Robust Mu-Synthesis Controller Design and Analysis for Load Frequency Control in an AC Microgrid System

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IFAC PapersOnLine 55-1 (2022) 547–554
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10.1016/j.ifacol.2022.04.090
10.1016/j.ifacol.2022.04.090 2405-8963
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2022 The Authors. This is an open access article under the CC BY-NC-ND license
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)
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-
based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
aff
ect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. T
he presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching
, etc.
arise instability issues in the microgrids [5]
. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard
(,∆)
configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling,

-Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust
−ℎ
controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust - Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization
Robust

- Synthesis Controller Design and Analysis for Load Frequency
Control in an AC Microgrid System
P Shambhu Prasad*, Alivelu M Parimi**
*Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: shambuprasad.13@gmail.com)
**Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus, Hyderabad
India. e-mail: alivelu@hyderabad.bits-pilani.ac.in)
Abstract: In this paper, a robust −ℎ controller has been designed for maintaining the frequency
regulation, in an AC microgrid, with exogenous disturbances as inputs and in the presence of parametric
uncertainties. Frequency stability has always been a core issue for seamless operation, specifically for
renewable energy-based microgrids. A robust approach to the problem is required due to the intermittent
nature of the sources, and uncertainties arise due to operational aspects. In this paper, a novel approach of
modeling the uncertainties as probability distribution functions has been adapted. Such an approach, gives
us insights into the robustness of the system, in terms of bounds and evaluating worst-case scenarios.
Investigating the structural singular values, calculating gap and υ-gap, designing the −
ℎ controller with  −  approach, and controller order reduction are other robust
aspects based on which, the robust stability and performance aspects have been analyzed. Further, the
stability studies were validated using real-time computer hardware in loop (CHIL) environment with
dSPACE RT1202. The simulation results were analyzed post real-time execution of the embedded code
generated from the MATLAB Simulink environment. The validation proved the efficacy of the proposed
controller with the modeled uncertainties and exogenous inputs.
Keywords: Robust Control, Power System Stability, State Space Modelling, -Synthesis Controller,
dSPACE RT1202
1. INTRODUCTION
The benefits of green energy have caused the rise of non-
conventional sources of energy in recent times. The economic
and ecological advantages of renewable energy have given a
path to large-scale commissioning of non-conventional
sources of energy. Microgrids are fundamentally small-scale
generating units equipped with renewable energy sources and
interfaced with power electronic-based converting units [1],
[2]. The presence of converting units, in addition to
characteristic aspects of microgrids, causes instability issues in
the system [3], [4]. Characteristic aspects such as the
intermittent nature of renewable energy sources, parametric
uncertainties, load deviations, impedance mismatching, etc.
arise instability issues in the microgrids [5]. Frequency
stability, which is one of the critical aspects of the generating
units is often deteriorated by the parametric uncertainties and
perturbations in the system [6]. In such scenarios, a robust
control design is need of the hour, to address the perturbations
and uncertainties to provide absolute solutions for flawless
operations.
1.1 Motivation and Incitement
Renewable energy-based sources are intermittent in their
behavioral aspects. The perturbations like solar irradiance for
PV modules, wind speed for wind turbine generators, etc.
affect the operational characteristics of the microgrid.
Similarly, parametric specifications change their values due to
different factors like, aging, wear, and tear, etc. which are
considered as parametric uncertainties [7]. The variability in
system specifications, affects the performance, other aspects
such as stability, reliability, etc. [8]. Investigating these factors
becomes an important aspect of stability studies. Investigating
the parametric uncertainties and perturbation include modeling
these within the dynamics of the system. They can be modeled
as structured and unstructured forms within the system
dynamics. Modeling them as structured uncertainties allows us
to investigate worst-case scenarios [9]. Figure 1 represents the
modeling of uncertainties, wherein the uncertainties are taken
out of the dynamics and arranged in linear fractional
transformation, (,∆) the form is also known as the standard
configuration.
Figure 1. Standard (,∆) configuration for modelling structured
uncertainties
Modeling the uncertainties in standard configuration form also
analyses the structured singular values in normalized form,
which are helpful to deal with when the robust stabilization

548 P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554
results are conserved. The configuration also performs well
against exogenous disturbance for which a closed feedback
structure is desired. In this paper, the parametric uncertainties
in the system specifications for an AC microgrid have been
considered, as structured uncertainties, and robust stability
investigation has been done in the presence of perturbations
and exogenous disturbances.
1.2 Literature Review
Several strategies have been reported concerning modeling the
uncertainties [10]–[14]. In [15] probability distribution-based
info-gap decision theory has been used to carry robust
modeling against uncertainties. A new approach to distribution
operation with multiple microgrids was proposed. In [16] the
effect of uncertainty caused due to time-delay and actuator
saturation, on load frequency control for an interconnected
power system was analyzed. However, in these works, the
worst-case scenarios of parametric uncertainties on the system
were not analyzed. In [17] the upper and lower bounds of
uncertainties were determined by a Lebesque-measurable
matrix, and norm reduction criterion was applied for
disturbance rejection. A novel approach of robust design
against nonlinear uncertainties by maximizing the Lipschitz
constant was proposed in [18]. A robust -synthesis controller
with exogenous disturbances is discussed in [19] in the
presence of parametric uncertainties. Other works with
uncertainties in DC microgrid and stand-alone systems were
discussed in [20], [21]. Robust control methods have also been
effective against Rate of Change of Frequency(RoCoF) and
improving inertia [22], [23]. The control methodology is also
being effective concerning conventional power systems [22].
1.3 Contributions and paper structure
The existing developments and the limitations therein
motivate us to design and analyze robust -synthesis
controllers and investigate the worst gain scenarios in terms of
bound limits. The contributions and the novelty of the paper
can be described as follows
• Modeling the uncertainties as probability distribution
functions, and evaluating the limits of upper and
lower bounds of the controller.
• Controller order reductions with the simulated
exogenous disturbances, and structured uncertainties.
• Presenting a robust methodology to investigate the
structured singular values, for the modeled system,
and validating the results with a computer hardware
simulation loop (CHIL) environment.
The paper is organized as follows: Section 2 describes the
state-space modeling, and control law formulation in section
3. The proposed controllers have been discussed in section 4,
followed by results and discussion.
2. SYSTEM DESCRIPTION
For the design of the proposed controller, an AC microgrid was
considered with renewable energy sources represented with
low order system. The proposed system is shown in Fig. 2.
Figure 2. Schematic of AC Microgrid
As shown in the figure, diesel engine generator (DEG),
microturbine (MT), wind turbine (WT), PV array (PV),
Battery energy system (BES), flywheel (FW), and fuel cell
(FC) have been interconnected to the utility grid via an electric
network. The power electronic-based converters are connected
to the sources, to interface the generating units to the grid. The
next subsection discusses the state-space modeling of the
proposed system.
2.1 State-space modeling of the system
The micro sources shown in Fig. 2 are modeled
mathematically as transfer functions models for steady-state
stability analysis as shown in Figure 3. For transfer function
representation, the models are represented by a low-order
system. For the design of the proposed controller, without loss
of generality, the schematic shown in Figure 2, can be
approximated as shown in Figure 3 neglecting the non-
linearities. The general form of first-order lag is represented as
()=
1+(1)
 is the gain and  represents the time constant for ℎ micro
source. The gain and time constant are important parameters
in evaluating the dynamic behavior of the micro sources. For
example, based on the time constant for the battery energy
storage, which is usually a few seconds, the time to charge and
discharge energy to its battery cells is calculated. The flywheel
whose time constant is less takes less time as compared to the
battery to release energy. The dynamic model is used to derive
a relationship between deviated power ∆ and deviated
frequency∆. The power imbalance ∆ can be represented as
∆=∆−∆(2)
∆ is the deviated total load power, and ∆ is the deviated
total generated power. The system frequency deviation can
now be expressed as ∆= ∆
 (3)
 is the microgrid frequency characteristic constant. The
transfer function representing the microgrid frequency
deviation and unit power deviation can be expressed as
()=∆
 =1
(1+)=1
+ (4)

P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554 549
results are conserved. The configuration also performs well
against exogenous disturbance for which a closed feedback
structure is desired. In this paper, the parametric uncertainties
in the system specifications for an AC microgrid have been
considered, as structured uncertainties, and robust stability
investigation has been done in the presence of perturbations
and exogenous disturbances.
1.2 Literature Review
Several strategies have been reported concerning modeling the
uncertainties [10]–[14]. In [15] probability distribution-based
info-gap decision theory has been used to carry robust
modeling against uncertainties. A new approach to distribution
operation with multiple microgrids was proposed. In [16] the
effect of uncertainty caused due to time-delay and actuator
saturation, on load frequency control for an interconnected
power system was analyzed. However, in these works, the
worst-case scenarios of parametric uncertainties on the system
were not analyzed. In [17] the upper and lower bounds of
uncertainties were determined by a Lebesque-measurable
matrix, and norm reduction criterion was applied for
disturbance rejection. A novel approach of robust design
against nonlinear uncertainties by maximizing the Lipschitz
constant was proposed in [18]. A robust -synthesis controller
with exogenous disturbances is discussed in [19] in the
presence of parametric uncertainties. Other works with
uncertainties in DC microgrid and stand-alone systems were
discussed in [20], [21]. Robust control methods have also been
effective against Rate of Change of Frequency(RoCoF) and
improving inertia [22], [23]. The control methodology is also
being effective concerning conventional power systems [22].
1.3 Contributions and paper structure
The existing developments and the limitations therein
motivate us to design and analyze robust -synthesis
controllers and investigate the worst gain scenarios in terms of
bound limits. The contributions and the novelty of the paper
can be described as follows
• Modeling the uncertainties as probability distribution
functions, and evaluating the limits of upper and
lower bounds of the controller.
• Controller order reductions with the simulated
exogenous disturbances, and structured uncertainties.
• Presenting a robust methodology to investigate the
structured singular values, for the modeled system,
and validating the results with a computer hardware
simulation loop (CHIL) environment.
The paper is organized as follows: Section 2 describes the
state-space modeling, and control law formulation in section
3. The proposed controllers have been discussed in section 4,
followed by results and discussion.
2. SYSTEM DESCRIPTION
For the design of the proposed controller, an AC microgrid was
considered with renewable energy sources represented with
low order system. The proposed system is shown in Fig. 2.
Figure 2. Schematic of AC Microgrid
As shown in the figure, diesel engine generator (DEG),
microturbine (MT), wind turbine (WT), PV array (PV),
Battery energy system (BES), flywheel (FW), and fuel cell
(FC) have been interconnected to the utility grid via an electric
network. The power electronic-based converters are connected
to the sources, to interface the generating units to the grid. The
next subsection discusses the state-space modeling of the
proposed system.
2.1 State-space modeling of the system
The micro sources shown in Fig. 2 are modeled
mathematically as transfer functions models for steady-state
stability analysis as shown in Figure 3. For transfer function
representation, the models are represented by a low-order
system. For the design of the proposed controller, without loss
of generality, the schematic shown in Figure 2, can be
approximated as shown in Figure 3 neglecting the non-
linearities. The general form of first-order lag is represented as
()=
1+(1)
 is the gain and  represents the time constant for ℎ micro
source. The gain and time constant are important parameters
in evaluating the dynamic behavior of the micro sources. For
example, based on the time constant for the battery energy
storage, which is usually a few seconds, the time to charge and
discharge energy to its battery cells is calculated. The flywheel
whose time constant is less takes less time as compared to the
battery to release energy. The dynamic model is used to derive
a relationship between deviated power ∆ and deviated
frequency∆. The power imbalance ∆ can be represented as
∆=∆−∆(2)
∆ is the deviated total load power, and ∆ is the deviated
total generated power. The system frequency deviation can
now be expressed as ∆= ∆
 (3)
 is the microgrid frequency characteristic constant. The
transfer function representing the microgrid frequency
deviation and unit power deviation can be expressed as
()=∆
 =1
(1+)=1
+ (4)
Figure 2. Control architecture for the proposed system
As shown, D is the damping constant and M is the inertia
constant of the microgrid respectively. Considering all
sources, the total power generation, for supplying demand load
can be expressed as
  
The power deviations can be expressed as

   
The linearized state-space model of the system shown in Fig.
3 can be obtained in the form of

 
 
 
The representation is adopted for robust control strategies,
where  is the state matrix,  is the input matrix, and  is the
output matrix. For the other variables,  is the state variable,
 is the control input signal,  is the disturbance vector, is
the measured output, and  is the controlled output. Other
variables are expressed as
      
      
  
 
The ,  and  are the disturbances in wind,
solar PV array, and load. In the robust control strategies, these
are taken as disturbance inputs. The rest of the state variables
are given by (13)-(14). The next section discuss the problem
formulation.















      

     
   
    
    
   
     
  
     


      


       



















































 


  
  
  
  
  
  













































       
3. PROBLEM FORMULATION
3.1 Structured Uncertainties
The uncertainties are modeled in structured and unstructured
form. In unstructured form, the uncertainties are perturbed in
different configurations like additive perturbation
configuration, multiplicative perturbation, etc. In structured
form, the uncertainties are taken out from the dynamics and
reconfigured in standard (upper) linear fractional
transformation form as shown in Figure 1. The uncertain block
 can be represented as [24]

In (15) 

 

 , where n represents the
dimension of the block . A set of  is defined as Δ. The 
represents scalar blocks and  represents full blocks. The
structured singular values are discussed in the next sub-
section.
3.2 Structured Singular Values
The design of -synthesis controller can be mathematically
represented as 

In (16)  is the feedback controller as shown in Fig.1 The
controller  stabilizes the plant under the influence of
exogenous disturbances and parametric uncertainties. It can be
equivalently stated as
 
Equation (17) defines the necessary and sufficient condition
for system stability in the presence of parametric uncertainties
. The uncertainties represented by (15) should have a small
value, to satisfy (17). Any uncertainty should not make 

550 P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554
 singular at any . The structured singular values
define the smallest size of uncertainty, which makes 
 singular at some frequency . For a normalized
set of structured uncertainty defined by
 
The structured singular value is defined as


 
If there is no  such that , then 
. As shown in Fig. 1, for an interconnected system , the
structured singular value, concerning  is given by


Hence, the system as described by Fig.1 is robustly stable, if
 is stable and  or
 
Equation (20) explains the condition for robust stability in the
presence of uncertainties and exogenous disturbances. The
proposed uncertainty modeling is being discussed in the next
sub-section.
3.3 Uncertainty Modelling as Probability Distribution
Function
The probability density function  for any random variable
, for a set of  real numbers, is defined as [25]
 
When  is considered as a normal random variable, with mean
 and standard deviation , the probability distribution
function is defined as


 
The distribution function when plotted around , attains a bell
shape. In this work, the parameters damping ( and inertia
constant (, which are parametric specifications for the
alternator, as shown in Figure 2, are modeled as a probability
distribution function. Modeling them as probability
distribution functions, the bounds of the norm, worst-case
scenarios, and structured singular values are evaluated, and
insights into the robustness of the system in terms of stability
and performance are investigated. In the next sub-section, the
Iterative method of obtaining the controller parameters is
being discussed.
3.4 D-K Iteration Method
The design of the robust controller is based on the D-K
iteration method. The feedback controller  is designed to
fulfill robust stability and performance criteria, which can be
expressed as 
 
The criteria (23) can be expressed as optimal control problem
as 

 
 Iteration method is used to solve (24) for a stabilizing
controller  and a diagonal constant scaling matrix 



 
Pertaining to equation (23), a stabilizing controller  is to
be found, such that

 
 
The D-K iteration, which effectively minimizes (26) is shown
as a flow chart in Figure 4. The next section deals with deriving
gap and υ-gap between a nominal and perturbed system.
3.5 Gap and υ-gap
The close loop behavior to two systems can be very close even
though the norm of the difference between the two open-loop
systems can be arbitrarily large. To characterize the distance
between two models, the norm gap  and ν-gap or
Vinnicombe gap () are evaluated. It investigates the
controller stability for a nominal system and perturbed system.
For two systems,  and, where 
and .  and  are the coprime
factorizations of  satisfying the relation 

 and




 
The gap between two systems  and  is defined as



And the ν-gap between  and  is defined by:


 
Where,
= number of unstable zeros in  – number of
unstable poles in 
A small value implies that any controller that stabilizes  will
likely stabilize , and the closed-loop gains of the two
systems would be similar.
The -gap satisfies the following properties



The ball of plants denoted by  centered on  and with
radius , is defined as
 

P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554 551
 singular at any . The structured singular values
define the smallest size of uncertainty, which makes 
 singular at some frequency . For a normalized
set of structured uncertainty defined by
 
The structured singular value is defined as


 
If there is no  such that , then 
. As shown in Fig. 1, for an interconnected system , the
structured singular value, concerning  is given by


Hence, the system as described by Fig.1 is robustly stable, if
 is stable and  or
 
Equation (20) explains the condition for robust stability in the
presence of uncertainties and exogenous disturbances. The
proposed uncertainty modeling is being discussed in the next
sub-section.
3.3 Uncertainty Modelling as Probability Distribution
Function
The probability density function  for any random variable
, for a set of  real numbers, is defined as [25]
 
When  is considered as a normal random variable, with mean
 and standard deviation , the probability distribution
function is defined as


 
The distribution function when plotted around , attains a bell
shape. In this work, the parameters damping ( and inertia
constant (, which are parametric specifications for the
alternator, as shown in Figure 2, are modeled as a probability
distribution function. Modeling them as probability
distribution functions, the bounds of the norm, worst-case
scenarios, and structured singular values are evaluated, and
insights into the robustness of the system in terms of stability
and performance are investigated. In the next sub-section, the
Iterative method of obtaining the controller parameters is
being discussed.
3.4 D-K Iteration Method
The design of the robust controller is based on the D-K
iteration method. The feedback controller  is designed to
fulfill robust stability and performance criteria, which can be
expressed as 
 
The criteria (23) can be expressed as optimal control problem
as 

 
 Iteration method is used to solve (24) for a stabilizing
controller  and a diagonal constant scaling matrix 



 
Pertaining to equation (23), a stabilizing controller  is to
be found, such that

 
 
The D-K iteration, which effectively minimizes (26) is shown
as a flow chart in Figure 4. The next section deals with deriving
gap and υ-gap between a nominal and perturbed system.
3.5 Gap and υ-gap
The close loop behavior to two systems can be very close even
though the norm of the difference between the two open-loop
systems can be arbitrarily large. To characterize the distance
between two models, the norm gap  and ν-gap or
Vinnicombe gap () are evaluated. It investigates the
controller stability for a nominal system and perturbed system.
For two systems,  and, where 
and .  and  are the coprime
factorizations of  satisfying the relation 

 and




 
The gap between two systems  and  is defined as



And the ν-gap between  and  is defined by:


 
Where,
= number of unstable zeros in  – number of
unstable poles in 
A small value implies that any controller that stabilizes  will
likely stabilize , and the closed-loop gains of the two
systems would be similar.
The -gap satisfies the following properties



The ball of plants denoted by  centered on  and with
radius , is defined as
 
Figure 4. Flow Chart for D-K Iteration
4.0 RESULTS AND DISCUSSION
To validate the effectiveness of the proposed control approach,
time-domain simulations were performed on the nonlinear
averaged model. The results are analyzed for robust stability
and performance in different sections. With each, the robust
stability norms are analyzed and an assessment is made by
plotting the singular values and by using the classical control
theory approach. The robustness of the system is also
evaluated by evaluating  gap. Then a comparison is made
with the other existing controller, on the Simulink platform.
The frequency response is analyzed for different controllers,
and suitable conclusions are made. The closed-loop structure
of the system is shown in Figure 5.  is the controller
function,  is the plant model,  is the uncertainties in
the system. and  are the input disturbances.  is
the perturbed model of the system including disturbances and
uncertainties? and  represent the error signals.
Figure 5. Block diagram for the closed-loop system
And and  being the output and input weighting
functions assigned to these error signals. They are expressed
as

 

 
  
  
  
For stability analysis, ,  and  are considered
as input disturbances to the system, and  is the measured
output fed to the controller. The output of the controller is fed
to plant model . The parametric specifications of the
system as shown in Fig. 2 are given in the following tabulation.
Table 1. Parametric Specifications
Parameter Value Parameter Value
D (pu/Hz) 0.012

2
M (pu/Hz) 0.2

2

4

1.5

0.1

1.8

0.1
The following table shows the considered parameters for the
modeling as a probability distribution function.
Table 2. Parametric Specifications
Parameter Mean Standard
Deviation
D (pu/Hz) 0.12 2
M (pu/Hz) 0.2 2
A variation of  is obtained from the nominal value with
the proposed uncertainty modeling.
4.1 Stability Analysis
For the evaluation of robust stability and performance, the
controller is designed by  method and bounds
the system is evaluated, as per equation (26). Before
synthesizing the controller, the parametric uncertainties are
modeled as a probability distribution function as per equation
(22). Then, for evaluating the norm, system transfer function
 is obtained by augmenting the uncertain system, along
with the weighting functions expressed in (31)-(33).

The modeled system was simulated on MATLAB Simulink,
using the Robust Control Toolbox. The obtained value of
bound (“bnd”) was 0.7963, which satisfies the robust stability
and performance criteria. Figure 6 shows the singular values
for the uncertainties when modeled as a probability
distribution function. With all variations, the singular values
have been found to have a magnitude less than the bound of

552 P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554
0.7963, indicating the robust performance in the presence of
exogenous disturbances.
Figure 6. Frequency Response with variations in D and M
4.2 Controller Order Reduction
One of the drawbacks of  is it results in a very
high-order controller. The order of the controller was found to
be 17, for the modeled uncertainties. The order of the
controller was reduced by the Hankel-Norm approximation
approach [24]. The reduced order of the controller was found
to be 3. The reduced-order controller transfer function derived
is  

4.3 Robustness Margins
The robustness margins were calculated for the modeled
uncertainty. Table 3 gives the robustness margins
Table 3. Robustness Margins
Stability Margins Margin
Lower Bound 1.5209
Upper Bound 1.5239
Critical Frequency 0.3956
The stability margin is relative to the uncertainty level as
defined in the uncertain system with the modeled parametric
uncertainties. A robust stability margin of greater than 1 means
the system is stable for all values of its modeled uncertainty.
A robust stability margin of less than one indicates the system
becomes unstable for some values of uncertain elements
within their specified range. For the proposed modeled system,
both the upper and lower limits of the bound are greater than
one, indicating the system is stable for all its uncertainty. A
critical frequency of 0.3956 rad/s indicates stability margin is
least at this frequency.
4.3 Worst-Case gains and Margins
The worst-case gains and margins for the system have been
evaluated. The worst-case peak gain of the system is found to
be 0.8038 with the modeled uncertainties. It refers to the
largest singular value of the frequency response matrix. One
important conclusion that can be made from this is the stability
of the modeled system with uncertainties is preserved as (34).
Table 4. Worst-Case Margins
Stability Margins Margin
Lower Bound 0.8021
Upper Bound 0.8038
Critical Frequency 0

 
4.4 Structured Singular Values
Structured singular values reveal the upper and lower bounds
of , as shown in Table 4. It can also be seen from equation
(19), that for all , modeled as parametric uncertainty with
probability distribution function, and with  less than
1.2440, the matrix , is not singular. Further,
there exists a matrix , with norm equal to 1.2467, for which
the matrix  is singular. The limits (1.2440
and 1.2467) are equivalent to the inverse of upper and lower
bounds.
4.5 Gap and υ-gap
The gap and υ-gap between the nominal system  and
perturbed system  have been tabulated below.
Table 5. Gap and υ-gap
Metrics Value
Gap 0.9864
υ-gap 0.9857
The value, close to 1, is due to the parametric uncertainties
modeled as a probability distribution function. If the value of
gap and υ-gap indicate lies close to zero, it indicates that any
controller which can stabilize  can also stabilize .
This essentially means that there is no need for any robust
control design. A value close to 1, indicates both plants are far
apart. In the proposed plant, as shown in Table 4, its value is
near to unity. This is due to the uncertainties modeled. Hence,
a robust control approach is a must for the exogenous
disturbances, and parametric uncertainties considered.
4.6 Comparative Assessment with Other Modeling
uncertainties methods
A comparative assessment with other modeling methods has
been made and presented in Table 5. As seen from the table,
when  controller is designed with additive
perturbation, it gives the norm of greater than one, which
indicates unstable for the system. The same can be concluded
for multiplicative perturbation also. The υ-gap is 0.56 and
0.65, for each case, which indicates any controller that can
stabilize the nominal system can stabilize the system with
perturbations. Its value, when near to unity, indicates difficulty
in obtaining the robust controllers, as is the case for
uncertainties modeled as probability distribution function, and
by Lebesque Measurable matrix. However, the upper and
lower bounds for Lebesque and Info-gap decision theory are

P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554 553
0.7963, indicating the robust performance in the presence of
exogenous disturbances.
Figure 6. Frequency Response with variations in D and M
4.2 Controller Order Reduction
One of the drawbacks of  is it results in a very
high-order controller. The order of the controller was found to
be 17, for the modeled uncertainties. The order of the
controller was reduced by the Hankel-Norm approximation
approach [24]. The reduced order of the controller was found
to be 3. The reduced-order controller transfer function derived
is  

4.3 Robustness Margins
The robustness margins were calculated for the modeled
uncertainty. Table 3 gives the robustness margins
Table 3. Robustness Margins
Stability Margins
Margin
Lower Bound
1.5209
Upper Bound
1.5239
Critical Frequency
0.3956
The stability margin is relative to the uncertainty level as
defined in the uncertain system with the modeled parametric
uncertainties. A robust stability margin of greater than 1 means
the system is stable for all values of its modeled uncertainty.
A robust stability margin of less than one indicates the system
becomes unstable for some values of uncertain elements
within their specified range. For the proposed modeled system,
both the upper and lower limits of the bound are greater than
one, indicating the system is stable for all its uncertainty. A
critical frequency of 0.3956 rad/s indicates stability margin is
least at this frequency.
4.3 Worst-Case gains and Margins
The worst-case gains and margins for the system have been
evaluated. The worst-case peak gain of the system is found to
be 0.8038 with the modeled uncertainties. It refers to the
largest singular value of the frequency response matrix. One
important conclusion that can be made from this is the stability
of the modeled system with uncertainties is preserved as (34).
Table 4. Worst-Case Margins
Stability Margins
Margin
Lower Bound
0.8021
Upper Bound
0.8038
Critical Frequency
0

 
4.4 Structured Singular Values
Structured singular values reveal the upper and lower bounds
of , as shown in Table 4. It can also be seen from equation
(19), that for all , modeled as parametric uncertainty with
probability distribution function, and with  less than
1.2440, the matrix , is not singular. Further,
there exists a matrix , with norm equal to 1.2467, for which
the matrix  is singular. The limits (1.2440
and 1.2467) are equivalent to the inverse of upper and lower
bounds.
4.5 Gap and υ-gap
The gap and υ-gap between the nominal system  and
perturbed system  have been tabulated below.
Table 5. Gap and υ-gap
Metrics
Value
Gap
0.9864
υ-gap
0.9857
The value, close to 1, is due to the parametric uncertainties
modeled as a probability distribution function. If the value of
gap and υ-gap indicate lies close to zero, it indicates that any
controller which can stabilize  can also stabilize .
This essentially means that there is no need for any robust
control design. A value close to 1, indicates both plants are far
apart. In the proposed plant, as shown in Table 4, its value is
near to unity. This is due to the uncertainties modeled. Hence,
a robust control approach is a must for the exogenous
disturbances, and parametric uncertainties considered.
4.6 Comparative Assessment with Other Modeling
uncertainties methods
A comparative assessment with other modeling methods has
been made and presented in Table 5. As seen from the table,
when  controller is designed with additive
perturbation, it gives the norm of greater than one, which
indicates unstable for the system. The same can be concluded
for multiplicative perturbation also. The υ-gap is 0.56 and
0.65, for each case, which indicates any controller that can
stabilize the nominal system can stabilize the system with
perturbations. Its value, when near to unity, indicates difficulty
in obtaining the robust controllers, as is the case for
uncertainties modeled as probability distribution function, and
by Lebesque Measurable matrix. However, the upper and
lower bounds for Lebesque and Info-gap decision theory are
Table 6. Comparative assessment with different uncertainty modeling
Modeling
Technique
‖‖
Lower
Bound
Upper
Bound
Worst
Case
υ-gap Remark
Additive Perturbation [24] 1.2048 1.019 1.0215 1.7656 0.56 Unstable
Multiplicative [24] 1.304 1.186 1.190 1.8 0.65 Unstable
Info Gap Decision Theory
[15]
1.146 0.9518 0.9176 1.93 0.78 Stable but not
robust
Lebesque Measurable Matrix
[17]
1.0254 1.0198 1.0219 1.308 0.85 Stable but not
robust
Probability Distribution
Function
0.79 1.5209 1.5239 0.8038 0.986 Stable and
robust
near to one, which is not a good metric for robustness. It can
be said that they are stable, but don’t satisfy robustness for the
proposed system.
4.6 dSPACE-RT1202 Results
The simulated results were validated on dSPACE-RT1202, the
real-time processor in a loop environment. The simulation
response as shown below was obtained in real-time.
Figure 7. Simulated Exogenous Disturbances
Figure 8. Frequency deviation response for −ℎ
controller
Figure 7 represents the step deviations in solar irradiation,
wind speed, and load modeled as input exogenous
disturbances. For the modeled disturbances, and parametric
uncertainties, as defined in (22), the frequency deviation
response is shown in Figure 8. The simulation response is
obtained and validated in real-time, and the output is obtained
from dSPACE control desk 7.3 post real-time execution of the
embedded code generated from the MATLAB Simulink
model. Both the response validates each other. The processor
speed of the dSPACE RT1202 is around 2 GHz, with an 800
MHz bus frequency
Figure 9. Frequency deviation response extracted from Dspace-
RT1202
5. CONCLUSIONS
In this paper, an approach was made to gain insights into the
robustness of the system, in the presence of parametric
uncertainties and exogenous disturbance inputs. The proposed
system was modeled mathematically with the state-space
approach, and parametric uncertainties were modeled as a
probability distribution function. It was found that, with the
modeled uncertainties, the designed −ℎ controller,
has satisfied the robust stability and robust performance norms
of the system. Modeling the uncertainties has allowed us to
investigate the upper and lower bounds of the uncertainties,
which can prove to be critical design input arguments for real-
time implementation. Evaluation gap and υ-gap analyses the
difference between nominal and perturbed systems. Similarly,
the worst-case margins would prove effective for analyzing the
robustness under different scenarios. A comparative
assessment with the existing techniques for modeling the
uncertainties proves that the proposed technique satisfies the
robust stability and performance criteria, for the proposed
system. The work could be extended by turning the gains of
the weighting functions, by treating the − as an

554 P Shambhu Prasad et al. / IFAC PapersOnLine 55-1 (2022) 547–554
objective function, and the controller constants as variables.
Also, investigating the role of uncertainty modeling, along
with time delays can be approached. Further, a machine
learning approach can also be used to tune the weighting
functions.
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