Optimal performance index, minimization

An optimization procedure is a way of maximizing or minimizing the performance index. There are many different procedures, some of which will be discussed later on in this chapter. To determine the best optimization procedure, a performance index for the procedures must first be established. It could be the procedure that reaches a point within 5% of the optimum in the shortest time. It could be the one that requires the fewest steps or costs the least to reach that point. It could have constraints like a maximum cost or a time limit. 

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

For more the details about the MPC CB software and the operations conditions, the reader can refer to [12]. The optimal minimization of the drying time rmder constraints may be equivalent to define the performance index as the maximization of the velocity of the sublimation interface. Since MPC CB solves a rninirnization problem, the objective function is ... 

In the last twenty years, various non-deterministic methods have been developed to deal with optimum design under environmental uncertainties. These methods can be classified into two main branches, namely reliability-based methods and robust-based methods. The reliability methods, based on the known probabiUty distribution of the random parameters, estimate the probability distribution of the system s response, and are predominantly used for risk analysis by computing the probability of system failure. However, variation is not minimized in reliability approaches (Siddall, 1984) because they concentrate on rare events at the tail of the probability distribution (Doltsinis and Kang, 2004). The robust design methods are commonly based on multiobjective minimization problems. The are commonly indicated as Multiple Objective Robust Optimization (MORO) and find a set of optimal solutions that optimise a performance index in terms of mean value and, at the same time, minimize its resulting dispersion due to input parameters uncertainty. The final solution is less sensitive to the parameters variation but eventually maintains feasibility with regards probabilistic constraints. This is achieved by the optimization of the design vector in order to make the performance minimally sensitive to the various causes of variation. 

In the optimization block, the control input applied to the smart structural system is obtained by minimizing a generalized linear quadratic (LQ) performance index with weights on the control moves. The performance index is given by... 

According to the LQG approach, a system, described by Equation (1) and the optimal control forces u t) should minimize the performance index (2). An additional assumption is that the optimal feedbackis afimction ofthe measurement vector, containing the noised floor accelerations... 

Step 2 Calculation of u(t), d(t), which minimize the performance index J = J(Q RJ. For the obtained optimal solution find the value of the global performance index... 

Suppose that the chief control objective is to maximize a production rate while satisfying inequality constraints on the inputs and the outputs. Assume that the production rate can be adjusted via a flow control loop whose set point is denoted as u sp in the MPC control structure. Thus, the optimization objective is to maximize u sp, or equivalently, to minimize u sp- Consequently, the performance index in (20-68) becomes = u sp-This expression can be derived by setting all of the weighting factors equal to zero except for c, the first element of c. It is chosen to be = -1. 

The chapter is organized as follows In section 2, we first review basic results on I/O-controllability of linear systems. In section 3, a new t)fpe of I/O-controllability index is introduced, the Robust Performance Number. In section 4, I/O-controllability analysis by optimization is presented. Sections 5 and 6 contain two case studies an air separation plant and a reactive distillation column where these tools are applied to select the best control structure and quantify the process I/O-controllability. The evaluations of the control structures are validated by simulations with low order controllers which can easily be obtained from the analysis, in particular from the computed or estimated attainable performance of the chosen structure, using the procedure described in [9, 42, 29]. So the construction of practically relevant controllers of minimal complexity is seamlessly integrated with the analysis. 

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