Chemical Process Control a First Course with Matlab

2 - 16
a n y (n) +a n – 1 y (n –1) + ... + a 1 y (1) +a o y=b m x (m) +b m-1 x (m –1) + ... + b 1 x (1) +b o x (2-28)
with n > m and zero initial conditions y (n–1) = ... = y = 0 at t = 0, the corresponding Laplace
transform is
Y(s)
X(s) = b ms m +b m – 1 s m – 1 + ... + b 1 s+b o
a n s n +a n – 1 s n – 1 =G(s) = Q(s)
+ ... + a 1 s+a o
P(s)
(2-29)
Generally, we can write the transfer function as the ratio of two polynomials in s. 1 When we
talk about the mathematical properties, the polynomials are denoted as Q(s) and P(s), but the same
polynomials are denoted as Y(s) and X(s) when the focus is on control problems or transfer
functions. The orders of the polynomials are such that n > m for physical realistic processes. 2
We know that G(s) contains information on the dynamic behavior of a model as represented by
the differential equation. We also know that the denominator of G(s) is the characteristic
polynomial of the differential equation. The roots of the characteristic equation, P(s) = 0: p 1 , p 2 ,...
p n , are the poles of G(s). When the poles are real and negative, we also use the time constant
notation:
p 1 = – 1 τ 1
,p 2 = – 1 τ 2
, ...
, p n = – 1 τ n
The poles reveal qualitatively the dynamic behavior of the model differential equation. The "roots
of the characteristic equation" is used interchangeably with "poles of the transfer function."
For the general transfer function in (2-29), the roots of the polynomial Q(s), i.e., of Q(s) = 0,
are referred to as the zeros. They are denoted by z 1 , z 2 ,... z m , or in time constant notation,
z 1 = – 1 τ a
,z 2 = – 1 τ b
, ... , z m = – 1 τ m
We can factor Eq. (2-29) into the so-called pole-zero form:
G(s) = Q(s)
P(s) = b m
a n
(s – z 1 )(s– z 2 ) ... (s – z m )
(s – p 1 )(s– p 2 ) ... (s – p n )
If all the roots of the two polynomials are real, we can factor the polynomials such that the
transfer function is in the time constant form:
G(s) = Q(s)
P(s) = b o
a o
(τ a
s +1) (τ b
s + 1) ... (τ m
s+1)
(τ 1
s +1) (τ 2
s + 1) ... (τ n
s+1)
(2-30)
(2-31)
Eqs. (2-30) and (2-31) will be a lot less intimidating when we come back to using examples in
Section 2.8. These forms are the mainstays of classical control analysis.
Another important quantity is the steady state gain. 3 With reference to a general differential
equation model (2-28) and its Laplace transform in (2-29), the steady state gain is defined as the
All the features about poles and zeros can be obtained from this simpler equation.
1 The exception is when we have dead time. We'll come back to this term in Chapter 3.
2 For real physical processes, the orders of polynomials are such that n ≥ m. A simple
explanation is to look at a so-called lead-lag element when n = m and y (1) + y = x (1) + x. The
LHS, which is the dynamic model, must have enough complexity to reflect the change of the
forcing on the RHS. Thus if the forcing includes a rate of change, the model must have the same
capability too.
3 This quantity is also called the static gain or dc gain by electrical engineers. When we talk
about the model of a process, we also use the term process gain quite often, in distinction to a
system gain.