Access to this full-text is provided by Springer Nature.
Content available from Scientific Reports
This content is subject to copyright. Terms and conditions apply.
1
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports
Parameter identication
of a delayed innite‑dimensional
heat‑exchanger process based
on relay feedback and root loci
analysis
Libor Pekař 1,2*, Mengjie Song 3, Subhransu Padhee 4, Petr Dostálek 1 &
František Zezulka 2
The focus of this contribution is twofold. The rst part aims at the rigorous and complete analysis of
pole loci of a simple delayed model, the characteristic function of which is represented by a quasi‑
polynomial with a non‑delay and a delay parameter. The derived spectrum constitutes an innite
set, making it a suitable and simple‑enough representative of even high‑order process dynamics.
The second part intends to apply the simple innite‑dimensional model for relay‑based parameter
identication of a more complex model of a heating–cooling process with heat exchangers. Processes
of this type and construction are widely used in industry. The identication procedure has two
substantial steps. The rst one adopts the simple model with a low computational eort using the
saturated relay that provides a more accurate estimation than the standard on/o test. Then, this
result is transformed to the estimation of the initial characteristic equation parameters of the complex
innite‑dimensional heat‑exchanger model using the exact dominant‑pole‑loci assignment. The
benet of this technique is that multiple model parameters can be estimated under a single relay
test. The second step attempts to estimate the remaining model parameters by various numerical
optimization techniques and also to enhance all model parameters via the Autotune Variation Plus
relay experiment for comparison. Although the obtained unordinary time and frequency domain
responses may yield satisfactory results for control tasks, the identied model parameters may not
reect the actual values of process physical quantities.
It is a well-known fact that dozens of industrial processes, including chemical ones, as well as social, economic,
and other everyday systems are aected by latencies and delays1–6. Delays appear mainly due to mass, energy, and
data transportation in the process and network interconnections, and their existence is closely related to distrib-
uted parameter systems. In modern discrete-time control systems, delays also arise from the human–machine
interaction and signal sampling and processing7. As complex systems include internal feedback loops, internal
delays must be considered along with the input–output ones; nevertheless, internal delays are oen ignored
when process modeling. However, such an approach can be unreasoning as the solution of partial dierential
equations (PDEs)—representing the reign of many industrial process models—oen results in functions with
lumped and distributed delays 8–10.
On the other hand, time-delay models (TDMs) may be very good estimators of some systems and processes
dynamics, even if any signicant physical delay is not supposed to appear in the process. TDMs have the form of
functional dierential equations, or more specically, delay dierential equations (DDEs), instead of PDEs. Even
a simple TDM can express the dynamics of a high-order non-delay model11,12 with sucient accuracy for control
OPEN
1Department of Automation and Control Engineering, Faculty of Applied Informatics, Tomas Bata University
in Zlín, nám. T. G. Masaryka 5555, 760 01 Zlín, Czech Republic. 2Department of Technical Studies, College of
Polytechnics Jihlava, Tolstého 1556/16, 586 01 Jihlava, Czech Republic. 3Department of Energy and Power
Engineering, School of Mechanical Engineering, Beijing Institute of Technology, Engine East Building 125,
Beijing 100081, China. 4Department of Electrical and Electronics Engineering, Sambalpur University Institute of
Information Technology, Burla, Sambalpur 769018, India. *email: pekar@utb.cz
Content courtesy of Springer Nature, terms of use apply. Rights reserved
2
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
design. However, these models are innite-dimensional because of the transcendental form of the characteristic
equation (CE)13. All innitely many solutions of the CE constitute the TDM spectrum of characteristic values
(or poles). e pole loci most signicantly determine the dynamic and stability properties of the model14. e
innite nature of the TDM spectrum yields its advantages and disadvantages when estimating the actual system
dynamics. e dominant (i.e., usually the rightmost) subset of poles can match the pole loci of a high order
system; however, one must be careful about other (uncontrolled) TDM poles, especially of high-frequency ones.
Various methods and techniques for the TDM spectrum analysis and its pole loci estimation exist; see a survey
by Pekař and Gao15. We do let provide the reader with just a few. If TDMs have the so-called commensurate delays
solely, pole loci can be determined analytically via the Lambert W function16. However, this is the only exact
method besides the direct solution of the CE via the analysis of the distribution of the roots of the corresponding
CE in the frequency domain, see, e.g., the work by Amrane etal.17. e family of numerical methods includes
a wide variety of approaches and techniques that are based on, e.g., the mapping of real and imaginary parts of
CEs solutions18, bifurcation analysis of DDEs19, full discretization of TDMs3, continuation property of TDMs
pole loci7 or on structural properties of a class of functional Vandermonde matrices20.
TDMs pole loci analysis is closely related to the inverse problem of the (partial) spectrum assignment, i.e.,
the determination of the model parameter (or even delay) values so that a subset of the characteristic values
match the prescribed positions while other poles are suciently away from the chosen loci. ere, again, exist
several computational methods as well as principles of how to select the desired loci. e direct root assignment
is essentially the most straightforward technique that gives rise to the solution of a set of algebraic (linear or
nonlinear) equations21, the dimension of which is given by the number of assigned roots, their multiplicity and
complexity. Its computational simplicity is, however, ransomed by a danger of a possible existence of model poles
located right from the prescribed ones, which yields a problem of the poles’ dominancy. Only a few tools ensure
that the desired TDMs poles are dominant, e.g., a modied Nyquist stability criterion can be applied22, yet it is
based on a graphical trial-and-reset procedure. Recently, several ad-hoc results for single17 and multiple real20,23
prescribed poles or even single a complex conjugate pair24 guarantying their dominancy have been derived;
however, they usually levy large computational burden. Alternatively, the root dominance can be a posteriori
checked using the argument principle (i.e., the Mikhailov curve based) approach25 or via the solution of a special
convolution integral26, which requires an advanced mathematical eort as well. Whenever the direct assignment
is not satisfactory, a numerical spectrum optimization can be made, e.g., by the quasi-continuous shiing of the
roots27,28 or using its combination with the minimization of a specic tness function reecting the remaining
spectrum, robustness issues, etc.29–31. Unfortunately, a non-convex optimization problem must be solved in many
cases, see, e.g.,32 and references therein.
e use of relay in the feedback control system represents a favorite system parameters’ identication and
automatic controller tuning framework that have received a great deal of attention since the pioneering work
by Åstrom and Hägglund33, where the ideal on/o relay was used to generate sustained (ultimate) oscillations.
is parameter estimation framework enables to prevent the process output from driing too far away from the
reference signal (setpoint), which is required for many industrial processes. During recent decades, a multitude
of derived techniques and methods have been developed34,35 that have found a great favor of practitioners, espe-
cially in chemical and process engineering36–41.
ree families of approaches to evaluate unknown model parameters34 exist. Namely, using a describing func-
tion (DF) represents the most common approach33,38,42. Roughly speaking, this function is a linear approximation
(usually based on the Fourier series expansion) of the nonlinear relay behavior. Second, the curve tting approach
attempts to t the feedback response in the time domain based on an analytic formulation of the response43–46.
As third, the frequency tting does the same yet in the frequency domain, which corresponds to the seeking of
multiple frequency points, besides the ultimate case47–49.
It is known that a suciently accurate process model can reduce errors in controller tuning50. e ultimate
gains obtained from the standard (on/o) relay feedback oscillation amplitudes can have errors of over 15%51.
Besides, Jeon etal.52 pointed out that model parameters obtained from the sustained relay oscillations can be
insucient if there is a mismatch in the model order and process dynamics, which gives rise to the need for more
complex models. However, as the original method suers from signicant errors in model parameters’ estima-
tion and only one (critical) point of the frequency characteristics (i.e., two model parameters) can be obtained
from the test, researchers have developed methods to t the parameters more precisely and/or to search for more
frequency response points under one or more relay tests.
Regarding the former group of methods, improved accuracy can be obtained by compensating for the phase
lag caused by the relay module53—which is suitable for higher-order processes or those with large input–output
delay, by using a biased relay54, a relay with two-band hysteresis to reduce the oscillation frequencies55 or relays
with multiple switching56,57 that prune the relay oscillation harmonics and the eect of the input nonlinearity.
Unfortunately, the reduction of the oscillation frequencies increases the experimental times. Other techniques,
e.g., attempt to obtain as sinus-like relay output as possible by using a relay with saturation35,58,59, to reduce the
eect of noises and disturbances50,60, to apply the so-called area methods that integrate specically modied
time responses61 or to use asymmetrical limit cycle62 or a shape factor63. However, most methods suer from
the sensitivity to plant–model structural mismatch58.
e simplest approach how to obtain more frequency points is to perform more than one relay test using
an additional integrator64,65, via the parasitic relay66, a relay with hysteresis67 or the biased relay42 that alter the
DF68. Li etal.69 proposed a well-applicable method called the Autotune Variation (ATV) that introduces an arti-
cial delay in an additional test following the standard relay experiment. e technique was further improved
by Kim70, Marchetti and Scali38, and Scali etal.71. ese methods are popular among practitioners due to their
computational simplicity. However, multiple experiments may be time consumptive; therefore, researchers have
attempted to gain information about more frequency points or other system dynamic features under a single relay
Content courtesy of Springer Nature, terms of use apply. Rights reserved
3
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
test. Even a purposefully induced asymmetry can be used to determine additional frequency response points
without performing additional relay feedback tests72. Nevertheless, the given asymmetry may yield the termina-
tion of oscillations, which fails the relay test. Numerous other techniques exist, such as the shiing method51,73,74,
the use of relay transient48,54,75 or the computation of weighting moments47,61.
Most of the linear process models used in literature when applying some of the relay-based identication
experiments include input–output delays within First-Order plus Dead Time (FOPDT) or Second-Order plus
Dead Time SOPDT models, either stable44–46,50,55,63,69,76 or unstable43,44,51,77–81 ones. e models are represented
simply by a series of a delay-free low-order submodel and the delay element. However, FOPDT models can suer
from poor performances for some low-order processes with fast parasitic dynamics61. Besides, the delay value is
hardly ever considered as signicant or dominant, even if such an assumption is oen unreasonable in practice82.
Surprisingly, not many results concern internal delays incorporated in TDMs. Parameter estimate of TDMs is
more complicated than parameter estimate of standard models47. e pioneering work in this eld was presented
by Vyhlídal and Zítek12. e authors comprised a rst-order-derivative model with one internal delay parameter
within the standard relay test33. We do let call this TDM the Simple First-Order one (SFOTDM) for further pres-
entation. e identied SFOTDM was further used for internal model control design. ese results were taken
as a starting point for parameter estimation of a TDM of the continuous stirred-tank reactor. e parameter
values are further enhanced by solving nonlinear objective functions governed by the dierence of the model and
measured responses in the time domain via an—optimization algorithm. Nevertheless, a relay was not used in
the proposed design; the authors only referred to the awkward need to determine the input–output delay value
before the relay test. Pekař and Prokop83 compared the use of the saturated relay42 and the limit-cycle evaluation
using the exponential decaying followed by the discrete-time Fourier transform54,66. e authors considered a
rst-order-derivative TDM that included three non-delay parameters, one internal delay parameter, and the dead
time as the model of a circuit system with heat exchangers84. An articial delay was used for the additional relay
test. In47, the method of moments64 was applied to the serial combination of the SFOTDM and a low-pass lter
represented by the delay-free rst-order submodel85, and to its high-order generalization that, however, does not
represent a universal TDM. Two versions of the shiing technique (or, the shi transformation) were applied to
the preceding model by Hofreiter74,86. e DF was based on the fundamental-harmonic (i.e., higher harmonics
are neglected) Fourier series expansion of the shied control (input) signal. A biased relay with hysteresis was
used because of practical reasons; however, it was not explicitly included in the algorithm.
As this study concerns a circuit process with heat exchangers, it should also be noted that relay-based identi-
cation experiments (followed by a controller design in many cases) have been applied to heat-exchanger systems.
For instance, a case study on an autotuning control method for a cross-ow heat exchanger was published in87.
Jin etal.88 presented a Ziegler-Nichols-based method based on using the ultimate gain (instead of on a nonlin-
ear element) to get the sustained oscillations. Some researchers use nonlinear models such as Wiener-type or
Hammerstein-type89. e reader is also referred to37 and references therein.
Let us provide the reader with the motivation to perform the presented research. Models of industrial pro-
cesses usually include a large number of unknown parameters. Hence, when applying the above-celebrated
relay-based identication tests in practice, multiple experiments and/or solutions of nonlinear optimization prob-
lems are usually needed (even for linear models). As mentioned above, we focus on a circuit process with heat
exchangers with large input–output and internal delays84. Considering the simplest single-input single-output
case, the derived model includes six non-delay parameters, one internal delay parameter, and two parameters
in the input–output relation. Let us call the model as Heat-Exchanger TDM (HETDM) for further presentation.
erefore, any attempt to t these nine parameter values (e.g., in the frequency domain) requires a good initial
guess. Hence, the main idea of the presented research is to perform a two-step relay-feedback identication
procedure. e SFOTDM is assumed in the rst step, which requires a relatively low computational eort when
estimating model parameters’ values. e second step adopts the identied SFOTDM in the sense that dominant
characteristic values (poles) of the model coincide with the dominant poles of the analytically-derived HETDM.
e pole assignment yields the determination of the characteristic function parameters. ese values can either
be xed while remaining model parameters are then set from the single experiment data or they constitute the
initial estimation that are further enhanced along with other undetermined parameters that do not depend on
poles. As the bridge between the two basic steps, a thorough analysis of pole loci of the SFOTDM resulting in
explicit and implicit formulae and a simple graphical procedure is made. is quasi-polynomial root analysis
constitutes a substantial contribution to the presented paper.
Relay feedback experiments for both the used models use the standard (i.e., on/o) biased relay for the initial
estimation of the ultimate gain and the computation of the process static gain90. However, the selected asymmetry
is small enough not to aect the applied DF yet sucient to determine the static gain. e relay with saturation42
is applied to enhance the DF evaluation. is nonlinear element can estimate the DF precisely in the ideal case.
In the second step (i.e., for the HETDM), single and multiple relay tests are made. e single test is performed to
identify the transfer function numerator parameters that are not aected by pole loci. Contrariwise, the multiple
test attempts to determine four frequency points via three additional experiments utilizing an articial delay as
per the ATV + technique38,69,71. e response ultimate data and the given DF are processed via the well-established
Levenberg–Marquardt (LM) method91 and the Nelder–Mead (NM) algorithm92 to solve a certain nonlinear
frequency-based constrained optimization problem. Several numerical scenarios are benchmarked. Namely, LM
and NM techniques are compared when using a single relay test to determine transfer function numerator param-
eters. In addition, the NM algorithm is used to estimate eight model parameters when applying the ATV + test.
Contributions of the presented research can be summarized as follows:
1. Exact analytic rules to determine pole loci of the SFOTDM are derived.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
4
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
2. Saturated relay feedback experiment is performed on the SFOTDM. e detected pole loci are then set to
the initial parameters’ guess for the HETDM (as a suciently accurate mathematical model of the circuit
system with heat exchangers), which is enhanced via the LM method, under the single relay-test data.
3. ree scenarios are compared to determine the remaining model parameters and further enhance the already
estimated ones via the ATV+ technique and the solution of a nonlinear optimization problem using the LM
and NM algorithms.
4. An independent determination of numerator and denominator transfer function coecients along with the
pole loci assignment enables to reduce the number of necessary relay test for some of the scenarios.
e rest of the paper is organized as follows. “Methods and techniques” summarizes theoretical fundamentals
of (retarded) TDM spectrum and feedback relay-based experiment, the model parameters’ identication using
a DF, and the LM and NM methods. “Results” has two fundamental subsections. e rst subsection provides
the reader with a detailed analysis of the SFOTDM pole loci. e second one presents the HETDM and all steps
of determining its parameter values. Namely, the mathematical model of the HETDM is introduced, then the
reader is acquainted with the poles assignment, the transfer function numerator estimation using a single relay
test, and the complete model parameters estimation via the ATV + technique. In “Discussion”, the obtained results
are discussed. Finally, “Conclusions” concludes the paper.
e standard notation is used throughout the paper, i.e.,
C,N,R
denote the sets of complex, natural (excluding
zero) and real numbers, respectively,
Rn
+
expresses the n-dimensional Euclidean space of positive real-valued
vectors,
Re(s)
and
Im(s)
mean the real and imaginary parts of some
s∈C
, respectively. Superscript
T
denotes
the vector (matrix) transpose.
Methods and techniques
Retarded quasi‑polynomial and its spectrum. Let us concisely introduce the Retarded Quasi-Polyno-
mial (RQP) form and its spectrum, i.e., the zero points9,14,15. A RQP has the following form
where
s∈C
is the Laplace transform variable,
qij ∈R
are non-delay parameters,
τij ∈R+
with
τi0=0
represent
delays, and
n∈N
means the RQP order (of derivative).
Denition 1 e RQP spectrum is the set of RQP zeros, i.e.,
Proposition 1 It holds for
that
1. If exist
i≥0, j>0
such that
qij,τij = 0
, then
||=∞
(i.e., the RQP spectrum is innite).
2. RQP zeros
sk∈
are isolated and function
Rn−1
i=0mi�
qij,
τ
ij
�→
sk
∈
C
is continuous.
3. For any nite
γ∈R
, the subset
�R={s∈�:Res>γ
}
is nite, while
�L={s∈�:Res≤γ}
is innite. □
Note that the relation
q
ij
,τ
ij
→ s
k
is not necessarily smooth; namely, in points where a multiple real root
bifurcates into a complex pair.
Denition 2 e RQP spectral abscissa is dened as
Relay‑based parameter identication. As introduced above, experimental plant identication using
the relay (or another simple nonlinear element) method represents a widely used technique in various engineer-
ing and industrial applications. Consider a plant (the model of which is to be identied) under a relay feedback
control, as depicted in Fig.1. In the gure,
r(t),e(t),u(t)
, and
y(t)
mean the reference, control error, manipu-
(1)
q
(s)=sn+
n−1
i=0m
i
j=0
qijsie−τij
s
(2)
�
:=
s:q(s)=0
(3)
α�:= sup {Res:s∈�}
Figure1. A framework scheme of the relay feedback identication experiment.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
5
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
lated input and controlled output variables, and
G(s)
stands for the actual plant (process) dynamics. e choice
of
r(t)
(usually of a constant value) enables to set the operating point.
If the relay parameters are appropriately set, the closed-loop system reaches sustained oscillations of period
Tosc
in a nite time. If the relay element does not cause a phase lag, the corresponding angular frequency
ωosc =2π/Tosc
is supposed to be close to the so-called ultimate frequency
ωu
for which
Im
G
jω
u
=
0
(more
precisely
∡
G
jω
u
=−
π
),
j2
=−
1
. However, as a model
Gm(s)
cannot express the true dynamics
G(s)
exactly,
it generally holds that
ωosc = ωu
. Whenever the relay exposes a phase lag,
ωosc <ω
u
.
By adopting the idea of the DF, one point
G
jω
osc
∈
C
can be estimated, i.e., two parameters of
Gm(s)
can
be determined. e relay DF,
N(·)∈C
, can be considered as a linear approximation of the relay gain. It is usu-
ally derived using a consideration that
e(t)
is a harmonic signal and
u(t)
is subject to a truncated Fourier series
expansion. en, for the sustained oscillations, it holds that
which enables to estimate parameters of
Gm(s)
. Note that (4) can be graphically interpreted as the intersection
of the Nyquist plot of
Gm(s)
with the horizontal line
−N−1(·)
. e DF depends on the amplitude
A
of
e(t)
oscil-
lations and some other relay setting parameters.
On/o relay test. Let us consider an asymmetrical biased two-level relay. Its static characteristics and the
corresponding sustained oscillations (limit cycles) are displayed in Figs.2 and 3, respectively.
In the gures,
B+,B−
are upper and lower relay output levels, respectively, for which the bias (shi) parameter
reads
δ=|B+−B−|/2
, and
ε≥0
expresses the hysteresis parameter. e particular DF is
where
B=(B++B−)/2
, see, e.g.67,68,73. en, the ideal on/o relay gives rise to
N(A,0,0
)=N(A)=4B/(πA)
.
In practice, the setting
ε = 0
is suitable when the feedback signal is aected by noise so that the switching
relay rate can be reduced. e advantage of the option
δ = 0
lies, i.a., in the possibility to estimate the process
static gain
k=Gm(0)=G(0)
as
(4)
N
(·)Gm
jωosc
=−1+0j ⇔
N(·)Gm
jωosc
=1
∡
N
(·)
Gmj
ω
osc
=−
π
(5)
N
(A,δ,ε)=
4B
πA
1−δ
A21−ε
A2
−jε
A
,
δ,ε<A
(6)
k
=
t
0
+T
osc
t0y(θ)dθ
t0+Tosc
t0
u(θ)dθ
Figure2. e static characteristics of the asymmetrical biased relay.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
6
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
for
t0
satisfying that sustained oscillations start for some
t<t0
35.
Purposefully induced asymmetry can also be used to estimate and attenuate the load disturbance42. However,
it may stop oscillations so that the relay test fails. In addition, model parameter identication with asymmetric
relay yields an estimation error of up to 40% in a FOPDT case50.
Relay with saturation. Estimating the critical point at
ωu
(or any other
ωosc
) does not provide an accurate
enough parameter estimation for some processes, e.g., for those with signicant time delays. For instance, an
error of 23% for FOPDT models was reported59.
Model parameters identication can be improved by using saturation relay35,59. Its static characteristics and
a sketch of the corresponding sustained oscillations (under the assumption of a harmonic output variable) are
depicted in Figs.4 and 5.
e saturation relay does not cause an abrupt step change at
e(t)=±ε
, yet it provides a smooth transient
around zero. e relay input
e(t)
is multiplied by
ksat
resulting in the relay output
u(t)
up to the limit
B=ksat A
. e corresponding DF reads
Ideally, if the gain
ksat
is set optimally (i.e.,
A=A
), input and output signals has the same shape; hence, the
DF
Nsat
A,A
=k
sat
is exact. However, in real conditions,
u(t)
has a shape of the truncated sinusoidal wave
with upper and lower limits. Not that the limit case
ksat →∞
yields the ideal relay, i.e.
Nsat (A,0
)=N(A,0,0
)
.
A saturation relay test should follow the standard relay experiment (see the preceding subsection). Once
kosc =N(A,·)
is found, then it is set
ksat =kminkosc ,kmin >1
. Originally, it was suggested to take
kmin =1.4
59.
e higher the value is, the closer to the two-level signal
u(t)
is. Contrariwise, smaller values of
kmin
force
u(t)
to be closer to sinus-like waves; however, the relay takes a longer time or even can fail to generate sustained
oscillation.
(7)
N
sat
A,A
=
2B
π
1
Asin−1
A
A
+
A2−A2
A2
Figure3. Sustained oscillations using the asymmetrical biased relay.
Figure4. e static characteristics of the saturation relay.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
7
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
ATV + technique. One of the fundamental drawbacks of the original relay feedback test is that only a single
point of the frequency characteristics can be determined, which allows estimating only two model parameters.
e ATV + technique38,69,71 introduces an articial delay
τa>0
to the serial link between the relay and the pro-
cess.
Every single value of
τa
causes the phase lag of
ϕa=ωoscτa
where
ωosc
means the corresponding angular
frequency of sustained oscillations when the delay applies here. en, the overall phase shi attributed to the
process reads
Hence, by detecting
ωosc
and the corresponding amplitude
A
, another point on the process (model) Nyquist
curve can be determined. Obviously, whenever a number of
N
model parameters are needed to be resolved,
then
⌈N/2−1⌉
distinct
τa
s is required, where
⌈·⌉
means the ceiling function (i.e., the rounding upward to the
nearest integer).
e original setting69 comes from the following idea. e goal is to identify a point located at 45° distance
from the negative real axis, i.e.,
ϕa=π/4
(under the assumption that
∡N(·)=0
). is point is expected to occur
at frequency
ωosc =3/5ωu
. It eventually yields the following condition and the setting result
e disadvantage of this technique is the prolongation of the relay feedback experiment. However, if the
initial conditions are sustained oscillations, it lasts a signicantly shorter time to restore the oscillations than
starting from a constant steady state.
Parameter optimization methods. To solve (2) for given roots
sk
, (4), and (8), two well-established
optimization algorithms are adopted. eir concise description to acquaint the reader follows.
Levenberg–Marquardt method. Consider a set of nonlinear dierentiable functions
f
=
f
1
,f
2
,...f
nT
,
C
×R
m
�
x,p
�→ f
i
x,p
∈
R
,
i=1, 2, ...,n
, where
x
is a function variable, and
p
=
p
1
,p
2
,...,p
mT
∈R
m
expresses the set of function parameters. en, the set of algebraic functions
where
0=(0, 0, ...,0
)T
, can be iteratively solved via
where
(8)
∡
G
jω
osc
=−π−∡N(·)+ϕ
a
(9)
τ
a
3
5
ωu=
π
4
⇒τa=
5π
12
ω
u
(10)
f=0
(11)
k+1
p=
k
p+
k
p
kp=−
k+kdiag
k
−1
J
x,kp
T
f
x,kp
Figure5. Sustained oscillations using the saturation relay.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
8
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
J
means the Jacobian of
f
with respect to
p
,
k
expresses the iteration step, and
>0
is an adjustable parameter
(the so-called damping factor)93. Solution (11) and (12) attempts to solve the nonlinear least-squares problem, i.e.,
e value of
(the so-called damping factor) may vary during iterations. One of the framework strategies is
to decrease its value as the residual sum on the right-hand side of (13) (
Res
p
) decreases, and vice versa. Let
us introduce the multiplicative factor
κ>0
as
k+1
=k
κ
. Particular choices of
1
,1p
, and
κ
are discussed in
“Relay-based parameter identication of heat-exchanger process”.
A disadvantage of the LM algorithm is that the solution may converge to a local minimum (as for other
Newton-type methods) or it may even diverge (especially, if
is set inappropriately).
Nelder–Mead method. Assume an unconstrained optimization problem, rst
e idea of the NM method92 is to iteratively search for the optimal solution by moving a variable-shape
simplex in the space of
p
. e simplex vertices represent the so-called test points. Once the initial simplex
1
S=
1
p
1
,
1
p
2
,...,
1
p
m+1
is selected, its vertices are re-ordered such that f
1
p
i
≤f
1
p
i+1
,i=1, 2, ...
m
(i.e.,
1p1
represents the best solution estimation) and set
k=1
. Then, the center of the subvector
k˜
S=
k
p
1
,
k
p
2
,...,
k
p
m
is computed
e worst-valued vertex is reected through
kpc
as
k
p
r
=
k
p
c
+γ
rk
p
c
−
k
p
n
+
1
,γ
r
>
0
. en, four sce-
narios can happen:
1. If
f
k
p
1
≤f
k
p
r
<f
k
p
m
, then set a new simplex
k+1
S=
k
p
1
,
k
p
2
,...,
k
p
m
,
k
p
r
.
2. If
f
k
p
r
<f
k
p
1
, compute the expanded point
k
p
e
=
k
p
c
+γ
ek
p
r
−
k
p
c
,γ
e
>
1
. On condition that
f
k
p
e
<f
k
p
1
, set
k+1
S=
k
p
1
,
k
p
2
,...,
k
p
m
,
k
p
e
, else
k+1
S=
k
p
1
,
k
p
2
,...,
k
p
m
,
k
p
r
.
3. If
f
k
p
m
≤f
k
p
r
<f
k
p
m+1
, the outer contraction is done as
k
p
oc
=
k
p
c
+γ
ock
p
r
−
k
p
c
,
0<γ
oc <1
.
Whenever f
k
p
oc
<f
k
p
r
, set
k+1
S=
k
p
1
,
k
p
2
,...,
k
p
m
,
k
p
oc
, else perform the shrinkage as
4. If
f
k
p
r
≥f
k
p
m+1
, compute the inner contraction
k
p
ic
=
k
p
c
+γ
ick
p
m+1
−
k
p
c
,
0<γ
ic <1
. On con-
dition that f
k
p
ic
<f
k
p
m+1
, set
k+1
S=
k
p
1
,
k
p
2
,...,
k
p
m
,
k
p
ic
, else shrink the simplex using (16).
en
k=k+1
, re-order simplex vertices, and calculate (15), etc.
If, however, inequality constraints
on a subset
˜p⊆p
are required, one may use the concept of barrier functions. at is, instead of the objective
function f
p
as in (14), the extended function
p
is subject to the optimization procedure
where
β>0
and
fb
˜p
>
0
must be suciently small as soon as all
gj
˜p
≪
0
; otherwise, the value of
fb
˜p
increases considerably until
fb
˜p
→∞
as
gj
˜p
→
0−
.
Results
Root loci analysis of the simple quasi‑polynomial. In this subsection, a thorough zero loci analysis of
the SFOTDM12 is provided. e derived results then serve for the pole assignment of the HETDM giving rise to
the initial parameters setting of its CE (see “Parameter estimation of the heat-exchanger process model via pole
assignment”). e model reads
where
0<T,ϑ,τ,k<∞
.
Although pole loci properties of the SFOTDM were studied in the past, according to the authors’ best knowl-
edge, a complete image and a thorough exact guide on nding the dominant subset of its spectrum has not been
(12)
k
=
J
f
x,kp
T
J
f
x,kp
(13)
p
opt =lim
k→∞
kp=arg
p
min
n
i=1
fi
x,p
2
=: arg
p
Res
p
(14)
min
f
p
∈R,p∈R
m
(15)
k
pc=
1
m
m
i=1
kp
i
(16)
k+1S=γs
kS,0<γ
s<1
(17)
gj
˜p
<
0, j
=
1, 2,
...
,n
(18)
�
p
=f
p
+βf
b
˜p
(19)
G
SFOTDM (s)=
k
Ts +e
−ϑse−τ
s
Content courtesy of Springer Nature, terms of use apply. Rights reserved
9
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
provided yet. For instance, Marshal, Gorecki, Walton, and Korytowski94 studied a generalized characteristic RQP
of the SFOTDM with relative parameters (
T=1, �=ϑ/T,θ=τ/T
), and they determined ranges in which
the model is asymptotically stable, aperiodic, and periodic. Moreover, intersections of pole loci trajectories with
the imaginary axis for the generalized model were determined. Analogous conditions for which the model is
stable, overdamped, critically damped, and underdamped were presented in95. Asymptotic behavior of pole loci
trajectories in innity and nearby the imaginary axis were also studied in96.
Hence, our aim is to analyze the solution (or its rightmost subset) of the CE
in
C
and provide the reader with a simple guide how to compute these pole loci.
Lemma 1 12,94,96 All solutions of (20) lie in the open le half complex plane (LHP) if and only if
□
Result (21) can also be formulated as
�∈(0, 0.5π)
.
Lemma 2 12,94,96. ere exists a double real root
s1,2 =−1/ϑ
in the spectrum of
qSFOTDM (s)
if and only if
In addition, there does not exist a solution of (20) or its pair
si=α+ωj, si=α−ωj
with
where the bar denotes the complex conjugate. □
Result (22) can also be formulate as
=e−1
. Let us introduce relative real and imaginary parts of a quasi-
polynomial root as
α=−ϑα
,
ω=ϑω
, respectively. en the range (23) becomes
Lemma 2 means that
qSFOTDM (s)
has the rightmost double real root at
α=1
for
=e−1
.
Lemma 3 96. e double dominant (i.e., rightmost) real root
s1,2 =−1/ϑ
(i.e.,
α=1
) becomes a complex con‑
jugate pair for
�lim δ→0
+
=e−1+δ
. Contrariwise, the double real root becomes a pair of single real roots for
�lim δ→0
+
=e−1−δ
. □
Theorem1
qSFOTDM (s)
has a real dominant zero in the LHP for
=
0, e
−1
and a complex conjugate rightmost
pair in the LHP for
�
=
e
−1
, 0.5π
, where the particular root abscissa is within the range (23) (or (24), equiva‑
lently). □
Proof From Lemma 1, a negative root abscissa exists only for (23). If
ranges from 0 to
e−1
, the rightmost real
root moves from
α=0
to
α=1
due to Lemma 2 and Lemma 3. From Lemma 2, it is also known that there is
the rightmost double real root for
=e−1
that bifurcates into a conjugate pair for
�
=
e
−1
, 0.5π
. Eventually,
this pair reaches the imaginary axis again for
�=0.5π
as the only (i.e., the rightmost) quasi-polynomial root
pair due to Lemma 1. ■
In the following part of the subsection, dominant (and other) SFOTDM pole loci are investigated.
Lemma 4 96. Whenever
s1,2 =±ωj, ω�= 0
, it holds that
ω
=
1
T
=
(2k+1)π
2ϑ
(i.e.,
ω
=�=
2k+1
2π
),
k=0, 1, 2, ...
.
□
Lemma 5 e double real
s1,2 =−1/ϑ
(i.e.,
α=1
) in
=e−1
is the only multiple real root of
qSFOTDM (s)
. □
Proof e double real root
s1,2 =α
must satisfy
where
q
′
SFOTDM
(s)=
dq
SFOTDM
(s)
ds
. Conditions (25) read
(20)
CESFOTDM
:
qSFOTDM
(
s
)=
Ts
+
e−ϑs
=
0
(21)
1
T
∈
0,
π
2ϑ
(22)
1
T
=
1
eϑ
(23)
α
∈
−
1
ϑ
,0
,ω≥
0
(24)
α∈(0, 1),ω≥0
(25)
q
SFOTDM (s)
s=α
=q
′
SFOTDM
(s)
s=α
=
0
Content courtesy of Springer Nature, terms of use apply. Rights reserved
10
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
e rst formula of (26) can be rewritten as
e latter condition in (26) agrees with
By comparison (27) and (28), it can be deduced that
α=1
is the only double real root.
Now we must show that it is the only multiple root for any nite
. Such a root must satisfy (25) and
simultaneously
Since
ϑ = 0
, only two solutions of (29) exist
e former solution in (30) yields the stability border due to Lemma 1 and
α→∞
from (27) and (28), which
indicates a root at innity. It is, however, in contradiction with (29). e latter possibility in (30) gives
→∞
from (27) but
→ −∞
from (28), which yields a contradiction again.
As an alternative of the proof, one can easily deduce that (28) and (29) are in contradiction. ■
Lemma 6 Equation(20) can have only two real solutions (counting multiplicity). □
Proof Lemma 6 implies from Lemma 5 directly due to the root continuity (see Proposition 1). at is, a complex
conjugate zeros of
qSFOTDM (s)
can bifurcate in a pair of distinct real roots only through a multiple pair.
Alternatively, distinct real solutions of (20) satisfy (27). Function
α → αe−α
is unimodal with local and global
maximum in
α=1
and the function value
1/e
. is point agrees with Lemma 5. Otherwise, the function has two
distinct intersections with the constant function
∈
0, e
−1
for
α∈(0, ∞]
. Hence, there is no real solution of
(20) for
�>e−1
. e situation is illustrated in Fig.6. ■
Theorem2 Let
has a positive nite value. en,
(a) If
�>e−1
, complex conjugate (single) zeros
si=α+ωj, si=α−ωj
,
ω>0
, of
qSFOTDM (s)
are given by all
solutions of the set of equations
(26)
T
α+e
−ϑα
=
0
T−ϑe
−ϑα
=0
(27)
α
+
1
Te−ϑα =0⇔−
α
ϑ+
1
Te−ϑα =0⇔−
α+�e−ϑα =
0
⇒�=
α
e
α
(28)
1
−�eα=0⇔�=
1
eα
(29)
q
(k)
SFOTDM (s)
s=α
=(−1)kϑk−1�eα=0, k=2, 3,
...
(30)
�=0 or α→ −∞
(31)
α
=
ω
tan ω
,ω�= kπ,k=1, 2,
...
Figure6. Intersections of a constant function
and
αe−α
(Lemma 6).
Content courtesy of Springer Nature, terms of use apply. Rights reserved
11
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
(b) If
∈
0, e
−1
, complex conjugate zeros of
qSFOTDM (s)
are given by (31) and (32), and single real roots are
given by the unique solution pair of
(c) If
=e−1
, complex conjugate zeros of
qSFOTDM (s)
are given by (31) and (32), and the multiple real root reads
s1,2 =−ϑ−1
(i.e.,
α=1
). □
Proof Consider item a) rst. From eorem1 and Lemma 6, there are no real solutions of (20). Complex con-
jugate ones have to satisfy
i.e., both the real and imaginary parts must be equal to zero
Aer some algebraic manipulation, conditions (35) become
By expressing
�eα
from one equation und substituting it into another one, formula (31) is obtained where
singularities are to be denied. Further, the latter equation in (36) gives
which yields (32) directly. Naturally, only positive right-hand sides of (37) are admissible to get real
α
values.
We know from Lemmas 5 and 6 and eorem1 that the only possible double real root bifurcates into a com-
plex conjugate pair for
�>e−1
and there cannot exist another real root of
qSFOTDM (s)
. Note that only one root
from the pair is sucient to take due to the symmetry.
Assuming item b), a pair of single real roots exists due to eorem1. However, it is the only such a pair
according to Lemma 6. A single real root must satisfy the rst condition in (26), giving rise to (27), the result of
which agrees with (33). However, complex conjugate zeros of
qSFOTDM (s)
must simultaneously exist due to its
transcendental manner.
Regarding item c), the existence of the double real root is given by Lemma 2. Besides, there is no other real
root of
qSFOTDM (s)
due to Lemma 5, yet
si∈C\R
as in (31) and (32) still exist. ■
eorem2 does not consider multiple quasi-polynomial roots
si∈C\R
. e following proposition veries
that such roots can be neglected.
Proposition 2 Equation(20) does not admit a multiple pair solution
si=α+ωj, si=α−ωj
,
ω>0
. □
Proof Any n-multiple
si∈C\R
must satisfy (35) and also
It is enough to show that a complex conjugate pair of multiplicity 2 does not exist, i.e.,
n=1
. We proof a
contradiction; hence, let there exists a double root
si∈C\R
that has to satisfy
which gives
e latter formula in (39) has two solutions:
�eα=0
or
sin ω=0
. e rst one is in the contradiction to the
former condition in (39), whereas the second one yields
(32)
α
=ln
ω
�sin ω
,α∈
R
(33)
�=αe−α,α∈R
(34)
q
SFOTDM (s)
s=α+ωj
=
0
(35)
T
α+e
−ϑα
cos (ϑω)=
0
Tω−e
−ϑα
sin (ϑω)=0
(36)
−
α+�e
α
cos ω=
0
ω−�e
α
sin ω=0
(37)
e
α
=
ω
�sin ω
(38)
q
(k)
SFOTDM (s)
s
=α+ω
j
=0, k=1, 2, ...,n−
1
0
=q
′
SFOTDM
s=α+ωj=T−ϑe
−ϑ
s=α+ωj
=T−ϑe−αϑ cos (ωϑ)−jϑe−αϑ sin (ωϑ
)
=
T
−ϑ
e
α
cos
ω−
j
ϑ
e
α
sin
ω
(39)
1
−�e
α
cos ω=
0
�e
α
sin ω=0
(40)
sin ω=kπ,k=0, 1, ...
Content courtesy of Springer Nature, terms of use apply. Rights reserved
12
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
By substituting (40) into the rst condition in (39), one gets
�eα=1
, which implies
ω=sin (ω)
from (36)
or (37). at is, the unique solution
ω=0
means that the quasi-polynomial root is real, which gives a contra-
diction. ■
Corollary 1
(a) If
e−1
<�<
π
2
, the rightmost zeros of
qSFOTDM (s)
form a complex conjugate pair
si
=−
α
ϑ
+
ω
ϑ
j, s
i
=−
α
ϑ
−
ω
ϑj
, given by the unique solution of the constrained optimization problem
(b) If
∈
0, e
−1
, the rightmost zero of
qSFOTDM (s)
is a single real root given by the unique solution of
(c) If
=e−1
, the rightmost zero of
qSFOTDM (s)
is the double real root
s1,2
=−
1
ϑ
(i.e.,
α=1
).□
Proof Regarding item a), the given range of
yields the rightmost complex solutions of (20) according to
eorem1. Besides, its abscissa is within the range (24). All complex roots have their loci given by (31) and (32)
of eorem2.
e only fact remaining to prove item a) is to show that
ω∈(0, 0.5π)
if and only if
α=(0, 1)
whenever
�
∈
e
1
, 0.5π
. From Lemma 3,
lim
�→0.5π
−
ω=0.5π−
and
lim
�→e
(−
1
)+
ω=0
due to Lemma 2 (i.e., the complex pair
becomes the double real
qSFOTDM (s)
zero).
(Necessity.) Now, we prove by contradiction that
ω
remains within the limit. Consider that
α=(0, 1)
. Let
exist
ω=0
or
ω=
π
2
such that the rst equation in (41) holds. e limit values are, respectively,
which is in contradiction to
α=(0, 1)
.
e case
ω<0
can be omitted due to the root loci symmetry in
C
. Whenever
ω>0.5π
, then there exists
tan ω<0
, which implies
α<0
from (31), and we have a contradiction again.
(Suciency). It holds that
ω<tan ω
for
ω∈(0, 0.5π)
, which gives
α=(0, 1)
directly.
Regarding item b), there exist two single real zeros of
qSFOTDM
(
s)
due to Lemma 6, the loci of which are given
by (33) in eorem2. In addition, the right root from the pair determines the spectral abscissa for
∈
0, e
−1
from eorem1, which implies (42).
Item c) represents a reformulation of Lemma 2. ■
(41)
min
0
<α<1
α=
ω
tan ω
s. t.
:α=ln
ω
�
sin
ω
,ω∈
0, π
2
(42)
min
0<α<1
α=�e
α
(43)
lim
ω
→π
2
α=lim
ω→π
2
ω
tan ω
=0
lim
ω
→0
α=lim
ω→0
ω
tan ω
=lim
ω→0
1
1
cos
2
ω
=
1
Figure7. Intersections of
α1(ω)
and
α2(ω)
to get
si∈C\R
(eorem2, Corollary 1).
Content courtesy of Springer Nature, terms of use apply. Rights reserved
13
Vol.:(0123456789)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
Remark 1 Corollary 1 can be extended to an unstable case; however, it is useless for this study. If needed, eo-
rem2 can be generally used. □
To conclude this subsection, a graphical procedure to nd all the roots of
qSFOTDM (s)
or the rightmost
spectrum of the SFOTDM poles follows. Whenever condition of item b) of eorem2 is satised, real poles are
found as per Fig.6. Complex conjugate poles are given by (31) and (32) or (41), which can be graphically inter-
preted as intersections of real-valued functions
α1(ω)=ω/ tan ω
and
α2(ω)=ln (ω/(�sin ω))
where values
α2(ω)∈C\R
determine “forbidden regions” of the function graph, see Fig.7 for illustration.
e example gure is done for
T=ϑ=�=1
. Forbidden regions are highlighted in red. e circle indicates
the position on the dominant complex conjugate pair according to (41).
Relay‑based parameter identication of heat‑exchanger process. Innite‑dimensional heat‑ex‑
changer process model. e HETDM serves as a simulation testbed. e mathematical model arises from heat
and mass balance equations that include delays and a thorough analysis of static and dynamic responses of the
particular laboratory appliance (see Fig.8). A concise description of the apparatus follows84 rst. Positions in the
gure correspond to the numbers in curly brackets.
e heat uid circulates in the closed loop owing through an instantaneous heater {1}, a long insulated
coiled pipeline {2}, and a cooler {3}. e power input to the heater (that can be viewed as a solid–liquid ow
heat exchanger) is continuously controlled in the pulse-width-modulation sense. Its maximum value is 750W.
e heated uid temperature on the heater output {4} is only slightly aected when owing through the 15m
long pipeline; however, the most signicant loop delay is caused therein. e outlet temperature of the pipeline
is measured by a platinum resistance Pt1000 thermometer {5}. e cooler is constructed as a radiator (i.e., a
Figure8. e HETDM appliance.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
14
Vol:.(1234567890)
Scientic Reports | (2022) 12:9290 | https://doi.org/10.1038/s41598-022-13182-5
www.nature.com/scientificreports/
plate-and-n heat exchanger) that can be considered as an indirect unmixed cross-ow heat exchanger from
the process point of view. It is equipped with two cooling fans {6} (one of them is continuously controlled, while
another is on/o type). e expansion tank compensates for the impact of the water thermal expansion {7}.
e outlet temperature from the cooler is measured by Pt1000 again {8}. Finally, the continuously controllable
magnetic drive centrifugal pump {9} serves for uid circulation.
Despite its simplicity, the mathematical formulation of the HETDM and especially its dynamic properties
are remarkable due to the model transcendental characteristic equation84. As the model is multivariable, the
relation between the heater power input
u(t)
and the cooler outlet heat uid temperature
y(t)
is selected as the
most interesting input–output pair. Note that both quantities are considered as their deviations from a steady
state. e analytically modelled linearized relation reads
which is a DDE where
b0,b0τ,a2,a1,a0,a0ϑ∈R
and
τ,τ0,ϑ∈R+
express input/output delays and the state
(internal) delay, respectively. e corresponding transfer function is
the denominator of which represents the model characteristic RQP (i.e.,
qHETDM
). In84, the following param-
eter values have been determined by a thorough and complex analysis of static and dynamic data
Let us take these data as a benchmark for the signicantly more straightforward relay-based experiment. As
the values arise from determined physical quantities of the process, they are assumed to be closed to the actual
(true) real-life values.
Remark 2 e used Pt1000 thermometers have the guaranteed time constant
T63
of 8s, i.e.,
T90 ≈18.4
s. is
additional dynamical latency has not been taken into account in analytically-derived model (44), and the true
temperature values can be dierent from the measured ones. However, such negligence does not pose a serious
problem with the model. First, plant delays
τ,ϑ
caused by the thermal uid transportation have signicantly
higher values, approx.
≈150
s. is means that possible sensor latencies have only a minor eect on the overall
dynamics. Second, sensors’ latencies do not aect the internal delay of the model itself since they act only in the
input/output relation; yet, they are included in the internal delay of the relay-feedback closed loop. If the system
is considered linear (indeed, model (44) is a linearized formulation valid in the vicinity of an operating point),
a sensor delay can be considered as the additional input/output delay of the model. As input/output delays are
not derived analytically but based on measurements, the relay experiment data’s evaluation also covers these
non-modeled latencies. erefore, once the model is used for plant control, the plant model and the output signal
for the feedback have the same value (in the ideal case).
Simple model parameter estimation using the relay‑based experiment. e rst step of the
identication chain is estimating the SFOTDM parameters, especially those of
qSFOTDM
(20). It has three sub-
steps. First, the on/o relay with
δ>0
(and
ε
being suciently small), see (5), is used to estimate the static gain
k
in (19) as per (6). Second, the ideal relay (
δ=0
) is applied to get the initial estimation of oscillation data and the
input/output delay value. Finally, the saturation relay is used to improve the accuracy of oscillation parameters,
which yields the SFOTDM parameters from (4) and (7). All the substeps can be done within a single experiment,
saving time since the transition from particular substantial oscillations to others takes less time than setting the
oscillations from a constant steady state.
Let us use the notation
τ→τs,ϑ→ϑs
for (19) to distinguish the SFOTDM parameters from those of the
HETDM (for which no subscript is used). e combination of (4) and (7) can be solved analytically yielding12
where
N·(A,·)
stands for either (5) or (7). However, it is inherently expected that the argument of
cos−1(·)
is
within the range
[−1, 1]
. Whenever it does not hold, a numerical solution of the combination of (4) and (7) have
to be used instead of (47).
Set
B=100
,
δ=0.05
, and
ε=10−5
. e relay-test responses are displayed in Fig.9. e arrows indicate
when a particular relay starts to be used.
e eventual data from Fig.9 are summarized in Table1. Formula (19) gives
k=3.22 ×10−2
. e value of
τs
can be estimated as the time interval between the switching point of
u(t)
and the peak time instant of
y(t)
.
Hence, it can be measured that
τs≈136.7
s. Note that
kmin =1.4
has been taken for the saturation relay setting,
which gives rise to
ksat =185.1
,
A=0.555
.
As
kN·(A,·)c