Optimal control policy

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Simulation results with different time interval (P) are reported in Table 4. Optimal control policy in reactor temperature for each case is shown in Fig. 4. As shown in Table 4, when one time interval (P = 1) is used, the amount of product C obtained at the final time (tf = 200 min) is 7.0171 kmol and the optimal temperature (isothermal operation) setpoint is 88.01 °C whereas usingP = 20, the amount of product C achieved is 7.0379 kmol. It was found from... 

Also note that if Ns is chosen sufficiently large, the piecewise constant optimal control policy will be sufficiently close to the continuous optimal control policy. 

It is clear from Table 5.9 that the results obtained are in very good agreement with the objectives set for each individual optimisation problem. Table 5.9 also deary shows the advantages of optimal reflux policies over the conventional constant reflux operation. Table 5.9 shows that the time optimal control policy (variable reflux) saves about 63% of the operation time compared to that required in the simulation (Table 4.6). Even the time optimal constant reflux policy saves about 33% of the operation time compared to the original simulation... 

At any time t, the true estimation of the state variables requires instantaneous values of the unknown mismatches eft). To find the optimal control policies in terms of any decision variables (say z) of a dynamic process using the model will require accurate estimation of ex(t) for each iteration on z during repetitive solution of the optimisation problem (see Chapter 5). Although estimation of process-model mismatches for a fixed operating condition (i.e. for one set of z variables) can be obtained easily, the prediction of mismatches over a wide range of the operating conditions can be very difficult. 

Mujtaba and Hussain (1998) implemented the general optimisation framework based on the hybrid scheme for a binary batch distillation process. It was shown that the optimal control policy using a detailed process model was very close to that obtained using the hybrid model. 

The optimal control policy is very simple. If the state of the reactor lies to the left of BC the heater should be used until the state reaches the boundary, after which Q should be controlled by Eq. (6) until the desired conversion is reached. If the state of the reactor is to the right of the boundary, maximum cooling is applied until the boundary is reached and followed. It only remains to consider what happens if the value of Q required by Eq. (6) exceeds the maximum Q. Let D be the point on BC where Eq. (6) first gives the value Q. Then clearly there is no difficulty with paths such as FGC or PRC which first meet BC above D. If the curve DE is the... 

Fig. 9.3. Optimal control policy as a function of time. (The curves are drawn to terminate at the same point of the time a.xis, and the beginnings of paths are marked with a dot.)...

An optimal control strategy for batch processes using particle swam optimisation (PSO) and stacked neural networks is presented in this paper. Stacked neural networks are used to improve model generalisation capability, as well as provide model prediction confidence bounds. In order to improve the reliability of the calculated optimal control policy, an additional term is introduced in the optimisation objective function to penalise wide model prediction confidence bounds. PSO can cope with multiple local minima and could generally find the global minimum. Application to a simulated fed-batch process demonstrates that the proposed technique is very effective. 

Once the optimal feed rates were obtained, they were applied to the actual process (i.e. simulation by the mechanistic model of the process). Table 2 shows the difference between the amounts of the final product and by-product on neural network model and the actual process. It can be seen from Table 2 that the actual amounts of product and by-product rmder these optimal control policies are quite different from the neural network model predictions. This indicates that the single neural network based optimal control policies are only optimal on the neural network model and are not optimal on the real process. Hence, they are not reliable. This is mainly due to the model plant mismatches, which is rmavoidable in data based modelling. 

The optimal temperature policy in a batch reactor, for a first order irreversible reaction was formulated by Szepe and Levenspiel (1968). The optimal situation was found to be either operating at the maximum allowable temperature, or with a rising temperature policy, Chou el al. (1967) have discussed the problem of simple optimal control policies of isothermal tubular reactors with catalyst decay. They found that the optimal policy is to maintain a constant conversion assuming that the decay is dependent on temperature. Ogunye and Ray (1968) found that, for both reversible and irreversible reactions, the simple optimal policies for the maximization of a total yield of a reactor over a period of catalyst decay were not always optimal. The optimal policy can be mixed containing both constrained and unconstrained parts as well as being purely constrained. 

Other recent work in the field of optimization of catalytic reactors experiencing catalyst decay includes the work of Romero e/ n/. (1981 a) who carried out an analysis of the temperature-time sequence for deactivating isothermal catalyst bed. Sandana (1982) investigated the optimum temperature policy for a deactivating catalytic packed bed reactor which is operated isothermally. Promanik and Kunzru (1984) obtained the optimal policy for a consecutive reaction in a CSTR with concentration dependent catalyst deactivation. Ferraris ei al. (1984) suggested an approximate method to obtain the optimal control policy for tubular catalytic reactors with catalyst decay. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

After the identification of a model, the open-loop optimal control policy for the nominal plant can be determined by solving the following optimization problem. 

Liapis and Litchfield [9] performed a quasisteady-state analysis for a system where 0 and = 0 and obtained general guidelines about the optimal control policy at the beginning of the drying process (when neither of the state constraints is active), as well as during operation, when the process may be heat or mass transfer limited [6,9]. 

The complete unsteady-state optimal control problem has been stndied by Litchfield and Liapis [12] for a system where 0 and q = 0 using turkey meat and nonfat-reconstituted milk as model foodstuffs. The results of the dynamic analysis for nonfat-reconstituted milk confirm the suggested control policies of the quasisteady-state analysis. At low chamber pressures the dynamic analysis with turkey meat showed control results similar to those obtained by the quasi-steady-state analysis. However, at higher pressures the assumed control policy based on the quasisteady-state analysis was not optimal. The optimal control dynamic study of Litchfield and Liapis [12] suggested that the policies of the quasisteady-state analysis may be useful guidelines but they should be interpreted with some caution. To obtain accurate optimal control policies on the heat input and chamber pressure of the freeze drying proeess, the complete unsteady-state optimal control problem should be solved [76,78,79,82,83]. 

Porter B. and Merzougui T., 1997. Evolutionary synthesis of optimal control policies for manufacturing systems. Proceedings of the IEEE 6th International Conference on Emerging Technologies and Factory Automation, pp. 304-309. 

Godorr, S., Hildebrandt, D., Glasser, D., McGregor, C., 1999. Choosing optimal control policies using the attainable region approach. Ind. Eng. Chem. Res. 38, 639-651. 

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