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Statistical and Numerical Methods for Chemical Engineers HS2015 Exercise 7 ASSIGNMENT 1 Consider the following lab data i ti qi 1 0 1 2 1 3 3 2 4 4 3 6 5 4 10 and the two models of the form a. q(t ) 0 1 t 2 b. q(t ) 0 exp(1 t ) where 0 and 1 are the model parameters. 1. Specify for a. and b. whether the model is linear or non-linear in the parameters. Since in both models the two parameters are overdetermined by the lab data, we want to obtain them in the least square sense. In the lecture the linear and non-linear least squares problems were introduced as linear : min Ax b xD 2 nonlinear : min f ( x ) b xD 2 where the symbols have been defined in the lecture. 2. For models a) and b) give expressions for A, x , b and f , x , b for the linear and nonlinear systems, respectively. Let us define the scalar function ϕ for the linear and non-linear cases: linear : ( x ) 1 Ax b 2 nonlinear : ( x ) 2 2 1 f ( x) b 2 2 2 The parameters in the least squares sense are then the solution of the following minimization problem: min ( x ) xD Turn page, v.s. A necessary condition for an extremum, i.e. minimum or maximum is that the gradient of ϕ with respect to the parameters vanishes / x1 , / x2 ,..., / xn 0 T (1) Realize that this is a system of n equations and n variables (the parameters x1, x2,…, xn). 3. Suggest a method to solve the system (1) from the lecture. In the lecture you will see that the solution of a linear least squares problem is given by the solution of the following linear system AT Ax AT b the so called normal equations. 4. With your expressions for A and b from 2. use MATLAB to determine x for the linear problem. Plot the result together with the data. Use the template lstsqr_linear.m For the remaining non-linear problem we will use Newton’s method. The Jacobian of the vectorvalued function is 2 / x12 ... 2 / x1xn H 2 / xnx1 ... 2 / xn2 This matrix is actually called the Hessian matrix of the scalar-valued function ϕ. Newton’s method would then be x ( k 1) x (k) H 1 x ( k ) 5. Implement Newton’s method to determine the parameters of the non-linear model. Use the template lstsqr_nolinear.m as well as the file phi.m where the tedious computations of ϕ, and H are already implemented. Statistical and Numerical Methods for Chemical Engineers HS2015 Exercise 7 ASSIGNMENT 2 Consider a batch reactor with second order reactions: dA 2k1A 2 dt dB k1A 2 2k2 B2 dt (2) Solve the batch reactor and plot the solutions Use k1 = 1000; k2 = 0.1; y0 = [1, 0.1] and tSpan = [0,1]; Do you expect the system to be stiff? Use the linearization method to check the stiffness ratio at y = y0 (by hand), by calculating the eigenvalues of the Jacobian of the system Use ode45 to solve the system and plot the solutions. Is it stiff? Looking at the Jacobian, can you say why? Note how y changes and how it changes the Jacobian and its eigenvalues! Turn page, v.s.