Nonsingular matrix approximations

Following a similar procedure to the one employed above, it is easy to verify that we obtain a model that approximates the fast dynamics of the system in Figure 4.2, in the form of Equation (4.20). Also, it can be verified that only 2N + 8 of the 2N + 9 steady-state constraints that correspond to the fast dynamics are independent. After controlling the reactor holdup Mr, the distillate holdup Md, and the reboiler holdup MB with proportional controllers using respectively F, D, and B as manipulated inputs, the matrix Lb (x) is nonsingular, and hence the coordinate change... 

Newton s Method The classical Newton s method is a technique that instead of specifying a step length at each iteration uses the inverse of the Hessian matrix, H(x)" to deflect the direction of steepest descent. The method assumes that /(x) may be approximated locally by a second order Taylor approximation and is derived quite easily by determining the minimum point of this quadratic approximation. Assuming that H(x ) is nonsingular, then the algorithmic process is defined by... 

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