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Optimal Signal Design in System Identification
for Model Predictive Control (MPC)
Md. Tanjil Sarker1, Gobbi Ramasamy2
1,2 PV Energy Storage Lab, Faculty of Engineering, Multimedia University, 63100 Cyberjaya, Malaysia
Corresponding author: Gobbi Ramasamy (e-mail: gobbi@mmu.edu.my).
The first author would like to thanks Multimedia University, Malaysia, for providing financial support under the post-doctoral research fellowship program.
ABSTRACT In control engineering, system identification is frequently used to create models from input-
output data. The MPC is a very well-known control strategy because of its capacity to deal with a system’s
limitations. MPC determines the optimal input based on optimization. The optimization for each time step
needs to be achieved in less time than the sampling time of the system. In this paper, the optimal signal design
in system identification for MPC is presented. The reference signal has derived from the input perturbation
signal in both time domain and frequency domain. An equation has derived to determine the relationship
between the input signal and the reference signal. Three dissimilar amplitude spectra, namely flat spectrum
signal is designed using the time-frequency swapping technique, MOOSE2 signal is designed from the
MOOSE2 programme and optexcit signal is designed from the Frequency Domain System Identification
Toolbox, has been examined under MPC. The time-frequency swapping procedure is employed to reduce the
crest factor for input signal for the constraints on the rate of change of control signal. A new method has been
proposed for mitigating the constraint on the rate of change of control signal. The crest factor is reduced by
13.42% for the input signal and 37.64% for the optimal parameter vector for the control sequence signal. This
research contributes to advancing the state-of-the-art in MPC by offering a systematic approach for system
identification that can lead to improved control performance and greater adaptability in real-world
applications.
INDEX TERMS Estimation, MPC, optimal spectrum design, perturbation signal, system identification and
constraints.
I. INTRODUCTION
This The MPC is an essential progressive control procedure
for tough multivariable control problems [1]. Model based
predictive control was first introduced for used in the field
of chemical and petrochemical. There was an enormous
interest for this methodology in view of its capability to
determine optimal control activities wherewith extensive
measure of cash could be spared. In the petrochemical
industry where items were estimated in kilo or megatons,
just a little enhancement could prompt massive reserve
funds. The principal explanation behind utilizing this sort
of control approach in chemical and petrochemical fields
was the fact that even though the technique was
computationally demanding, the system dynamics were
slow [1]. MPC determines the optimal input based on
optimization. The optimization for each time step needs to
be achieved in less time than the sampling time of the
system. In this time, the stability was not hypothetically
decided [2]. The earlier version of MPC process was not
steady state and automatic. On the other hand, stable
systems and extensive prediction horizon were selected to
improve this sort of control method. Usually, the
optimization problem is predefined and the optimized
control signal can be computed with the actual state values.
In practice, if some states are not measurable, an observer
or estimator can be applied. At the start of this century
another methodology was presented, referred to as the
Explicit MPC. This technique allows the MPC to control
electrical and mechanical systems with fast dynamics. The
transfer function models are applicable to both stable and
unstable plants and the controller gives a more
parsimonious explanation of process dynamics. Currently,
the use of state-space models for MPC system design
procedures is growing in greater popularity than other
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
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8 VOLUME XX, 2017
models [3]. Control algorithms based on MPCs numerically
solve an optimization problem based on user-defined
prediction horizon and cost function, making use of the
receding horizon concept [4].
In the dynamic landscape of control systems
engineering, MPC stands as a powerful approach for
optimizing complex systems in real-time. At the heart of
MPC's effectiveness lies the accurate representation of the
system's behavior through model identification. However,
the quality of this identification heavily depends on the
input signals used, including the reference signals that drive
the system. The input multisine signals can precisely meet
a user-defined frequency domain requirement. The design
of multisine is typically done in two stages. The first stage
consists of defining the required spectrum [5]. The second
stage then realises the spectrum arbitrarily while taking any
power constraints into account. The input multisine signal
is ideal for identification designs that require the spectrum
to be carefully shaped either or both the number of
harmonics to be precisely defined [5]. The multisine is a
good choice for system identification because it is very
flexible in terms of creating arbitrary power spectrums,
often with a small crest factor. It also saves time due to its
low crest factor. This explores the critical role of reference
signal design in the context of systems identification under
MPC. In dynamic systems, reference signals may vary over
time to adapt to changing conditions or achieve specific
control goals. Designing reference signals is not a trivial
task. It requires an understanding of the system's dynamics,
the control objectives, and often the consideration of
constraints. Effective reference signal design has a
profound impact on industries ranging from manufacturing
and robotics to energy management and aerospace. It can
lead to improved product quality, reduced energy
consumption, enhanced safety, and increased operational
efficiency.
The remainder of the paper is structured as follows,
Section II delivers a basic concept of MPC with including
state-space models with embedded integrator, the
configuration of MPC and investigation of the effects of
constraints. Derivation of reference signal under MPC and
the method for mitigating the constraint are described in
Sections III and IV, respectively. Finally, some concluding
thoughts and ideas for future work are included in Section
V.
II. BASIC CONCEPT OF MPC
The MPC estimations are based on current calculations and
forecasts for future output measurements. The MPC control's
purpose is to choose control actions such that the predicted
response progresses to the set point in the best possible way
[6]. Figure 1 displays the actual output y, modified input u, and
predicted output for the SISO control system. The control
horizon M is the finite future time duration over which the
controller plans and optimises the control inputs in MPC. The
control horizon is an important parameter that influences the
MPC algorithm's performance and computational complexity.
It is one of the elements that make up the prediction horizon
P. The control horizon is the precise time period inside the
prediction horizon over which control inputs are actively
adjusted to maximise the system's performance. The
prediction horizon specifies the overall time duration over
which the future states are predicted.
.
FIGURE 1. Basic concept of MPC.
The MPC technique computes a set of M values of the
input at the current sampling
moment, denoted by k. In the collection includes the future
inputs M-1 as well as the current input u(k). The input is
maintained constant after M control movements, where M
is the size of the control horizon [1]. The inputs are
measured so that a set of P projected outputs
reaches the set point in the optimum way. The
control measurements are constructed by optimizing an
objective function. P is the number of predictions defined
as the prediction horizon. Receding horizon method is one
of the typical features of MPC [1]. At each sampling
moment, M control moves are computed, but only the first
move is actually performed. At the next sampling moment,
a fresh set of control movements is computed, with just the
initial input change being applied this time.
A. State-space Models with Embedded Integrator
There are many types of MPC models available in the field of
control engineering. Integral action is a feature of the majority
of industrial control systems, such as PID controllers [7]. The
proportional controller can be used to shorten the system's rise
time, but it will never be able to completely remove steady
state inaccuracy. The integrated controller can completely get
rid of the steady state error. The state-space model is used for
simplicity of the design structure and the direct connection to
the classical linear quadratic regulation [8-9]. Here, a state-
space model with an embedded integrator is used to construct
the MPC. The state-space technique from the book of [10]
served as the inspiration for the creation of the design models.
By providing a dynamic representation of the system, enabling
accurate prediction of future states, forming the basis for the
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objective function, incorporating constraints, and supporting
real-time adaptation, state-space models contribute to
optimising signal design in MPC [11]. These models serve as
the MPC framework's foundation, allowing controllers to
make informed decisions for optimal system control in a
variety of applications.
In a state-space model, the ability to forecast the future
depends on the current data, which is simultaneously
represented by the state variable. If the plant is SISO, the state-
space model equation is described by [10]
(1)
(2)
where, xm is the state variable vector, u is the input variable or
manipulated variable and y is the process output. Am, Bm and
Cm are the model parameters of a state-space form. Note that
this plant model has u(k) as its input. Doing a difference
operation on both sides of (1),
(3)
The following is the state variable's difference equation:
(4)
The following is the control variable difference equation:
State variable and control variable are the increments of the
variables and . Converting the difference equation
to the state-space form,
. (5)
The state-space model input is . Connecting
to the output a new state variable vector is
(6)
From (5) and (6),
(7)
(8)
where
, where is the assumed dimension
of .
B. The Configuration of MPC
The general objective of MPC design is to calculate a
manipulated variable u to optimise the plant output y as the
future behaviour. The optimization is executed inside a limited
time window by giving plant data at the beginning of the time
window [12]. In The block diagram of the MPC is given in Fig
2 where the input signal is u, output signal is y and the optimal
parameter vector is Δu. Am, Bm and Cm are the model
parameters of a state-space form and the state variable vector
is xm. The backward shift operator is and the discrete time
integrator is
. Kmpc is the state feedback control gain using
MPC and the Ky is the state feedback control gain related to y.
The estimated state variable vector is and r is the reference
or set point signal.
FIGURE 2. Block diagram of the MPC.
The optimal parameter vector ΔU is defined by
(9)
The pair of matrices are used in the prediction
equation, where
(10)
and
(11)
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In the optimization literature,
is called the
Hessian matrix.
is a diagonal matrix, where
. is used as a tuning parameter and is
the control horizon. The data vector contains the set point
information expressed as
(12)
where
is correlated
with the set point change, whereas
is
correlated with the state feedback control. These rely on the
system parameters; consequently, they are constant matrices
for a time-invariant system. The control principle of this MPC
is receding horizon. So it can take only the first element of ΔU
at time ki as the incremental control. Thus
(13)
(14)
where the is the first element of
(15)
and is the first row of
. (16)
The inverse matrix operation is a critical step in solving the
optimisation problem associated with predicting future states
and optimising control inputs in practical Model Predictive
Control (MPC) applications for real-world systems. However,
performing this operation in practise necessitates taking into
account computational efficiency, numerical stability, and
real-time implementation constraints [13]. For optimisation
problems, quasi-Newton techniques like the Broyden–
Fletcher–Goldfarb–Shanno (BFGS) algorithm can be applied
without explicitly forming the Hessian matrix [14]. Without
requiring a direct inversion, these techniques yield an
approximate inverse. To obtain the inverse of a positive
definite symmetric matrix, one well-known and efficient
technique is the Cholesky decomposition (CD) inversion [15].
Inverse matrix has been applied in to the High-Speed Finite
Control Set Model Predictive Control for Power Electronics
[16].
C. Investigation of the Effects of Constraints
Constraints define the boundaries within which the system
must operate. These limits can include bounds on control
inputs, state variables, or outputs. MPC enforces these limits
to prevent the system from reaching conditions that could lead
to equipment damage, environmental harm, or compromised
product quality. Three types of constraints are frequently used
in the design of control application. These are
i) amplitude of control signal,
ii) rate of change of control signal and
iii) amplitude of output
The first two constraints are imposed on the control variables
u(k), and the last one on the output y(k) [10].
Amplitude of control signal: This is determined by the
operational range for the plant input signal. If the upper and
lower limits of the control signal are and ,
respectively, this constraint is stated in the form
(17)
where u(k) is represents the deviation of the control signal
from the nominal value.
Rate of change of control signal: The constraints are
developed in a set of linear equations, which is developed on
the parameter vector ΔU. In the optimization literature, the
vector ΔU is frequently known as the decision variable. The
receding horizon control strategy is applied to express and
solve the predictive control problem, for every moving
horizon window, with the constraints taken into account. The
rate of change of the control signal is Δu(k), the upper and
lower limits are and , respectively. The
constraints at sample time ki in terms of Δu are specified in the
form
(18)
The constraints at future samples, for instance the first three
samples, are enforced
as
(19)
Amplitude of output: This is determined by the functioning
range of the plant output. If output is , its upper and lower
limits are and , respectively. The output constraint
is stated in the form
(20)
A larger slack variable is added as a ‘soft’ constraint for the
output to solve the conflict problem between control and
incremental control variables. The conflict problem occurs
when the control or incremental control variables infringe their
individual constraints. The form of output constraint after
adding the slack variable is
where (21)
In summary, constraints are a fundamental aspect of
MPC, serving multiple purposes, including safety assurance,
disturbance rejection, and performance optimization. Their
effects on the control strategy and the controlled system must
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
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8 VOLUME XX, 2017
be carefully considered during the design and implementation
of MPC to strike the right balance between competing
objectives while ensuring the overall stability and reliability of
the system.
III. DERIVATION OF REFERENCE SIGNAL UNDER MPC
In Model Predictive Control (MPC), the reference signal is a
key component used to design the control law. The reference
signal represents the desired trajectory or setpoint that the
system should follow over a finite prediction horizon. The
control algorithm then computes the control inputs to drive the
system towards this reference signal while optimizing a
specified cost function. The specifics of the MPC formulation
may vary depending on the system and application, including
the choice of cost function, constraints, and prediction horizon
length. In this Section, the theory was developed for the
known system. The practical case where the system is
unknown (and thus need to be identified). Based on the Figure
2 can write the control variable signal,
(22)
(23)
The reference or set point signal r in the time domain
approach is
, (24)
where is the state variable vector in the time domain and
In the frequency domain
, (25)
where and is the z-
transformation of , the state variable vector in the frequency
domain,
, (26)
. (27)
The reference signal can be directly obtained from the
Equation (24) and (25). Equation (27) can be used to
determine the relationship between the input signal and the
reference signal. The relationship in MPC is established
through the optimization process that seeks to minimize the
error or deviation between the system's predicted behavior and
the desired reference trajectory. MPC adjusts the control
inputs based on the reference signal and the current state of the
system, enabling it to track the reference signal while
considering system dynamics and constraints.
An example of lowpass systems is shown in below, where
(28)
In this experiment, the MOOSE2 signal is used as an input
signal according to guideline in [17]. The MOOSE2 spectrum
is obtained from MOOSE2 toolbox [18-20]. The signal design
procedure is in [21-23]. The MOOSE2 suitable signal period,
constructed on the procedures in [23] has N = 60, and excited
harmonics from 1 to 24. The shape of the input signal typically
reflects how the control algorithm is adjusting the control
inputs to drive the system's behavior closer to the desired
reference signal.
In the frequency domain, the shape of
signal relies on
the shape of. The shape of
for System is
shown in Fig. 3. The magnitude of
increases from low to
high frequency due to the shape of. This general
trend is the same for all systems. This can help avoid excessive
control effort and ensure that the system can track the
reference signal effectively. Figs. 4 and 5 illustrate the plots
obtained for System . The reference signal r is smaller than
the input signal u. The relative amplitudes of r and u will
depend on the system parameters.
The frequency response plots in the form of Bode plots are
shown in Fig. 6 for . Errors are typically higher at points
where the frequency response magnitude of the actual
system is higher. This is because for the same percentage
error, portions of the magnitude response with a larger gain
will give a larger absolute error. In this particular run, the
MOOSE2 signal gives very low error at low frequencies
showing that the steady state gain has been identified with
high accuracy.
FIGURE 3. Shape of
for G1.
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FIGURE 4. Time domain signals for G1, where the solid line with
circles and the dashed-dotted line with asterisks represent the
reference or set point signal r and input signal u, respectively.
FIGURE 5. Frequency domain signals for G1, where the solid line with
circles and the dashed-dotted line with asterisks represent the
reference or set point signal R(z) and input signal U(z), respectively.
FIGURE 6. Frequency Bode plots of G1, where the solid, dashed and
dashed-dotted lines represent the actual system frequency response,
the estimated system frequency response and the error, respectively.
An example of bandpass systems is shown in below, where
(29)
In this experiment, the MOOSE2 signal is used as an input
signal according to guideline in [18]. The proper signal
period, built on the guiding principle in [23] has N = 48, and
excited harmonics from 1 to 20. Figs. 6 and 7 illustrate the
plots obtained for System using the MOOSE2 signal as
an input signal. A band-pass system allows a specific range
of frequencies to pass while attenuating others. Here,
reference signal contains frequency components within the
passband of the system. The reference signal has proper
energy within the band-pass frequency range. Figs. 6 and 7
illustrate the plots obtained for System . The reference
signal of r is bigger than the input signal u.
FIGURE 7. Time domain signals for G2, where the solid line with
circles and the dashed-dotted line with asterisks represent the
reference or set point signal r and input signal u, respectively.
FIGURE 8. Frequency domain signals for G2, where the solid line with
circles and the dashed-dotted line with asterisks represent the
reference or set point signal R(z) and input signal U(z), respectively.
The peak power in the MOOSE2 signals in the magnitude
response, at harmonic 6. The Bode plots for are shown in
Fig. 9. It can be seen that the errors are large at low frequencies
due to the small gain resulting in a low SNR.
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
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FIGURE 9. Frequency domain signals for G2, where the solid, dashed
and dashed-dotted lines represent the actual system frequency
response, the estimated system frequency response and the error,
respectively.
IV. METHOD FOR MITIGATING THE CONSTRAINT
The constraint for rate of change of control signal is used
in the MPC for plant friendly identification [24-26]. A low
crest factor for the rate of change of control signal is suitable
for plant identification, as it provides the necessary
information with shorter signal length and smaller maximum
move sizes. The other two constraints are used in the MPC
for limiting the input and output respectively. These two
constraints can be mitigated by appropriate scaling of the
input signal. Hence, in this research, only the constraint on
the rate of change of control signal is considered. It plays a
significant role in the MPC. It can prevent rapid changes of
the process states, ensure the closed loop stability, and
increase the economic performance of the plant [27]. It can
also ensure the optimal use of energy in the plant.
A new method is proposed for mitigating the constraint on
the rate of change of control signal. Basically, it follows two
steps, which are given below:
Step 1: Converting time domain signal to frequency domain
The original input signal is
(30)
where is the amplitude, f and are the frequency and
phase of harmonic k and F is the number of harmonics. This
signal is generated using the time-frequency swapping
algorithm [20]. Hence, the original parameter vector for the
control sequence in time domain is defined by
. (31)
Rewrite (31) as
(32)
where and are the amplitude and phase of parameter
vector for the control sequence, respectively. The parameter
vector for the control sequence in frequency domain is
defined by
(33)
Step 2: Converting frequency domain signal to time domain
The values of are fed to the time-frequency
swapping algorithm [20] for generating the optimal
parameter vector for the control sequence in the time domain,
which is
(34)
where and are the optimal amplitude and phase,
respectively. is equal to the . However, the phase
angles and are not same. The time-frequency swapping
algorithm can provide the same magnitude and lower crest
factor with optimal phase [28-29] because this algorithm
uses only the phases as free variables. The optimal input
signal in time domain is defined by
(35)
Rewrite the Equation (35)
(36)
where and are the optimal amplitude and phase of .
Here, and are scaled to have the same power for fair
comparison. The input signal in frequency domain is
defined by
(37)
The and are expected to give lower crest factors
than and , respectively. The as an input signal or
as the optimal parameter vector for the control
sequence signal can be used in the MPC for mitigating the
rate of change of control signal constraints. The above
mention methodology is applied to several systems. The
reduction in crest factor for both the control signal and the
rate of change of control signal is observed as a general trend
for all the systems tested.
Results for system are shown in below. The MOOSE2
signal is used as an input signal. The general information of
signal for fair comparison is given below in Table 1. The
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
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8 VOLUME XX, 2017
crest factor is reduced by 13.42% for the input signal and
37.64% for the optimal parameter vector for the control
sequence signal.
TABLE I
GENERAL INFORMATION ON MOOSE2 SIGNAL.
SL
Signal
Crest Factor
Signal Power
1
2.414
0.013
2
2.298
0.013
3
2.090
0.013
4
1.433
0.017
Fig. 10 illustrates the plots obtained for System using the
parameter vector for the control sequence. has smaller
amplitude than consistent with the reduction in crest
factor. The constraint is not activated for the amplitude of
.
FIGURE 10. Parameter vector for the control sequence signals
for G1 in the time domain, where the solid line and the dashed-dotted
line represent and , respectively.
FIGURE 11. Frequency domain parameter vector for the control
sequence signals for G1, where the solid line with circles and the
dashed-dotted line with asterisks represent the and ,
respectively.
Fig. 11 illustrates the plots obtained for System using the
parameter vector for the control sequence. and are
well matched in the frequency domain. Some of the power
outside the specified harmonics from 1 to F in has been
redistributed to within the specified harmonics in .
V. CONCLUSION
In conclusion, a method has been proposed to the optimal
spectrum design to systems under MPC in closed loop. The
reference signal is derived from the input perturbation signal.
When the rate of change of control signal is constrained, the
time-frequency swapping procedure is employed to reduce the
crest factor for this signal. A new method is proposed for
mitigating the constraint on the rate of change of control
signal, where the crest factor is reduced for the input signal.
This inside is very useful for signal design under MPC.
Suggestions for future work, investigated the iterative
system identification, where the input signals will be adjusted
iteratively with the current estimation of the model
parameters. Iterative adjustment of input signals may
encounter convergence issues. It can be difficult to ensure that
the iterative process converges to a stable solution in a
reasonable timeframe, especially when dealing with complex
and nonlinear systems.
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
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MD. TANJIL SARKER received the
B.Sc. degree in Electrical and Electronics
Engineering (EEE) and Master of
Business Administration (MBA) degree
in Human Resource Management (HRM)
from the Bangladesh University, Dhaka,
Bangladesh, in 2013 and 2015,
respectively. He obtained the M.Sc.
degree in Computer Science and
Engineering (CSE) from the Jagannath
University, Dhaka, Bangladesh, in 2018.
He was awarded his Ph.D. degree in Engineering in the Faculty of
Engineering, Multimedia University, Malaysia in 2022. Now he is working
as post-doctoral research fellow in the Faculty of Engineering, Multimedia
University, Malaysia. He has conducted many research works in relevant
fields. His research interests are in system identification, signal processing
and control, renewable energy, power system analysis and high voltage
engineering.
GOBBI RAMASAMY received the
Bachelor degree in Electrical Engineering
from University of Technology,
Malaysia, and the Master degree in
technology management from the
National University of Malaysia, and the
Ph.D. degree in the area of torque control
of switched reluctance motors from
Multimedia University, Malaysia. He has
been associated with technical education
for more than ten years.
He was an R&D Engineer in an electronics company before became a
Lecturer in electrical and electronics engineering. He has supervised many
research projects on power electronics, variable-speed drives, automation,
and domestic electrical installations. He is a project leader and member of
various government research projects related to electric motors and drives
system. He is a consultant providing solutions for many problems associated
to electric motors and drives system for various industries. He is an associate
professor in Faculty of Engineering, Multimedia University, Malaysia.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3342024
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/
