147
7
MEASUREMENT AND
CONTROL INSTRUMENTATION
ERROR ANALYSIS
7-0 INTRODUCTION
Systems engineering considerations increasingly require that real-time I/O systems
fully achieve necessary data accuracy without overdesign and its associated costs.
In pursuit of those goals, this chapter assembles the error models derived in previ-
ous chapters for computer interfacing system functions into a unified instrumenta-
tion analysis suite, including the capability for evaluating alternate designs in over-
all system optimization. This is especially of value in high-performance
applications for appraising alternative I/O products.
The following sections describe a low data rate system for a digital controller
whose evaluation includes the influence of closed-loop bandwidth on intersample
error and on total instrumentation error. Video acquisition is then presented for a
high data rate system example showing the relationship between data bandwidth,
conversion rate, and display time constant on system performance. Finally, a high-
end I/O system example combines premium performance signal conditioning with
wide-range data converter devices to demonstrate the end-to-end optimization goal
for any system element of not exceeding 0.1%FS error contribution to the total in-
strumentation error budget.
7-1 LOW-DATA-RATE DIGITAL CONTROL INSTRUMENTATION
International competitiveness has prompted a renewed emphasis on the develop-
ment of advanced manufacturing processes and associated control systems whose
complexity challenge human abilities in their design. It is of interest that conven-
tional PID controllers are beneficially employed in a majority of these systems at
Multisensor Instrumentation 6
␴
Design. By Patrick H. Garrett
Copyright © 2002 by John Wiley & Sons, Inc.

= Hz dominant-pole closed-loop bandwidth
(7-2)
BW
CL
= Hz universal closed-loop bandwidth (7-3)
For simplicity of analysis, the product of combined controller, actuator, and
process gains K is assumed to approximate unity, common for a conventionally
tuned control loop, and an example one-second process time constant enables the
choice of an unconditionally stable controller sampling period T of 0.1 sec (f
s
= 10
Hz) by the development of Figure 7-2. The denominator of the z-transformed trans-
fer function defines the joint influence of K and T on its root solutions, and hence
stability within the z-plane unit circle stability boundary. Inverse transformation
and evaluation by substitution of the controlled variable c(n) in the time domain an-
alytically reveals a 10–90% amplitude rise time t
r
value of 10 sampling periods, or
1 sec, for unit step excitation. Equation (7-3) then approximates a closed-loop band-
width BW
CL
value of 0.35 Hz. Table 7-1 provides definitions for symbols employed
in this example control system.
2.2
ᎏ
2
␲
t
r
1 + K

C
΂
1 +
ᎏ
2
␲
1
Is
ᎏ
+
ᎏ
2
␲
s
D
ᎏ
΃
K
P
K
C
΂
1 +
ᎏ
2
␲
1
Is
ᎏ
+

R
148
MEASUREMENT AND CONTROL INSTRUMENTATION ERROR ANALYSIS
149
FIGURE 7-1. Digital control system instrumentation.
150
Forward path = ·
␶
0
= 1.0 sec
= K · z-transformed
= transfer function
=
C(z) = · unit-step input
= T = 0.1 sec, K = 1.0
= partial fraction expansion
= +
C(z) = +
c(n) = [(–0.5)(0.8)
n
+ (0.5)(1)
n
]·U(n) inverse transform
BW
CL
= = 0.35 Hz t
r
= nT = 1.0 sec
2.2
ᎏ

ᎏ
z –1
K(1 – e
–T
)
ᎏᎏ
z – e
–T
(1 + K) + K
K(1 – e
–T
)
ᎏᎏ
z – e
–T
(1 + K) + K
Forward path
ᎏᎏ
1 + Forward path
C(z)
ᎏ
R(z)
(1 – e
–T
)
ᎏ
(z – e
–T
)
K

than full-scale signal amplitude V
s
encountered at steady-state, as described by the
included error models. Largest individual error contributions are attributable to the
differential-lag signal conditioning filter and controller D/A-output interpolation. It
is notable that the total instrumentation error
␧
C
value defines the residual variabili-
ty between the true temperature and the measured controlled variable C, including
when C has achieved equality with the setpoint R, and this error cannot further be
reduced by skill in controller tuning.
Tuning methods are described in Figure 7-3 that ensure stability and robustness
to disturbances by jointly involving process and controller dynamics on-line. Con-
troller gain tuning adjustment outcomes generally result in a total loop gain of ap-
proximately unity when the process gain is included. The integrator equivalent val-
ue I provides increased gain near 0 Hz to obtain zero steady-state error for the
7-1 LOW DATA RATE DIGITAL CONTROL INSTRUMENTATION
151
TABLE 7-1. Process Control System Legend
Symbol Dimension Comment
R °C Controller setpoint input
C °C Process controlled variable
E °C Controller error signal
K
C
watts/°C Controller proportional gain
I sec Controller integral time
D sec Controller derivative time
U watts Controller output actuation

Analog Multiplexer
Transfer error 0
ෆ
.
ෆ
0
ෆ
1
ෆ
%
Leakage 0.001
Crosstalk 0.00005
␧
AMUX
⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ
+ l␴ RSS 0
ෆ
.
ෆ
0
ෆ
1
ෆ

.
ෆ
0
ෆ
0
ෆ
6
ෆ
%
Noise + distortion (–80 dB) 0.010
Temperature coefficients (
1
–
2
LSB) 0.003
␧
D/A
⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ
+ 1␴ RSS 0.016%FS
152
MEASUREMENT AND CONTROL INSTRUMENTATION ERROR ANALYSIS
TABLE 7-2. Digital Control Instrumentation Error Summary
Element

1
ෆ
0
ෆ
0
ෆ
Signal conditioning (Table 3-5)
Signal Quality 0.009 60 Hz
␧
coh
(Table 4-5)
Multiplexer 0
ෆ
.
ෆ
0
ෆ
1
ෆ
1
ෆ
Average transfer error
A/D 0.020 14-bit successive approximation
D/A 0.016 14-bit actuation output
Noise aliasing 0.000049 –85 dB AMUX crosstalk from 40 mV @ 20 kHz
Sinc 0
ෆ
.
ෆ
1

␧
C
0.204%FS 1␴ RSS
0.458%FS ⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ
+ 1␴ RSS
1.478%FS ⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ
+ 6␴ RSS
Noise Aliasing
␧
coherent alias
= Interference · AMUX crosstalk · sinc · 100%
= · –85 dB · sinc
΂΃
· 100% m defined at f
coh
= · (0.00005) · sinc

O
FS
–1/2
V
S
2
·
Ά
sinc
2
΂
1 –
΃
·
΄
1 +
΂΃
2
΅
–1
␧
⌬V
= ·100%
+ sinc
2
΂
1 +
΃
·
΄

΅
–1
·
10 Hz + 0.35 Hz
ᎏᎏ
0.35 Hz
sin
␲
΂
1 +
ᎏ
0
1
.3
0
5
H
H
z
z
ᎏ
΃
ᎏᎏᎏ
␲
΂
ᎏ
1 +
1
0
0

ᎏ
0
1
.3
0
5
H
H
z
z
ᎏ
΃
f
s
+ BW
CL
ᎏᎏ
BW
CL
BW
CL
ᎏ
f
s
f
s
– BW
CL
ᎏᎏ
BW

2000 · 10 Hz – 20 kHz
ᎏᎏᎏ
10 Hz
40 mV
ᎏ
4096 mV
mf
s
– f
coh
ᎏ
f
s
V
coh
ᎏ
V
o
FS
7-1 LOW DATA RATE DIGITAL CONTROL INSTRUMENTATION
153
΅
΄
΅
΄
=
΄΅
–1/2
· 100%
= 0.174%FS

4
ᎏ
΃
2
· (0.001142)
154
MEASUREMENT AND CONTROL INSTRUMENTATION ERROR ANALYSIS
Quarter Decay PID Parameters Trapezoidal PID Parameters
P = 1.2 adjusted quarter decay P = 100% · Process Gain
trapezoidal tuning
I = period
quarter decay
, sec I = Process Period, sec
D =
quarter decay
, sec D = 0.44 (Process Lag + Process Period), sec
period
ᎏ
4
100%
ᎏᎏ
Controller K
c
Process Gain
trapezoidal tuning
=
FIGURE 7-3. Process controller tuning algorithms.
͵
area
output pulse power · dt