Control horizon

The MPC control problem illustrated in Eqs. (8-66) to (8-71) contains a variety of design parameters model horizon N, prediction horizon p, control horizon m, weighting factors Wj, move suppression factor 6, the constraint limits Bj, Q, and Dj, and the sampling period At. Some of these parameters can be used to tune the MPC strategy, notably the move suppression faclor 6, but details remain largely proprietary. One commercial controller, Honeywell s RMPCT (Robust Multivariable Predictive Control Technology), provides default tuning parameters based on the dynamic process model and the model uncertainty. 

Rawlings and co-workers proposed to carry out parameter estimation using Newton s method, where the gradient can be cast in terms of the sensitivity of the mean (Haseltine, 2005). Estimation of one parameter in kinetic, well-mixed models showed that convergence was attained within a few iterations. As expected, the parameter values fluctuate around some average values once convergence has been reached. Finally, since control problems can also be formulated as minimization of a cost function over a control horizon, it was also suggested to use Newton s method with relatively smooth sensitivities to accomplish this task. The proposed method results in short computational times, and if local optimization is desired, it could be very useful. 

For this MFC system, Eqs. (118) through (125), Vuthandam et al. (1995) developed sufficient conditions for robust stability with zero offset. These conditions can be used directly for calculation of minimum values for the prediction and control horizon lengths, p and m, respectively, as well as for the move suppression coefficients rji, which are not equal over the... 

A second proposition relies on the idea that the on-line optimization problem is unconstrained after a certain time step in the finite moving horizon. Where in the finite horizon that happens is determined by examining whether the state has entered a certain invariant set (Mayne, 1997). Once that happens, then closed-form expressions can be used for the objective function from that time point the end of the optimization horizon, p. The idea is particularly useful for MFC with nonlinear models, for which the computational load of the on-line optimization is substantial. A related idea was presented by Rawlings and Muske (1993), where the on-line optimization problem has a finite control horizon length, m, and infinite prediction horizon length, p, but the objective function is truncated, because the result of the optimization is known after a certain time point. 

In the second case the setpoints for the average concentrations of A at extract and that of B at raffinate are changed simultaneously after 20 switching times as shown in Figure 3. For the control purpose, the prediction and control horizons are set equal to S and 2 switching periods, respectively. The weighting matrices are tuned by the trial and error method. Here it is noticed that the control inputs act predictively to bring the control output to their new respective setpoints. It is clearly seen that the control performance is quite satisfactorily. 

In order to implement a MPC strategy to optimize the operations of the FISC, the system has to be conceptualized as a dynamic entity in terms of states, input and outputs [5]. Some inputs will constitute disturbances to the model and some others manipulated variables for control purposes. A subset of the output variables will be controlled outputs whose values will be desired to follow some predefined trajectory or assume particular values in certain periods of the control horizon. For the FISC system, the state variables are the inventories of the different goods in the storage facilities fresh fruit (NPFS), packed fruit (PFS, PPFS) and concentrated juice (CJS, PCJS). The manipulated variables are the flows of all the streams of the system (Fig. 1). The FISC is considered to be a centralized system [6]. For the MPC implementation, the overall profit of the business is maximized in each time period for a certain planning horizon, subject to the mass balance model of system. 

For the MPC multivariable controller a sampling time of T=5 min has been used. The prediction horizon has been of p=200 and the control horizon of m=10. The appropriate MPC controller tuning has been achieved by simulation. 

In the classical concept of predictive control, the trajectory (or set-point) of the process is assumed to be known. Control is implemented in a discrete-time fashion with a fixed sampling rate, i.e. measurements are assumed to be available at a certain frequency and the control inputs are changed accordingly. The inputs are piecewise constant over the sampling intervals. The prediction horizon Hp represents the number of time intervals over which the future process behavior will be predicted using the model and the assumed future inputs, and over which the performance of the process is optimized (Fig. 9.1). Only those inputs located in the control horizon H, are considered as optimization variables, whereas the remaining variables between Hr+1 and Hp are set equal to the input variables in the time interval Hr. The result of the optimization step is a sequence of input vectors. The first input vector is applied immediately to the plant. The control and the prediction horizon are then shifted one interval forward in time and the optimization run is repeated, taking into account new data on the process state and, eventually, newly estimated process parameters. The full process state is usually not measurable, so state estimation techniques must be used. Most model-predictive controllers employed in industry use input-output models of the process rather than a state-based approach. 

The objective of meeting the product specifications is reflected by the purity constraint over the control horizon Hn which is corrected by a bias term 8Pue1 resulting from the difference between the last simulated and the last measured process output to compensate un-modeled effects ... 

Enzyme activity is assumed to decay exponentially over the experiment. Fast controller response in both directions can be observed. Compared with the uncontrolled case, the controller controls the product purity and compensates the drift in the enzyme activity. The evolution of the results of the optimization algorithm during each cycle is plotted as a dashed line, shifted by one cycle to the right in order to vitalize the convergence. This shows that a feasible solution is found rapidly and that the controller can be implemented under real-time conditions. In this example, the control horizon was set to two cycles and the prediction horizon was set to ten cycles. A diagonal matrix i j = 0.02 I (3,3) was chosen for regularization. 

The desired purity for the experiment reported below was set to 55.0% and the controller was started at the 60th period. As in the simulation study, a diagonal matrix R = 0.02 I (3,3) was chosen for regularization. The control horizon was set to Hr = 1 and the prediction horizon was Hv = 60 periods. Figure 9.10 shows the evolution of both the product purity and the controlled variables. In the open-loop mode, where the operating point was calculated based on the initial model, the product purity constraint was violated at periods 48 and 54. After one cycle, the controller drove the purity above 55.0% and kept it there. The controller first reduces the desorbent consumption. This action seems to contradict the intuitive idea that more desorbent injection should enhance separation. However, in the presence of a reaction, this is not true, as shown by this experiment. The controlled variables converge towards a steady state, but they still change from period to period, due to the non-ideality of the plant. 

Model predictive control, 238 control horizon, 239 prediction horizon, 239 tuning parameters, 242... 

For the considered rectification process the numerical and explicit DMC algorithms were designed. In the case of the numerical implementation the sampling period Tp=2 min was assumed. The dynamics and prediction horizons were assumed equal to D=N=50. Other values of the parameters of the controller are the control horizon A =25 and the values of coefficients X. =X. =10. The simulation... 

The AFM algorithm can be easily incorporated into an MFC scheme, where in each time step k a rigorous nonlinear optimization problem is formulated. The objective is to calculate the optimal values of the manipulated variables v over a control horizon M, so that the error between the RBF model predictions and the desired set-point over a prediction horizon N is minimized. As soon as the optimization problem is solved, the first control move (k) is implemented, and then the RBF model is updated using the AFM algorithm. The procedure is shown in figure 2. Assuming one controlled variable, the optimization problem can be described by the following set of equations ... 

These equations are then used to predict the effect of future changes in the MV. The prediction horizon is the number (M) of sample periods used in predicting the value of the CV. The control horizon is the number (AO of control moves (AMVO that are calculated into the future. In matrix form Equation (8.35), for example, therefore becomes... 

Future moves are calculated to minimise the sum of the squares of the predicted deviations from target over the control horizon. While the next N moves are calculated, only the first (AMVq) is implemented. At the next cycle the controller recalculates a new set of control moves. This wiU account for any prediction errors or unmeasured load disturbances. 

In Equations 18.60 and 18.61, N is the open-loop settling time of the process and in Equation 18.64, M is the control horizon (number of necessary control moves). The quantity P=N+M is called the prediction horizon. In the case of a batch or semibatch process, P is a receding horizon and as such, at every control execution, the horizon is reduced by one interval. 

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