Discrete performance index

Our goal is to estimate the function P(r) from the set of discrete observations Y(tj). We use a nonparametric approach, whereby we seek to estimate the function without supposing a particular functional form or parameterization. We require that our estimated function be relatively smooth, yet consistent with the measured data. These competing properties are satisfied by selecting the function that minimizes, for an appropriate value of the regularization parameter X, the performance index ... 

Using Eq. (4.1.13), the estimation problem is cast in discrete form as the determination of the coefficients c = [c1 . .., cns] that minimize the quadratic performance index ... 

When only the control is discretized as described in section 5.7, integration of the model equations is required to evaluate the performance index and to obtain the gradients. The evaluation of such gradients will consume a significant part of the total computational time needed to solve the optimisation problem. 

Prett and Garcia (1988) pose the validation problem as a discrete time linear optimal control problem under uncertainty. The uncertainty is defined by simple bounds, giving a polyhedral set of uncertain parameters V. For this problem, certain forms of uncertainty, e.g., in gains only, together with a quadratic performance index can be shown to satisfy the convexity requirements for the worst-case parameters to lie at vertices of V. This allows the algorithm of Gross-mann et al, based on examination only of vertices of V, to be applied (see Section II.A.l). The mathematical formulation is... 

Using a direct search technique on the performance index and the steepest ascent method, Seinfeld and Kumar (1968) reported computational results on non-linear distributed systems. Computational results were also reported by Paynter et al. (1969). Both the gradient and the accelerated gradient methods were used and reported (Beveridge and Schechter, 1970 Wilde, 1964). All the reported computational results were carried out through discretization. However, the property of hyperbolic systems makes them solvable without discretization. This property was first used by Chang and Bankoff (1969). The method of characteristics (Lapidus, I962a,b) was used to synthesize the optimal control laws of the hyperbolic systems. 

With the feasible path approach the optimization algorithm automatically performs case studies by variing input data. There are several drawbacks the process equations (32c) have to be solved every time the performance index is evaluated. Efficient gradient-based optimization techniques can only be used with great difficulties because derivatives can only be evaluated by perturbing the entire flowsheet with respect to the decision variables. This is very time consuming. Second, process units are often described by discrete and discontinuous relations or by functions that may be nondifferentiable at certain points. To overcome these problems quadratic module models can be... 

Fourier analysis of descriptors is performed by the DFT, which is the sum of the descriptor g r) over all distances r multiplied by a complex exponential. With a descriptor consisting of n components expressed in its discrete form g[x] x is the index of a discrete component), the DFT can be written as... 

If the dynamic behaviour of a system can be described by a switched LTI system, a linear implicit DAE system can be derived from the bond graph. The entries of its matrices depend on the discrete switch states. As long as no structural changes occur, i.e. no residual sinks are switched on, the DAE system is of index <1. For system modes in which residuals sinks are switched on, the DAE system is of index 2. There are solvers available for its direct numerical computation that are based on the BDF-method. An alternative may be to perform a DEVS simulation that uses quantised based integration. 

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