MATLAB functions exponential

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

There are numerous other examples of two-box models. For instance, a two-box epilimnion/hypolimnion model was discussed in Chapter 21, and additional examples are given as problems at the end of this chapter. We must remember that as long as these models are linear, their solutions can be constructed with the help of Box 21.6. They always consist of the sum of not more than two exponential functions and are thus fairly simple. This situation changes drastically if we allow the differential equations to become nonlinear. A system of two or more nonlinear differential equations rarely can be solved analytically, yet the available computer tools (such as MATLAB) make their solution easy. 

The left-hand-side function g(y, K) in (4.87) represents a line in the y-g plane with slope 1 + K and g intercept — find values of yj and yrn with associated multiple (or a unique) steady states of the system, we need to find instances of multiple (or unique) crossing points of this line and the exponential function f(y, K) on the right-hand side of equation (4.87). For this purpose we use the MATLAB m function hetcontbifrange.m. 

We do not design our own algorithm here but use the fin Insearch. m function supplied by Matlab. It is based on the original Nelder, Mead simplex algorithm. As an example, we re-analyse our exponential decay data Data Decay. m (see p. 106], this time fitting both parameters, the rate constant and the amplitude. Compare the results with those from the linearisation of the exponential curve, followed by a linear least-squares fit, as performed in Linearisation of Non-Linear Problems, (p.127). 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

Note that the matrix exponential can be computed by the function expm in MATLAB [171] At is the sampling time step. In order to simplify the symbols, the following notations are defined ... 

Direct time integration is performed using the Newmark total equilibrium method (Carr 2007). MATLAB codes were developed for the time domain analysis incorporating both classical Rayleigh viscous and non-viscous damping models. In the case of non-viscous damping, a single exponential model called Biot s relaxation function is used as the Kernel function. Biot s relaxation function is of the form... 

NOTE Here we used. / and instead of / and respectively, to tell MATLAB that the y vector is a function of the x vector mapping each point of x by the corresponding point of y. This kind of dot division, dot multiplication, and dot exponentiation should be applied for matrix (or vector as a subset of matrix) arguments. 

The functions exppdf(x,mu) and expcdf(x,mu) are available in the MATLAB Statistics Toolbox for calculating the exponential probability density and exponential cumulative distribution function, respectively. 

Note that in MATLAB, the functions exp(A), c. o.r(A), sin A), are element-by-element functions and do not obey the above definitions. The MATLAB functions expm A), expmJ A), expm2 A), and expmSiA) calculate exponential of the matrix A by different algorithms. The function expm2(A) calculates exponential of the matrix A as in Eq. (2.83). 

Program Description The MATLAB function LinearODE.m solves a set of linear ordinary differential equations. The first part of the function checks the number of inputs and their sizes, or values. The next section of the function performs the solution of the set of ordinary differential equations, which can be done by either the matrix exponential method [Eq. (5.40)] or the eigenvector method [Eq. (5.53)]. The method of. solution may be introduced to the function through the fifth input argument. The default method of solution is the matrix exponential method. 

---