MATLAB polynomials

In MATLAB, polynomials are stored exactly the same as vectors. Functions in MATLAB will interpret them properly if we follow the convention that a vector stores the coefficients of a polynomial in descending order—it begins with the highest order term and always ends with a constant even if it is zero. Some examples ... 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

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. 

We first take the painful route just so we better understand the results from MATLAB. If we have to do the chore by hand, we much prefer the completing the perfect square method in Example 2.8. Even without MATLAB, we can easily find that the roots of the polynomial s2 + 4s +13 are -2 3j, and F(s) can be written as the sum of... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

To evaluate the matrix polynomial in Eq. (9-23), we use the MATLAB function polyvalm () which applies the coefficients in p2 to the matrix A. 

Note The factor k is 6 here, and in the MATLAB manual, it is referred to as the "gain." This factor is really the ratio of the leading coefficients of the two polynomials q(s) and p(s). Make sure you understand that the "k" here is not the steady state gain—which is the ratio of the last constant coefficients. (In this example, the steady state gain is -12/-4 = 3.) MATLAB actually has a function named degain to do this. 

MATLAB is object-oriented. Linear time-invariant (LTI) models are handled as objects. Functions use these objects as arguments. In classical control, LTI objects include transfer functions in polynomial form or in pole-zero form. The LTI-oriented syntax allows us to better organize our problem solving we no longer have to work with individual polynomials that we can only identify as numerators and denominators. 

We have expected q(s) = -0.1s + 1, and p(s) = 0.1s + 1. Obviously, MATLAB normalizes the polynomials with the leading coefficients. On second thought, the Pade approximation is so simple that there is no reason why we cannot do it ourselves as in a textbook. For the first order approximation, we have... 

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. 

The example below shows a short Matlab program that fits the function y =tan(x) with a polynomial of degree 3 defined by 4 linear parameters, i.e. the elements of a. 

Defying Matlab elegance, one could write equation (4.46) as a loop, but it is certainly faster to vectorise the equation. The vectorised Matlab code (note that the polynomial degree equals the number of parameters minus one, nd=np-1) ... 

MATLAB s polynomial-roots finder roots does not handle repeated or clustered roots very well, but otherwise it is the best 0(n3) root finder available. Note that an operations count of 0(nP) for an algorithm signifies that the algorithm performs K n additions and multiplications (for some algorithm specific constants K and j, but depending on n) to obtain its output from n input data. Most of the polynomial-root finders of the last century unfortunately were even slower 0(n4) algorithms and all in all much too slow and inaccurate. 

MATLAB has a built-in root finder for scalar equations f(x) = 0 in one real variable x that are in standard form. The built-in MATLAB function is fzero. The use of fzero hinges on a user-defined function, such as the function f inside the following fzero tester, called fzerotryl, that we apply to our previously studied third degree polynomial. 

The first call of fzero inside fzerotryl takes 24 iterations to arrive at the real root x = -1.1304 of our trial polynomial p(x) x — 2x2 + 4 when starting at x0 = —2, while the second call converges after 9 iterations when looking for real roots of p inside the interval [—2, —1]. Please look up help fzero to learn more about this MATLAB function and how it was used. 

The following experiments validate our assessment of troubles with Newton or bisection root finders for multiple roots. First we use the bisection method based MATLAB root finder f zero, followed by a simple Newton iteration code, both times using the chosen polynomial p x) of degree 9 in its extended form (1.6). 

MATLAB s 0(n3) polynomial-root finder roots, used for the same polynomialp, encounters different problems and computes 4 complex conjugate root pairs instead. These lie on a small radius circle around the ninefold root 2. As input for roots, we represent our polynomial (x — 2)9 of degree 9 in extended form by its coefficient vector [1 -18. .. 2304 -512],... 

Only the earlier mentioned faster 0(n2) polynomial-root finder pzero discovers the ninefold real root 2 of p correctly see the Resources appendix for a quote of the literature for pzero and the folder pzero on our CD for the actual MATLAB code of pzero. 

Next consider MATLAB with a polynomial of higher order than a straight line. Put the data into the vectors t and p. The only change is in the polyfit command - just type what order, n, you want ... 

To use nonlinear regression, you minimize Eq. (E.3) with respect to the unknown parameters. Polynomial and multiple regression do this too (behind the scenes), but for nonlinear curve fits it is necessary to use functions such as Solver in Excel and fminsearch in MATLAB. This is demonstrated using the same example given above for multiple regression. 

MATLAB is able to do regressions up to very large polynomial orders, using the "polyfit" function. The syntax for this function is ... 

The Hemiite polynomials usually are defined by an integral equation. None the less, they are typically computed by annother series expansion. For the computation of the Hermite polynomials we use the built in functions of Matlab which are very accurate and computationally fast. 

---