This paper investigates the problem of output regulation of Shimizu-Morioka chaotic system, which is one of the classical chaotic systems proposed by T. Shimizu and N. Morioka (1980). Explicitly, for the constant tracking problem, new state feedback control laws regulating the output of the Shimizu-Morioka chaotic system have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Shimizu-Morioka chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Shimizu-Morioka chaotic system.
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLijait
This paper investigates output regulation of the Shimizu-Morioka chaotic system, a classical three-dimensional chaotic system, using state feedback control. The paper reviews the output regulation problem for nonlinear control systems and the regulator equations approach. It then applies this approach to solve the constant output regulation problem for the Shimizu-Morioka system. Explicit state feedback control laws are derived to regulate the first and third states to constant references by solving the regulator equations. Numerical simulations illustrate the effectiveness of the proposed control schemes.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
Output Regulation of SPROTT-F Chaotic System by State Feedback ControlAlessioAmedeo
This paper solves the output regulation problem of Sprott-F chaotic system, which is one of the classical chaotic systems discovered by J.C. Sprott (1994). Explicitly, for the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Sprott-F chaotic system. Our controller design has been carried out using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott-F chaotic system has important applications in many areas of Science and Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-F chaotic system.
ACTIVE CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF THE WANG-CHEN-YUAN SYSTEMecij
In this paper, we design an active controller for regulating the output of the Wang-Chen-Yuan system (2009), which is one of the recently discovered 3-D chaotic systems. For the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Wang-Chen-Yuan system. Numerical simulations using MATLAB are exhibited to validate and demonstrate the usefulness of the active controller design for the output regulation of the Wang-Chen-Yuan system.
Active Controller Design for Regulating the Output of the Sprott-P Systemijccmsjournal
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This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLijait
This paper investigates output regulation of the Shimizu-Morioka chaotic system, a classical three-dimensional chaotic system, using state feedback control. The paper reviews the output regulation problem for nonlinear control systems and the regulator equations approach. It then applies this approach to solve the constant output regulation problem for the Shimizu-Morioka system. Explicit state feedback control laws are derived to regulate the first and third states to constant references by solving the regulator equations. Numerical simulations illustrate the effectiveness of the proposed control schemes.
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This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
Output Regulation of SPROTT-F Chaotic System by State Feedback ControlAlessioAmedeo
This paper solves the output regulation problem of Sprott-F chaotic system, which is one of the classical chaotic systems discovered by J.C. Sprott (1994). Explicitly, for the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Sprott-F chaotic system. Our controller design has been carried out using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott-F chaotic system has important applications in many areas of Science and Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-F chaotic system.
ACTIVE CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF THE WANG-CHEN-YUAN SYSTEMecij
In this paper, we design an active controller for regulating the output of the Wang-Chen-Yuan system (2009), which is one of the recently discovered 3-D chaotic systems. For the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Wang-Chen-Yuan system. Numerical simulations using MATLAB are exhibited to validate and demonstrate the usefulness of the active controller design for the output regulation of the Wang-Chen-Yuan system.
Active Controller Design for Regulating the Output of the Sprott-P Systemijccmsjournal
This document summarizes research on controlling the output of the Sprott-P chaotic system. It begins by introducing the output regulation problem for nonlinear control systems and the solution approach of Byrnes and Isidori. It then applies this approach to regulate the output of the Sprott-P system for constant tracking problems. Explicit state feedback control laws are derived to regulate each of the three state variables x1, x2, and x3 to track a constant reference signal ω. Theorems are provided stating the control laws solve the output regulation problem for each tracking case.
HYBRID SLIDING SYNCHRONIZER DESIGN OF IDENTICAL HYPERCHAOTIC XU SYSTEMS ijitjournal
This document summarizes a research paper on using sliding mode control to achieve hybrid synchronization between identical hyperchaotic Xu systems. Hybrid synchronization means the odd states are completely synchronized while the even states are anti-synchronized. The paper derives stability results using Lyapunov theory and designs a sliding mode controller to drive the slave system states to track the master system states. Numerical simulations using MATLAB demonstrate the hybrid synchronization scheme works for identical hyperchaotic Xu systems.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
STABILITY ANALYSIS AND CONTROL OF A 3-D AUTONOMOUS AI-YUAN-ZHI-HAO HYPERCHAOT...ijscai
This document summarizes a research paper that analyzes the stability and control of a 3D autonomous hyperchaotic system using fuzzy logic modeling. The paper presents the equations that define the hyperchaotic Ai-Yuan-Zhi-Hao system. A Takagi-Sugeno fuzzy logic controller is designed to control the system. Lyapunov stability analysis is used to derive control laws for each fuzzy subsystem and conditions for asymptotic stability. Simulation results show the controlled trajectories converge when initial conditions are between -50 and 50, demonstrating controllability of the hyperchaotic attractor.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
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GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
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The document provides an overview of state-space representation of linear time-invariant (LTI) systems. It defines the state of a dynamical system and explains that the state-space approach models systems using sets of first-order differential equations rather than transfer functions. The key advantages of the state-space approach include its ability to model more complex multi-input multi-output systems and incorporate internal system behavior. Examples are provided to demonstrate how higher-order systems can be converted to state-space form by defining state variables and writing the corresponding state equations.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERSijscai
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Controllability of Linear Dynamical SystemPurnima Pandit
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The document presents an exponential state observer design for uncertain chaotic and non-chaotic systems. It introduces a class of uncertain nonlinear systems and explores the state observation problem for such systems. Using an approach with integral and differential equalities, an exponential state observer is established that guarantees global exponential stability of the error system. Numerical simulations exhibit the feasibility and effectiveness of the observer design.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
This document summarizes research on adaptive chaos control and synchronization of a hyperchaotic Liu system. The system is a 4-dimensional hyperchaotic system with unknown parameters. Adaptive control laws are designed to stabilize the hyperchaotic system to the origin based on Lyapunov stability theory. Adaptive control laws are also derived to achieve synchronization between identical hyperchaotic Liu systems with unknown parameters. Numerical simulations demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU ...ijait
This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). In the hybrid synchronization design of master and slave systems, one
part of the systems, viz. their odd states, are completely synchronized (CS), while the other part, viz. their
even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process of
synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
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Chapter 5
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Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
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Text Books:
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4. BCP, Surveying volume 1
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Output Regulation of SHIMIZU-MORIOKA Chaotic System by State Feedback Control
1. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
DOI : 10.5121/ijait.2011.1201 1
OUTPUT REGULATION OF SHIMIZU-MORIOKA
CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
This paper investigates the problem of output regulation of Shimizu-Morioka chaotic system, which is one
of the classical chaotic systems proposed by T. Shimizu and N. Morioka (1980). Explicitly, for the
constant tracking problem, new state feedback control laws regulating the output of the Shimizu-Morioka
chaotic system have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The
output regulation of the Shimizu-Morioka chaotic system has important applications in Electronics and
Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the
control schemes proposed in this paper for the output regulation of the Shimizu-Morioka chaotic system.
KEYWORDS
Chaos; feedback control; Shimizu-Morioka system; nonlinear control systems; output regulation.
1. INTRODUCTION
The output regulation problem is one of the core problems in control systems theory. Basically,
the output regulation problem is to control a fixed linear or nonlinear plant in order to have its
output tracking reference signals produced by some external generator (the exosystem). For
linear control systems, the output regulation problem has been solved by Francis and Wonham
([1], 1975). For nonlinear control systems, the output regulation problem was solved by Byrnes
and Isidori ([2], 1990) generalizing the internal model principle obtained by Francis and
Wonham [1]. Using Centre Manifold Theory [3], Byrnes and Isidori derived regulator
equations, which characterize the solution of the output regulation problem of nonlinear control
systems satisfying some stability assumptions.
The output regulation problem for nonlinear control systems has been studied extensively by
various scholars in the last two decades [4-14]. In [4], Mahmoud and Khalil obtained results on
the asymptotic regulation of minimum phase nonlinear systems using output feedback. In [5],
Fridman solved the output regulation problem for nonlinear control systems with delay using
centre manifold theory. In [6-7], Chen and Huang obtained results on the robust output
regulation for output feedback systems with nonlinear exosystems. In [8], Liu and Huang
obtained results on the global robust output regulation problem for lower triangular nonlinear
systems with unknown control direction.
In [9], Immonen obtained results on the practical output regulation for bounded linear infinite-
dimensional state space systems. In [10], Pavlov, Van de Wouw and Nijmeijer obtained results
on the global nonlinear output regulation using convergence-based controller design. In [11], Xi
and Dong obtained results on the global adaptive output regulation of a class of nonlinear
systems with nonlinear exosystems. In [12-14], Serrani, Isidori and Marconi obtained results on
the semi-global and global output regulation problem for minimum-phase nonlinear systems.
2. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
2
In this paper, we solve the output regulation problem for the Shimizu-Morioka chaotic system
([15], 1980). We derive state feedback control laws solving the constant regulation problem of
the Shimizu-Morioka chaotic system using the regulator equations of Byrnes and Isidori (1990).
The Shimizu-Morioka chaotic system is a classical three-dimensional chaotic system studied by
Shimizu and Morioka (1980). It has important applications in Electronics and Communication
Engineering.
This paper is organized as follows. In Section 2, we provide a review the problem statement of
output regulation problem for nonlinear control systems and the regulator equations of Byrnes
and Isidori [2], which provide a solution to the output regulation problem under some stability
assumptions. In Section 3, we present the main results of this paper, namely, the solution of the
output regulation problem for the Shimizu-Morioka chaotic system for the important case of
constant reference signals (set-point signals). In Section 4, we describe the numerical results
illustrating the effectiveness of the control schemes derived in Section 3 for the constant
regulation problem of the Shimizu-Morioka chaotic system. In Section 5, we summarize the
main results obtained in this paper.
2. REVIEW OF THE OUTPUT REGULATION PROBLEM FOR NONLINEAR
CONTROL SYSTEMS
In this section, we consider a multi-variable nonlinear control system described by
( ) ( ) ( )
x f x g x u p x ω
= + +
& (1a)
( )
s
ω ω
=
& (1b)
( ) ( )
e h x q ω
= − (2)
Here, the differential equation (1a) describes the plant dynamics with state x defined in a
neighbourhood X of the origin of n
R and the input u takes values in m
R subject to te effect of a
disturbance represented by the vector field ( ) .
p x ω The differential equation (1b) describes an
autonomous system, known as the exosystem, defined in a neighbourhood W of the origin of
,
k
R which models the class of disturbance and reference signals taken into consideration. The
equation (2) defines the error between the actual plant output ( ) p
h x R
∈ and a reference signal
( ),
q ω which models the class of disturbance and reference signals taken into consideration.
We also assume that all the constituent mappings o the system (1) and the error equation (2),
namely, , , , ,
f g p s h and q are continuously differentiable mappings vanishing at the origin, i.e.
(0) 0, (0) 0, (0) 0, (0) 0, (0) 0
f g p s h
= = = = = and (0) 0.
q =
Thus, for 0,
u = the composite system (1) has an equilibrium ( , ) (0,0)
x ω = with zero error (2).
A state feedback controller for the composite system (1) has the form
( )
,
u x
ρ ω
= (3)
where ρ is a continuously differentiable mapping defined on X W
× such that (0,0) 0.
ρ =
3. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
3
Upon substitution of the feedback control law (3) into (1), we get the closed-loop system
( ) ( ) ( , ) ( )
( )
x f x g x x p x
s
ρ ω ω
ω ω
= + +
=
&
&
(4)
The purpose of designing the state feedback controller (3) is to achieve both internal stability
and output regulation of the given nonlinear control system (1). Formally, we can summarize
these requirements as follows.
State Feedback Regulator Problem [2]:
Find, if possible, a state feedback control law ( )
,
u x
ρ ω
= such that the following conditions
are satisfied.
(OR1) [Internal Stability] The equilibrium 0
x = of the dynamics
( ) ( ) ( ,0)
x f x g x x
ρ
= +
&
is locally exponentially stable.
(OR2) [Output Regulation] There exists a neighbourhood U X W
⊂ × of ( )
, (0,0)
x ω =
such that for each initial condition ( )
(0), (0) ,
x U
ω ∈ the solution ( )
( ), ( )
x t t
ω of the
closed-loop system (4) satisfies
[ ]
lim ( ( )) ( ( )) 0.
t
h x t q t
ω
→∞
− =
Byrnes and Isidori [2] solved the output regulation problem stated above under the following
two assumptions.
(H1) The exosystem dynamics ( )
s
ω ω
=
is neutrally stable at 0,
ω = i.e. the exosystem
is Lyapunov stable in both forward and backward time at 0.
ω =
(H2) The pair ( )
( ), ( )
f x g x has a stabilizable linear approximation at 0,
x = i.e. if
0
x
f
A
x =
∂
=
∂
and
0
,
x
g
B
x =
∂
=
∂
then ( , )
A B is stabilizable.
Next, we recall the solution of the output regulation problem derived by Byrnes and Isidori [2].
Theorem 1. [2] Under the hypotheses (H1) and (H2), the state feedback regulator problem is
solvable if and only if there exist continuously differentiable mappings ( )
x π ω
= with (0) 0
π =
and ( )
u ϕ ω
= with (0) 0,
ϕ = both defined in a neighbourhood of 0
W W
⊂ of 0
ω = such that
the following equations (called the regulator equations) are satisfied:
4. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
4
(1) ( ) ( ( )) ( ( )) ( ) ( ( ))
s f g p
π
ω π ω π ω ϕ ω π ω ω
ω
∂
= + +
∂
(2) ( )
( ) ( ) 0
h q
π ω ω
− =
When the regulator equations (1) and (2) are satisfied, a control law solving the state feedback
regulator problem is given by
( )
( )
u K x
ϕ ω π ω
= + −
where K is any gain matrix such that A BK
+ is Hurwitz.
3. OUTPUT REGULATION OF THE SHIMIZU-MORIOKA CHAOTIC SYSTEM
In this section, we solve the output regulation problem for the Shimizu-Morioka chaotic system
([15], 1980), which is one of the paradigms of the three-dimensional chaotic systems described
by the dynamics
1 2
2 3 1 2
2
3 1 3
(1 )
x x
x x x ax u
x x bx
=
= − − +
= −
(5)
where 1 2 3
, ,
x x x are the states of the system, ,
a b are positive constant parameters of the system
and u is the scalar control.
T. Shimizu and N. Morioka ([15], 1980) showed that the system (5) has chaotic behaviour when
0.75,
a = 0.45
b = and 0.
u = The chaotic portrait of (5) is illustrated in Figure 1.
Figure 1. Chaotic Portrait of the Shimizu-Morioka System
5. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
5
In this paper, we consider the output regulation problem for the tracking of constant reference
signals (set-point signals).
In this case, the exosystem is given by the scalar dynamics
0
ω =
(6)
We note that the assumption (H1) of Theorem 1 holds trivially.
Linearizing the dynamics of the Shimizu-Morioka system (5) at 0,
x = we obtain
0 1 0
1 0
0 0
A a
b
= −
−
and
0
1 .
0
B
=
Using Kalman’s rank test for controllability ([16], p738), it can be easily seen that the pair
( , )
A B is not completely controllable. However, we can show that the pair ( , )
A B is
stabilizable, as follows.
Suppose we take [ ]
1 2 3 .
K k k k
=
Then the characteristic equation of A BK
+ is given by
2
2 1
( ) ( ) (1 ) 0
b a k k
λ λ λ
+ + − − + =
(7)
From Eq. (7), it is clear that 0
b
λ = − is always an eigenvalue of A BK
+ and that the other
two eigenvalues of A BK
+ will be also stable provided that
2 0
a k
− and 1
(1 ) 0
k
− +
or equivalently that
1 1
k − and 2 .
k a
(8)
Since 3
k does not play any role in the above calculations, we can take 3 0.
k =
Thus, we assume that
[ ]
1 2 0 ,
K k k
=
where 1
k and 2
k satisfy the inequalities (8).
Thus, the assumption (H2) of Theorem 1 also holds.
Hence, Theorem 1 can be applied to solve the constant regulation problem for the Shimizu-
Morioka system (5).
6. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
6
3.1 The Constant Tracking Problem for 1
x
Here, the tracking problem for the Shimizu-Morioka chaotic system (5) is given by
1 2
2 3 1 2
2
3 1 3
1
(1 )
x
x
x
x
x x ax u
x bx
e x ω
−
=
= − +
= −
= −
(9)
By Theorem 1, the regulator equations of the system (9) are obtained as
2 ( ) 0
π ω =
[ ]
3 1 2
1 ( ) ( ) ( ) ( ) 0
a
π ω π ω π ω ϕ ω
− − + = (10)
2
1 3
( ) ( ) 0
b
π ω π ω
− =
1( ) 0
π ω ω
− =
Solving the regulator equations (10) for the system (9), we obtain the unique solution as
2
1 2 3
( ) , ( ) 0, ( )
b
ω
π ω ω π ω π ω
= = = and
2
( ) 1
b
ω
ϕ ω ω
= −
(11)
Using Theorem 1 and the solution (11) of the regulator equations for the system (9), we obtain
the following result which provides a solution of the output regulation problem for (9).
Theorem 2. A state feedback control law solving the output regulation problem for the
Shimizu-Morioka chaotic system (9) is given by
[ ]
( ) ( ) ,
u K x
ϕ ω π ω
= + − (12)
where ( ), ( )
ϕ ω π ω are defined as in (11) and [ ]
1 2 0
K k k
= satisfy the inequalities (8).
3.2 The constant Tracking Problem for 2
x
Here, the tracking problem for the Shimizu-Morioka chaotic system (5) is given by
1 2
2 3 1 2
2
3 1 3
2
(1 )
x
x
x
x
x x ax u
x bx
e x ω
−
=
= − +
= −
= −
(13)
It is easy to show that the regulator equations for the system (13) are not solvable. Thus, by
Theorem 1, the output regulation problem is not solvable for this case.
7. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
7
3.3 The Constant Tracking Problem for 3
x
Here, the tracking problem for the Shimizu-Morioka chaotic system (5) is given by
1 2
2 3 1 2
2
3 1 3
3
(1 )
x
x
x
x
x x ax u
x bx
e x ω
−
=
= − +
= −
= −
(14)
By Theorem 1, the regulator equations of the system (9) are obtained as
2 ( ) 0
π ω =
[ ]
3 1 2
1 ( ) ( ) ( ) ( ) 0
a
π ω π ω π ω ϕ ω
− − + = (15)
2
1 3
( ) ( ) 0
b
π ω π ω
− =
3 ( ) 0
π ω ω
− =
Solving the regulator equations (15) for the system (14), we obtain the unique solution as
1 2 3
( ) , ( ) 0, ( )
b
π ω ω π ω π ω ω
= = = and ( )
( ) 1 b
ϕ ω ω ω
= − (16)
Using Theorem 1 and the solution (16) of the regulator equations for the system (14), we obtain
the following result which provides a solution of the output regulation problem for (14).
Theorem 3. A state feedback control law solving the output regulation problem for the
Shimizu-Morioka chaotic system (14) is given by
[ ]
( ) ( ) ,
u K x
ϕ ω π ω
= + − (17)
where ( ), ( )
ϕ ω π ω are defined as in (16) and [ ]
1 2 0
K k k
= satisfy the inequalities (8).
4. NUMERICAL SIMULATIONS
For simulation, the parameters are chosen as the chaotic case of the Shimizu-Morioka system,
viz. 0.75
a = and 0.45.
b =
For achieving internal stability of the state feedback regulator problem, a feedback gain matrix
K must be chosen so that A BK
+ is Hurwitz.
As noted in Section 3, 0.45
b
λ = − = − is always an eigenvalue of .
A BK
+
Suppose we wish to choose a gain matrix [ ]
1 2 0
K k k
= such that the other two
eigenvalues of A BK
+ are 3, 3.
− −
By Eq. (7), the values of 1
k and 2
k are determined by equating terms of the equation
2 2
2 1
+6 9 ( ) (1 )=0
a k k
λ λ λ λ
+ = + − − +
8. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
8
A simple calculation yields
1= 10
k − and 2 5.25.
k = −
For the numerical simulations, the fourth order Runge-Kutta method with step-size 6
10
h −
= is
deployed to solve the systems of differential equations using MATLAB.
4.1 Constant Tracking Problem for 1
x
Here, the initial conditions are taken as
1 2 3
(0) 10, (0) 5, (0) 12
x x x
= = = and 2.
ω =
The simulation graph is depicted in Figure 2 from which it is clear that the state trajectory 1( )
x t
tracks the constant reference signal 2
ω = in 7 seconds.
Figure 2. Constant Tracking Problem for 1
x
4.2 Constant Tracking Problem for 2
x
As pointed out in Section 3, the output regulation problem is not solvable for this case because
the regulator equations for this case do not admit any solution.
9. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
9
4.3 Constant Tracking Problem for 3
x
Here, the initial conditions are taken as
1 2 3
(0) 6, (0) 8, (0) 12
x x x
= = = and 2.
ω =
The simulation graph is depicted in Figure 3 from which it is clear that the state trajectory
3 ( )
x t tracks the constant reference signal 2
ω = in 11 seconds.
Figure 3. Constant Tracking Problem for 3
x
5. CONCLUSIONS
In this paper, the output regulation problem for the Shimizu-Morioka chaotic system (1980) has
been investigated in detail and a complete solution for the output regulation problem for the
Shimizu-Morioka chaotic system has been derived for the tracking of constant reference signals
(set-point signals). The state feedback control laws achieving output regulation proposed in this
paper were derived using the regulator equations of Byrnes and Isidori (1990). Numerical
simulation results were presented in detail to illustrate the effectiveness of the proposed control
schemes for the output regulation problem of Shimizu-Morioka chaotic system to track constant
reference signals.
10. International Journal of Advanced Information Technology (IJAIT) Vol. 1, No. 2, April 2011
10
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