and a disturbance variable (usually a load change). 

To design a feedforward controller, that is, to find F, we must know both and The objective of most feedforward controllers is to hold the controlled variable constant at its stea dystate value. Therefore the change or perturbation in should be zero. The output is given by the equation 

Setting equal to zero and solving for the relationship between and L, give the feedforward controller transfer function 

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Linear Feedforward Control / 9.2.2 Nonlinear Feedforward Control Openloop-Unstable Processes... 

In practice, many feedforward control systems are implemented by using ratio control systems, as discussed in Chap. 8. Most feedforward control systems are installed as combined feedforward-feedback systems. The feedforward controller takes care of the large and frequent measurable disturbances. The feedback controller takes care of any errors that come through the process because of inaccuracies in the feedforward controller or other unmeasured disturbances. Figure 11.4d shows the block diagram of a simple linear combined fe forward-/ feedback system. The manipulated variable is changed by both the feedforward controller and the feedback controller. 

The addition of the feedforward controller has no effect on the closedloop stability of the system for linear systems. The denominators of the closedloop transfer functions are unchanged. 

There are no inherent linear limitations in feedforward control. Nonlinear feedforward controllers can be designed for nonlinear systems. The concepts are illustrated in Example 11.3. 

The above nonlinear feedforward controller equations were found analytically. In more complex systems, analytical methods become too complex, and numerical techniques must be used to find the required nonlinear changes in manipulated variables. The nonlinear steadystate changes can be found by using the nonlinear algebraic equations describing the process. The dynamic portion can often be approximated by linearizing around various steadystates. 

Feedforward control provides a linear correction and therefore can provide only partial compensation to a nonlinear process. Nevertheless, feedforward control can be effective when properly implemented, since it can reduce the amount of feedback correction required. When tuning a feedforward controller for a nonlinear process, care should be taken to ensure that the feedforward controller is tuned with consideration to both increases and decreases in the disturbance level. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

V.20 In Example 6.4 we developed the linearized model of a nonisothermal CSTR. Develop a nonlinear steady-state feedforward controller which maintains the value of c A at the desired set point in the presence of changes in cAp Tt. The coolant temperature Tc is the manipulated variable. 

The nonlinear simulation was used to illustrate the closed-loop response of the controlled variable X2 following a 30 percent increase in feed composition. The results are shown in Figure 21.4b with the feedback-only dual and PID algorithms. Control is immensely improved with the feedforward action. The slight deviation in X2 with feedforward control is due to inaccuracies in the linear model and the long sampling time relative to the process dead time. The... 

In this section, we consider the design of feedforward control systems based on dynamic, rather than steady-state, process models. We will restrict our attention to design techniques based on linear dynamic models. But nonlinear process models can also be used (Smith and Corripio, 2006). 

A comparison of Eqs. 15-37 and 15-39 indicates that if Ad is small, Xioc K, If the steady-state feedforward control law of Eq. 15-38 is indeed linear, then = K and the gains for the two design methods are equivalent. 

Approximation of nonlinearity effect on mixing time constant. Inclusion of nonlinearity effect on control loop errors. Linearization of controlled variable by signal characterization. Calculation of continuous pH feedforward signal. Specification of ratio factor for flow feedforward. 

Based on the linearized models around the equilibrium point, different local controllers can be implemented. In the discussion above a simple proportional controller was assumed (unity feedback and variable gain). To deal with multivariable systems two basic control strategies are considered centralized and decentralized control. In the second case, each manipulated variable is computed based on one controlled variable or a subset of them. The rest of manipulated variables are considered as disturbances and can be used in a feedforward strategy to compensate, at least in steady-state, their effects. For that purpose, it is t3q)ical to use PID controllers. The multi-loop decoupling is not always the best strategy as an extra control effort is required to decouple the loops. 

From the stability of the inverse (or zero) dynamics motion (Eq. 15) and of the linear filter (Eq. 16b) (tuned sufficiently fast), the stability of the closed-loop system motion follows. The regulated-measured outputs (z, ) have quasi(q)LNPA tracking dynamics, and the unmeasured regulated outputs (Zf) have the Eb-stability property (7) of the ZD motion (Eq. 15). As the controller gain is tuned faster, the controller approaches the behavior of its feedforward counterpart (Eq. 14), and this feature in turn constitutes the behavior recovery target of the measurement -driven controller that will be developed next. 

In terms of composition control the performance of this version of the scheme is unlikely to be distinguishable from that shown in Figure 12.64. The additional complexity therefore might not be justified. But while it is still not correct to keep the modified reflux ratio constant as feed composition changes. Figure 12.70 shows a more linear relationship. This would be helpful if feedforward on feed composition was being contemplated. 

If Coriolis and magnetic flowmeters are installed to measure the reagent and influent flows, cascade control, flow feedforward, and online identification of a titration curve should be utilized to facilitate start-up and recovery from the flat portions of the titration curve and improve over system performance and knowledge. If the titration curve is relatively fixed or can be calculated or identified online, linear reagent demand control should be used to suppress oscillations and noise and restore... 

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