MATLAB equation

A solver tool can be used to solve the MATLAB equations simultaneously. 

The tutorial demonstrates how MATLAB is used to generate root loeus diagrams, and lienee how to design eontrol systems in the. v-plane. Examples given in Chapter 5 are used to illustrate the MATLAB eommands. The roots of the eharaeteristie equation (or any polynomial) ean be found using the roots eommand. 

This example uses the MATLAB eommand Iqr to provide the eontinuous solution of the redueed matrix Rieeati equation (9.25)... 

The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

We will not write out the entire closed-loop function C/R, or in this case, T/Tsp. The main reason is that our design and analysis will be based on only the characteristic equation. The closed-loop function is only handy to do time domain simulation, which can be computed easily using MATLAB. Saying that, we do need to analysis the closed-loop transfer function for several simple cases so we have abetter theoretical understanding. 

Consider Example 5.3 again and let s pick xp to be 5 min, Kp be 0.8 [unit]. Instead of using the equation that we derived in Example 5.3, we can use the following statements in MATLAB to generate a simulation for the case of a unit step change in the set point. This approach is much faster than using Simulink. 

Note 2 The iterative solution in solving the ultimate frequency is tricky. The equation has poor numerical properties—arising from the fact that tan9 "jumps" from infinity at 9 = (ir/2) to negative infinity at 9 = (ir/2)+. To better see why, use MATLAB to make a plot of the function (LHS of the equation) with 9 < co < 1. With MATLAB, we can solve the equation with the f zero () function. Create an M-file named f. m, and enter these two statements in it ... 

Example 7.6 Construct the root locus plots of some of the more common closed-loop equations with numerical values. Make sure you try them yourself with MATLAB. 

The points at which the loci cross the imaginary axis can be found by the Routh-Hurwitz criterion or by substituting s = jco in the characteristic equation. (Of course, we can also use MATLAB to do that.)... 

To determine the shape of a root locus plot, we need other rules to determine the locations of the so-called breakaway and break-in points, the corresponding angles of departure and arrival, and the angle of the asymptotes if the loci approach infinity. They all arise from the analysis of the characteristic equation. These features, including item 4 above, are explained in our Web Support pages. With MATLAB, our need for them is minimal. 

However, MATLAB allows us to get the answer with very little work—something that is very useful when we deal with more complex systems. Consider a numerical problem with values of the process gain Kp = 1, and process time constants X = 2 and x2 = 4 such that the closed-loop equation is... 

From here on, we will provide only the important analytical equations or plots of asymptotes in the examples. You should generate plots with sample numerical values using MATLAB as you read them. 

To find the new gain margin, we need to, in theoiy, reverse the calculation sequence. We first use the phase equation to find the new crossover frequency (Dcg. Then we use the magnitude equation to find the new GOL, and the new GM is of course 1/ G0lI However, since we now know the values of td, xp, and KcKvKpKm, we might as well use MATLAB. These are the statements ... 

With MATLAB, we can easily prepare the root locus plots of this equation for the cases of Xj = 0.05, 0.5, and 5 s. (You should do it yourself. We ll show only a rough sketch in Fig E10.1. Help can be found in the Review Problems.)... 

The fact that the algorithm used by MATLAB does not return a normalized output matrix C can create problems when we do feedback calculations in Chapter 9. The easy solution is to rescale the model equations. The output equation can be written as... 

What if we click a point not exactly on a root locus When we select a point s, MATLAB calculates the value k = -p(s )/q(s ), which will only be a real positive number if s satisfies the closed-loop equation. Otherwise, k is either complex, or negative if the pole is a real number. In this case, MATLAB calculates the magnitude of k, uses it as the gain and computes the corresponding closed-loop poles. Thus we find the chosen points are always right on the root loci no matter where we click. 

The Wegstein method is a secant method applied to g(x) = x - Fix). In Microsoft Excel, roots are found by using Goal Seek or Solver. Assign one cell to be x, put the equation for/(x) in another cell, and let Goal Seek or Solver find the value of x that makes the equation cell zero. In MATLAB, the process is similar except that a function (m-file) is defined and the command fzeroCf .xO) provides the solution x, starting from the initial guess xO. 

The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations [Gottlieb, D., and S. A. Orszag, Numerical Analysis of Spectral Methods Theory and Applications, SIAM, Philadelphia (1977) Trefethen, L. N., Spectral Methods in Matlab, SIAM, Philadelphia (2000)]. Suppose we have a grid of equidistant points... 

Table 5-4 Results after substituting into the original equations and calculating the differences between predicted and actual results (using MATLAB calculations)...

Table 21-1 Matrix operations in MATLAB to compute equations 21-1-21-4...

Table 26-1 Simple SVD performed on matrix A using MATLAB other matrix relationships are also shown (see equations 26-1 through 26-4)...

We now report on the evaluation of the integral in equation 43-63, which was done numerically by computer. The numerical computations were carried out using MATLAB. Here we examine the conditions and the results obtained for this exercise. 

An empirical test of this possibility was performed by computing values of the variance of the two terms in equation 44-77. The Normal random number generator of MATLAB was used to create multiple values of Normally distributed random numbers for Er and Es these were plugged into the two expressions of equation 44-77 and the variance computed. Values between 100 and 106 were used in each computation of the variance. 

To do the computations, we again use the random number generator of MATLAB to produce Normally-distributed random numbers with unity variance to represent the noise values of Er will then directly represent the S/N ratio of the data being evaluated. For the computations reported here, we use 100,000 synthetic values of the expression on the RHS of equation 44-76a to calculate the variance of, for each combination of conditions we investigate. 

Equation 51-77 was programmed in MATLAB, using the Poisson distribution for both AEr and A s the actual distribution used corresponded to the value of Er and Es, respectively. The computations were done for 0.01 [Pg.319]

With the advent of modern software tools, however, tools such as MATLAB and even the older language, APL, matrix operations can be coded directly from the matrix-math expressions, and then it becomes near-trivial to create and solve the matrix equations on-the-fly, so to speak, and calculate the coefficients for any derivative using any desired polynomial, and computed over any odd number of data points. 

Then we compute the matrix inverse of equation 56-25 (in MATLAB, this is just ... 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

To do this, the product of the mass flowrate and the specific enthalpy was substituted by the corresponding enthalpy flow. Results of the reconciliation procedure using the Q-R factorization are given in Table 7. Table 8 compares the residuum of the balance equations, the value of the objective function, and the computing time of the MATLAB implementation for both approaches (Q-R factorization and use of SQP with the reduced set of balance equations). These results show the improvement and the efficiency achieved using Q-R decomposition when the system can be represented as bilinear. 

An implementation of this algorithm, using the sequential procedure within the MATLAB environment, was proposed by Figueroa and Romagnoli (1994). To solve step 2, the constr function from the MATLAB Optimization Toolbox has been used. The numerical integration necessary in this step has been performed via the function ode45 for the solution of ordinary differential equations. 

The equivalent to the law of mass action in equilibria are the sets of differential equations in kinetics. They are defined by the chemical reaction scheme. Again, there are explicit solutions for very simple models but most other models lead to sets of differential equations that need to be integrated numerically. Matlab supplies an extensive collection of functions for... 

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