Main properties of root-loci of time-delay control systems and a method of finding the stable interval of gain

Abstract

In this paper, we will discuss the extended graphic representation of a characteristic equation of the type 1 + KG(s)H(s)e^-τs = 0,where G(s)H(s) is a rational fraction of the complex variable s (≡ x + iy) with real coefficients. Because.we have deduced the equation of the root-locus and the expression for the curve (K, xs), the main characteristics of such curves are easily described qualitatively. It follows that much unnecessary work can be omitted. When we chose y as the assigned variable, the equation of root-locus of the above expression is an algebraic equation of the (m + n)-th degree in x, where (n + m) is the sum of degrees of the numerator and denominator of the fraction G(s)H(s). In order to obtain the stable interval (SI) of gain K, we must solve a transcendental equation of y. Finally, the figures of root-loci and curves (K, x) of the expression G(s)H(s) of the types

$$\frac{1}{{s + c}},\frac{1}{{s(s + c)}},\frac{{s + A}}{{s + B}},$$

are given here for illustration, and a method of finding the SI of K of the expression

$$GH = {{(1 + {T_A}s)} \mathord{\left/{\vphantom {{(1 + {T_A}s)} {s(1 + }}} \right. \kern-\nulldelimiterspace} {s(1 + }}{T_L}s)(1 + {T_N}s)$$

is calculated by a direct numerical method.