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v(t) P(s)
r(t) + e(t) − + u(t) u(t-h) y(t)
C(s) e -hs P (s)
0
−

FIGURE 26.18 Feedback system a with time delay.

Stability of Delay Systems
Stability of the feedback system shown in Fig. 26.18 is equivalent to having all the roots of

c s() = Ds() + e Ns() (26.20)
−hs

in the open left half plane, − , where D(s) = D c (s)D p (s) and N(s) = N c (s)N p (s). We assume that there is
no unstable pole–zero cancellation in taking the product P 0 (s)C(s), and that deg(D) > deg(N) (here N
and D need not be monic polynomials). Strictly speaking, χ(s) is not a polynomial because it is a
transcendental function of s. The functions of the form (26.20) belong to a special class of functions
called quasi-polynomials. The closed-loop system poles are the roots of (26.20).
Following are known facts (see [1,10]):
(i) If r k is a root of (20), then so is (i.e., roots appear in complex conjugate pairs as usual).r k
(ii) There are infinitely many poles r k ∈ , k = 1, 2,…,satisfying χ(r k ) = 0.
(iii) And r k ’s can be enumerated in such a way that Re(r k + 1) ≤ Re(r k ) ; moreover, Re(r k ) → −∞ as
k → . ∞

Example
−hs
If G h (s) = e /s, then the closed-loop system poles r k , for k = 1, 2,…, are the roots of

−hs k −jhw k
e
1 + e -----------------------e ± j2kp = 0 (26.21)
s k + jw k
where r k = σ k + jω k for some s k ,w k ∈ . Note that e ±j2kπ = 1 for all k = 1, 2,…. Equation (26.1) is equivalent
to the following set of equations:

−hs
e k = σ k + jw k (26.22)

( ± 2k 1)p = hw k + ∠ ( σ k + jw k ), k = 1,2,… (26.23)
–

It is quite interesting that for h = 0 there is only one root r = −1, but even for infinitesimally small h > 0 there
are infinitely many roots. From the magnitude condition (26.22), it can be shown that

σ k ≥ ⇒ w k ≤ 1 (26.24)
0

Also, for s k ≥ 0 , the phase (s k + jw k ) is between −p/2 and +p/2, therefore (26.23) leads to

σ k ≥ ⇒ h w k ≥ p (26.25)
---
0
2

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