In this paper, a novel robust adaptive type-2 fuzzy nonsingular sliding mode controller is proposed to
stabilize the unstable periodic orbits of uncertain perturbed chaotic system with internal parameter
uncertainties and external disturbances. This letter is assumed to have an affine form with unknown
mathematical model, the type-2 fuzzy system is used to overcome this constraint. A global nonsingular
terminal sliding mode manifold is proposed to eliminate the singularity problem associated with normal
terminal sliding mode control. The proposed control law can drive system tracking error to converge to
zero in finite time. The adaptive type-2 fuzzy system used to model the unknown dynamic of system is
adjusted on-line by adaptation law deduced from the stability analysis in Lyapunov sense. Simulation
results show the good tracking performances, and the efficiently of the proposed approach.
Adaptive Type-2 Fuzzy Second Order Sliding Mode Control for Nonlinear Uncerta...rinzindorjej
In this paper, a robust adaptive type-2 fuzzy nonsingular sliding mode controller is designed to stabilize the unstable periodic orbits of uncertain perturbed chaotic system with internal parameter uncertainties and external disturbances. In Higher Order Sliding Mode Control (HOSMC),the chattering phenomena of the control effort is reduced, by using Super Twisting algorithm. Adaptive interval type-2 fuzzy systems are proposed to approximate the unknown part of uncertain chaotic system and to generate the Super Twisting signals. Based on Lyapunov criterion, adaptation laws are derived and the closed loop system stability is guaranteed. An illustrative example is given to demonstrate the effectiveness of the proposed controller.
Robust Adaptive Controller for Uncertain Nonlinear SystemsIJITCA Journal
The document presents a comparison of adaptive backstepping control and adaptive backstepping sliding mode control for uncertain nonlinear systems transformed into semi-strict feedback form. It describes the design of an adaptive backstepping controller using a recursive algorithm to stabilize the system and ensure output tracking. An adaptive backstepping sliding mode controller is also designed combining the backstepping technique with sliding mode control. Simulation results show that the adaptive backstepping sliding mode control has better tracking performance and parameter estimation ability, especially when the unknown parameters are time-varying, demonstrating its greater robustness.
Big Bang- Big Crunch Optimization in Second Order Sliding Mode ControlIJMTST Journal
In this article, Second order sliding mode with Big Bang- Big Crunch optimization technique is employed
for nonlinear uncertain system.The sliding surface describes the transient behavior of a system in sliding
mode. Frequently, PD- type sliding surface is chosen as a hyperplane in the system state space.An integral
term incorporated in the sliding surface expression that resulted in a type of PID sliding surface as hyperbolic
function for alleviating chattering effect. The sliding mode control law is derived using direct Lyapunov
stability approach and asymptotic stability is proved theoretically. Here, novel tuning scheme is introduced for
estimation of PID sliding surface coefficients, due to which it reduces the reaching time as well as disturbance
effect.The simulation results are presented to make a quantitative comparison with the traditional sliding
mode control. It is demonstrated that the proposed control law improves the tracking performance of system
dynamic model in case of external disturbances and parametric uncertainties.
Quasi sliding mode control of chaos in fractional order duffing systemkishorebingi
This document summarizes a research paper presented at the Second International Conference on Power, Control and Embedded Systems in 2014. The paper proposes a quasi-sliding mode controller to control chaos in a fractional-order Duffing system. A fractional-order Duffing system model is introduced to exhibit chaotic behavior. A sliding surface is defined, and a switching control law is designed to stabilize the system states on the sliding surface. Numerical simulations demonstrate the effectiveness of the proposed controller in eliminating chaos and stabilizing the fractional-order Duffing 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.
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.
This document describes the design methodology for first, second, and third order sliding mode control. It presents the main concepts of sliding mode control and discusses how to design controllers for single-input single-output systems of different orders. Simulation results in MATLAB and Simulink demonstrate the robustness of sliding mode control in the presence of uncertainties and its ability to drive system states and outputs to zero.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and 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 anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
Adaptive Type-2 Fuzzy Second Order Sliding Mode Control for Nonlinear Uncerta...rinzindorjej
In this paper, a robust adaptive type-2 fuzzy nonsingular sliding mode controller is designed to stabilize the unstable periodic orbits of uncertain perturbed chaotic system with internal parameter uncertainties and external disturbances. In Higher Order Sliding Mode Control (HOSMC),the chattering phenomena of the control effort is reduced, by using Super Twisting algorithm. Adaptive interval type-2 fuzzy systems are proposed to approximate the unknown part of uncertain chaotic system and to generate the Super Twisting signals. Based on Lyapunov criterion, adaptation laws are derived and the closed loop system stability is guaranteed. An illustrative example is given to demonstrate the effectiveness of the proposed controller.
Robust Adaptive Controller for Uncertain Nonlinear SystemsIJITCA Journal
The document presents a comparison of adaptive backstepping control and adaptive backstepping sliding mode control for uncertain nonlinear systems transformed into semi-strict feedback form. It describes the design of an adaptive backstepping controller using a recursive algorithm to stabilize the system and ensure output tracking. An adaptive backstepping sliding mode controller is also designed combining the backstepping technique with sliding mode control. Simulation results show that the adaptive backstepping sliding mode control has better tracking performance and parameter estimation ability, especially when the unknown parameters are time-varying, demonstrating its greater robustness.
Big Bang- Big Crunch Optimization in Second Order Sliding Mode ControlIJMTST Journal
In this article, Second order sliding mode with Big Bang- Big Crunch optimization technique is employed
for nonlinear uncertain system.The sliding surface describes the transient behavior of a system in sliding
mode. Frequently, PD- type sliding surface is chosen as a hyperplane in the system state space.An integral
term incorporated in the sliding surface expression that resulted in a type of PID sliding surface as hyperbolic
function for alleviating chattering effect. The sliding mode control law is derived using direct Lyapunov
stability approach and asymptotic stability is proved theoretically. Here, novel tuning scheme is introduced for
estimation of PID sliding surface coefficients, due to which it reduces the reaching time as well as disturbance
effect.The simulation results are presented to make a quantitative comparison with the traditional sliding
mode control. It is demonstrated that the proposed control law improves the tracking performance of system
dynamic model in case of external disturbances and parametric uncertainties.
Quasi sliding mode control of chaos in fractional order duffing systemkishorebingi
This document summarizes a research paper presented at the Second International Conference on Power, Control and Embedded Systems in 2014. The paper proposes a quasi-sliding mode controller to control chaos in a fractional-order Duffing system. A fractional-order Duffing system model is introduced to exhibit chaotic behavior. A sliding surface is defined, and a switching control law is designed to stabilize the system states on the sliding surface. Numerical simulations demonstrate the effectiveness of the proposed controller in eliminating chaos and stabilizing the fractional-order Duffing 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.
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.
This document describes the design methodology for first, second, and third order sliding mode control. It presents the main concepts of sliding mode control and discusses how to design controllers for single-input single-output systems of different orders. Simulation results in MATLAB and Simulink demonstrate the robustness of sliding mode control in the presence of uncertainties and its ability to drive system states and outputs to zero.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and 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 anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and 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 Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
This document describes the design of a type-III control loop using an explicit analytical PID tuning technique. It begins with definitions of closed-loop control systems and describes the characteristics of type-I, type-II, and type-III control loops. It then presents the conventional symmetrical optimum design criterion for PID tuning and derives equations for tuning PID controller parameters. The document proposes using a revised symmetrical optimum criterion to design a PID controller and compares it to the conventional criterion, claiming the proposed technique provides better control.
1) Control theory deals with analyzing and designing closed-loop control systems to achieve desired output behaviors.
2) The document provides examples of modeling control systems using transfer functions and state-space representations. These include modeling an RF control system and passive and active filter circuits.
3) State-space representation involves expressing higher-order differential equations as a set of first-order equations and representing the system using matrix equations that can be analyzed and simulated on computers. This allows visualization and analysis of dynamic systems.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Modern Control - Lec 03 - Feedback Control Systems Performance and Characteri...Amr E. Mohamed
The document summarizes key concepts about feedback control systems including:
- It defines the order of a system as the highest power of s in the denominator of the transfer function. First and second order systems are discussed.
- Standard test signals like impulse, step, ramp and parabolic are introduced to analyze the response of systems.
- The time response of systems has transient and steady-state components. Poles determine the transient response.
- For first order systems, the responses to unit impulse, step, and ramp inputs are derived. The step response reaches 63.2% of its final value after one time constant.
- For second order systems, the natural frequency, damping ratio, and poles are defined.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on linear stochastic processes as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
2013.06.18 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on nonlinear analysis of time series as part of the Summer School on Modern Statisitical Analysis and Computational Methods hosted by the Social Sciences Compuing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
DTC Method for Vector Control of 3-Phase Induction Motor under Open-Phase FaultRadita Apriana
Three-phase Induction Motor (IM) drives are widely used in industrial equipments. One of the
essential problems of 3-phase IM drives is their high speed and torque pulsation in the fault conditions.
This paper shows Direct Torque Control (DTC) strategy for vector control of a 3-phase IM under opencircuit
fault. The objective is to implement a solution for vector control of 3-phase IM drives which can be
also used under open-phase fault. MATLAB simulations were carried out and performance analysis is
presented.
AN INVESTIGATION ON FUZZY LOGIC CONTROLLERS (TAKAGI-SUGENO & MAMDANI) IN INVE...ijfls
The concept of controlling non-linear systems is one the significant fields in scientific researches for the purpose of which intelligent approaches can provide desirable applicability. Fuzzy systems are systems with ambiguous definition and fuzzy control is an especial type of non-linear control. Inverse pendulum system is one the most widely popular non-linear systems which is known for its specific characteristics such as being intrinsically non-linear and unsteady. Therefore, a controller is required for maintaining stability of the system Present study tries to compare the obtained results from designing fuzzy intelligent controllers in similar conditions and also identify the appropriate controller for holding the inverse pendulum in vertical position on the cart.
This document provides an overview of time-domain analysis and design of control systems. It discusses block diagrams and their simplification through reduction rules. Transient response specifications for control systems are defined, including delay time, rise time, peak time, maximum overshoot, and settling time. The transient response of second-order systems to a unit step input is analyzed. Factors that affect transient response such as damping ratio are explained. Formulas for calculating delay time and rise time of a second-order system based on its damping ratio and natural frequency are also presented.
State-Space Analysis of Control System: Vector matrix representation of state equation, State transition matrix, Relationship between state equations and high-order differential equations, Relationship between state equations and transfer functions, Block diagram representation of state equations, Decomposition Transfer Function, Kalman’s Test for controllability and observability
Design and Implementation of Sliding Mode Controller using Coefficient Diagra...IOSR Journals
This document describes the design and implementation of a sliding mode controller using the coefficient diagram method for a nonlinear spherical tank process. It begins with background on sliding mode control and introduces a second order sliding mode control algorithm. It then describes using the coefficient diagram method to design a two degree of freedom adaptive sliding mode controller that satisfies time and frequency domain specifications. Simulation and real-time results are presented comparing the sliding mode controller to a PI controller, showing the sliding mode controller has better performance in terms of settling time, overshoot, and error criteria. In conclusion, the coefficient diagram method is shown to be an effective way to optimize sliding mode controller parameters for the nonlinear spherical tank process.
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.
This document summarizes a research paper that presents a continuous sliding mode controller (CSMC) for balancing an inverted pendulum system. The CSMC improves upon integral sliding mode control (ISMC) by replacing the discontinuous term, eliminating chattering while maintaining robustness. A super twisting algorithm is used as a disturbance estimator. Simulation results demonstrate the effectiveness of the proposed CSMC at stabilizing the pendulum and depicting its robustness. The document provides mathematical modeling of the inverted pendulum system and formulates the control objective of designing a nonlinear robust controller to balance the pendulum using a continuous sliding mode approach.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on time series analysis as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
This document discusses and compares the classical/transfer function approach and the state space/modern control approach for modeling dynamical systems. The classical approach uses Laplace transforms and transfer functions in the frequency domain, while the state space approach uses matrices to represent systems of differential equations directly in the time domain. The state space approach can model nonlinear, time-varying, and multi-input multi-output systems and considers initial conditions, while the classical approach is limited to linear time-invariant single-input single-output systems. The document provides examples of modeling circuits using the state space representation.
This document provides an analysis of the time response of control systems. It defines time response as the output of a system over time in response to an input that varies over time. The time response analysis is divided into transient response, which decays over time, and steady state response. Different types of input signals are described, including step, ramp, and sinusoidal inputs. Methods for analyzing the first and second order systems are presented, including determining the transient and steady state response. Static error coefficients like position, velocity and acceleration constants are defined for different system types and inputs. Examples are provided to illustrate the analysis of first and second order systems.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
The International Journal of Computational Science, Information Technology an...rinzindorjej
The International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) is an open access peer-reviewed journal that publishes quality articles which make innovative contributions in all areas of Computational Science, Mathematical Modeling, Information Technology, Networks, Computer Science, Control and Automation Engineering. IJCSITCE is an abstracted and indexed journal that focuses on all technical and practical aspects of Scientific Computing, Modeling and Simulation, Information Technology, Computer Science, Networks and Communication Engineering, Control Theory and Automation. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on advanced techniques in computational science, information technology, computer science, chaos, control theory and automation, and establishing new collaborations in these areas.
In this paper, a new 3D chaotic system is controlled by generalized backstepping method. Generalized
backstepping method is similarity to backstepping method but generalized backstepping method is more
applications in systems than it. Backstepping method is used only to strictly feedback systems but
generalized backsteppingmethod expand this class. New 3D chaotic system is controlled in two participate
sections; stabilization and tracking reference input. Numerical simulations are presented to demonstrate
the effectiveness of the controlschemes.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and 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 Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
This document describes the design of a type-III control loop using an explicit analytical PID tuning technique. It begins with definitions of closed-loop control systems and describes the characteristics of type-I, type-II, and type-III control loops. It then presents the conventional symmetrical optimum design criterion for PID tuning and derives equations for tuning PID controller parameters. The document proposes using a revised symmetrical optimum criterion to design a PID controller and compares it to the conventional criterion, claiming the proposed technique provides better control.
1) Control theory deals with analyzing and designing closed-loop control systems to achieve desired output behaviors.
2) The document provides examples of modeling control systems using transfer functions and state-space representations. These include modeling an RF control system and passive and active filter circuits.
3) State-space representation involves expressing higher-order differential equations as a set of first-order equations and representing the system using matrix equations that can be analyzed and simulated on computers. This allows visualization and analysis of dynamic systems.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Modern Control - Lec 03 - Feedback Control Systems Performance and Characteri...Amr E. Mohamed
The document summarizes key concepts about feedback control systems including:
- It defines the order of a system as the highest power of s in the denominator of the transfer function. First and second order systems are discussed.
- Standard test signals like impulse, step, ramp and parabolic are introduced to analyze the response of systems.
- The time response of systems has transient and steady-state components. Poles determine the transient response.
- For first order systems, the responses to unit impulse, step, and ramp inputs are derived. The step response reaches 63.2% of its final value after one time constant.
- For second order systems, the natural frequency, damping ratio, and poles are defined.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on linear stochastic processes as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
2013.06.18 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on nonlinear analysis of time series as part of the Summer School on Modern Statisitical Analysis and Computational Methods hosted by the Social Sciences Compuing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
DTC Method for Vector Control of 3-Phase Induction Motor under Open-Phase FaultRadita Apriana
Three-phase Induction Motor (IM) drives are widely used in industrial equipments. One of the
essential problems of 3-phase IM drives is their high speed and torque pulsation in the fault conditions.
This paper shows Direct Torque Control (DTC) strategy for vector control of a 3-phase IM under opencircuit
fault. The objective is to implement a solution for vector control of 3-phase IM drives which can be
also used under open-phase fault. MATLAB simulations were carried out and performance analysis is
presented.
AN INVESTIGATION ON FUZZY LOGIC CONTROLLERS (TAKAGI-SUGENO & MAMDANI) IN INVE...ijfls
The concept of controlling non-linear systems is one the significant fields in scientific researches for the purpose of which intelligent approaches can provide desirable applicability. Fuzzy systems are systems with ambiguous definition and fuzzy control is an especial type of non-linear control. Inverse pendulum system is one the most widely popular non-linear systems which is known for its specific characteristics such as being intrinsically non-linear and unsteady. Therefore, a controller is required for maintaining stability of the system Present study tries to compare the obtained results from designing fuzzy intelligent controllers in similar conditions and also identify the appropriate controller for holding the inverse pendulum in vertical position on the cart.
This document provides an overview of time-domain analysis and design of control systems. It discusses block diagrams and their simplification through reduction rules. Transient response specifications for control systems are defined, including delay time, rise time, peak time, maximum overshoot, and settling time. The transient response of second-order systems to a unit step input is analyzed. Factors that affect transient response such as damping ratio are explained. Formulas for calculating delay time and rise time of a second-order system based on its damping ratio and natural frequency are also presented.
State-Space Analysis of Control System: Vector matrix representation of state equation, State transition matrix, Relationship between state equations and high-order differential equations, Relationship between state equations and transfer functions, Block diagram representation of state equations, Decomposition Transfer Function, Kalman’s Test for controllability and observability
Design and Implementation of Sliding Mode Controller using Coefficient Diagra...IOSR Journals
This document describes the design and implementation of a sliding mode controller using the coefficient diagram method for a nonlinear spherical tank process. It begins with background on sliding mode control and introduces a second order sliding mode control algorithm. It then describes using the coefficient diagram method to design a two degree of freedom adaptive sliding mode controller that satisfies time and frequency domain specifications. Simulation and real-time results are presented comparing the sliding mode controller to a PI controller, showing the sliding mode controller has better performance in terms of settling time, overshoot, and error criteria. In conclusion, the coefficient diagram method is shown to be an effective way to optimize sliding mode controller parameters for the nonlinear spherical tank process.
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.
This document summarizes a research paper that presents a continuous sliding mode controller (CSMC) for balancing an inverted pendulum system. The CSMC improves upon integral sliding mode control (ISMC) by replacing the discontinuous term, eliminating chattering while maintaining robustness. A super twisting algorithm is used as a disturbance estimator. Simulation results demonstrate the effectiveness of the proposed CSMC at stabilizing the pendulum and depicting its robustness. The document provides mathematical modeling of the inverted pendulum system and formulates the control objective of designing a nonlinear robust controller to balance the pendulum using a continuous sliding mode approach.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on time series analysis as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
This document discusses and compares the classical/transfer function approach and the state space/modern control approach for modeling dynamical systems. The classical approach uses Laplace transforms and transfer functions in the frequency domain, while the state space approach uses matrices to represent systems of differential equations directly in the time domain. The state space approach can model nonlinear, time-varying, and multi-input multi-output systems and considers initial conditions, while the classical approach is limited to linear time-invariant single-input single-output systems. The document provides examples of modeling circuits using the state space representation.
This document provides an analysis of the time response of control systems. It defines time response as the output of a system over time in response to an input that varies over time. The time response analysis is divided into transient response, which decays over time, and steady state response. Different types of input signals are described, including step, ramp, and sinusoidal inputs. Methods for analyzing the first and second order systems are presented, including determining the transient and steady state response. Static error coefficients like position, velocity and acceleration constants are defined for different system types and inputs. Examples are provided to illustrate the analysis of first and second order systems.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
The International Journal of Computational Science, Information Technology an...rinzindorjej
The International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) is an open access peer-reviewed journal that publishes quality articles which make innovative contributions in all areas of Computational Science, Mathematical Modeling, Information Technology, Networks, Computer Science, Control and Automation Engineering. IJCSITCE is an abstracted and indexed journal that focuses on all technical and practical aspects of Scientific Computing, Modeling and Simulation, Information Technology, Computer Science, Networks and Communication Engineering, Control Theory and Automation. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on advanced techniques in computational science, information technology, computer science, chaos, control theory and automation, and establishing new collaborations in these areas.
In this paper, a new 3D chaotic system is controlled by generalized backstepping method. Generalized
backstepping method is similarity to backstepping method but generalized backstepping method is more
applications in systems than it. Backstepping method is used only to strictly feedback systems but
generalized backsteppingmethod expand this class. New 3D chaotic system is controlled in two participate
sections; stabilization and tracking reference input. Numerical simulations are presented to demonstrate
the effectiveness of the controlschemes.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and 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 Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
This document presents a second order integral sliding mode control approach for speed control of a DC motor system. The DC motor system is modeled as a second order system with uncertainty and disturbances. Three controllers are designed and compared - a second order integral sliding mode controller, a conventional sliding mode controller, and a PID controller. Simulation results show that the proposed second order integral sliding mode controller has the fastest response time, no overshoot, smooth control input, and drives the sliding surface to zero, performing better than the other controllers in terms of robustness and disturbance rejection.
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.
Back-Stepping Control of Free-Floating Space Robot based on Adaptive Neural N...TELKOMNIKA JOURNAL
Trajectory tracking control problems of the free-floating space robot are considered by the paper,
back-stepping control method based on adaptive neural network is put forward. The complex system is
decomposed into several simple sub-systems. The control laws are designed by derived, so that closedloop
stability can be obtained by each subsystem; Because of the influence of interference and the
measurement level limitation, accurate mathematical model is difficult to be obtained. Neural network
controller of good nonlinear approximation ability is designed to compensate the uncertainty of system
model. Adaptive learning laws are designed to ensure that weights can be adjusted online real-time. The
system uniformly ultimately bounded (UUB) is proved based on the Lyapunov theory. Simulation
experiments show that the control method can fast track the desired trajectory, and has a good application
value for space robotic manipulators with uncertainty.
Integral Backstepping Sliding Mode Control of Chaotic Forced Van Der Pol Osci...ijctcm
ABSTRACT
Forced Van der Pol oscillator exhibits chaotic behaviour and instability under certain parameters and this poses a great threat to the systems where it has been applied hence, the need to develop a control method to stabilize and control chaos in a Forced Van der Pol oscillator so as to avoid damage in the controlled system and also to prevent unmodeled dynamics from being introduced in the system. Sliding Mode control makes use of the regulatory variables derived from the controlled Lyapunov function to bring the new variables to stability. The essence of using Integral Backstepping was to prevent chattering which can occur in the control input and can cause instability to the system by igniting unmodeled dynamics. Simulation was done using MATLAB and the results were provided to show the effectiveness of the proposed control method. Integral Backstepping Sliding Mode control method was effective towards stability and chaos control. It was also robust towards matched and unmatched disturbance.
Controller Design Based On Fuzzy Observers For T-S Fuzzy Bilinear Models ijctcm
This article is devoted to the design of a fuzzy-observer-based control for a class of nonlinear systems with bilinear terms. The class of systems considered is the Takagi-Sugeno (T-S) fuzzy bilinear model. A new procedure to design the observer-based fuzzy controller for this class of systems is proposed. The aim is to design the fuzzy controller and the fuzzy observer of the augmented system separately in order to guarantee that the error between the state and its estimation converges faster to zero. By using the Lyapunov function, sufficient conditions are derived such that the closed-loop system is globally asymptotically stable. Moreover, sufficient conditions for controller design based on state estimation for robust stabilization of TS FBS with parametric uncertainties is proposed. The observer and controller gains are obtained by solving some linear matrix inequalities (LMIs). An example is given to illustrate the effectiveness of the proposed approach.
An Exponential Observer Design for a Class of Chaotic Systems with Exponentia...ijtsrd
In this paper, a class of generalized chaotic systems with exponential nonlinearity is studied and the state observation problem of such systems is explored. Using differential inequality with time domain analysis, a practical state observer for such generalized chaotic systems is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to demonstrate the validity, effectiveness, and correctness of the obtained result. Yeong-Jeu Sun "An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38233.pdf Paper URL : https://www.ijtsrd.com/engineering/electrical-engineering/38233/an-exponential-observer-design-for-a-class-of-chaotic-systems-with-exponential-nonlinearity/yeongjeu-sun
Hierarchical robust fuzzy sliding mode control for a class of simo under-actu...TELKOMNIKA JOURNAL
The development of the algorithms for single input multi output (SIMO) under-actuated systems with mismatched uncertainties is important. Hierarchical sliding-mode controller (HSMC) has been successfully employed to control SIMO under-actuated systems with mismatched uncertainties in a hierarchical manner with the use of sliding mode control. However, in such a control scheme, the chattering phenomenon is its main disadvantage. To overcome the above disadvantage, in this paper, a new compound control scheme is proposed for SIMO under-actuated based on HSMC and fuzzy logic control (FLC). By using the HSMC approach, a sliding control law is derived so as to guarantee the stability and robustness under various environments. The FLC as the second controller completely removes the chattering signal caused by the sign function in the sliding control law. The results are verified through theoretical proof and simulation software of MATLAB through two systems Pendubot and series double inverted pendulum.
In this paper, the tracking control scheme is presented using the framework of finite-time sliding mode control (SMC) law and high-gain observer for disturbed/uncertain multi-motor driving systems under the consideration multi-output systems. The convergence time of sliding mode control is estimated in connection with linear matrix inequalities (LMIs). The input state stability (ISS) of proposed controller was analyzed by Lyapunov stability theory. Finally, the extensive simulation results are given to validate the advantages of proposed control design.
this is the basic slide for sliding mode controller and how it works. for, any control engineer this this the most important technique to control the non linearity of a system and bring it back to a aymptotically stable system.
Investigation of auto-oscilational regimes of the system by dynamic nonlinear...IJECEIAES
The paper proposes a method for the analysis and synthesis of self-oscillations in the form of a finite, predetermined number of terms of the Fourier series in systems reduced to single-loop, with one element having a nonlinear static characteristic of an arbitrary shape and a dynamic part, which is the sum of the products of coordinates and their derivatives. In this case, the nonlinearity is divided into two parts: static and dynamic nonlinearity. The solution to the problem under consideration consists of two parts. First, the parameters of self-oscillations are determined, and then the parameters of the nonlinear dynamic part of the system are synthesized. When implementing this procedure, the calculation time depends on the number of harmonics considered in the first approximation, so it is recommended to choose the minimum number of them in calculations. An algorithm for determining the self-oscillating mode of a control system with elements that have dynamic nonlinearity is proposed. The developed method for calculating self-oscillations is suitable for solving various synthesis problems. The generated system of equations can be used to synthesize the parameters of both linear and nonlinear parts. The advantage is its versatility.
Discretizing of linear systems with time-delay Using method of Euler’s and Tu...IJERA Editor
Delays deteriorate the control performance and could destabilize the overall system in the theory of discretetime
signals and dynamic systems. Whenever a computer is used in measurement, signal processing or control
applications, the data as seen from the computer and systems involved are naturally discrete-time because a
computer executes program code at discrete points of time. Theory of discrete-time dynamic signals and systems
is useful in design and analysis of control systems, signal filters, state estimators and model estimation from
time-series of process data system identification. In this paper, a new approximated discretization method and
digital design for control systems with delays is proposed. System is transformed to a discrete-time model with
time delays. To implement the digital modeling, we used the z-transfer functions matrix which is a useful model
type of discrete-time systems, being analogous to the Laplace-transform for continuous-time systems. The most
important use of the z-transform is for defining z-transfer functions matrix is employed to obtain an extended
discrete-time. The proposed method can closely approximate the step response of the original continuous timedelayed
control system by choosing various of energy loss level. Illustrative example is simulated to demonstrate
the effectiveness of the developed method.\
T-S Fuzzy Observer and Controller of Doubly-Fed Induction GeneratorIJPEDS-IAES
This paper aims to ensure a stability and observability of doubly fed induction generator DFIG of a wind turbine based on the approach of fuzzy control type T-S PDC (Parallel Distributed Compensation) which determines the control laws by return state and fuzzy observers.First, the fuzzy TS model is used to precisely represent a nonlinear model of DFIG proposed and adopted in this work. Then, the stability analysis is based on the quadratic Lyapunov function to determine the gains that ensure the stability conditions.The fuzzy observer of DFIG is built to estimate non-measurable state vectors and the estimated states converging to the actual statements. The gains of observatory and of stability are obtained by solving a set of linear matrix inequality (LMI).Finally, numerical simulations are performed to verify the theoretical results and demonstrate satisfactory performance.
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.
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.
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROLijccmsjournal
This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu, Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while their even states are antisynchronized. The stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Liu systems have been proved 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 anti- synchronization of the identical hyperchaotic Liu systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Liu systems.
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROLijccmsjournal
This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu,
Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems,
the odd states of the two systems are completely synchronized, while their even states are anti-
synchronized. The stability results derived in this paper for the hybrid synchronization of identical
hyperchaotic Liu systems have been proved 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 anti- synchronization of the identical hyperchaotic Liu systems. Numerical
simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper
for the identical hyperchaotic Liu systems.
On tracking control problem for polysolenoid motor model predictive approach IJECEIAES
The Polysolenoid Linear Motor (PLM) have been playing a crucial role in many industrial aspects due to its functions, in which a straight motion is provided directly without mediate mechanical actuators. Recently, with several commons on mathematic model, some control methods for PLM based on Rotational Motor have been applied, but position, velocity and current constraints which are important in real systems have been ignored. In this paper, position tracking control problem for PLM was considered under state-independent disturbances via min-max model predictive control. The proposed controller forces tracking position errors converge to small region of origin and satisfies state including position, velocity and currents constraints. Further, a numerical simulation was implemented to validate the performance of the proposed controller.
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-
offs in the intensity
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metrics such as the shortest paths, computation time stamps obtained eventually and the planne
d
schedules.
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In this paper we consider an approach to optimiz
e
manufacturing elements of emitter
-
coupled logic. The
optimization
leads
to decreas
ing
dimensions of these elements.
Framework this pa
per we consider man
u-
facturing
these
elements of emitter
-
coupled logic
framework
a heterostructure with required configur
a
tion
.
After grown the heterostructure
the
required areas
have been doped
by diffusion or ion implant
a
tion
.
It is
attracted an interest
optimization of annealing of dopant and/or radiation defects
to decrease dimensions
of the considered elements
.
We co
n
sider
an approach
to make the optimization.
The modeling part based
on modified method of functional corrections, which
gives us possibili
ty to analyzed manufacturing the
elements of emitter
-
coupled logic
without crosslinking of solutions on interfaces between layers of heter
o-
stru
c
ture
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CHAOS CONTROL VIA ADAPTIVE INTERVAL TYPE-2 FUZZY NONSINGULAR TERMINAL SLIDING MODE CONTROL
1. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
19
CHAOS CONTROL VIA ADAPTIVE INTERVAL TYPE-2
FUZZY NONSINGULAR TERMINAL SLIDING MODE
CONTROL
Rim Hendel1
, Farid Khaber1
and Najib Essounbouli2
1
QUERE Laboratory, Engineering Faculty, University of Setif 1, 19000 Setif, Algeria
2
CReSTIC of Reims Champagne-Ardenne University, IUT de Troyes, France
ABSTRACT
In this paper, a novel robust adaptive type-2 fuzzy nonsingular sliding mode controller is proposed to
stabilize the unstable periodic orbits of uncertain perturbed chaotic system with internal parameter
uncertainties and external disturbances. This letter is assumed to have an affine form with unknown
mathematical model, the type-2 fuzzy system is used to overcome this constraint. A global nonsingular
terminal sliding mode manifold is proposed to eliminate the singularity problem associated with normal
terminal sliding mode control. The proposed control law can drive system tracking error to converge to
zero in finite time. The adaptive type-2 fuzzy system used to model the unknown dynamic of system is
adjusted on-line by adaptation law deduced from the stability analysis in Lyapunov sense. Simulation
results show the good tracking performances, and the efficiently of the proposed approach.
KEYWORDS
Chaotic System, Type-2 Fuzzy Logic System, Nonsingular Terminal Sliding Mode Control, Lyapunov
Stability.
1. INTRODUCTION
Chaos is a particular case of nonlinear dynamics that has some specific characteristics such as
extraordinary sensitivity to initial conditions and system parameter variations. The study of chaos
can be introduced in several applications as: medical field, fractal theory, electrical circuits and
secure communication [1]. Nowadays, the scientific community has identified two problems in
chaos control: suppression and synchronization. The chaos suppression problem can be defined as
the stabilization of unstable periodic orbits (UPO's) of a chaotic attractor in equilibrium points or
periodic orbits with period n embedded into the chaotic attractor [2]. Many nonlinear control
techniques have been applied for chaos elimination and chaos synchronization such as linear and
nonlinear control techniques based on feedback [3-6], variable structure control [7-8], nonlinear
control [9-11], active control [12-14], backstepping design [15-17], fuzzy logic control [18-19],
and adaptive control [20-21].
Unfortunately, most of the above approaches mentioned have not considered the uncertainties and
unknown parameters of the chaotic system, internal and external disturbances. Then, a useful and
effective control scheme to deal with uncertainties, time varying properties, nonlinearities and
bounded externals disturbances is the sliding mode control (SMC). Since then, different
controllers based on sliding mode control schemes have been proposed to control chaotic systems
[22-23]
However, its major drawback in practical applications is the chattering problem. A lot of works
have proceeded to solve this problem by using adaptive control [24-26], intelligent approach [27-
29], and higher order sliding mode control [30]. In general, the sliding surface is designed as a
2. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
20
linear dynamic equation s e ce= +& . However, the linear sliding surface can only guarantee the
asymptotic error convergence in the sliding mode, i.e., the output error cannot converge to zero in
finite time. The terminal sliding mode TSM has a nonlinear surface pq
ees β+= & , while reaching
the terminal sliding mode, the system tracking error can be converged to zero in finite time.
Furthermore, TSM controller design methods have a singularity problem. Moreover, the known
bounds of uncertainties is required. Based on TSM, some nonsingular terminal sliding mode
(NTSM) control systems have been proposed to avoid the singularity in TSM [31-33].
The objective of this paper is to force the n-dimensional chaotic system to a desired state even if
it has uncertainties system, external and internal disturbances, by incorporation the fuzzy type-2
approach and nonsingular terminal sliding mode (NTSM) control. We introduced an adaptive
type-2 fuzzy system for model the unknown dynamic of system, and we use boundary layer
method to avoid a chattering phenomenon.
The organization of this paper is as follows. After a description of system and problem
formulation in section II, the adaptive type-2 fuzzy nonsingular terminal sliding mode control
scheme is presented in section III. Simulation example demonstrate the efficiently of the
proposed approach in section IV. Finally, section V gives the conclusions of the advocated design
methodology.
2. DESCRIPTION OF SYSTEM AND PROBLEM FORMULATION
Consider n-order uncertain chaotic system which has an affine form:
++∆+=
−≤≤=
−−
+
,)()(),(),(
,11,1
tutdtxftxfx
nixx
n
ii
&
&
(1)
where 1 2[ ( ) ( ) ( )] n
nx x t x t x t= … ∈ℜ is the measurable state vector, ( , )f x t is unknown nonlinear
continuous and bounded function, ( )u t ∈ℜ is control input of the system, ( )d t is the external
bounded disturbance, and ( , )f x tD represents the uncertainties,
df tdtxfFtxf ∆≤∆≤∆< )(,),(,),( (2)
where fF ∆, and d∆ are positive constants.
The control problem is to get the system to track an n- dimensional desired vector ( )dy t which
belong to a class of continuous functions on 0[ , ]t ∞ . Let’s the tracking error as;
)](...)()([
])()()()()()([
)()()(
)1(
)1()1(
tetete
tytxtytxtytx
tytxte
n
n
d
n
dd
d
−
−−
−−−
=
−−−=
−=
&
K&& (3)
Therefore, the dynamic errors of system can be obtained as;
3. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
21
++∆+−=
=
=
)()(),()(),(
,
)(
32
21
tutdtxftytxfe
ee
ee
n
dn&
M
&
&
(4)
The control goal considered is that;
lim ( ) lim ( ) ( ) 0,d
t t
e t x t y t
→∞ →∞
= − → (5)
2.1. Terminal Sliding Mode Control
We consider a second order nonlinear system (1), the conventional TSM is described by the
following first order terminal sliding variable;
q p
s e eβ= +& (6)
where 0β > is a design constant, and ( ), ,p q p q> are positive odd integers. The sufficient
condition to ensure the transition trajectory of the tracking error from approaching phase to the
sliding one is:
),(),(),(),(
2
1 2
testestestes
dt
d
−−−−
−≤= η& (7)
where 0η > is a constant.
If ( , )f x t is known and free of uncertainties and external disturbances, and when the system (1)
is restricted to the ( , ) 0s e t = , it will be governed by an equivalent control equ obtained by:
+−−= −
−
ee
p
q
ytxfu pq
deq &&& 1
),( β (8)
The global control is composed of the equivalent control and discontinuous term, such that;
sgn( )dis su k s= − (9)
where ( 0)s sk k > is switching gain, by adding this term to (8), we obtain the global control:
++−−= −
−
)(),( 1
ssignkee
p
q
ytxfu s
pq
dTSM &&& β (10)
which ensures that TSM occurs. Then, we can choose switching gain as follows:
sk Dη= + (11)
Where fdD ∆+∆= . If (0) 0s ≠ , it’s clear that the tracking errors will reach the sliding mode
( 0s = ) within the finite time rt , which satisfies;
(0)
r
s
t
η
≤ (12)
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22
Suppose the attaining time is st from ( ) 0re t ≠ to ( ) 0s re t t+ = . In this phase, the sliding mode
( )0s = is reached, i.e., the system dynamics is determined by the following nonlinear
differential equation:
0=+ pq
ee β& (13)
By integrating the differential equation pq
ee β−=& , we have:
( )
p
q
rs te
qp
p
t
−
−
=
1
)(
β
(14)
From TSM control (10), the term containing ee pq
&1−
may cause a singular problem.
2.2. Non Singular Terminal Sliding Mode Control
In order to overcome the singularity problem in the conventional TSM systems, the proposed
NTSM model is described as follows:
qp
ees &
β
1
+= (15)
where ,qβ and ( )1 2p p q< < have been defined in (6). For system (1) with the nonsingular
sliding mode manifold (15), the control is designed as;
++−−= −
−
)()(),( 2
ssignke
p
q
tytxfu s
qp
dNTSM &&& β (16)
Thus to satisfy the transition condition (7), the time derivative of s is:
( )
11
( , ) ( ) ( ) ( , ) ( )
p
q
d
p
s e e f x t y t u t f x t d t
qβ
−
= + − + + ∆ +& & & && (17)
Using control law (16),
( )
11
sgn( ) ( , ) ( )
p
q
s
p
s e k s f x t d t
qβ
−
= − + ∆ +& &
After some manipulations, we obtain:
( )
1
1
1
sgn( )
1
p
q
s f d
p
q
p
ss e k s s
q
p
e s
q
β
η
β
−
−
≤ − + ∆ + ∆
≤ −
& &
&
(18)
Since 0β > , p and q are positive odd integers ( )1 2p q< < , we have 1
0p q
e −
>& (when 0e ≠& ),
then;
1
( )
,1
( ) 0
p
q
ss e s
p
e e
q
ρ
ρ η
β
−
≤ −
= >
& &
& &
for 0e ≠& (19)
Therefore, the condition for Lyapunov stability is satisfied when 0e ≠& , and the tracking errors
can reach the sliding mode s=0 within finite time. Substituting the control (16) into system (4)
yields;
5. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
23
2
( , ) ( ) sgn( )
p
q
s
q
e e f x t d t k s
p
β
−
= − + ∆ + −&& &
When 0e =& , we obtain ,
( , ) ( ) sgn( )e f x t d t D sη= ∆ + − +&& ,
0
0
e for s
e for s
η
η
≤ − >
≥ <
&&
&&
Then, for a small ( 0),ε ε > there exists a vicinity of 0e =& , such that e ε<& , therefore, it is
concluded that the NTSM manifold (15) s=0 can be reached in the phase plane in finite time.
Note that in control law (16), the nonlinear function ( , )f x t is unknown. Then, the purpose of
this paper is to approximate ( , )f x t by interval type-2 fuzzy logic system and to eliminate
chattering, a saturation function can be used to replace the sign function in switching term. The
adaptation law of adjustable parameter of the fuzzy system is deduced from the Lyapunov
stability.
3. ADAPTIVE INTERVAL TYPE-2 FUZZY NON-SINGULAR TERMINAL SLIDING
MODE CONTROL
In this section, the adaptive fuzzy system used to approximate the unknown function ( , )f x t has
the same structure as the output fuzzy system using the center of set method [34], then we replace
( , )f x t by ),(ˆ
f
xf
−−
θ , such as:
)(),(ˆ
−−−−−
= xxf
f
T
ff
ξθθ (20)
where
f−
θ is adjustable vector parameters.
In order to guarantee the global stability of closed loop system (1) with the convergence of
tracking error to zero, we propose the following control law:
++−−= −
−−
)()(),(ˆ 2
ssignke
p
q
tyxfu s
qp
d
f
NTSM &&& βθ (21)
To derive the adaptive laws of fq , we define the optimal parameter vector *
f−
θ as;
,),(),(ˆsupargmin
*
−=
−Ω∈Ω∈−
txfxf
fxf xff
θθ
θ
where fΩ and xΩ are constraint sets of suitable bounds on fθ and x , respectively, they are
defined as;
{ },:,: xxf
ff
f MxxM ≤=Ω
≤=Ω
−−
θθ
where fM and xM are positive constants.
We define the minimum approximation error as;
6. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
24
),(ˆ)t,(
*
f
xfxfw
−−−
−= θ
We can write,
f
f
T
f
MF
xxfw
−≤
−≤
−−−−
)()t,( *
ξθ
By using α=− fMF , it can be easily concluded that w is bounded ,w α≤ (i.e. w L∞∈ ).
To study the closed loop stability and to find the adaptation law of adjustable parameter, we
consider the following Lyapunov function:
f
T
f
f
f
ssV
−−−
+= θθ
γ
θ
~~
2
1
2
1
)
~
,( 2
(22)
where ,
~ *
fff −−−
−= θθθ and fγ is arbitrary positive constant, so the time derivative of (22) is:
f
T
f
f
ssV
−−
+= θθ
γ
&&& ~1
(23)
Using the control law (21), and (20), the time derivative of the NTSM manifold (15) becomes:
))(),()sgn(),(ˆ),(ˆ),(ˆ),(()(
)(),()sgn(),(ˆ),()(
**
tdtxfskxfxfxftxfe
tdtxfskxftxfes
sff
f
s
f
+∆+−−+−′=
+∆+−−′=
−−−−−−−−
−−−−
θθθρ
θρ
&
&&
))(),()sgn()()(()(
*
tdtxfskxwes sff
f
+∆+−−−′=
−−−−
ξθθρ && (24)
such that
11
( )
p
qp
e e
q
ρ
β
−
′ =& & .
The substitution of (24) in (23) will be:
( )
′−++∆+−′=
−−−−
)()(
~1
)(),()sgn()( xestdtxfskwesV ff
ff
f
s ξργθθ
γ
ρ &&&& (25)
By choosing the following adaptation law:
)()( xes
f
f
f −−
′= ξργθ && (26)
where
ff −−
=θθ &&~
, therefore, we obtain:
( )
( )
( )
( ) sgn( ) ( , ) ( )
( ) ( , ) ( )
( )
s
s
s
V s e w k s f x t d t
e sw k s s f x t sd t
e w k D s
ρ
ρ
ρ
′= − + ∆ +
′= − + ∆ +
′= − +
& &
&
&
7. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
25
Then,
( )
( )
( )
( )
V e w s
e s
ρ η
ρ α η
′≤ −
′≤ −
& &
&
(27)
From the universal approximation theorem, it is expected that α will be very small (if not equal
to zero) in the adaptive fuzzy system, and ( ) 0eρ′ >& . So, we have 0.V <&
The overall scheme of the adaptive type-2 fuzzy nonsingular terminal sliding mode control in
presence of uncertainties, external disturbance and the training data is corrupted with internal
noise is shown in Figure 1.
Figure1. Overall scheme of the adaptive type-2 fuzzy nonsingular terminal sliding mode control system.
4. SIMULATION EXAMPLE
The above described control scheme is now used to control the states of chaotic system which is
defined as follows;
1 2
3
2 2 1 1
,
0.4 1.1 2.1cos(1.8 )
x x
x x x x t
=
= − − − −
&
&
(28)
With initial states: (0) [0.1 0]T
x = .
For free input, the simulation results of system are shown in Figure 2.
Chaotic system
1
1 2
, 1 1,
( , ) ( , ) ( ) ( ),
[ ( ) ( ) ( )]
i i
n
n
n
x x i n
x f x t f x t d t u t
x x t x t x t
+= ≤ ≤ −
= +∆ + +
= … ∈ℜ
&
&
x
y
e
)()( xes ff
f
ξργθ && ′=
−
s(e)
),(ˆ
fxf θ
)(
−
x
f
ξ
++−−= −
−−
)()(),(ˆ 2
ssignke
p
q
tyxfu s
qp
d
f
NTSM
&&& βθ
20 dB
+ -
IT-2FLS
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0 5 10 15 20 25 30
-3
-2
-1
0
1
2
3
time(s)
Stateresponse
x1
x2
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-3
-2
-1
0
1
2
3
x1
x2
Figure 2. Time response (x1, x2) and typical chaotic behavior of duffing oscillator
In order to force the states system ( ), 1,2ix t i = to track the reference trajectories ( )dy t and
( )dy t& in finite time, such as ( )( ) ( / 30) sin( ) 0.3sin(3 )dy t t tπ= + , the adaptive interval type-2
fuzzy nonsingular terminal sliding mode control ( )u t is added into the system as follows:
1 2
3
2 2 1 1
,
0.4 1.1 2.1cos(1.8 ) ( , ) ( ) ( )
x x
x x x x t f x t d t u t
=
= − − − − + ∆ + +
&
&
(29)
We choose 15, 3, 5f q pγ = = = and 1β = , the TSM and NTSM manifolds are selected as,
/q p
TSMs e eβ= +& and ( ) /
1/ p q
NTSMs e eβ= + & , respectively. To design the fuzzy system ˆ( , )ff x θ ,
we define seven type-2 Gaussian membership functions depending ( ), 1,2ix t i = selected as
, 1,...,7l
iF l = are shown in table. 1, with variance 0.5σ = and initial values 2 7(0)fθ ×= Ο .
Table 1. Interval Type-2 Fuzzy Membership Functions For ( 1,2).ix i =
Mean Mean
m1 m2 m1 m2
1 ( )
i
iF
xµ -3.5 -2.5 5 ( )
i
iF
xµ 0.5 1.5
2 ( )
i
iF
xµ -2.5 -1.5 6 ( )
i
iF
xµ 1.5 2.5
3 ( )
i
iF
xµ -1.5 -0.5 7 ( )
i
iF
xµ 2.5 3.5
4 ( )
i
iF
xµ -0.5 0.5
In this section, two control laws are adopted, adaptive type-2 fuzzy nonsingular terminal sliding
mode control (AT- 2FNTSM) described in (21), and adaptive type-2 fuzzy terminal sliding mode
control (AT-2FTSM), which is designed as follow;
++−−= −
−−
)(),(ˆ 1
ssignkee
p
q
yxfu s
pq
d
f
TSM
&&& βθ (1)
9. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
27
The simulation results are presented in the presence of uncertainties
( ) ( )1 2( , ) sin 2 ( ) sin 3 ( )f x t x t x tπ π∆ = , external disturbance ( ) sin( )d t t= , and white Gaussian
noise is applied to the measured signal ( ), 1,2ix t i = with Signal to Noise Ratios (SNR=20dB). A
boundary layer method is used to eliminate chattering.
4.1. Adaptive Interval Type-2 Fuzzy Terminal Sliding Mode Control (AT-2FTSM)
The tracking performance of states ( )x t is shown in Figure 3. The control input ( )u t and the
phase-plane trajectories of system are represented in Figures 4-5.
0 2 4 6 8 10 12 14 16 18 20
-0.1
-0.05
0
0.05
0.1 x1
yd
0 2 4 6 8 10 12 14 16 18 20
-0.2
0
0.2
time(s)
x2
yd'
Figure 3. The output trajectories of (x1, x2).
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
e1
e2
ideal sliding mode
practical trajectory
-0.1 -0.05 0 0.05 0.1
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
x1,x2
yd,yd'
Figure 4. Phase-plane of tracking error and typical chaotic behavior of duffing oscillator
11. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
29
0 2 4 6 8 10 12 14 16 18 20
-6
-4
-2
0
2
4
time(s)
u
Figure8. Control input u(t)
According to the above simulation results, we can see that both controller provide a good tracking
of outputs system 1 2( , )x x to their trajectories in finite time. Furthermore, a singularity problem
occurs in the case of AT-2FTSM control as shown in Figure 5. The proposed approach allows
obtaining a smooth control signal (Figure 8), then, the NTSM manifold (15) can eliminate the
singularity problem associated with conventional TSM manifold.
5. CONCLUSION
In this paper, the problem of stabilization orbit of nonlinear uncertain chaotic system in the
presence of external, internal disturbances and disturbances is solved by incorporation of interval
type-2 fuzzy approach and non-singular terminal sliding mode control. In order to eliminate the
chattering phenomenon efficiently, a boundary layer method is used, and an adaptive interval
type-2 fuzzy system is introduced to approximate the unknown part of system. Based on the
Laypunov stability criterion, the adaptation law of adjustable parameters of the type-2 fuzzy
system and the stability of closed loop system are ensured. A simulation example has been
presented to illustrate the effectiveness and the robustness of the proposed approach.
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13. International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015
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Authors
Rim Hendel received here engineering and Master degrees in Automatic from Setif
University (Setif 1), Algeria, in 2009 and 2012 respectively. From November 2012, she is
Ph.D. student in the Engineering Faculty with the QUERE laboratory at the University of
Setif 1. Here research interests are higher order sliding mode control, fuzzy type-1 and
type-2 systems, nonlinear systems
Farid khaber received his D.E.A in 1990 and his Master in 1992 degrees in industrial
control, and his PhD in 2006 from Setif University (Setif 1), Algeria, in automatic control.
He is currently a Professor in the Engineering Faculty from the same university. His
research interests include multivariable adaptive control, LMI control and type-2 fuzzy
control of renewable energy systems
Najib Essounbouli received his Maitrise from the University of Sciences and
Technology of Marrakech (FSTG) in Morocco, his D.E.A. in 2000, his Ph.D. in 2004,
and its Habilitation from Reims University of Champagne- Ardenne, all in Electrical
Engineering. From September 2005 to 2010, he has been an Assistant Professor with IUT
of Troyes, Reims Champagne Ardenne University. He is a currently a Professor and Head
of the Mechanical Engineering Department of IUT at Troyes, Reims University. His
current research interests are in the areas of fuzzy logic control, robust adaptive control,
renewable energy and control drive.