Minimum variance control

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. 

Comparison of the system behavior using three different model predictive controllers (a) minimum variance, (b) input constraint, (c) input penalty. 

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). 

Other self-tuning controllers have been designed which overcome these difficulties, e.g. the Generalised Minimum Variance self-tuning controller (GMV)m and the Generalised Predictive Controller (GPC)m. 

Minimum Variance Purity Control of Preparative Chromatography with Simultaneous Optimization of Yield An On-Line Species-Specific Detector... 

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). 

The major emphasis will be on the minimum variance stochastic control schemes of Box and Jenkins ( b) and Astrom (l ), and on modifications of them. These schemes have seen successful application in the polymer industry, and they are intuitively appealing and yet simple enough to be implemented by the plant operators using either a programmable hand calculator or control charts and tables. More powerful adaptive versions can be implemented if a small online mini-computer is available. 

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... 

Even starting off with poor initial controller parameter values, the self-tuning algorithm usually converges quite rapidly to a stable controller. After a certain amount of data has been collected one can test whether or not the assumed controller form is optimal (in the sense of the minimum variance) and then change it if necessary. 

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