Cost function matrix

If the matrix A is positive definite, i.e. it is symmetric and has positive eigenvalues, the solution of the linear equation system is equivalent to the minimization of the bilinear form given in Eq. (64). One of the best established methods for the solution of minimization problems is the method of steepest descent. The term steepest descent alludes to a picture where the cost function F is visualized as a land-... 

The idea of selecting waveforms adaptively based on tracking considerations was introduced in the papers of Kershaw and Evans [3, 4], There they used a cost function based on the predicted track error covariance matrix. 

Simulations were performed for both cost functions. Target trajectories in range and Doppler were randomly created. The maneuvers for the trajectories were generated using a given transition probability matrix. We identified four maneuvers 0 acceleration 10m/s2 acceleration 50m/s2 acceleration —10m/s2 acceleration. 

However, Eq. (6.9) cannot be solved for the m actuator forces because m> n (i.e., the matrix of actuator moment arms is nonsquare). Static optimization theory is usually used to solve this indeterminate problem (Seireg and Arvikar, 1975 Hardt, 1978 Crowninshield and Brand, 1981). Here, a cost function is hypothesized, and an optimal set of actuator forces is found, subject to the equality constraints defined by Eq. (6.9) plus additional inequality constraints that bound the values of the actuator forces. If, for example, actuator stress is to be minimized, then the static optimization problem can be stated as follows (Seireg and Arvikar, 1975 Crowninshield and Brand, 1981) Find the set of actuator forces that minimizes the sum of the squares of actuator stresses ... 

The stable transfer matrices Tu s), T is) and T2i s) are determined by the plant, the Youla parameter Q s) is a free stable transfer matrix. In the optimization, the Youla parameter 5) is expressed as a finite series expansion in terms of suitable fixed transfer matrices qi s) and variable coefficients x [4], In the evaluation of the cost function, the time-domain equivalent of (16) is needed which for a better numerical efficiency can be reformulated via Laplace-transforms to avoid time intensive computations during the optimizations (see e.g. [44, 45]). The advantage of this formulation is that it is linear in the unknown x, and all other quantities can be computed before the iterations performed in the optimization process. This leads to... 

The new molecular orbitals may be obtained as follows. As trial functions, 4>j t) are given by the linear combination of 4>i t) so that the overlap error estimation function may be minimized. The cost function for the overlap matrix is defined as the square of the difference from the unit matrix... 

We now consider the origin of the excellent performance of the conjugate gradient method for a quadratic cost function, defined in terms of a symmetric, positive-definite matrix A and a vector b,... 

The gradient vector provides information about the local slope of the cost function surface. Further improvement in efficiency is gained by using knowledge of the local curvature of the surface, as encoded in the real, symmetric Hessian matrix, with the elements... 

LTSA seeks to minimise a cost function that minimises the distances between points in the low-dimensional space and the tangent space. As shown in [16], the solution to this minimisation problem is formed by the d smallest eigenvectors of an alignment matrix F. The alignment matrix is found by iteratively summing over all local information matrices ... 

The reason for this is simple. If the reaction chemistry is not "clean" (meaning selective), then the desired species must be separated from the matrix of products that are formed and that is costly. In fact the major cost in most chemical operations is the cost of separating the raw product mixture in a way that provides the desired product at requisite purity. The cost of this step scales with the complexity of the "un-mixing" process and the amount of energy that must be added to make this happen. For example, the heating and cooling costs that go with distillation are high and are to be minimized wherever possible. The complexity of the separation is a function of the number and type of species in the product stream, which is a direct result of what happened within the reactor. Thus the separations are costly and they depend upon the reaction chemistry and how it proceeds in the reactor. All of the complexity is summarized in the kinetics. 

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