Discrete feedback control

There is a special type of controller, called a Smith predictor or deadtime compensator, that can be applied in either continuous or discrete form. It is basically a special type of model-based controller, in the same family as IMC. Figure 20.6a gives a sketch of a conventional feedback control system. Let s break up the total openloop process into the portion without any deadtime G j,(s) nd deadtime e... 

Discrete Time (Digital) Fixed Peremeter Feedback Controllers... 

Discrete time controllers will not normally be stand-alone units but will be simulated within the software of a digital computer. The capacity of the computer can be used if necessary to produce more complex forms of feedback control than those provided by the standard algorithms of the classical fixed parameter controller. 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

The structures operative within a discrete state of consciousness make up a system whose parts stabilize each other s functioning by means of feedback control, so that the state maintains its overall pattern of functioning in spite of some changes in the environment. Yet when certain key environmental stimuli come along, the pattern can break down and be replaced by another, as when some personal remark causes a transition from one identity state to another. 

This work presents the on-line level control of a batch reactor. The on-line strategy is required to accommodate the reaction rate disturbances which arise due to catalyst dosing uncertainties (catalyst mass and feeding time). It is concluded that the implemented shrinking horizon on-line optimization strategy is able to calculate the optimal temperature profile without causing swelling or sub-optimal operation. Additionally, it is concluded that, for this process, a closed-loop formulation of the model predictive controller is needed where an output feedback controller ensures the level is controlled within the discretization intervals. 

Analysis of full sheet data is useful for process performance evaluations and product value calculations. For feedback control or any other on-line application, it is necessary to continuously convert scanner data into a useful form. Consider the data vector Y ,k) for scan number k. It is separated into its MD and CD components as Y( , A ) = yM )( )+Yc )( , k) where Ymd ) s the mean of Y ,k) as a scalar and YcD -,k) is the instantaneous CD profile vector. MD and CD controllers correspondingly use these calculated measurements as feedback data for discrete time k. Univariate MD controllers are traditional in nature with only measurement delay as a potential design concern. On the other hand, CD controllers are multivariate in form and must address the challenges of controller design for large dimensional correlated systems. 

The PID control results presented so far have been based on a continuous application of the controller. The digital apphcation of feedback control is applied at discrete points in time. DCSs use sequential microprocessors that perform control calculations for many control loops. Typical control loops are executed every 0.2 to 0.5 s for regulatory loops and 30 to 120 s for supervisory loops. The time between control applications is the control interval. At. PID control is apphed industrially on DCSs using digital formulas that are applied at discrete control intervals [Equations... 

Application of one-channel feedback control to spiral waves in the light-sensitive BZ system allows to observe the discrete set of stable resonant attractors experimentally [21, 30, 43, 46]. Note, that Eq. (9.23) for the radius of the resonance attractor contains only one medium dependent parameter (p, which specifies the direction of the resonance drift. To avoid a rather complicated experimental procedure to determine this value, the obtained experimental data were fitted to the theoretically predicted linear dependence (9.23) using p = —0.31. The results are shown in Fig. 9.3 by dashed lines. Then, the boundaries of the basin of attraction were specified in accordance with Eq. (9.24) (solid lines in Fig. 9.3). 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

Proportional-integral-derivative (PID) controllers are derived for the feedback control of inventory levels. The PID control law in discrete velocity form is given by the following relationship (Marlin, 1995) ... 

Suppose that the process to be identified is placed under relay feedback control and oscillates with some period T. Using a sampling interval of At, the number of samples within a period is AT =. The periodic square wave u k) generated by the relay output can be completely described over this period [0, T] using a discrete Fourier expansion (Godfrey, 1993)... 

In this section, the Direct Synthesis (DS) method presented in Section 12.2 is extended to the design of digital controllers. We begin with special cases that lead to a PID controller, and then show how other types of digital feedback controllers can be derived using the Direct Synthesis technique. Both Gc and G must be expressed as discrete-time in the closed-loop transfer function (17-59). In Direct Synthesis, the designer specifies the desired closed-loop transfer function YlYg d- The controller that yields the desired performance is obtained from (17-59)... 

We have presented a number of different approaches for designing digital feedback controllers. Digital controllers that emulate continuous-time PID controllers can include a number of special features to improve operability. Controllers based on Direct Synthesis or IMC can be tuned in continuous or discrete time, avoid ringing, eliminate offset, and provide a high level of performance for set-point changes. Minimum variance control can be very effective if a disturbance model is available. 

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