Chemical Process Control a First Course with Matlab

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C. Complex conjugate poles
c i
Terms of the form +
c* i
, where p
s – p i s – p* i = α + jβ and p* i = α – jβ are the complex
i
poles, have time-domain function c i e p i t +c*i e p* i t of which form we seldom use. Instead,
we rearrange them to give the form [some constant] x e αt sin(βt + φ) where φ is the phase
lag.
It is cumbersome to write the partial fraction with complex numbers. With complex
conjugate poles, we commonly combine the two first order terms into a second order term.
With notations that we will introduce formally in Chapter 3, we can write the second order
term as
as + b
τ 2 s 2 +2ζτ s+1 ,
where the coefficient ζ is called the damping ratio. To have complex roots in the
denominator, we need 0 < ζ < 1. The complex poles p i
and p* i
are now written as
p i , p* i = – ζ τ ± j 1 – ζ 2
τ
with 0 < ζ < 1
and the time domain function is usually rearranged to give the form
[some constant] x e – ζt/τ sin
1 – ζ 2
τ
t+φ
where again, φ is the phase lag.
D. Poles on the imaginary axis
If the real part of a complex pole is zero, then p = ±ωj. We have a purely sinusoidal
behavior with frequency ω. If the pole is zero, it is at the origin and corresponds to the
integrator 1/s. In time domain, we'd have a constant, or a step function.
E. If a pole has a negative real part, it is in the left-hand plane (LHP). Conversely, if a pole
has a positive real part, it is in the right-hand plane (RHP) and the time-domain solution is
definitely unstable.