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182 5. Approximation and Location of Zeros

(a) Show that, if all the zeros of f(t) have negative real part, then

a, b, c, d are all positive. Henceforth, we will assume that this

condition holds.

(b) Show that, if all the zeros of f(t) are real, then all must be

negative and bc - ad > 0.

(c) Show that, if bc 5 ad, then f(t) must have a nonreal zero.

(d) Suppose that f(t) has two nonreal zeros u f vi (v # 0). Verify

that, for t = u f vi, f(t) = 0 is equivalent to

uu3 + bu2 + cu + d = v2(3au + b)

and

3au2+2bu+c=av2;

eliminate v2 to obtain

8a2u3 + 8abu2 + 2(b2 + ac)u + (bc - ad) = 0.

Show that

(i) if bc > ad, then u < 0;

(ii) if bc = ad, then u = 0;

(iii) if bc < ad, then u > 0.

(e) Show that a cubic polynomial at3 + bt2 + ct + d is stable if and

only if all its coefficients are nonzero with the same sign and

bc > ad.

[This condition has a nice generalization, known as the Routh-Hurwitz

Criteria, to polynomials of degree n. These can be conveniently ex-

pressed using determinants.]

11. Nyquist diagram. To test the stability of a given polynomial, first
    determine a positive real M which is greater than the absolute values
    of its zeros. Let C be the curve consisting of that portion of the
    imaginary axis consisting of points yi for which ]y] 5 M and the
    semicircle of center 0 and radius M lying to the right of the imaginary
    axis. The image p(C) of this curve is called the Nyquist diagram for
    the polynomial.

(a) Show that the polynomial is stable if and only if its Nyquist

diagram does not make a circuit of the origin.

(b) Verify that all of the zeros of ,z2 + 22 - 3 lie inside the circle of

center 0 and radius 6. Show that this polynomial maps the y-axis

(imaginary axis) onto the parabola y2 + 4x + 12 = 0 with vertex

(-3,0) (the point (x, y) is identified with the complex number

x + yi). Verify that the image .of the point G(cosB + isine) is