Minimum variance controller

Constraint parameter In constrained minimum variance controller... 

Appendix Converting a Constrained Minimum Variance Controller to a PIP... 

The MPC controller that minimizes the variance of the output (minimum variance controller) during a setpoint change corresponds to the controller setting w = 1, A = 0, and no bounds on the input. The response for this controller design for m = 2 and p = 4 is given in Figure E16.3 by the solid line. 

The input for most chemical processes is normally constrained, (e.g., a valve ranges between 0 and 100 percent open). An unconstrained minimum variance controller might not be able to achieve the desired input trajectory for the response. The controller design should take the process input constraints into account. The results of a simulated setpoint change for such a controller with bounds of —40 and 40 for the input and controller parameters w = 1 and A = 0 is given by the dashed line in Figure El6.3. 

In the last decade several other multivariable controllers have been proposed. We will briefly discuss two of the most popular in the sections below. Other multivariable controllers that will not be discussed but are worthy of mention are minimum variance controllers (see Bergh and MacGregor, lEC Research, Vol. 26, 1987, p. 1558) and extended horizon controllers (see Ydstie, Kershenbaum, and Sargent, AIChE J., Vol. 31, 1985, p. 1771). 

Figure 4. Internal model control representation of a minimum variance control method for product purity with simultaneous optimization of product yield.

Physical state space models are more attractive for use with the LQP (especially when state variables are directly measurable), while multivariable black box models are probably better treated by frequency response methods (22) or minimum variance control (discussed later in this section). 

Recently there has been great interest in discrete-time optimal control based on a one-step ahead optimization criterion, also known as minimum variance control. A number of different approaches for minimum variance control has been developed in the last decade. MacGregor (51) and Palmor and Shinnar (52) have provided overviews of these minimum variance controller design techniques. 

Both Astrom (53) and Box and Jenkins (54) have developed modeling approaches for equation (13), which involve obtaining maximum likelihood estimates of the parameters in the postulated model followed by diagnostic checking of the sum of the residuals. The Box and Jenkins method also develops a detailed model for the process disturbance. Both of the above references include derivations of the minimum variance control. 

The minimum variance control for an SISO system finds the unrestricted minimum of the expected value of a quadratic objective function ... 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

Minimum Variance Control Given that the objective of most continuous flow polymer reactor systems is to maintain the output... 

O.Sz Ma-t with - 1. The rederived minimum variance controller would be... 

The most popular tool for monitoring single-loop feedback and feedforward/feedback controllers is based on relative performance with respect to minimum variance control (MVC) [53, 102[. The idea is not to implement MVC but to use the variance of the controlled output variable that would be obtained if MVC were used as the reference point. The variation of the inflation of the controlled output variance indicates if the process is operating as expected or not. Furthermore, if the variance with a MVC is larger than what could be tolerated, this indicates the need for modification of operating conditions or process. 

CPM of multivariable control systems has attracted significant attention because of its industrial importance. Several methods have been proposed for performance assessment of multivariable control systems. One approach is based on the extension of minimum variance control performance bounds to multivariable control systems by computing the interactor matrix to estimate the time delay [103, 116]. The interactor matrix [103, 116] can be obtained theoretically from the transfer function via the Markov parameters or estimated from process data [114]. Once the interactor matrix is known, the multivariate extension of the performance bounds can be established. 

If, in continuous polymerization, the samples are sufficiently infrequent that the process settles between samples, and if the samples are not autocorrelated, the procedures outlined above can be used. This amounts to manual steady-state control with the need for control identified by the control chart. If the process is fast compared with the sampling interval, but the samples are autocorrelated (as is often the case), a controller can be developed which specifies the correction to be made to the process. A process model is needed for a process which is fast compared with the sampling interval, this can be simply a gain between the manipulated input and the process output. A disturbance model is also necessary a simple but effective model for continuous process systems is that of an integrated white noise sequence. With these assumptions, a minimum variance controller may be derived [45]. This takes the form of the following discrete pure integral controller ... 

The aim of the minimum-variance control algorithm is the minimization of the following performance index (Astrom and Wittenmark, 1995 Astrom and Wittenmark, 1990). 

This chapter develops the control system with an adaptive minimum variance controller. The plant and the controller are simulated as discrete in time (Phillips and Nagle, 1995). The plant parameters are polynomials 5(z ) and standard deviation of white noise of... 

The minimum-variance controller serves as an example of a self-tuner. The optimal value of the output signal standard deviation is obtained in the steady state. The plant is assmned to be of ARX type (AutoRegressive with exogenous input). 

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