Prediction horizon

A dynamic model of the process is used to predict the future outputs over a prediction horizon consisting of the next p samphng periods. 

A reference trajectoiy is used to represent the desired output response over the prediction horizon. 

At each sampling instant, a control policy consisting of the next m control moves is calculated. The control calculations are based on minimizing a quadratic or linear performance index over the prediction horizon while satisfying the constraints. 

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. 

In order to investigate the performance of a deterministic online scheduler, we apply it to the example problem under demand uncertainty for three periods. The model of the scheduling problem used in the scheduler considers a prediction horizon of H = 2 periods. Only the current production decision Xi(ti) is applied... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

In MPC a dynamic model is used to predict the future output over the prediction horizon based on a set of control changes. The desired output is generated as a set-point that may vary as a function of time the prediction error is the difference between the setpoint trajectory and the model prediction. A model predictive controller is based on minimizing a quadratic objective function over a specific time horizon based on the sum of the square of the prediction errors plus a penalty... 

To minimize /, you balance the error between the setpoint and the predicted response against the size of the control moves. Equation 16.2 contains design parameters that can be used to tune the controller, that is, you vary the parameters until the desired shape of the response that tracks the setpoint trajectory is achieved (Seborg et al., 1989). The move suppression factor A penalizes large control moves, but the weighting factors wt allow the predicted errors to be weighted differently at each time step, if desired. Typically you select a value of m (number of control moves) that is smaller than the prediction horizon / , so the control variables are held constant over the remainder of the prediction horizon. 

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

Example 9—A finite prediction horizon may not be a good approximation of an infinite onefor nonlinear processes. In the second example, consider the single-state, single-input system... 

Rawlings and Muske (1993) have shown that this idea can be extended to unstable processes. In addition to guaranteeing stability, their approach provides a computationally efficient method of on-line implementation. Their idea is to start with a finite control (decision) horizon but an infinite prediction (objective function) horizon, i.e., m < < and p = , and then use the principle of optimality and results from optimal control theory to substitute the infinite prediction horizon objective by a finite prediction horizon objective plus a terminal penalty term of the form... 

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. 

A fourth proposition was discussed by Vuthandam et al. (1995). Their idea is that the values of the MFC tuning parameters must satisfy robust stability requirements. It turns out that for the robust stability requirements developed by these authors, performance improves as the prediction horizon length, p, increases from its minimum value to larger values, but after a certain point performance deteriorates as p increases further. This happens because for very large p the input move terms in the on-line objective function must be penalized so much that the controller becomes very sluggish and performance suffers. Results such as these depend on the form of the robust stability conditions. If such conditions are only sufficient, as is the case with Vuthandam et al (1995), then performance-related results may be conservative. 

The controller receives the on-line composition measurement of the product outlets (extract and raffinate) as feedback data from the plant. These measurements are filtered through a periodic Kalman filter and used together with the simplified SMB model results to estimate the state of the system and to remove the possible moidel errors. The formulation of RMPC is based on the assumption that possible errors or disturbances are likely to repeat and will have a periodic effect on the output, which is the most likely correlation between disturbances and output in a SMB unit. The estimated future concentration profile in the SMB is used to optimize the future behaviour of the plant over a predefined prediction horizon. The controller implements the calculated optimal plant input by changing the external flow rates in order to control the internal flow rates, which are the manipulated variables. Time lags, e.g. between online concentration measurements and optimizer or between optimizer and SMB plant, are insignificant relative to the process dynamics and sampling time for the planned scheme. 

Hence, there is a close link between the model- and modifier-adaptation methods in that the parameterization and the update procedure are both intended to match the KKT conditions. Essentially, modifier-adaptation schemes use a model-predictive control with a one-step prediction horizon. Such a short horizon is justified because the system is static. However, since the updated modifiers are valid only locally, modifier-adaptation schemes require some amount of filtering/regularization (either in the modifiers or in the inputs) to avoid too aggressive corrections that may destabilize the 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. 

The second level is the advanced and predictive control. These are two different control schemes that work at the same level. Information is transmitted horizontally and vertically in this (and upper) level. More elaborated control strategies as selective control, ratio control, feedforward control are implemented. In this second level implicit as well as explicit (heuristic and first principles based) models are used to generate the action. The action is the set point (goal) to achieve at the lowest level. Prediction horizon is (in the case of model predictive control) of tens of movements. 

Once the solution of the dynamic optimisation problem is foimd (with 8 collocation points for state variables, and considering the control variable as piecewise constant within 1 element with a length of 0.06 h hich is the prediction horizon), the computed optimal input within the first sampling period is applied both to the actual plant and to the model. The whole procedure is repeated with the moving horizon strategy in each sampling instant. 

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