SME
Process Dynamics & Control
Feedback control
Use this for “A P controller with Kc = 3 runs a process with Kₚ = 1.5 and τ = 8 min. After a setpoint step of +4, what offset is left, and what is the closed-loop τ?”
Diagram
Loop gain KcKₚ = 4 × 2 = 8.
The
output
settles
at
R
×
KcKₚ
1 + KcKₚ
=
5
×
8
9
=
4.44,
an
offset
of
0.556.
τcl
=
τ
1 + KcKₚ
=
10
9
=
1.11
min:
the
loop
is
faster
than
the
process.
The sensor’s signal enters the comparator with a minus: negative feedback.
This is an example. Type your own values to start a new problem.
KcKₚ
=
Kc
×
Kₚ
Y∞
=
R
×
KcKₚ
÷
(1
+
KcKₚ
)
τcl (min)
=
τ (min)
÷
(1
+
KcKₚ
)
eₛₛ
Offset
calculated
Formulas
KcKₚ = Kc × Kₚ
8 = 4 × 2
Y∞
=
R
×
KcKₚ
1 + KcKₚ
4.444
=
5
×
8
1 + 8
eₛₛ
=
R
−
Y∞
0.5556 = 5 − 4.444
τcl
=
τ
1 + KcKₚ
1.111
=
10
1 + 8
Assumptions
•
Proportional-only control of a first-order process Kₚ ÷ (τs + 1), in deviation variables.
•
An ideal sensor and valve (gain 1); Kc and Kₚ both positive, so the loop is stable.
•
The setpoint steps up by R; the final value is where the output settles.
Step-by-step
WE KNOW
Kc = 4, Kₚ = 2, τ = 10 min, R = 5
FIND
loop gain (KcKₚ), final value (Y∞), offset (eₛₛ), closed-loop time constant (τcl)
Step 1 · Find loop gain (KcKₚ)
Use KcKₚ = Kc × Kₚ
Around the loop the gains multiply: controller, then process (valve and sensor are 1).
KcKₚ = 4 × 2
KcKₚ = 8
Step 2 · Find final value (Y∞)
Use Y∞ = R × KcKₚ ÷ (1 + KcKₚ)
The
closed
loop’s
steady
gain
is
K
1 + K
(set
s
=
0),
so
the
output
settles
short
of
the
setpoint.
Y∞
=
5
×
8
1 + 8
Y∞
=
5
×
8
9
Y∞
=
40
9
Y∞ = 4.444
Step 3 · Find offset (eₛₛ)
Use eₛₛ = R ÷ (1 + KcKₚ)
What
the
output
never
reaches:
R
−
RK
1 + K
is
R
1 + K
,
smaller
but
never
0
as
K
grows.
eₛₛ
=
5
1 + 8
eₛₛ
=
5
9
eₛₛ = 0.5556
Step 4 · Find closed-loop time constant (τcl)
Use τcl = τ ÷ (1 + KcKₚ)
The
closed
loop
is
still
first
order,
with
τ
divided
by
1
+
K:
feedback
makes
it
faster.
τcl
=
10
1 + 8
τcl
=
10
9
τcl = 1.111 min
Check
Put the numbers back in. Both sides should match.
8 = 4 × 2 ✓
4.444
=
5
×
8
1 + 8
✓
0.5556 = 5 − 4.4444 ✓
1.111
=
10
1 + 8
✓
Topic check
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