Optimal control horizon time

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. 

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 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. 

This paper presents an on-line model based level control of a batch reactor with reaction rate uncertainties. The analyzed chemical batch process is catalyzed by a catalyst which decomposes in the reactor therefore it is fed several times during the batch. The chemical reaction produces a vapour phase by-product which causes level change in the system. The on-line control method is based on the shrinking horizon optimal control methodology based on the detailed model of the process. The results demonstrate that the on-line optimization based control strategy provides good control performance despite the disturbances. 

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. 

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. 

Simultaneous dynamic optimization. In order to validate this approach, the sjm-thesis of MTBE is used as case study. Three spatial variables and sixteen control-related parameters were optimized during a time horizon of 14400 s. A tradeoff between control and economic performances exists in the design of RD processes, as shown in table 6.5. The design optimized sequentially with respect to dynamic behavior led to a RD process... 

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 ... 

It is common knowledge that for time invariant systems and cost criteria and an infinite horizon T —> 00 the resulting optimal control can be formulated as a statie state feedback control law k x) [26], i.e. the optimal control depends only on the eurrent state vector of the plant. In aecordanee with what has been said above, the eontroller nonlinearity is influeneed by (1) the plant dynamics Eq. (18), (2) the region of operation eharacterized by the set of initial eonditions B and (3) the performanee eriterion Eq. (19). 

SIM is a data structure that contains information about how the optimization calculation is to be performed. SIM.tO is the initial time and SIM.tH is the horizon time. SIM.NS is the number of subintervals. SIM.xO is the initial state. SIM.isRestart is 0 if the simulation is to start at the initial guess supplied by FUN.uO and is nonzero if the information in the optional input parameter TRAJO sets the initial input trajectory. SIM.constraint is 0 if the control inputs are not constrained, and is nonzero if FUN.constraint is to be used to define a set of input control constraints. SIM.verbose is 0 if no information is to be printed to the screen and is nonzero if the status of the calculation is to be displayed. 

MPC (The Model Predictive Control) uses predictive control methods with a dynamic model (linear or non-linear) to compute control signal trajectory that minimize quality indicator for a given time horizon. In each step of the algorithm, the control vector in consecutive moments is computed x(fe),x(fe+1),..., x k + Ns — 1) (k - actual time, Ns - control horizon). In each discrete time step k first control vector x(fe) from optimized control trajectory is used—then, when the prediction and control time horizon are moved one step forward, the whole procedure is repeated. 

In the recent years Simulated Moving Bed (SMB) technology has become more and more attractive for complex separation tasks. To ensure the compliance with product specifications, a robust control is required. In this work a new optimization bas adaptive control strategy for the SMB is proposed A linearized reduced order model, which accounts for the periodic nature of the SMB process is used for online optimization and control purposes. Concentration measurements at the raffinate and extract outlets are used as the feedback information together with a periodic Kalman filter to remove model errors and to handle disturbances. The state estimate from the periodic Kalman filter is then used for the prediction of the outlet concentrations over a pre-defined time horizon. Predicted outlet concentrations constitute the basis for the calculation of the optimal input adjustments, which maximize the productivity and minimize the desorbent consumption subject to constraints on product purities. 

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. 

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. 

In this study, the optimization of a WWTP control law by means of a multi-objectives genetic algorithm proved to be a reliable technique. To be feasible, this optimization however needs to be performed on a limited time horizon (typically two or three sludge retention times). This study showed that the addition of a long term simulation of final optimized settings allows the decision maker to check the quality of the results produced, as well as to have more insight into the performances of the control law. 

During the last two decades, model-predictive control (MPC) (Allgower et ah, 1999, Mayne, 2000 and Rao and Rawlings, 2000) has increasingly been applied to the control of processes with interacting dynamics. The basic idea of MPC is to employ a plant model that predicts the reaction of the plant to the past and future inputs, and to optimize a number of future inputs such that the predicted outputs follow the desired trajectory over a certain period of time (called the prediction horizon). This process is iterated, only the next input is applied to the plant, and new inputs are computed for a prediction horizon that is shifted one step into the future, taking new measurements into account. Thus, the behavior of the real process and disturbances are taken into account. 

Inventory control and pricing optimization Consider a retailer that maintains an inventory of a particular product. The retailer s objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite, or infinite, time horizon. 

As is well-known in practice, the protein folding properties and, hence the product quality is dependent on the cultivation conditions of the host organism. Thus, in order to guarantee products within narrow specification limits, the production process must be kept under tight control (4). An essential prerequisite is accurate monitoring of the process state and an optimization of the trajectories of the key process variables. Advanced control strategies require to predict the process behaviour at least over time horizons which are needed to influence the process so that the state variables will not escape from the acceptable intervals. Prediction, however, means that the process has to be modelled. 

In multiple shooting, the integration horizon is divided into time intervals, with the control variables approximated by polynomials in each control interval and differential variables assigned initial values at the beginning of each interval. The DAE system is solved separately within each control interval, as shown in Fignre 14.2b. Profiles for partial derivatives with respect to the optimization variables, as well as the initial conditions of the state variables in each time interval, are obtained through integration of the sensitivity eqnations. These state and sensitivity profiles are solved independently over each time interval and can even be computed in parallel. Additional equations are inclnded in the NLP to enforce continuity of state variables at the time interval boundaries. 

A two-layered control strategy was described for supply chain management purposes. The strategy combines feedback controllers to account for the fast dynamics at the inventory nodes, while utilising the power of a fiilly-centralised optimisation-based model predictive controller to achieve an optimal operating policy for the supply chain network over a selected time horizon. 

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