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.

ISBNs: 0-471-20506-0 (Print); 0-471-22155-4 (Electronic)
the process interface level to obtain industry standard functions useful for integrat-
ing process operations, such as control tuning regimes and distributed communica-
tions. In fact, for many applications, these controllers are deployed to acquire
process measurements, absent control actuation, owing to the utility of their sensor
signal conditioning electronics. More significant is an illustration of how control
performance is influenced by the controller instrumentation.
Figure 7-1 illustrates a common digital controller instrumentation design. For
continuity, the thermocouple signal conditioning example of Figure 4-5 is em-
ployed for the controller feedback electronics front end that acquires the sensed
process temperature variable T, including determination of its error. Further, the
transfer function parameters described by equation (7-1) are for a generic dominant
pole thermal process, also shown in Figure 7-1, that can be adapted to other
processes as required. When the process time constant
␶
0
is known, equation (7-2)
can be employed to evaluate the analytically significant closed-loop bandwidth
BW
CL
–3 dB frequency response. Alternately, closed-loop bandwidth may be evalu-
ated experimentally from equation (7-3) by plotting the controlled variable C rise
time t
r
resulting from setpoint step excitation changes at R.
= ·
΄΅
(7-1)
BW
CL

= 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

P
K
C
΂
1 +
ᎏ
2
␲
1
Is
ᎏ
+
ᎏ
2
␲
s
D
ᎏ
΃
ᎏᎏᎏ
2
␲␶
0
␶
0
s
ᎏᎏᎏ
1 + K
P
K

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

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

2
␲
t
r
0.5 z
ᎏ
(z – 1)
–0.5 z
ᎏ
(z – 0.8)
B
ᎏ
z – 1
A
ᎏ
z – 0.8
(0.1)
ᎏᎏ
(z – 0.8)(z – 1)
C(z)
ᎏ
z
(1 – e
–0.1
)z
ᎏᎏᎏ
(z – e
–0.1
(2) + 1)(z – 1)
z

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

ᎏ
s + 1
1 – e
–sT
ᎏ
s
FIGURE 7-2. Closed-loop bandwidth evaluation.
Examination of Figure 7-1 reveals Analog Devices linear and digital conversion
components with significant common-mode interference attenuation associated
with the signal conditioning amplifier demonstrated in Figure 4-5. The corollary
presence of 40 mV of 20 KHz power converter noise at an analog multiplexer input
is also shown to result in negligible crosstalk interference as coherent noise sam-
pled data aliasing. A significant result is the influence of the closed-loop bandwidth
BW
CL
on interpolating the controller D/A output by attenuating its sampled data,
image frequency spectra. Owing to the dynamics of parameters included in this in-
terpolation operation, intersample error is the dominant contribution to total instru-
mentation error shown Table 7-2. The 0.45%FS 1␴ total controller error approxi-
mates eight-bit accuracy, consisting of a 0
ෆ
.
ෆ
2
ෆ
5
ෆ
%FS static mean component plus
0.20%FS RSS uncertainty.
Error magnitude declines with reduced electronic device temperatures and less

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

s rad/sec Complex variable
K
P
°C/watts Process gain
␶
0
sec Process time constant
t
r
sec Process response rise time
BW
CL
Hz System closed-loop bandwidth
T °C Process sensed variable
V
CJC
mV/°C Cold junction compensation
V
O
FS
4.096 V
pk
Full-scale process variable value
V
s
volts Process variable signal value
controlled variable C. This effectively furnishes a control loop passband for accom-
modating the bandwidth of the error signal E. The lead element derivative time D
value enhances the transient response for both set point and process load changes to
achieve reduced time required for C to equal R.

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

1
ෆ
%FS
14-Bit A/D
Mean integral nonlinearity (1 LSB) 0.006%
Noise + distortion (–80 dB) 0.010
Quantizing uncertainty (
1
–
2
LSB) 0.003
Temperature Coefficients (
1
–
2
LSB) 0.003
␧
A/D
⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ
+ 1␴ RSS 0.020%FS
14-Bit D/A
Mean integral nonlinearity (1 LSB) 0
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

␧
%FS
Comment
Sensor 0
ෆ
.
ෆ
0
ෆ
1
ෆ
1
ෆ
Linearized thermocouple (Table 4-5)
Interface 0
ෆ
.
ෆ
0
ෆ
3
ෆ
2
ෆ
CJC sensor (Table 4-5)
Amplifier 0.103 OP-07A (Table 4-4)
Filter 0
ෆ
.
ෆ

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

ෆ
0
ෆ
0
ෆ
Average attenuation over BW
CL
Intersample 0.174 Interpolated by BW
CL
from process
␶
0
0
ෆ
.
ෆ
2
ෆ
5
ෆ
4
ෆ
%FS ⌺m
ෆ
e
ෆ
a
ෆ
n
ෆ

␧
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

΂΃
· 100%
= 0.000049%FS
Sinc
␧
sinc
=
΂
1 –
΃
· 100%
=
΂
1 –
΃
· 100%
= 0
ෆ
.
ෆ
1
ෆ
0
ෆ
0
ෆ
%FS
Controlled Variable Interpolation
V
2

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

1 +
΂΃
2
΅
–1
·
4.096 V
2
–1/2
(4.096 V)
2
·
Ά΄ ΅
2
·
΄
1+
΂΃
2
΅
–1
= ·100%
+
΄΅
2
·
΄
1 +
΂΃
2

΅
–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

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

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

CL
BW
CL
ᎏ
f
s
sin
␲
0.35 Hz/10 Hz
ᎏᎏᎏ
␲
0.35 Hz/10 Hz
1
ᎏ
2
sin
␲
BW
CL
/f
s
ᎏᎏ
␲
BW
CL
/f
s
1
ᎏ
2

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

7-2 HIGH-DATA-RATE VIDEO ACQUISITION
Industrial machine vision, laboratory spectral analysis, and medical imaging in-
strumentation are all supported by advances in digital signal processing, frequent-
1
ᎏᎏᎏᎏᎏᎏ
΂
ᎏ
0
3
.1
.0
1
3
0
ᎏ
΃
2
· (0.001313) +
΂
ᎏ
–0
3
.
.
1
2
0
5
9
1

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

ᎏᎏᎏ
͵
area
input pulse power · dt
Ί