ap-4-12.dvi


Acta Polytechnica Vol. 52 No. 4/2012

An Application of the Individual Channel Analysis and Design Approach

to Control of a Two-input Two-output Coupled-tanks System

David J. Murray-Smith1

1
School of Engineering, Rankine Building, University of Glasow, Glasgow G12 8AT, Scotland, United Kingdom

Correspondence to: David.Murray-Smith@glasgow.ac.uk

Abstract

Frequency-domain methods have provided an established approach to the analysis and design of single-loop feedback

control systems in many application areas for many years. Individual Channel Analysis and Design (ICAD) is a more

recent development that allows neo-classical frequency-domain analysis and design methods to be applied to multi-input

multi-output control problems. This paper provides a case study illustrating the use of the ICAD methodology for an

application involving liquid-level control for a system based on two coupled tanks. The complete nonlinear dynamic

model of the plant is presented for a case involving two input flows of liquid and two output variables, which are the

depths of liquid in the two tanks. Linear continuous proportional plus integral controllers are designed on the basis of

linearised plant models to meet a given set of performance specifications for this two-input two-output multivariable

control system and a computer simulation of the nonlinear model and the controllers is then used to demonstrate

that the overall closed-loop performance meets the given requirements. The resulting system has been implemented in

hardware and the paper includes experimental results which demonstrate good agreement with simulation predictions.

The performance is satisfactory in terms of steady-state behaviour, transient responses, interaction between the controlled

variables, disturbance rejection and robustness to changes within the plant. Further simulation results, some of which

involve investigations that could not be carried out in a readily repeatable fashion by experimental testing, give support

to the conclusion that this neo-classical ICAD framework can provide additional insight within the analysis and design

processes for multi-input multi-output feedback control systems.

Keywords: multivariable, feedback, control, coupled tanks, frequency domain, nonlinear.

1 Introduction

Problems of liquid level control arise in many indus-
tries. Common examples include control of levels in
blending and reaction vessels within chemical pro-
cesses. This paper describes the application of the
Individual Channel Analysis and Design (ICAD) ap-
proach to the development, implementation and test-
ing of conventional diagonal controllers for a system
involving liquid levels in a pair of coupled tanks [1].
The objective of the paper is to provide a detailed
case-study to illustrate use of the ICAD methodol-
ogy and to demonstrate some of the benefits of this
neo-classical frequency-domain approach to problems
involving multivariable control.

The coupled-tanks equipment around which the
control system is designed has two inputs, which are
external liquid flow-rates into each of the tanks. The
two outputs are the resulting levels of liquid in the
tanks. There is only one outflow and this is from the
second tank. Figure 1 is a schematic diagram of this
system.

Established frequency-domain methods, such as
Bode/Nyquist analysis, are of central importance as

Figure 1: Schematic diagram of the coupled-tanks
equipment

a basis for design tools for single-input single-output
feedback control systems in many application areas.
The success of the frequency-domain approach is due,
in part, to the graphical nature of these techniques
which provides transparency and flexibility in satisfy-
ing design specifications in the presence of practical
constraints. The extension of classical methods of
analysis and design to multivariable systems involv-

121



Acta Polytechnica Vol. 52 No. 4/2012

ing more than one input and more than one output
can introduce difficulties. It may still be possible, in
cases where cross-coupling is not strong, to design the
control system using approaches involving one loop
at a time. However, in cases where dynamic interac-
tions between loops are significant, more skill and ex-
perience is necessary in order to produce a successful
design, and a process of tuning using trial-and-error
methods may be needed.

The Individual Channel Analysis and Design
(ICAD) methodology was developed in the early
1990s and allows frequency-domain methods to be
applied to the problems of analysis and design of
multi-input multi-output feedback control systems
(see, e.g., [2–6]). This approach allows an m-input
m-output feedback control problem to be split into
m single-input single-output problems without loss
of structural information. Each controlled output is
paired with a specific reference input to form what
is termed a “Channel”. The ICAD approach makes
direct use of the customer performance specifica-
tion for different channels to provide a framework
within which classical single-input single-output con-
trol engineering concepts can be extended to multi-
input multi-output cases involving significant levels
of cross-coupling. Traditional methods for appli-
cations involving single-input single-output (SISO)
systems can thus be applied to multi-input multi-
output (MIMO) problems. This includes the use of
open-loop system information in the form of Nyquist
and Bode plots for system analysis and design, along
with simple measures of robustness, such as gain and
phase margins.

It must be emphasized that the ICAD approach
is not, itself, a design method but should be viewed
as a framework through which useful insight may be
gained about the dynamics of the plant and the char-
acteristics of the complete controlled system. Perfor-
mance can be assessed for any chosen form of linear
controller (which may be designed using any suit-
able approach), and limitations of a design can be
investigated. Thus, compared with some other ap-
proaches to multivariable control, it can be based on
traditional analysis and design methods familiar to
all control system designers. Each channel has its
own customer-defined performance specifications and
these may be expressed in a simple way in terms of
SISO requirements.

2 The coupled-tanks system

The two-input coupled-tanks laboratory system of
Figure 1 consists of a container (of volume 6 liters),
with a central partition which divides it into two sep-
arate tanks. Coupling between these tanks is pro-
vided by a number of holes of various diameters po-
sitioned near the base of the partition. The strength

of coupling may be adjusted through the insertion of
plugs into one or more of these holes. The system
is equipped with a drain tap which is under man-
ual control and this allows the output flow rate from
one of the tanks to be adjusted. Both tanks have
inflows from electrically driven variable-speed pumps
and are equipped with sensors that can detect the
level of liquid and provide a proportional electrical
output voltage.

The hardware is based around a single-input com-
mercial product intended for teaching applications
(TecQuipment Ltd) [1]. This had a flow input only
to Tank 1 and was modified at the University of
Glasgow through the addition of the second pump to
provide the inflow to Tank 2. The original resistive
level sensors have been replaced using more accurate
differential-pressure based depth sensors.

The derivation of a detailed nonlinear model of
the system may be found in [7] and in a more recent
conference paper [8] which also includes a very brief
account of the application of the ICAD approach to
the design of a control system for this process.

The model is based on the application of the prin-
ciple of conservation of mass to the liquid within each
tank. Bernoulli’s equations provide the basis for de-
termining the flow from one tank to the other and
from the second tank to the external environment.
This leads to the following pair of equations, involv-
ing variables shown in Figure 1:

A1
dH1
dt

= Qi1 − Cd1a1
√
2g(H1 − H2) (1)

A2
dH2
dt

= Qi2 + Cd1a1
√
2g(H1 − H2) − (2)

Cd2a2
√
2g(H2 − H3)

These equations describe the dynamics of the cou-
pled tanks system, in nonlinear form, for all cases for
which the level in Tank 2 is below that in Tank 1.
It is, of course, possible to derive a similar set of
nonlinear equations to describe the system for cases
involving a liquid level in Tank 2 which is greater
than the level in Tank 1.

Parameter values for the laboratory system are as
follows:

Cross-sectional area of tanks, A1 = A2 = 9.7 ×
10−3 m2.

Cross-sectional area of the orifice between Tank 1
and Tank 2, a1 = 3.956 × 10−5 m2.

Cross-sectional area of the orifice representing the
outlet drain of Tank 2, a2 = 3.85 × 10−5 m2.

Height of the outlet drain above the base of
Tank 2, H3 = 0.03 m.

Gravitational constant, g = 9.81 ms−2.
Maximum flow rates for inputs to Tank 1 and

Tank 2:
Qi1 max = Qi2 max = 5 × 10−5 m3 s−1.

122



Acta Polytechnica Vol. 52 No. 4/2012

The minimum flow rates for these inputs are zero
since the pumps are not reversible and thus negative
inputs are not possible.

Maximum levels of liquid in Tank 1 and Tank 2:
H1 max = H2 max = 0.3 m.

The minimum level possible in each tank is 0.03 m
which corresponds to the height of the outlet drain.

Values of the discharge coefficients Cd1 and Cd2
have to be determined empirically and the values ob-
tained depend on the system operating point. The
value used for Cd1 in the design calculations was 0.63
and the value used for Cd2 was 0.58.

In addition to the above parameters, electrical
signals in the system are related to the variables of
the model (as described by Equations (1) and (2))
through the following two parameters:

Pump flow-rate calibration constant, Gp = 7.2 ×
10−6 m3s−1V−1.

Liquid depth sensor calibration constant, Gd =
33.33 Vm−1.

It should be noted that no dynamic representa-
tion is included for the two pumps and the associ-
ated electrical drives as it was found, through hard-
ware testing, that they show a very fast response to
changes of electrical input compared with the level
changes within the tanks themselves. Time constants
that are associated with the pumps were therefore ne-
glected for the purposes of the control system design.

For the preliminary stages of the design it is also
appropriate to consider a linearised model, within
which the variables represent small variations of sys-
tem variables about steady state values.

h̃1(t) = H̄1 − H1(t) (3)
h̃2(t) = H̄2 − H2(t) (4)
qi1(t) = Q̄i1 − Qi1(t) (5)
qi2(t) = Q̄i2 − Qi2(t) (6)
q23(t) = Q̄23 − Q23(t) (7)

In Equations (3)–(7) the variables that have a hori-
zontal bar above them denote values at the chosen op-
erating point, which is normally defined by a steady-
state condition.

If Equations (1) and (2) are re-arranged in the
standard state-space form, we get a pair of nonlinear
equations:

dH1
dt

= f1(H1, H2, Qi1) (8)

dH2
dt

= f2(H1, H2, H3, Qi2) (9)

Then, since the level H3 may be assumed con-
stant, linearisation produces the standard linear state
space model:⎡

⎢⎣
dh̃1
dt
dh̃2
dt

⎤
⎥⎦ =

⎡
⎢⎣

∂f1
∂H1

∂f1
∂H2

∂f2
∂H1

∂f2
∂H2

⎤
⎥⎦
[
h̃1

h̃2

]
+ (10)

⎡
⎢⎣

∂f1
∂Q1

∂f1
∂Q2

∂f2
∂Q1

∂f2
∂Q2

⎤
⎥⎦
[
qi1

qi2

]

In Equation (10) all the partial derivatives must be
evaluated at the operating point (H̄1, H̄2, Q̄i1, Q̄i2).

The resulting linearised equation, after evaluation
of the partial derivatives, has the form:

[
˙̃
h1
˙̃
h2

]
=

⎡
⎢⎣
−α1
A1

α1
A1

α1
A2

−(α1 + α2)
A2

⎤
⎥⎦
[
h̃1

h̃2

]
+ (11)

⎡
⎢⎣

1

A1
0

0
1

A2

⎤
⎥⎦
[
qi1

qi2

]

where

α1 =
Cd1a1

2

√
2g

H̄1 − H̄2
(12)

and

α2 =
Cd2a2

2

√
2g

H̄2 − H3
(13)

The individual block transfer functions that de-
scribe the plant in Figure 1 may then be derived from
Equation (11) and are as follows:

g11(s) =

(α1+α2)
α1α2

[
1 + s A2

(α1+α2)

]
1 +

(A1α1+A1α2+A2α1)
α1α2

s + A1A2
α1α2

s2
(14)

g21(s) =
1
α1

1 +
(A1α1+A1α2+A2α1)

α1α2
s + A1A2

α1α2
s2

(15)

g12(s) =
1
α2

1 +
(A1α1+A1α2+A2α1)

α1α2
s + A1A2

α1α2
s2

(16)

g22(s) =
1
α2

(1 + sA1
α1

)

1 +
(A1α1+A1α2+A2α1)

α1α2
s + A1A2

α1α2
s2

(17)

3 An outline of the ICAD
approach

A linear time-invariant MIMO plant may be mod-
elled using a transfer function matrix G. If a control
matrix K is positioned in the forward path in cas-
cade with the plant transfer function matrix G and
immediately before it, a feedback loop can be created
around the combined system described by the prod-
uct KG. The essential feature of the ICAD approach
is that loops are considered individually, by opening
one loop while all other loops remain closed.

Details of the ICAD methodology and applica-
tions that have been considered previously may be
found in papers by Leithead and O’Reilly (e.g. [2–4]

123



Acta Polytechnica Vol. 52 No. 4/2012

and [5]), who were responsible for the initial develop-
ment of the approach. A bibliography of published
papers and reports relating to ICAD methods has
been made available by Kocijan [6].

The ICAD methodology allows a controller to be
assessed in a very direct fashion, for a given plant and
given design specifications, in terms of performance
and in terms of compromises and possible trade-offs.
The design goals typically may involve:
1. Steady state response
2. Transient response
3. Disturbance rejection
4. Closed-loop stability
5. Robustness to changes in plant characteristics
6. Protection of actuators from high-frequency sig-

nals that might lead to excessive wear
In the ICAD approach the significance of the ‘struc-
ture’ of the plant in translating the given MIMO sys-
tem into the equivalent set of channels is given special
emphasis [2].

Figure 2: Block diagram of a general two-input two-
output closed-loop system of the type being consid-
ered in this application. (Adapted from a diagram
in [2])

As is clear from the plant model, the coupled-
tanks application described in this paper involves a
two-input two-output system with feedback involv-
ing two channels. Figure 2 is a block diagram that
illustrates the type of system being investigated. If
we consider the forward signal transmission from the
reference signal r1 to the associated output y1, it may
be seen that the signal follows two pathways. One
path involves a direct link through the block g11 and
the other is through the blocks g21, g12 and a block
involving k2 and its associated feedback loop through
g22. This diagram may be simplified to give the struc-
ture shown in Figure 3 for the Channel C1. From
considerations of symmetry, the Channel C2 may be
handled in the same way to produce the simplified
block diagram of Figure 4.

Figure 3: Block diagram for Channel 1. (Adapted
from a diagram in [2])

These block diagrams can be used to show that,
ignoring the disturbance signal, each channel can be
described using a single-input single-output transfer
function:

C1 = k1g11(1 − γh2) (18)
and

C2 = k2g22(1 − γh1) (19)
where

γ =
g12g21
g11g22

(20)

h2 =
k2g22

1 + k2g22
(21)

and

h1 =
k1g11

1 + k1g11
(22)

Figure 4: Block diagram for Channel 2. (Adapted
from a diagram in [2])

In Equations (18)–(22) and in Figures 3 and 4, the
effects of coupling are represented as additive distur-
bance terms at the outputs of each channel and this
does not involve any loss of information. However, it
must be emphasised that ICAD is not a single-loop
design method since loop interactions are preserved.

It can be shown (e.g., [2, 4]) that, for robustness
to parameter uncertainties of the closed-loop system
stability, the Nyquist plots of (1−γh1) and (1−γh2)
must not lie close to the origin of the polar plane at
frequencies near to or below the open-loop gain cross-
over frequency. Hence, if the corresponding plots of
γh1(jω) or γh2(jω) come close to the point (1, 0) in

124



Acta Polytechnica Vol. 52 No. 4/2012

the polar plane the conventional SISO gain margins
for the effective transfer functions of C1 and C2 do
not provide robust measures of stability. In such a
case it may not be appropriate to attempt to use the
ICAD approach unless some form of pre-compensator
is introduced to modify the plant characteristics in an
appropriate way [5].

It can also be stated [4] that the quantities h1(jω)
and h2(jω) have magnitude values that are generally
close to one below the gain crossover frequency, and
the quantity γ(jω), which is termed the multivariable
structure function, provides a measure of the strength
of any inter-loop coupling in the system and can in-
dicate whether or not this is benign. For the case of a
two-input two-output system there is only one multi-
variable structure function. However, in general, for
systems having a larger number of input-output pairs
there will be more than one structure function.

In all cases it can be stated that when a mul-
tivariable structure function is small the interaction
effects are small. In the two-input two-output case,
if the multivariable structure function is small over
the complete frequency range of interest, the two
channels behave, more or less, as two independent
loops. On the other hand, if the multivariable struc-
ture function is shown to have a large magnitude, at
a frequency within the range that is important for
the application being considered, the loop interac-
tions become significant [2].

If the multivariable structure function has an ap-
propriate form, as discussed above, frequency re-
sponse information for each channel can be used in
the analysis of the nominal system in exactly the
same way as for the analysis of a conventional feed-
back loop in a SISO control system application. How-
ever, the multivariable structure function provides
additional information about potential interactions
and the stability robustness of the closed-loop sys-
tem.

It is important to note that, for successful applica-
tion of the ICAD approach to a two-channel system,
the closed-loop bandwidth specification for one chan-
nel must not be too similar to the equivalent specifi-
cation for the second channel. If this is not the case
the problem of the transfer function of one channel
depending on the controller of the other channel be-
comes a significant obstacle in the processes of anal-
ysis and design [2].

4 Design of the controller for
the coupled-tanks system

using ICAD

For the coupled-tanks system, it may be shown that
the multivariable structure function is given by:

γ(s) =
g12g21
g11g22

=
α2

α1+α2

(1 + sA1
α1

)(1 + s A2
α1+α2

)
(23)

The expressions for h1(s) and h2(s) for this sys-
tem may also be derived directly, from Equations (21)
and (22).

4.1 Design requirements

The specifications for the closed-loop system were
based on equivalent requirements for a SISO version
of the coupled-tanks system, for which considerable
previous design experience had been accumulated.
The requirements for the two-input two-output case
were as follows:
a) Zero steady-state errors in the liquid levels in

both tanks.
b) A maximum overshoot of 30 % in liquid levels.
c) A damping factor of at least 0.7 which corre-

sponds, approximately, to a phase margin of at
least 70 degrees for both channels.

d) The gain cross-over frequency should be at least
0.05 rad/s for both channels. This value is based
on previous experience with the design of PID
controllers for the SISO case for control of the
liquid level in Tank 2 (with controlled input flow
to Tank 1 only),

e) For successful application of the ICAD design
methodology, it is important to ensure that the
polar plots of the multivariable structure func-
tions γ(jω), γh1(jω) and γh2(jω) (in terms
of the magnitude and phase values at different
frequencies over the frequency range of signifi-
cance) never approach the point (1, 0).

4.2 An outline of the design process

The requirements outlined above provide a basis for
design using the ICAD methodology for the linearised
plant model, for selected operating conditions. De-
sign, in this case, has involved the use of Matlab R©

software and has led to continuous and digital con-
trollers involving proportional plus integral controller
structures for each channel.

The design process was carried out for parameter
values of the linearised model which correspond to an
operating point in the lower half of the depth range
in both tanks (H1 = 0.115 m and H2 = 0.071 m).
This is a typical operating point for the system un-
der open-loop conditions. The values used for the two
discharge coefficients are those given in Section 2.

The first step in the design process involves es-
tablishing that the gain cross-over frequency of the
open-loop transmittance of one channel will be sig-
nificantly different from the gain cross-over frequency

125



Acta Polytechnica Vol. 52 No. 4/2012

of the other. In this case it was decided, on the basis
of physical reasoning, that the gain cross-over fre-
quency of Channel 1 should be higher than that for
Channel 2. From the design requirements, this latter
value should be chosen to be at least 0.05 rad/s, so a
value of at least 0.5 rad/s was required for the gain
crossover frequency of Channel 1.

The next step involves evaluation of the magni-
tude and phase of the multivariable structure func-
tion over the range of frequencies that are of impor-
tance for the intended application. Figure 5 is a typi-
cal plot of the multivariable structure function in po-
lar form showing the magnitude and phase of γ(jω)
for the complete range of relevant frequencies, and it
is clear that the resulting plot involves small values
of magnitude and does not come close to the (1, 0)
point. This is satisfactory for the operating point
considered but similar plots should be considered for
a range of different operating conditions.

Figure 5: Plot of the multivariable structure function
in polar form for the coupled-tanks system showing
the magnitude and phase of γ(jω) for the complete
range of relevant frequencies for one operating point

Next, it is necessary to design the controller k2(s)
since the requirements in terms of gain cross-over fre-
quency for Channel 2 are less demanding than for
Channel 1. Equation (19) shows that the equation
for Channel 2 involves the transfer function h1(s)
and the known multivariable structure function γ(s).
The first step is to assume either that h1(s) = 0 or
that h1(s) = 1 and design the controller k2(s) ini-
tially on that basis [2]. In this application it was as-
sumed that h1(s) = 0, but an initial assumption that
h1(s) = 1 would have been equally appropriate. The
transfer function for g22(s) given in Equation (17) has
a magnitude at low frequencies of 1/α2 and at high
frequencies the magnitude decreases in an approxi-
mately linear fashion at −20 dB per decade. In order
to meet the design requirement of zero steady-state

closed-loop error this suggests use of a proportional
plus integral type of controller of the form:

β1 =
(1 + β2s)

s
(24)

where β1 and β2 are constants.
This controller will produce infinite gain at zero

frequency and thus eliminate any steady state error in
the closed-loop system for this channel. The choice
of parameter values for the controller involves, ini-
tially, the selection of a gain factor β1 to give a suit-
able gain cross-over frequency which is at least the
required minimum of 0.05 rad/s. The integral action

is then considered and the frequency ω =
1

β2
is cho-

sen to be sufficiently smaller than the gain cross-over
frequency to ensure that the overall phase margin is
not influenced to any large extent. Application of
this procedure gives the following controller:

k2(s) = 0.56
(1 + 8.929s)

s
(25)

Through the use of simulation, closed-loop step re-
sponses can be examined (usually on the basis of lin-
earised models) and further adjustments can be made
in the values for these controller parameters if this is
judged to be necessary.

After obtaining that first approximation to k2(s)
an initial single-input single-output design can be
carried out for the controller k1(s) on the basis of
h2(s), which is now available (from Equation (21)).
The procedure followed is essentially the same as for
Channel 1 but with the higher value of gain crossover
frequency that is required for this channel. The pro-
portional plus integral controller resulting initially
from this process has the form:

k1(s) = 4.676
(1 + 5.988s)

s
(26)

Having found an initial k1(s), this transfer function
can then be used to determine h1(s) by substitution
into Equation (22). The resulting gain and phase
margins must then be checked and adjustments made
to k1(s), if necessary. The process may have to be
repeated once or twice. Then, using the revised con-
troller transfer function for Channel 1, the design
can be completed for Channel 2 using a similar it-
erative procedure. Final checks must then be made
on both channels to compare the gain cross-over fre-
quencies with the design specifications and check that
the gain and phase margins are all satisfactory. This
process also involves re-examination of the Nyquist
plots of the multivariable structure functions γ(jω),
γh1(jω) and γh2(jω) to ensure that none of them
approaches the point (1, 0) and thus establish that
the gain and phase margins are valid measures of ro-
bustness [2].

126



Acta Polytechnica Vol. 52 No. 4/2012

Following the application of the above procedures
the optimised controller transfer functions were as
follows:

k1(s) = 5.0
(1 + 6.2s)

s
(27)

k2(s) = 0.56
(1 + 10.0s)

s
(28)

Figure 6: Bode diagram showing magnitude (dB)
and phase (deg) for Channel 1 with the compensation
provided by the controller transfer function of Equa-
tion (27). The gain cross-over frequency is indicated
by the vertical line at frequency of about 0.8 rad/s

Figure 7: Bode diagram showing magnitude (dB)
and phase (deg) for Channel 2 with the compensation
provided by the controller transfer function of Equa-
tion (28). The gain cross-over frequency is indicated
by the vertical line at frequency of about 0.2 rad/s

Figures 6 and 7 show the open-loop Bode plots for
Channels 1 and 2, respectively, for these optimised
controllers. From these Bode plots it may be seen
that the gain crossover frequencies for Channel 1 and
Channel 2 are approximately 0.8 rad/s and 0.2 rad/s,
respectively. The corresponding phase margins are
more than the required value of 70 degrees, in both

cases. From the gain-crossover frequencies it is clear
that the speed of response for Channel 2 is likely to
be about four times slower than for Channel 1, which
is consistent with the specifications.

Discrete equivalents of these continuous con-
trollers have been found and an ICAD-based con-
trol system has been implemented with a digital con-
troller using a general-purpose computer equipped
with analogue-to-digital and digital-to-analogue con-
verters. However, all experimental results presented
in this paper are for the continuous control case
where the controllers have been implemented us-
ing a small general-purpose electronic analogue com-
puter equipped with comparators and switches that
can provide limiting integrator action, if required, to
avoid integrator saturation.

For purposes of comparison, proportional plus in-
tegral controllers have also been designed empirically
using the Ziegler-Nichols reaction curve method (see,
e.g. [13]). This well-known approach to controller
design is based on measurements obtained from sim-
ple open-loop tests on the plant. Application of this
approach gave the following controller transfer func-
tions:

k1(s) = 6.98
(1 + 3.96s)

s
(29)

k2(s) = 8.9
(1 + 3.3s)

s
(30)

It should be noted that the two controller transfer
functions found from the application of the Ziegler-
Nichols approach (Equations (29) and (30)) are very
much closer in terms of parameter values than the
two controllers found using the ICAD approach, as
given in Equations (27) and (28). This is because of
the requirement within the ICAD methodology that
the bandwidth values for the two channels should be
significantly different.

5 Results

Extensive analysis and simulation studies have been
performed using Matlab R© and Simulink R© to inves-
tigate the performance of the system, especially in
terms of interactions between the two channels and
overall robustness of the control systems.

In the case of the control systems derived us-
ing the ICAD approach the performance of the con-
trollers has also been the subject of detailed ex-
perimental investigation in the laboratory using the
coupled-tanks system hardware. Interactions be-
tween the two channels have been investigated both
by experiment and through simulation.

For the simulation studies the full nonlinear
model has been used, with parameter values as given
in Section 2.

127



Acta Polytechnica Vol. 52 No. 4/2012

5.1 Simulation results

Figure 8 shows typical simulation results for a test
in which simultaneous step changes are applied to
the reference inputs determining the required levels
in the two tanks, using the continuous controllers of
Equations (27) and (28). The resulting simulated
changes in liquid levels in Tanks 1 and 2 are shown by
the upper and lower traces respectively. In the case
of Channel 1 the step change of reference imposed is
from 199 mm to 228 mm, while for Channel 2 the
change is from 165 mm to 198 mm. This test in-
volves input flow values for Tank 1 and Tank 2 which
both reach their upper limits for this magnitude of
demanded level change.

It can be seen from these simulation results that,
although the operating point considered is signifi-
cantly different from the design point, the design re-
quirements have been satisfied and also that the re-
sponse of Tank 2 is slower than that of Tank 1, as
expected.

Figure 8: Simulation results found using the non-
linear model with the controllers designed using the
ICAD approach for a test in which simultaneous step
changes in required levels for Tank 1 and Tank 2 are
applied at time t = 100 s. The vertical axis represents
liquid level (m) while the horizontal axis represents
time (s). The level in Tank 1 is represented by the
continuous line while the dashed line represents the
level in Tank 2

Investigation, through simulation, of interactions
between the two channels have involved introducing
a step change of the desired level in one channel while
maintaining the original set level in the other.

The upper set of simulation results presented in
Figure 9a shows the level of liquid in Tank 1 fol-
lowing the application of a step change of reference
for Channel 1 at time t = 100 s, together with the
record for the level in Tank 2. In Figure 9b the lower
plot shows the liquid level in Tank 2 following the
application, at time t = 100 s, of a step change of
reference for Channel 2 while the upper trace shows
the corresponding level in Tank 1.

a)

b)

Figure 9: a) Simulated responses of levels (m) in
Tank 1 (continuous line) and in Tank 2 (dashed line)
versus time (s) when the reference level for Channel 1
is changed. The horizontal axis represents time (s).
This test involved use of the nonlinear model with
controllers designed using the ICAD approach
b) Simulated responses of levels (m) in Tank 1 (con-
tinuous line) and in Tank 2 (dashed line) when the
reference level for Channel 2 is changed. This test
involved use of the nonlinear model with controllers
designed using the ICAD approach. The horizontal
axis represents time (s)

These results show that a transient disturbance
occurs in the level of Tank 2 when the set level of
Channel 1 is changed but that a negligible transient
is found in the level of Tank 1 when the set level of
Channel 2 is altered by a similar amount. This dif-
ference is due to the different bandwidths in the two
channels.

Results of an additional simulation test are shown
in Figure 10. This involves the simultaneous appli-
cation of negative step changes of reference for both
channels at time t = 100 s. The demanded changes
lead, transiently, to a complete cut-off of input flow
for both tanks for a period of about 20 s at the time
when the reference values are changed, as can be seen
from the almost straight-line form of the negative-
going responses in that part of the record.

128



Acta Polytechnica Vol. 52 No. 4/2012

Figure 10: Simulation results showing liquid lev-
els (m) for Tank 1 (continuous line) and Tank 2
(dashed line) for large negative step changes of refer-
ence. The horizontal axis represents time (s)

The closed-loop performance of the system with
the proportional plus integral controllers designed us-
ing the Ziegler-Nichols reaction curve approach (see,
e.g. [13]) was investigated through simulation. It
was found that for small changes of the reference
inputs the two-input two-output system with these
controllers behaved very much in accordance with ex-
pectations (as shown in Figure 11).

Figure 11: Simulation results found using the nonlin-
ear model with controllers designed using the Ziegler-
Nichols approach for a test involving relatively small
changes of reference level. The vertical axis repre-
sents liquid level (m) while the horizontal axis repre-
sents time (s). The level in Tank 1 is represented by
the continuous line while the dashed line represents
the level in Tank 2

For large positive changes of reference (similar to
those applied in obtaining the results shown in Fi-
gure 8 for the ICAD design) the responses for the
control system designed using the Ziegler-Nichols ap-
proach are found to be much more oscillatory, as
shown in Figure 12. This is also the case for the
transients found for large negative reference changes,
as shown in Figure 13.

Figure 12: Simulation results found using the nonlin-
ear model with controllers designed using the Ziegler-
Nichols approach for a test similar to that of Figure 8.
The vertical axis represents liquid level (m) while
the horizontal axis represents time (s). The level in
Tank 1 is represented by the continuous line while
the dashed line represents the level in Tank 2

Figure 13: Simulation results for the nonlinear model
with controllers designed using the Ziegler-Nichols
approach for a test involving large negative step
changes of reference. The vertical axis represents
liquid level (m) while the horizontal axis represents
time (s). The level in Tank 1 is represented by the
continuous line, whereas the dashed line is the level
in Tank 2

Responses found for simulated situations involv-
ing interactions between the two tanks were also more
oscillatory in nature and varied more with operating
point than those found using the controllers designed
using the ICAD approach.

5.2 Experimental results

As implemented using operational amplifiers and the
associated passive components, the two controllers
corresponded to the transfer functions (as given in
Equations (27) and (28)) that resulted from the final
optimisation stage of the design process. These are,
of course, also the controller transfer functions used
in the simulation studies discussed in Section 5.1.

129



Acta Polytechnica Vol. 52 No. 4/2012

Experimental results for the control system, when
implemented with these controllers, are shown in
Figure 14 for the case involving two simultaneous
changes of reference. The results are almost identical
in terms of steady state performance to the simulated
results of Figure 8 for the same test conditions, and
are very similar in terms of the settling time of the
transients. As in the simulation results, the inputs
both reach their limits during the transients.

The main difference observed between the experi-
mental results of Figure 14 and the simulation results
of Figure 8 is that the transients found experimen-
tally (especially for the level in Tank 1) are more os-
cillatory than those found through simulation. Sim-
ilar findings have been obtained for equivalent tests
at other operating points, and this suggests strongly
that there are imperfections within the model of the
two-tank system. Exactly what the modelling errors
might be is not, of course, clear from the information
from these closed-loop system tests alone.

Although the results shown in Figures 8 and 14
are for one specific operating condition, comparison
of experimental and simulation results for a range
of different conditions has shown good overall agree-
ment.

Figure 14: Experimental results for conditions equiv-
alent to those of the simulation results of Figure 8.
The continuous line shows the measured liquid
level (m) in Tank 1 while the dashed line shows the
measured level in Tank 2. The horizontal axis repre-
sents time (s)

The experimental investigation of interactions be-
tween channels produced results shown in Figures 15a
and 15b, which can be seen to correspond closely to
the corresponding simulation results shown in Fig-
ures 9a and 9b.

Experimental results for a test involving simul-
taneous large negative changes of reference value for
both channels simultaneously are shown in Figure 16.
These results are very similar in character to the sim-
ulated results of Figure 10. As in the simulation, the
controlled flows for Tank 1 and Tank 2 reach limiting
values (zero) during the transient period.

a)

b)

Figure 15: a) Experimental results, equivalent to the
simulation results of Figure 9a, involving application
of a step change of reference for Channel 1 while the
reference input for Channel 2 is held constant. The
record for liquid depth (m) in Tank 1 is shown by a
continuous line and for Tank 2 by the dashed line.
The horizontal axis represents time (s)
b) Experimental results, equivalent to the simulation
results of Figure 9, involving application of a step
change of reference for Channel 2 while the reference
for Channel 1 is held constant. The record for liquid
depth (m) in Tank 1 is shown by a continuous line
and for Tank 2 by the dashed line. The horizontal
axis represents time (s)

Figure 16: Experimental results showing liquid lev-
els (m) for Tank 1 (continuous line) and Tank 2
(dashed line) for large negative step changes of refer-
ence. The horizontal axis represents time (s)

130



Acta Polytechnica Vol. 52 No. 4/2012

Figure 17: Experimental results for a test involving
the addition of a small volume of water to Tank 1
(continuous trace) and to Tank 2 (dashed line) in
turn. The horizontal axis represents time (s)

a)

b)

Figure 18: a) Results of a simulated test involving the
addition of a small volume of water to Tank 1 (con-
tinuous line). The level in Tank 2 is shown by the
dashed line. The horizontal axis represents time (s)
b) Results of a simulated test involving the addition
of a small volume of water to Tank 2 (dashed line).
The level in Tank 1 is shown by the continuous line.
The horizontal axis represents time (s)

The behavior of the control system when sub-
jected to external disturbances is also of great practi-
cal importance. Figure 17 shows some typical experi-
mental results where external disturbances have been
introduced by adding, in as short a period of time as
possible, a disturbance in the form of a measured vol-
ume of water to each of the tanks in turn, with feed-
back control loops applied. The upper plot shows the
level in Tank 1 for a reference input of 227 mm, while
the lower plot shows the level in Tank 2 for a refer-
ence input of 198 mm. The disturbance inputs are
applied by the rapid addition of water, from a beaker,
to Tank 1 and addition of a similar volume to Tank 2
at about time t = 225 s. The effects of each of these
disturbances on each channel are clearly visible in
these records.

The results show the distinctive actions of the
two channels in countering the effects of the distur-
bance inputs. The levels for both channels return
to their set values after the disturbances, with tran-
sients of acceptable magnitude and duration. As with
the tests involving changes of reference, it is clear (as
would be expected) that the speed of response to dis-
turbances is influenced by the choice of bandwidths
for the two channels.

Similar results have been obtained through simu-
lation, but quantitative comparisons are difficult for
this type of test because it is hard to reproduce the
detailed time-course of the disturbance input within
the simulation. Figures 18a and 18b show typical
simulation results for disturbance tests which are
approximately equivalent to the experiments of Fi-
gure 17. The experimental and simulation results
are seen to be qualitatively consistent.

5.3 Results of additional
simulation-based investigations

One interesting practical finding, which has been
fully supported by simulation results, is that control
of the level in Tank 1 can only be achieved for condi-
tions in which the demanded level in Tank 1 is equal
to or greater than that in Tank 2. This is understand-
able in terms of the physics of the system since Tank 1
has only one outlet (to the second tank), whereas
Tank 2 has two outlets (one to the first tank and
the second through the drain pipe). If the demanded
value of H2 is greater than the demanded level of
H1, liquid will flow into Tank 1 from Tank 2 as well
as from the input but no liquid will flow out. Since
the input flow cannot become negative, satisfactory
control of the level in Tank 1 is impossible in these
conditions. Typical experimental results are shown
in Figure 19 and these demonstrate, in this specific
case, that a demanded level of 0.105 m in Tank 1
cannot be achieved in combination with a larger de-
manded level (in this case 0.131 m) in Tank 2. What

131



Acta Polytechnica Vol. 52 No. 4/2012

happens in practice is that the final levels in both
tanks become equal to the final demanded level in
Tank 2. Figure 20 shows results obtained from simu-
lation for a very similar set of conditions. Simulation
investigations have confirmed that the addition of a
drain pipe to the first tank eliminates this problem
and would allow independent control of the two levels
for any combination of reference values.

Figure 19: Levels (m) found in Tank 1 (continu-
ous line) and Tank 2 (dashed line) for a demanded
change of the reference for Channel 1 from 0.082 m to
0.105 m and for Channel 2 from 0.048 m to 0.131 m.
The horizontal axis represents time (s)

Figure 20: Simulated results for a test which involves
conditions which are very similar to those of the ex-
periment of Figure 16. In this case, following the step
changes of reference inputs, the demanded level in
Tank 2 is again greater (0.131 m) than the demanded
level in Tank 1 (105 mm). Here the continuous line
again represents the liquid level in Tank 1 and the
dashed line the level in Tank 2. The horizontal axis
represents time (s)

Another area for further investigation through
simulation relates to tests of robustness to changes
within the plant. These have involved, for exam-
ple, the introduction of sudden changes of the cross-
sectional area of the outlet drain orifice, or of the

cross-sectional area of the orifice responsible for the
coupling between Tank 1 and Tank 2.

Experimental testing is straightforward in the
case of the outlet from Tank 2, for which the drain
tap can be used, but investigation of changes of the
inter-tank orifice area presents practical difficulties
since the variation of cross-sectional area normally
requires the removal or insertion of a rubber bung for
one of the three orifices in the partition that separates
the two tanks. Even partial closing and re-opening
of the outlet drain tap is difficult to achieve manu-
ally in a precise and repeatable fashion. Simulation
can therefore be used to advantage to investigate the
performance of the system for this type of change.

Figure 21: Results of an experiment involving partial
closure and re-opening of the drain tap from Tank 2.
The action of closure occurs at about t = 60 s and
the re-opening takes place at about t = 150 s. The
continuous line shows the level (m) in Tank 1 and the
dashed line represents the level (m) in Tank 2. The
horizontal axis represents time (s)

Figure 22: Results from a simulation involving in-
stantaneous changes of the cross-sectional area of the
orifice representing the drain tap from Tank 2. Par-
tial closure occurs at t = 100 s and the re-opening
takes place at t = 300 s. The continuous line shows
the level (m) in Tank 1 and the dashed line represents
the level (m) in Tank 2

132



Acta Polytechnica Vol. 52 No. 4/2012

Inevitably, the results of simulation tests differ
slightly from tests carried out on the real system.
Typical experimental results showing the effects of
changes of drain tap opening and changing the cross-
sectional area of the inter-tank orifice are given in Fi-
gure 21. Results from simulation, for instantaneous
changes of the cross-sectional area of the orifice rep-
resenting the drain-tap and outlet pipe, are shown
in Figure 22 and these are broadly similar to the
experimental findings of Figure 21. Both the sim-
ulation results and the experimental findings confirm
that the robustness properties of the two-input two-
output control system, as displayed in this test, are
satisfactory. Transients for Tank 2 are significantly
larger than for Tank 1 as could be expected from the
bandwidths of the two channels.

6 Discussion and conclusions

The work reported in this paper illustrates the use of
the ICAD analysis and design approach for a prac-
tical application that involves significant nonlineari-
ties in terms both of control input limits and inherent
nonlinearity of the plant model. Analysis of the two-
input two-output system within the ICAD framework
provides helpful insight which can be used in the de-
sign and implementation of the control system.

Comparisons between simulation and experimen-
tal results also provide useful information about the
system performance and about limitations of the
plant model. A previous journal paper [9] report-
ing the application of the ICAD methodology to the
same system was concerned with issues of controller
parameter tuning and did not consider the response
to disturbances or address robustness issues.

It can be concluded that the coupled tanks equip-
ment provides a useful test-bed for investigation of is-
sues of nonlinear system modeling, multivariable con-
trol system design and implementation. The avail-
ability of a comprehensive nonlinear model of the
system, together with linearised representations ap-
propriate for control system design, also makes this
system suitable for the teaching of practical aspects
of multi-input multi-output control system analysis
and design using ICAD or other approaches. It is be-
lieved that the work reported in this paper could pro-
vide the basis for a useful case-study (most probably
for use at postgraduate level) on the ICAD methodol-
ogy. This could also illustrate the benefits of bringing
together simulation and experimental testing within
the processes of control system design and implemen-
tation.

Differences between simulation results and exper-
imental results are believed to relate mainly to limita-
tions in the representation of the plant and especially
the relationship used to describe the output flow from
the second tank within the nonlinear model. This as-

pect of the coupled-tanks system model has been dis-
cussed in previous model validation studies for this
system (e.g. [7,10,11]) and is the subject of ongoing
investigations.

Simulation results show that broadly satisfac-
torily results can also be obtained for this plant
with proportional plus integral controllers designed
empirically using the Ziegler-Nichols reaction curve
method. However, results found using that approach
have given responses, for the types of tests described
in this paper, that tend to be more oscillatory than
those found using the ICAD methodology, and indi-
cate some issues of closed-loop system robustness to
changes of operating point, variation of the magni-
tude of reference changes and the magnitude of dis-
turbances.

Claims that the ICAD approach can provide en-
hanced performance compared with other available
design methods would be inappropriate on the ba-
sis of the limited results presented in this single ap-
plication. However, it is believed that the ICAD
methodology brings more physical insight to the de-
sign process for multi-input multi-output systems,
and it must also be remembered that this approach
is not restricted to one single form of controller.

Acknowledgement

This paper is a modified and extended version
of a paper [8] presented at EUROSIM 2010 (the 7th

EUROSIM Congress on Modelling and Simulation),
which was organized by staff of the Department of
Computer Science and Engineering, Czech Techni-
cal University in Prague (CTU). The Congress took
place in Prague during the period 6-10 September,
2010.

The author must thank Jin Lin Chiang who, dur-
ing undergraduate project work at the University of
Glasgow [12], carried out experiments from which
some of the results in this paper have been derived.

The author also wishes to thank Professor John
O’Reilly of the University of Glasgow for many valu-
able discussions about ICAD methods.

References

[1] Wellstead, P. E.: Coupled Tanks Apparatus:
Manual. TecQuipment Ltd., UK, 1981.

[2] O’Reilly, J., Leithead, W. E.: Multivariable con-
trol by ‘individual channel design’. International
Journal of Control, 54, 1991, p. 1–46.

[3] Leithead, W. E., O’Reilly, J.: Performance is-
sues in the individual channel design of 2-input
2-output systems. 1. Structural issues. Interna-
tional Journal of Control, 54, 1991, p. 47–82.

133



Acta Polytechnica Vol. 52 No. 4/2012

[4] Leithead, W. E., O’Reilly, J.: Performance is-
sues in the Individual Channel Design of 2-input
2-output systems 2. Robustness issues. Interna-
tional Journal of Control, 55, 1992, p. 3–47.

[5] O’Reilly, J., Leithead, W. E.: Frequency-domain
approaches to multivariable feedback control
system design: An assessment by individual
channel design for 2-input 2-output systems.
Control Theory and Advanced Technology, 10,
1995, p. 1 913–1940.

[6] Kocijan, J.: Bibliography on Individual Channel
Analysis and Design Framework.
http://dsc.ijs.si/jus.kocijan/ICAD. (Accessed suc-
cessfully 1. 11. 2011).

[7] Gong, M., Murray-Smith, D. J: A practical exer-
cise in simulation model validation, Mathemati-
cal and Computer Modelling of Dynamical Sys-
tems, 4, 1998, p. 100–117.

[8] Murray-Smith, D. J.: The Individual Channel
Analysis and Design method applied to control
of a coupled-tanks system: simulation and ex-
perimental results. In “Proceedings of the 7th
EUROSIM Congress on Modelling and Simula-
tion. (Editors: M. Šnorek, Z. Buk,, M. Čepek,
and J. Drchal), Prague : Department of Com-
puter Science and Engineering, Czech Techni-

cal University in Prague (CTU), 2010. (ISBN
9788001045893).

[9] Murray-Smith, D. J., Kocijan, J., Gong, M.: A
signal convolution method for estimation of con-
troller parameter sensitivity functions for tun-
ing of feedback control systems by an itera-
tive process. Control Eng. Practice, 11, 2003,
p. 1 087–1 094.

[10] Tan, K. C., Li, Y., Murray-Smith, D. J. ,Shar-
man, K. C.: System identification and lineari-
sation using genetic algorithms with simulated
annealing. In Proc. Conf. on Genetic Algorithms
in Eng. Syst.: Innovations and Applications,
12–14 September 1995, (IEE Conference Publi-
cation No. 414), London : IEE, 1995, p. 164–189.

[11] Gray, G. J., Murray-Smith, D. J., Li, Y.,
Sharman, K. C., Weinbrenner, T.: Nonlin-
ear model structure identification using genetic
programming, Control Eng. Practice, 6, 1998,
p. 1 341–1 352.

[12] Chiang, J. L.: Control System for a Two-Input
Two-Output Liquid Level System. Project Re-
port 94-179, University of Glasgow, Department
of Electronics and Electrical Engineering, 1994.

[13] Golten, J., Verwer, A.: Control System Design
and Simulation, McGraw-Hill, London, 1991.

134