Department of Chemical Engineering I.I.T.
Bombay, India
Lecture 8: Dynamics of higher order
systems
Department of Chemical Engineering I.I.T. Bombay, India
Dynamic behaviour of higher order systems
Number of dynamic elements =n
K1 K2 K3 Ki Kn
u 1s + 1 2s +1 3s + 1 is +1 ns +1 y
Overall dynamics
n Ki
y N ( s ) = u ( s )
i =1 s + 1
i
The system would have n poles, each equal to the negative
reciprocal of the time constant.
Department of Chemical Engineering I.I.T. Bombay, India
Step Response of the nth order system
n Ki 1
y N ( s ) =
i =1 s + 1 s
i
Expanding by partial fractions,
A0 N Ai
y N ( s ) = K +
s i =1 i s + 1
It turns out that A0 =1 and Ai = -I and yN(t) is given by,
N
Ai t / i
y N (t ) = K 1 + e
i =1 i
Department of Chemical Engineering I.I.T. Bombay, India
Limiting case of the nth order system
Consider the special
p case when :
All the N steady state gains are equal to 1
All the time constants are identical and equal to /N
1
y N (s) = u (s)
N
s + 1
N
In the limiting case as N goes to infinity,
f G(s)
G( ) =e-s. This is the
case of a pure time delay.
Department of Chemical Engineering I.I.T. Bombay, India
Step responses for the limiting case
Department of Chemical Engineering I.I.T. Bombay, India
Higher order systems with zeros
q
( i s + 1)
y(s) = K i =1
p
u ( s)
( i s + 1)
i =1
This would be a (p,q)th order transfer function with p poles
and q zeros.
For realizability, q must be less than or equal to p. The
transfer function would then be called as a proper transfer
f
function.
i
As before, lead terms or zeros generally have an effect of
speeding up the response of the system.
system
Department of Chemical Engineering I.I.T. Bombay, India
Illustrative example of a (2,1) system
K (1s + 1)
g ( s) =
( 1s + 1)( 2 s + 1)
The step response expression for this plant can be written as,
as
1 1 t / 1 2 1 t / 2
y (t ) = K 1 e e
1 2 2 1
If 1 =0, the response
p is that of a ppure second order system.
y
Depending on the value of 1, different responses varying from first
order to second order with lead, can be realized.
Department of Chemical Engineering I.I.T. Bombay, India
Effect of the lead for the (2,1) system
Lead time constants: 0.05,1,4 and 8
Lag time constants: 1 and 4
Department of Chemical Engineering I.I.T. Bombay, India
Summary of the Low and higher order dynamics
First order : Overall response is a result of an equilibrium
between the driving and the opposing forces.
Response
espo se iss characterized
c a ac e ed by a ssteady
eady sstage
age ga
gain K aandd thee
time constant , which characterizes the speed of response.
Special cases of the response are the pure gain and the pure
integrator processes.
Sinusoidal response shows an output of the same frequency as
the input but differing in amplitude and phase.
Lead-lag dynamics are a result of a pure gain element in paraller
with
ith a pure-lag
l element.
l t
Department of Chemical Engineering I.I.T. Bombay, India
Summary of the Low and higher order dynamics
Second order : Overall response is a result of a two dynamic
elements in an interacting or non-interacting configuration.
Response is characterized by a steady stage gain K, the
damping factor which characterizes the oscillatory nature and
the natural time constant ,
which characterizes the speed of
response.
Higher order variants essentially show similar behaviour but
approach the pure time delay in the limiting case.
Higher
g order variants can also have numerator terms ((zeros)) and
higher denominator terms (poles) which can exhibit complex
dynamics.
Lead term in generally has a speed-up property over the lag
terms.
