A Polynomials and Polynomial Equations This appendix summarizes some results from classical algebra concerning univariate or multivariate polynomials. Only those properties of polynomials are mentioned that are important for understanding and applying the methods of robust control in this book. More details can be found in standard textbooks like [51] and [93] . Introduction The parameter space method [15] has as its basis the closed-loop characteristic poly­ nomial p(s, q, k), where q are the uncertain plant parameters, and k are the controller parameters. The aim of the control design is to find the controller parameters k such that the roots of the uncertain polynomial (the parameters q vary in a given operating domain) are located in a prescribed region I' of the s-plane. Fixing all but two param­ eters, the feasible region in the parameter plane is represented. The parameter space method can also be used in an analysis step. In this case, a test can be done in the plant parameter plane to determine if the entire operating domain is I'-stable. In both analysis and design, the basic equation is p(s, x, y) = 0, where x, y are parameters and s varies on the boundary of of the region f . Substituting s = u(Q)+ j v(Q) E of in the uncertain polynomial and separating into real and imaginary parts leads to the system of polynomial equations Rep(x,y,Q) = 0, Imp(x,y,Q) = 0, (A.I) which has to be solved for varying Q. The solutions represent parameter combinations that lead to a polynomial with a root or a root pair on the boundary of. So, after we have mapped the boundary of into the parameter plane, the next step is to find the active boundaries that are the points (or boundaries) that contribute to the boundary of the I'-stable area. The complexity of the system of equations (A.I) depends on how the parameters appear in the characteristic polynomial. We distinguish three types of dependence with increasing complexity: affine, bilinear and polynomial dependence. In the following, we discuss methods that can be used to solve the system (A.I) and related problems. To aid in this discussion, we first introduce some definitions and theorems from the theory of algebraic curves. We then present procedures for solving various systems like (A.I) 'and discuss the applications in which the procedures are useful. We demonstrate the methods presented using simple examples. 449 Algebraic Curves If f(x, y) is a bivariate polynomial in x, y: f(x,y) = L I aijXiyi (A.2) i,j?O (the prime denotes that the sum is finite) with real coefficientsaij, then the set {(x ,y) E ]R2 : f(x, y) = O} is called an algebraic curve. The degree of f is the highest sum of the exponents, and is equal to the number of common points with a straight line. We call f(x,y) = 0 (A.3) the implicit representation of the curve. Curves may have also an explicitrepresentation x = f(o:), y = g(o:), (AA) where x(o:) and y(o:) are rational functions of the real parameter 0: . Equivalent is the representation nl(O:) n2(0:) x = d1(0:) , y = d , (A.5) 2(0:) with ni(O:), di(o:), i = 1,2, polynomials in 0:. These curves are called rational curves. Nyquist or Popov plots are trivial examples of rational curves. The root loci have, in general, no parametric representation. Elimination Theory. The Resultant In this section, we follow the representation used in [121). More details of the algebraic theory can be found in classical textbooks like [42) , [54) or [194). For the theory of curves, see [59). Consider the two polynomials f(y) = anyn + an_lyn-l + an_2yn-2 + ... + ao } (A.6) g(y) = bmym + bm_1ym-l + bm_2ym-2 + ... + ho ,' (an # 0, bm # 0), whose coefficients ai, b, are real numbers. The theory of the resultant originates in the following: Problem Give a necessary and sufficient condition on the coefficients of (A.6), such that f(y) = 0 and g(y) = 0 have a common root. 0 Theorem 1.1 f and g have a common root, if and only if the determinant of the matrix R, in (A.8) is zero. o 450 A Polynomials and Polynomial Equations This determinant is called the resultant: of f and 9 with respect to y, i.e. Res (f,g,y) = det R . (A.7) Assume now that f and 9 have a common root , then the degree of the greatest common divisor of f and 9 (gcd(f, g)) is at least 1. For finding this greatest common divisor, an additional definition is given. Definition 1.2 The matrix R I of dimension m + n - 2, obtained by deleting the first and the last rows and the first and the last columns in the Matrix R, is called the first inner of R . Continuing on the deletion process, we obtain the inners R 2 , R 3 , ... of dimension m +n-4, m+n-6, . respectively. The determinants of inners are called subresultants. o Example 1.3 For n = 4, m = 3 we have the matrix R and 3 inners: a4 a3 a2 al an 0 0 0 a4 a3 a2 al ao 0 0 0 a4 a3 a2 al ao R = 0 0 0 b3 bz bl bo (A.S) 0 0 b3 ~ bl bo 0 0 bs ~ bl bo 0 0 b3 ~ bl bo 0 0 0 a4 a3 a2 al ao 0 a4 a3 a2 al R I = 0 0 b3 b2 bl 0 b3 b2 bl bo bs b2 bl bo 0 [ ~ a3 ~ ] R2 = 0 b3 b2 b3 b2 bl lIn the literature the matrix is often defined in a slightly different mann er, th e rows of the b, are exchanged , so only the sign of the resultant may change depending on th e number of exchanges. 451 o Theorem 1.4 If det R = det R, = det R k - l = 0 and det R k i- 0, then the degree of gcd(f, g) is k. In this case, gcd(f, g) equals the determinant of the matrix obtained from R k by replacing the last column in it by the column [ym-k-lf( y), ym-k-2f() y , . .. , (A.9) f(y) ,g(y) , yg(y),. .., yn-k-lg(y)]'. o an an-l an- 2 .. al ao 0 0 0 0 0 0 an 0 0 0 0 (m rows) an- l al ao 0 0 0 R al ao (A.lO) 0 0 0 bl bo (n rows) 0 bm bm-l b, bo 0 0 0 0 bm bm- 1 bm - 2 .. bl bo 0 0 0 0 0 Example 1.5 Suppose in Example 1.3 that k = deg(gcd(f, g)) = 1, then a4 a3 a2 al yf(y) 0 a4 a3 a2 f(y) gcd(f,g) = det 0 0 b3 b2 g(y) 0 b3 b2 bl yg(y) b3 ~ b1 bo y2g(y ) Using the fact that gcd(f, g) is linear in y, gcd(f, g) = Ay+B, all quadrat ic and higher 452 A Polynomials and Polynomial Equations terms in Y must vanish so we can simplify the determinant: a4 a3 a2 al aoy 0 a4 a3 a2 alY + an gcd(J,g) = det 0 0 b3 ~ bly + bo 0 b3 b2 bl boY ~ bz bl bo 0 Thus, gcd(J, g) = Ay + B, where A = det R I and a4 a3 a2 al 0 0 a4 a3 a2 an B=det 0 0 b3 ~ bo 0 b3 ~ bl 0 b3 ~ bl bo 0 For k = 1 in Theorem 1.4 we have 0 Corollary If f and 9 have a single common root Ya (det R = 0, det R I i= 0), then Ya = - det Rd det R I , (A.H) where R1 equals the matrix obtained from R 1 by replacing the last column in it by the column ~, an , bo, ~jT . 0 m-2 n-2 Application Let p(s) be any real polynomial. If we set s = jw, we can write P(JW.) = aO - a2W2+ a4W 4 + ... +JW. (at - a3w 2+ asw4)... = Peven + jw Podd · 2 Setting f2 = w , we can rewrite these polynomials as Peven(f2) = ao - a2f2 + a4f2 2. .. , 2 podd(f2) = at - a3f2 + asf2 •••• The resultant of the polynomials Podd(f2) and Peven(f2) with respect to f2 is the deter­ minant of the well-known Hurwitz matrix. Apart from the sign of some entries, R is the Hurwitz matrix (1.6.2). 453 Example 1.6 For n = 4, we have f = ao- a2n + a4n2 and 9 = a l - a3n, the matrix R is R = [ a~ =:: :~] . - a3 al 0 o The Discriminant A special case of the resultant is the discriminant Dt- The second polynomial 9 is the derivative of the first polynomial I, i.e, D/ := 2- ResU ,1', Y). (A.12) ao Let Yi, i = 1 . .. n, the roots of f(y) = O. Another definition of the discriminant is n n a~n-2 D/ = (-1) n(n2- 1) IIII(Yi - Yk)2 (i =1= k). (A.13) i=1 k=1 Theorem 1.7 The equation D] = 0 is a necessary and sufficient condition that f(y) has a root of multiplicity of at least 2. o Application The breakaway points (real) and saddle points (complex) of the root locus can be determined by computing the discriminant and finding the zeros. Example 1.8 2 2 Let p(s) = (s + 2)(s + 1)(s - 1) + k = S3 + 2s - S - 2+ k, then p'(s) = 3s +4s - 1 and + k 12o 1-1 2-2 - 1 -2 + k0] D; = det 0 0 3 4 -1 o 3 4 -1 0 3 4 -1 0 0 2 = 27k - 40k - 36 = 0 .