Optimal control cost functional

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

In this subsection we analyse an optimal control problem. The exterior forces f,g) are chosen to minimize the cost functional... 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

Figure 20.9 shows a typical optimisation trajectory found by the model for one load and power price combination. The starting point for the raw and recycle brine is based on current plant operating rules and appears as 100% on the flow axes. Subsequently the model varies the brine flows and runs to a new steady-state solution. The modelling package has a built-in optimisation routine which controls the searching process as dictated by the cost function, which is the price per unit of chlorine. In the case illustrated by Fig. 20.9 it requires eight runs to find the optimal solution. 

Optimal control theory A method for determining the optimum laser field used to maximize a desired product of a chemical reaction. The optimum field is derived by maximizing the objective function, which is the sum of the expectation value of the target operator at a given time and the cost penalty function for the laser field, under the constraint that quantum states of the reactants satisfy the Schrodinger equation. 

Since the classical treatment has its restrictions and the applicability of the quantum OCT is limited to low-dimensional systems due to its formidable computational cost, it would be very desirable to incorporate the semiclassical method of wavepacket propagation like the Herman-Kluk method [20,21] into the OCT. Recently, semiclassical bichromatic coherent control has been demonstrated for a large molecule [22] by directly calculating the percent reactant as a function of laser parameters. This approach, however, is not an optimal control. 

Fig. 7.10. Evolution of cost function during iterative control by manipulating individual 128 segments of the SLM. The optimized ion yield ratio is roughly as same as that obtained by varying the linear chirp rate...

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. 

In this chapter we have advocated the use of online model-based optimization for the automatic control of SMB plants. This approach has the advantage that the process is automatically operated at its economic optimum while meeting all relevant constraints on purities and flow rates. Application to a pilot-plant-scale reactive SMB process for glucose isomerization showed that implementation at a real plant is feasible - the requirements for additional hardware are moderate (a high-level PC and online concentration measurements in the recycle line). The experiments confirmed the excellent properties of the proposed control scheme. The scheme is extremely versatile, and the cost function and constraints can easily be adapted to any specific separation task. 

At the heart of an model predictive control (MPC) application is the optimization of a variable subject to constraints. A typical MPC cost functional is given as follows ... 

Model predictive control is based on real-time optimization of a cost function. Consequently, CPM methods that focus on the values of this cost function can be developed. The MPC cost function T(A ) is... 

Design Case Approach. Patwardhan et al. [222] have suggested the comparison of the achieved performance with the performance in the design case that is characterized by inputs and outputs given by the model. The design cost function Jdes has the same form as Eq. 9.18 where e k) and Au(A ) are substituted for e k) and Au(A ) to indicate the predicted deviations of model outputs from the set-points (an estimate of the disturbance is included) and the optimal control moves, respectively. Jack is the same as that in historical benchmark Eq. 9.18 and is calculated using plant data. Performance variation between the real plant (Jack) and model (Jdes) is expressed by... 

The essential step in the LQG benchmark is the calculation of various control laws for different values of A and prediction (P) and control (M) horizons (P = M). This is a case study for a special type of MPC (unconstrained, no feedforward) and a special parameter set (M = P) to find the optimal value of the cost function and an optimal controller parameter set. Using the same information (plant and disturbance model, covariance matrices of noise and disturbances), studies can be conducted for any t3q>e of MPC and the influence of any parameter can be examined. These studies... 

The values y and y are the current set point values for the two control loops (Figure 25.11a). But when the disturbances change, say dx = d" and d2 = d, the values of yt and y2 that minimize the cost function also change. Let the new optimum values be y" and y. Again they satisfy the following necessary conditions for optimality ... 

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