Designing a controller for nonlinear systems is difficult to be applied. Thus, it is usually based on a linearization around their equilibrium points. The state dependent Riccati equation control approach is an optimization method that has the simplicity of the classical linear quadratic control method. On the other hand, the singular perturbation theory is used for the decomposition of a high-order system into two lower-order systems. In this study, the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the singular perturbation theory and the state dependent Riccati equation technique together is addressed. In the proposed method, first, the Hamiltonian equations are described as a state-dependent Hamiltonian matrix, from which, the reduced-order subsystems are obtained. Then, these subsystems are converted into outerlayer, initial layer correction and final layer correction equations, from which, the separated state dependent Riccati equations are derived. The optimal control law is, then, obtained by computing the Riccati matrices.
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
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.
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.
Sensor Fusion Study - Ch1. Linear System [Hayden]AI Robotics KR
This document provides an overview of key topics in linear systems theory, including:
- Matrix algebra and calculus concepts needed to understand linear systems such as matrices, vectors, transposes, and ranks
- Definitions and properties of linear and nonlinear systems
- Discretization of continuous-time systems for simulation on digital computers
- Concepts of stability, controllability, and observability of linear systems
The document appears to be the first chapter of a textbook on linear systems and optimal state estimation, introducing fundamental concepts before addressing more advanced topics covered in later chapters.
Stability and stabilization of discrete-time systems with time-delay via Lyap...IJERA Editor
The stability and stabilization problems for discrete systems with time-delay are discussed .The stability and
stabilization criterion are expressed in the form of linear matrix inequalities (LMI). An effective method
allowing us transforming a bilinear matrix Inequality (BMI) to a linear matrix Inequality (LMI) is developed.
Based on these conditions, a state feedback controller with gain is designed. An illustrative numerical example
is provided to show the effectiveness of the proposed method and the reliability of the results.
Time Delay and Mean Square Stochastic Differential Equations in Impetuous Sta...ijtsrd
This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. G. Pramila | S. Ramadevi"Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11062.pdf http://www.ijtsrd.com/humanities-and-the-arts/other/11062/time-delay-and-mean-square-stochastic-differential-equations-in-impetuous-stabilization/g-pramila
This document contains a quiz on the topic of finite element analysis (FEA).
1. FEA involves discretizing a structure into smaller elements. Basic ideas of FEA were developed by aircraft engineers, and modern development occurred first in structural analysis.
2. Common methods of FEA include the force method and displacement method. Elements include 1D, 2D, and 3D elements like bars, beams, triangles, and quadrilaterals. Polynomial interpolation is often used.
3. FEA includes preprocessing like element discretization, solving using methods like the weighted residual or variational approach, and postprocessing of results. Software includes NISA and COSMOS. Analysis can be static or dynamic.
1. The document discusses numerical criteria for robust control of continuous systems when the system model is not precisely known.
2. It considers two classes of uncertain systems - systems with structured perturbations defined by state equations, and systems with unstructured disturbances involving unknown nonlinear functions.
3. For each class, sufficient conditions for asymptotic stability are established and illustrated with numerical examples, showing improvements over past works. The results provide guidance for robust controller synthesis.
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
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.
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.
Sensor Fusion Study - Ch1. Linear System [Hayden]AI Robotics KR
This document provides an overview of key topics in linear systems theory, including:
- Matrix algebra and calculus concepts needed to understand linear systems such as matrices, vectors, transposes, and ranks
- Definitions and properties of linear and nonlinear systems
- Discretization of continuous-time systems for simulation on digital computers
- Concepts of stability, controllability, and observability of linear systems
The document appears to be the first chapter of a textbook on linear systems and optimal state estimation, introducing fundamental concepts before addressing more advanced topics covered in later chapters.
Stability and stabilization of discrete-time systems with time-delay via Lyap...IJERA Editor
The stability and stabilization problems for discrete systems with time-delay are discussed .The stability and
stabilization criterion are expressed in the form of linear matrix inequalities (LMI). An effective method
allowing us transforming a bilinear matrix Inequality (BMI) to a linear matrix Inequality (LMI) is developed.
Based on these conditions, a state feedback controller with gain is designed. An illustrative numerical example
is provided to show the effectiveness of the proposed method and the reliability of the results.
Time Delay and Mean Square Stochastic Differential Equations in Impetuous Sta...ijtsrd
This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. G. Pramila | S. Ramadevi"Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11062.pdf http://www.ijtsrd.com/humanities-and-the-arts/other/11062/time-delay-and-mean-square-stochastic-differential-equations-in-impetuous-stabilization/g-pramila
This document contains a quiz on the topic of finite element analysis (FEA).
1. FEA involves discretizing a structure into smaller elements. Basic ideas of FEA were developed by aircraft engineers, and modern development occurred first in structural analysis.
2. Common methods of FEA include the force method and displacement method. Elements include 1D, 2D, and 3D elements like bars, beams, triangles, and quadrilaterals. Polynomial interpolation is often used.
3. FEA includes preprocessing like element discretization, solving using methods like the weighted residual or variational approach, and postprocessing of results. Software includes NISA and COSMOS. Analysis can be static or dynamic.
1. The document discusses numerical criteria for robust control of continuous systems when the system model is not precisely known.
2. It considers two classes of uncertain systems - systems with structured perturbations defined by state equations, and systems with unstructured disturbances involving unknown nonlinear functions.
3. For each class, sufficient conditions for asymptotic stability are established and illustrated with numerical examples, showing improvements over past works. The results provide guidance for robust controller synthesis.
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
Presentasi fisika komputasi invers matrik gauss jordan dengan pythonFidiyatiUmaroh
1) The document discusses invers matrices and the Gauss-Jordan elimination method. It defines an invers matrix and explains how to find the invers of square matrices using elementary row operations or the adjoint matrix.
2) The Gauss-Jordan elimination method and Gauss elimination method have the same number of operations. Both methods begin by reducing the augmented matrix into row echelon form.
3) The Gauss-Jordan elimination method can be used to calculate the invers of a square matrix by adding the identity matrix and performing row operations to transform the matrix into the identity matrix with the invers on the right side.
Design of second order linear time invariant systems for deadbeat responseIAEME Publication
This document summarizes a research paper that proposes a new method for analyzing the stability and deadbeat response of linear time-invariant systems using a modified Routh's table approach. The method first converts the system's characteristic equation with real coefficients into a complex coefficient equation using Romonov's transformation. It then forms a Routh-like table using the complex coefficients. Stability is determined by analyzing the sign pairs in the first column based on a proposed "Sign Pair Criterion." The method is demonstrated on an example system, verifying the number of complex roots and ability to design for a deadbeat response. The proposed approach provides an algebraic way to analyze stability and deadbeat response without determining the system's roots.
M.G.Goman, A.V.Khramtsovsky (1997) - Global Stability Analysis of Nonlinear A...Project KRIT
М.Г.Гоман, А.В.Храмцовский "Анализ устойчивости "в большом" в нелинейных задачах динамики", доклад на конференции AIAA Guidance, Navigation, and Control Conf., AIAA Paper 97-3721, 1997 г.
В статье описываются численные методы анализа устойчивости "в большом". Представлены результаты расчетов областей устойчивости стационарных установившихся режимов движения для различных нелинейных задач динамики пол\"ета, таких как инерционное вращение, сваливание, штопор и др.
M.G.Goman, A.V.Khramtsovsky "Global Stability Analysis of Nonlinear Aircraft Dynamics", presented at AIAA Guidance, Navigation, and Control Conf., AIAA Paper 97-3721, 1997.
Global stability analysis of dynamical system implies determination of all the system's attractors and their regions of asymptotic stability. Numerical methods necessary for global stability analysis are outlined: SSNE algorithm for systematic search of all the solutions of nonlinear system of equations, and two algorithms for computing cross-sections of asymptotic stability region boundaries. Results obtained during investigation of aircraft roll-coupling problem are presented as an example of global stability analysis.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
The synchronization problem of chaotic systems using active modified projective non- linear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.
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.
The International Journal of Engineering and Science (The IJES)theijes
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.
This document summarizes a project on recognizing handwritten digits using machine learning classifiers. The researchers used the MNIST dataset and preprocessed the images before extracting features. They then applied Naive Bayes and Logistic Regression classifiers and evaluated their performance based on accuracy and confusion matrices. Logistic Regression significantly outperformed Naive Bayes. Regularization was also investigated for Logistic Regression, with cross-validation used to select the optimal regularization parameter.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
In this paper, block-oriented systems with linear parts based on Laguerre functions is used to
approximation of a cone crusher dynamics. Adaptive recursive least squares algorithm is used to
identification of Laguerre model. Various structures of Hammerstein, Wiener, Hammerstein-Wiener models
are tested and the MATLAB simulation results are compared. The mean square error is used for models
validation.It has been found that Hammerstein-Wiener with orthonormal basis functions improves the
quality of approximation plant dynamics. The mean square error for this model is 11% on average
throughout the considered range of the external disturbances amplitude. The analysis also showed that
Wiener model cannot provide sufficient approximation accuracy of the cone crusher dynamics. During the
process it is unstable due to the high sensitivity to disturbances on the output.The Hammerstein-Wiener
model will be used to the design nonlinear model predictive control application.
This document reviews testing for causality between variables. It begins by defining Granger causality, which tests whether including one time series helps forecast another. For bivariate systems, causality can be tested by examining coefficients in a vector autoregression (VAR) model. For multivariate systems, causality is more complex and graphical models may help. The document outlines procedures for testing causality between stationary and nonstationary time series using impulse responses, vector autoregressive moving average (VARMA) models, and other techniques. It provides examples and discusses challenges like potential omitted common factors.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
AN EFFICIENT PARALLEL ALGORITHM FOR COMPUTING DETERMINANT OF NON-SQUARE MATRI...ijdpsjournal
One of the most significant challenges in Computing Determinant of Rectangular Matrices is high time
complexity of its algorithm. Among all definitions of determinant of rectangular matrices, Radic’s
definition has special features which make it more notable. But in this definition, C(N
M
) sub matrices of the
order m×m needed to be generated that put this problem in np-hard class. On the other hand, any row or
column reduction operation may hardly lead to diminish the volume of calculation. Therefore, in this paper
we try to present the parallel algorithm which can decrease the time complexity of computing the
determinant of non-square matrices to O(N).
This document summarizes a research article from the International Journal of Electrical Engineering and Technology that proposes a decentralized stabilization method for large-scale bilinear interconnected systems. The method involves aggregating the interaction terms of each subsystem with its state matrix, resulting in decoupled subsystems. Optimal control gains are then computed by solving the algebraic Riccati equation for each decoupled subsystem. These decentralized feedback controls are shown to stabilize the overall system with near-optimal performance and minimum control energy cost. The method is demonstrated numerically on a system of three second-order bilinear subsystems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
Effect of stability indices on robustness and system response in coefficient ...eSAT Journals
Abstract
In Control systems, designing a robust controller such that a desired system response is obtained despite plant parameter
variations is ubiquitous problem. In this context, Coefficient diagram method is an effective method and one of the recent design
methods based on the polynomial approach introduced by Shunji Manabe. In CDM, stability indices, stability limits and time
constant are the main design parameters. The stability indices and stability limits are indicative of stability and equivalent time
constant is indicative of speed of system response. A semi-log diagram known as coefficient diagram is the design tool using
which one can analyse the important features of a design such as stability, speed of response and robustness, all in one diagram.
The right choice of the stability indices is of paramount importance in the controller design. This paper deals with the effect of
variation in the stability indices upon the system response and robustness. A type 2 fourth order plant has been considered as an
example to analyse the effects of stability indices. The stability indices are varied one by one relative to the standard Manabe form
and in each case response is observed. The transient response of the system is sensitive to lower order indices. Also, robustness in
the design is analysed by coefficient diagrams of the perturbed plant.
Keywords: Coefficient Diagram Method (CDM), robustness; Stability Indices, Coefficient Diagram
M.G.Goman and A.V.Khramtsovsky (1993) - Textbook for KRIT Toolbox usersProject KRIT
M.G.Goman and A.V.Khramtsovsky "Methodology of the qualitative investigation. Theory and numerical methods. Application to Aircraft Flight Dynamics", textbook for KRIT Toolbox users, 1993, 81 p.
The textbook covers theory of qualitative analysis of nonlinear systems, basic numerical methods supported by KRIT package and some aircraft flight dynamics applications.
- 86 former Downing's Bakery employees won a $251,545 arbitration award against the company for unpaid salaries and wages in 2011.
- The award was registered with the Bulawayo High Court, which ordered Downing's Bakery to pay or have its property attached and auctioned.
- When Downing's Bakery did not pay, its property was attached by then Deputy Sheriff Regina Gumbo and sold at auction. However, the proceeds from the sale have disappeared, and Gumbo is now evasive about the matter.
- Lawyers for the workers and court officials have been unable to determine what happened to the money or inventory from the auction despite contacting Gumbo. This has left
Presentasi fisika komputasi invers matrik gauss jordan dengan pythonFidiyatiUmaroh
1) The document discusses invers matrices and the Gauss-Jordan elimination method. It defines an invers matrix and explains how to find the invers of square matrices using elementary row operations or the adjoint matrix.
2) The Gauss-Jordan elimination method and Gauss elimination method have the same number of operations. Both methods begin by reducing the augmented matrix into row echelon form.
3) The Gauss-Jordan elimination method can be used to calculate the invers of a square matrix by adding the identity matrix and performing row operations to transform the matrix into the identity matrix with the invers on the right side.
Design of second order linear time invariant systems for deadbeat responseIAEME Publication
This document summarizes a research paper that proposes a new method for analyzing the stability and deadbeat response of linear time-invariant systems using a modified Routh's table approach. The method first converts the system's characteristic equation with real coefficients into a complex coefficient equation using Romonov's transformation. It then forms a Routh-like table using the complex coefficients. Stability is determined by analyzing the sign pairs in the first column based on a proposed "Sign Pair Criterion." The method is demonstrated on an example system, verifying the number of complex roots and ability to design for a deadbeat response. The proposed approach provides an algebraic way to analyze stability and deadbeat response without determining the system's roots.
M.G.Goman, A.V.Khramtsovsky (1997) - Global Stability Analysis of Nonlinear A...Project KRIT
М.Г.Гоман, А.В.Храмцовский "Анализ устойчивости "в большом" в нелинейных задачах динамики", доклад на конференции AIAA Guidance, Navigation, and Control Conf., AIAA Paper 97-3721, 1997 г.
В статье описываются численные методы анализа устойчивости "в большом". Представлены результаты расчетов областей устойчивости стационарных установившихся режимов движения для различных нелинейных задач динамики пол\"ета, таких как инерционное вращение, сваливание, штопор и др.
M.G.Goman, A.V.Khramtsovsky "Global Stability Analysis of Nonlinear Aircraft Dynamics", presented at AIAA Guidance, Navigation, and Control Conf., AIAA Paper 97-3721, 1997.
Global stability analysis of dynamical system implies determination of all the system's attractors and their regions of asymptotic stability. Numerical methods necessary for global stability analysis are outlined: SSNE algorithm for systematic search of all the solutions of nonlinear system of equations, and two algorithms for computing cross-sections of asymptotic stability region boundaries. Results obtained during investigation of aircraft roll-coupling problem are presented as an example of global stability analysis.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
The synchronization problem of chaotic systems using active modified projective non- linear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.
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.
The International Journal of Engineering and Science (The IJES)theijes
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.
This document summarizes a project on recognizing handwritten digits using machine learning classifiers. The researchers used the MNIST dataset and preprocessed the images before extracting features. They then applied Naive Bayes and Logistic Regression classifiers and evaluated their performance based on accuracy and confusion matrices. Logistic Regression significantly outperformed Naive Bayes. Regularization was also investigated for Logistic Regression, with cross-validation used to select the optimal regularization parameter.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
In this paper, block-oriented systems with linear parts based on Laguerre functions is used to
approximation of a cone crusher dynamics. Adaptive recursive least squares algorithm is used to
identification of Laguerre model. Various structures of Hammerstein, Wiener, Hammerstein-Wiener models
are tested and the MATLAB simulation results are compared. The mean square error is used for models
validation.It has been found that Hammerstein-Wiener with orthonormal basis functions improves the
quality of approximation plant dynamics. The mean square error for this model is 11% on average
throughout the considered range of the external disturbances amplitude. The analysis also showed that
Wiener model cannot provide sufficient approximation accuracy of the cone crusher dynamics. During the
process it is unstable due to the high sensitivity to disturbances on the output.The Hammerstein-Wiener
model will be used to the design nonlinear model predictive control application.
This document reviews testing for causality between variables. It begins by defining Granger causality, which tests whether including one time series helps forecast another. For bivariate systems, causality can be tested by examining coefficients in a vector autoregression (VAR) model. For multivariate systems, causality is more complex and graphical models may help. The document outlines procedures for testing causality between stationary and nonstationary time series using impulse responses, vector autoregressive moving average (VARMA) models, and other techniques. It provides examples and discusses challenges like potential omitted common factors.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
AN EFFICIENT PARALLEL ALGORITHM FOR COMPUTING DETERMINANT OF NON-SQUARE MATRI...ijdpsjournal
One of the most significant challenges in Computing Determinant of Rectangular Matrices is high time
complexity of its algorithm. Among all definitions of determinant of rectangular matrices, Radic’s
definition has special features which make it more notable. But in this definition, C(N
M
) sub matrices of the
order m×m needed to be generated that put this problem in np-hard class. On the other hand, any row or
column reduction operation may hardly lead to diminish the volume of calculation. Therefore, in this paper
we try to present the parallel algorithm which can decrease the time complexity of computing the
determinant of non-square matrices to O(N).
This document summarizes a research article from the International Journal of Electrical Engineering and Technology that proposes a decentralized stabilization method for large-scale bilinear interconnected systems. The method involves aggregating the interaction terms of each subsystem with its state matrix, resulting in decoupled subsystems. Optimal control gains are then computed by solving the algebraic Riccati equation for each decoupled subsystem. These decentralized feedback controls are shown to stabilize the overall system with near-optimal performance and minimum control energy cost. The method is demonstrated numerically on a system of three second-order bilinear subsystems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
Effect of stability indices on robustness and system response in coefficient ...eSAT Journals
Abstract
In Control systems, designing a robust controller such that a desired system response is obtained despite plant parameter
variations is ubiquitous problem. In this context, Coefficient diagram method is an effective method and one of the recent design
methods based on the polynomial approach introduced by Shunji Manabe. In CDM, stability indices, stability limits and time
constant are the main design parameters. The stability indices and stability limits are indicative of stability and equivalent time
constant is indicative of speed of system response. A semi-log diagram known as coefficient diagram is the design tool using
which one can analyse the important features of a design such as stability, speed of response and robustness, all in one diagram.
The right choice of the stability indices is of paramount importance in the controller design. This paper deals with the effect of
variation in the stability indices upon the system response and robustness. A type 2 fourth order plant has been considered as an
example to analyse the effects of stability indices. The stability indices are varied one by one relative to the standard Manabe form
and in each case response is observed. The transient response of the system is sensitive to lower order indices. Also, robustness in
the design is analysed by coefficient diagrams of the perturbed plant.
Keywords: Coefficient Diagram Method (CDM), robustness; Stability Indices, Coefficient Diagram
M.G.Goman and A.V.Khramtsovsky (1993) - Textbook for KRIT Toolbox usersProject KRIT
M.G.Goman and A.V.Khramtsovsky "Methodology of the qualitative investigation. Theory and numerical methods. Application to Aircraft Flight Dynamics", textbook for KRIT Toolbox users, 1993, 81 p.
The textbook covers theory of qualitative analysis of nonlinear systems, basic numerical methods supported by KRIT package and some aircraft flight dynamics applications.
- 86 former Downing's Bakery employees won a $251,545 arbitration award against the company for unpaid salaries and wages in 2011.
- The award was registered with the Bulawayo High Court, which ordered Downing's Bakery to pay or have its property attached and auctioned.
- When Downing's Bakery did not pay, its property was attached by then Deputy Sheriff Regina Gumbo and sold at auction. However, the proceeds from the sale have disappeared, and Gumbo is now evasive about the matter.
- Lawyers for the workers and court officials have been unable to determine what happened to the money or inventory from the auction despite contacting Gumbo. This has left
La aplicación permite usar el teléfono móvil como walkie talkie, permitiendo comunicación privada en tiempo real con contactos al pulsar un botón para hablar. Ofrece transmisión en tiempo real y acceso a contactos y mensajes de texto.
The document discusses the history and importance of chocolate in human civilization. It notes that chocolate originated in Mesoamerica over 3000 years ago and was prized by the Aztecs and Mayans for its taste. Cocoa beans were used as currency and their cultivation was tightly regulated. The Spanish brought cocoa beans back to Europe in the 16th century, starting chocolate's global spread and popularity as both a drink and confection.
Malcoln decidió ir a un pequeño pueblo para estudiar sin distracciones. Aceptó un trabajo como chofer de autobús que transportaba niños con discapacidades mentales a un centro. Más tarde recibió una oferta anónima para arrojar el autobús y los niños por un barranco a cambio de dinero, lo cual hizo. Tres años después, cuando regresó al lugar del accidente borracho con amigos, fue atacado y descuartizado por los fantasmas de los niños asesinados.
El artículo habla sobre la apertura de un nuevo kiosko en la ciudad que ofrecerá productos de conveniencia a bajo precio las 24 horas del día. El dueño mencionó que espera llenar una necesidad en el mercado local al proveer artículos esenciales a precios accesibles incluso durante la noche.
This poem describes a vision of an ideal country where the mind is free from fear, knowledge is freely shared, and people are united rather than divided. It envisions a place where words reflect deep truths, constant self-improvement is valued, rational thinking is not hindered by dead habits, and the mind is led to think and act in increasingly broad and liberating ways. The poet hopes his country will awaken and achieve this heaven of freedom.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen 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 hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen 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.
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.
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.
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.
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.
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZA...ijait
This paper discusses the design of active controllers for achieving generalized projective synchronization (GPS) of identical hyperchaotic Lü systems (Chen, Lu, Lü and Yu, 2006), identical hyperchaotic Cai systems (Wang and Cai, 2009) and non-identical hyperchaotic Lü and hyperchaotic Cai systems. The synchronization results (GPS) for the hyperchaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for achieving the GPS of the
hyperchaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHEN...ijscai
This summary provides the key details about the document in 3 sentences:
The document describes the design of an active controller to achieve hybrid synchronization between two hyperchaotic systems - the hyperchaotic Zheng system and the hyperchaotic Yu system. Hybrid synchronization means the odd states of the systems are completely synchronized while the even states are completely anti-synchronized. The active controller is designed using Lyapunov stability theory and numerical simulations demonstrate the hybrid synchronization between the hyperchaotic 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.
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.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
1) The document describes using active nonlinear control to achieve global chaos synchronization between identical and non-identical hyperchaotic systems.
2) It specifically derives new results for synchronizing identical hyperchaotic Qi and Jha systems, as well as non-identical hyperchaotic Qi and Jha systems.
3) The synchronization is achieved through designing appropriate active nonlinear controllers and proving global exponential stability of the synchronization error dynamics using Lyapunov stability theory. Numerical simulations validate the theoretical results.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun 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.
ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
Approximate Solution of a Linear Descriptor Dynamic Control System via a non-...IOSR Journals
This document discusses approximate solutions for linear descriptor dynamic control systems using a non-classical variational approach. It begins by introducing descriptor systems and their importance in applications. It then discusses making irregular systems regular through computational algorithms. The paper focuses on consistent initial conditions and their characterization. It proposes using a non-classical variational approach to obtain approximate solutions with a high degree of accuracy and freedom of choice for the bilinear form.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. 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. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
Similar to Designing SDRE-Based Controller for a Class of Nonlinear Singularly Perturbed Systems (20)
The Use of Java Swing’s Components to Develop a WidgetWaqas Tariq
Widget is a kind of application provides a single service such as a map, news feed, simple clock, battery-life indicators, etc. This kind of interactive software object has been developed to facilitate user interface (UI) design. A user interface (UI) function may be implemented using different widgets with the same function. In this article, we present the widget as a platform that is generally used in various applications, such as in desktop, web browser, and mobile phone. We also describe a visual menu of Java Swing’s components that will be used to establish widget. It will assume that we have successfully compiled and run a program that uses Swing components.
3D Human Hand Posture Reconstruction Using a Single 2D ImageWaqas Tariq
Passive sensing of the 3D geometric posture of the human hand has been studied extensively over the past decade. However, these research efforts have been hampered by the computational complexity caused by inverse kinematics and 3D reconstruction. In this paper, our objective focuses on 3D hand posture estimation based on a single 2D image with aim of robotic applications. We introduce the human hand model with 27 degrees of freedom (DOFs) and analyze some of its constraints to reduce the DOFs without any significant degradation of performance. A novel algorithm to estimate the 3D hand posture from eight 2D projected feature points is proposed. Experimental results using real images confirm that our algorithm gives good estimates of the 3D hand pose. Keywords: 3D hand posture estimation; Model-based approach; Gesture recognition; human- computer interface; machine vision.
Camera as Mouse and Keyboard for Handicap Person with Troubleshooting Ability...Waqas Tariq
Camera mouse has been widely used for handicap person to interact with computer. The utmost important of the use of camera mouse is must be able to replace all roles of typical mouse and keyboard. It must be able to provide all mouse click events and keyboard functions (include all shortcut keys) when it is used by handicap person. Also, the use of camera mouse must allow users troubleshooting by themselves. Moreover, it must be able to eliminate neck fatigue effect when it is used during long period. In this paper, we propose camera mouse system with timer as left click event and blinking as right click event. Also, we modify original screen keyboard layout by add two additional buttons (button “drag/ drop” is used to do drag and drop of mouse events and another button is used to call task manager (for troubleshooting)) and change behavior of CTRL, ALT, SHIFT, and CAPS LOCK keys in order to provide shortcut keys of keyboard. Also, we develop recovery method which allows users go from camera and then come back again in order to eliminate neck fatigue effect. The experiments which involve several users have been done in our laboratory. The results show that the use of our camera mouse able to allow users do typing, left and right click events, drag and drop events, and troubleshooting without hand. By implement this system, handicap person can use computer more comfortable and reduce the dryness of eyes.
A Proposed Web Accessibility Framework for the Arab DisabledWaqas Tariq
The Web is providing unprecedented access to information and interaction for people with disabilities. This paper presents a Web accessibility framework which offers the ease of the Web accessing for the disabled Arab users and facilitates their lifelong learning as well. The proposed framework system provides the disabled Arab user with an easy means of access using their mother language so they don’t have to overcome the barrier of learning the target-spoken language. This framework is based on analyzing the web page meta-language, extracting its content and reformulating it in a suitable format for the disabled users. The basic objective of this framework is supporting the equal rights of the Arab disabled people for their access to the education and training with non disabled people. Key Words : Arabic Moon code, Arabic Sign Language, Deaf, Deaf-blind, E-learning Interactivity, Moon code, Web accessibility , Web framework , Web System, WWW.
Real Time Blinking Detection Based on Gabor FilterWaqas Tariq
The document proposes a new method for real-time blinking detection based on Gabor filters. It begins by reviewing existing methods and their limitations in dealing with noise, variations in eye shape, and blinking speed. The proposed method uses a Gabor filter to extract the top and bottom arcs of the eye from an image. It then measures the distance between these arcs and compares it to a threshold: a distance below the threshold indicates a closed eye, while a distance above indicates an open eye. The document claims this Gabor filter-based approach is robust to noise, variations in eye shape and blinking speed. It presents experimental results showing the method can accurately detect blinking across different users.
Computer Input with Human Eyes-Only Using Two Purkinje Images Which Works in ...Waqas Tariq
A method for computer input with human eyes-only using two Purkinje images which works in a real time basis without calibration is proposed. Experimental results shows that cornea curvature can be estimated by using two light sources derived Purkinje images so that no calibration for reducing person-to-person difference of cornea curvature. It is found that the proposed system allows usersf movements of 30 degrees in roll direction and 15 degrees in pitch direction utilizing detected face attitude which is derived from the face plane consisting three feature points on the face, two eyes and nose or mouth. Also it is found that the proposed system does work in a real time basis.
Toward a More Robust Usability concept with Perceived Enjoyment in the contex...Waqas Tariq
Mobile multimedia service is relatively new but has quickly dominated people¡¯s lives, especially among young people. To explain this popularity, this study applies and modifies the Technology Acceptance Model (TAM) to propose a research model and conduct an empirical study. The goal of study is to examine the role of Perceived Enjoyment (PE) and what determinants can contribute to PE in the context of using mobile multimedia service. The result indicates that PE is influencing on Perceived Usefulness (PU) and Perceived Ease of Use (PEOU) and directly Behavior Intention (BI). Aesthetics and flow are key determinants to explain Perceived Enjoyment (PE) in mobile multimedia usage.
Collaborative Learning of Organisational KnolwedgeWaqas Tariq
This paper presents recent research into methods used in Australian Indigenous Knowledge sharing and looks at how these can support the creation of suitable collaborative envi- ronments for timely organisational learning. The protocols and practices as used today and in the past by Indigenous communities are presented and discussed in relation to their relevance to a personalised system of knowledge sharing in modern organisational cultures. This research focuses on user models, knowledge acquisition and integration of data for constructivist learning in a networked repository of or- ganisational knowledge. The data collected in the repository is searched to provide collections of up-to-date and relevant material for training in a work environment. The aim is to improve knowledge collection and sharing in a team envi- ronment. This knowledge can then be collated into a story or workflow that represents the present knowledge in the organisation.
Our research aims to propose a global approach for specification, design and verification of context awareness Human Computer Interface (HCI). This is a Model Based Design approach (MBD). This methodology describes the ubiquitous environment by ontologies. OWL is the standard used for this purpose. The specification and modeling of Human-Computer Interaction are based on Petri nets (PN). This raises the question of representation of Petri nets with XML. We use for this purpose, the standard of modeling PNML. In this paper, we propose an extension of this standard for specification, generation and verification of HCI. This extension is a methodological approach for the construction of PNML with Petri nets. The design principle uses the concept of composition of elementary structures of Petri nets as PNML Modular. The objective is to obtain a valid interface through verification of properties of elementary Petri nets represented with PNML.
Development of Sign Signal Translation System Based on Altera’s FPGA DE2 BoardWaqas Tariq
The main aim of this paper is to build a system that is capable of detecting and recognizing the hand gesture in an image captured by using a camera. The system is built based on Altera’s FPGA DE2 board, which contains a Nios II soft core processor. Image processing techniques and a simple but effective algorithm are implemented to achieve this purpose. Image processing techniques are used to smooth the image in order to ease the subsequent processes in translating the hand sign signal. The algorithm is built for translating the numerical hand sign signal and the result are displayed on the seven segment display. Altera’s Quartus II, SOPC Builder and Nios II EDS software are used to construct the system. By using SOPC Builder, the related components on the DE2 board can be interconnected easily and orderly compared to traditional method that requires lengthy source code and time consuming. Quartus II is used to compile and download the design to the DE2 board. Then, under Nios II EDS, C programming language is used to code the hand sign translation algorithm. Being able to recognize the hand sign signal from images can helps human in controlling a robot and other applications which require only a simple set of instructions provided a CMOS sensor is included in the system.
An overview on Advanced Research Works on Brain-Computer InterfaceWaqas Tariq
A brain–computer interface (BCI) is a proficient result in the research field of human- computer synergy, where direct articulation between brain and an external device occurs resulting in augmenting, assisting and repairing human cognitive. Advanced works like generating brain-computer interface switch technologies for intermittent (or asynchronous) control in natural environments or developing brain-computer interface by Fuzzy logic Systems or by implementing wavelet theory to drive its efficacies are still going on and some useful results has also been found out. The requirements to develop this brain machine interface is also growing day by day i.e. like neuropsychological rehabilitation, emotion control, etc. An overview on the control theory and some advanced works on the field of brain machine interface are shown in this paper.
Exploring the Relationship Between Mobile Phone and Senior Citizens: A Malays...Waqas Tariq
There is growing ageing phenomena with the rise of ageing population throughout the world. According to the World Health Organization (2002), the growing ageing population indicates 694 million, or 223% is expected for people aged 60 and over, since 1970 and 2025.The growth is especially significant in some advanced countries such as North America, Japan, Italy, Germany, United Kingdom and so forth. This growing older adult population has significantly impact the social-culture, lifestyle, healthcare system, economy, infrastructure and government policy of a nation. However, there are limited research studies on the perception and usage of a mobile phone and its service for senior citizens in a developing nation like Malaysia. This paper explores the relationship between mobile phones and senior citizens in Malaysia from the perspective of a developing country. We conducted an exploratory study using contextual interviews with 5 senior citizens of how they perceive their mobile phones. This paper reveals 4 interesting themes from this preliminary study, in addition to the findings of the desirable mobile requirements for local senior citizens with respect of health, safety and communication purposes. The findings of this study bring interesting insight to local telecommunication industries as a whole, and will also serve as groundwork for more in-depth study in the future.
Principles of Good Screen Design in WebsitesWaqas Tariq
Visual techniques for proper arrangement of the elements on the user screen have helped the designers to make the screen look good and attractive. Several visual techniques emphasize the arrangement and ordering of the screen elements based on particular criteria for best appearance of the screen. This paper investigates few significant visual techniques in various web user interfaces and showcases the results for better understanding and their presence.
This document discusses the progress of virtual teams in Albania. It provides context on virtual teams and how they differ from traditional teams in their reliance on technology for communication across distances. The document then examines the use of virtual teams in Albania, noting the growing infrastructure and technology usage that enables virtual collaboration. It highlights some virtual team examples in Albanian government and academic projects.
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Robot Arm Utilized Having Meal Support System Based on Computer Input by Huma...Waqas Tariq
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In recent decades speech interactive systems have gained increasing importance. Performance of an ASR system mainly depends on the availability of large corpus of speech. The conventional method of building a large vocabulary speech recognizer for any language uses a top-down approach to speech. This approach requires large speech corpus with sentence or phoneme level transcription of the speech utterances. The transcriptions must also include different speech order so that the recognizer can build models for all the sounds present. But, for Telugu language, because of its complex nature, a very large, well annotated speech database is very difficult to build. It is very difficult, if not impossible, to cover all the words of any Indian language, where each word may have thousands and millions of word forms. A significant part of grammar that is handled by syntax in English (and other similar languages) is handled within morphology in Telugu. Phrases including several words (that is, tokens) in English would be mapped on to a single word in Telugu.Telugu language is phonetic in nature in addition to rich in morphology. That is why the speech technology developed for English cannot be applied to Telugu language. This paper highlights the work carried out in an attempt to build a voice enabled text editor with capability of automatic term suggestion. Main claim of the paper is the recognition enhancement process developed by us for suitability of highly inflecting, rich morphological languages. This method results in increased speech recognition accuracy with very much reduction in corpus size. It also adapts Telugu words to the database dynamically, resulting in growth of the corpus.
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Parameters Optimization for Improving ASR Performance in Adverse Real World N...Waqas Tariq
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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
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Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
How to Fix the Import Error in the Odoo 17Celine George
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Designing SDRE-Based Controller for a Class of Nonlinear Singularly Perturbed Systems
1. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 1
Designing SDRE-Based Controller for a Class of Nonlinear
Singularly Perturbed Systems
Seyed Mostafa Ghadami m.gadami@srbiau.ac.ir
Department of Electrical Engineering,
Science and Research Branch,
Islamic Azad University, Tehran, Iran
Roya Amjadifard amjadifard@tmu.ac.ir
Assistant Professor,
Kharazmi University,
Tehran, Iran
Hamid Khaloozadeh h_khaloozadeh@kntu.ac.ir
Professor,
K.N. Toosi University of Technology,
Tehran, Iran
Abstract
Designing a controller for nonlinear systems is difficult to be applied. Thus, it is usually based on
a linearization around their equilibrium points. The state dependent Riccati equation control
approach is an optimization method that has the simplicity of the classical linear quadratic control
method. On the other hand, the singular perturbation theory is used for the decomposition of a
high-order system into two lower-order systems. In this study, the finite-horizon optimization of a
class of nonlinear singularly perturbed systems based on the singular perturbation theory and the
state dependent Riccati equation technique together is addressed. In the proposed method, first,
the Hamiltonian equations are described as a state-dependent Hamiltonian matrix, from which,
the reduced-order subsystems are obtained. Then, these subsystems are converted into outer-
layer, initial layer correction and final layer correction equations, from which, the separated state
dependent Riccati equations are derived. The optimal control law is, then, obtained by computing
the Riccati matrices.
Keywords: Singularly Perturbed Systems, State-Dependent Riccati Equation, Nonlinear Optimal
Control, Finite-Horizon Optimization Problem, Single Link Flexible Joint Robot Manipulator.
1. INTRODUCTION
Designing regulator systems is an important class of optimal control problems in which optimal
control law leads to the Hamilton-Jacobi-Belman (HJB) equation. Various techniques have been
suggested to solve this equation. One of these techniques, which are used for optimizing in
infinite horizon, is based on the state-dependent Riccati equation (SDRE). In this technique,
unlike linearization methods, a description of the system as state-dependent coefficients (SDCs)
and in the form f(x)=A(x)x must be provided. In this representation, A(x) is not unique. Therefore,
the solutions of the SDRE would be dependent on the choice of matrix A(x). With suitable choice
of the matrix, the solution to the equation is optimal; otherwise, the equation has suboptimal
solutions. Bank and Mhana [1] proposed a suitable method for the selection of SDCs. Çimen [2]
provided the condition for the solvability and local asymptotic stability of the SDRE closed-loop
system for a class of nonlinear systems. Khaloozadeh and Abdolahi converted the nonlinear
regulation [3] and tracking [4] problems in the finite-horizon to a state-dependent quasi-Riccati
equation. They also provided an iterative method based on the Piccard theorem, which obtains a
solution at a low convergence rate but good precision. On the other hand, the system discussed
in this study is a class of nonlinear singularly perturbed systems. Naidu and Calise [5] dealt with
2. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 2
the use of the singular perturbation theory and the two time scale (TTS) method in satellite and
interplanetary trajectories, missiles, launch vehicles and hypersonic flight, space robotics. For LTI
singularly perturbed systems, Su et al. [6] and Gajic et al. [7] performed the exact slow-fast
decomposition of the linear quadratic (LQ) singularly perturbed optimal control problem in infinite
horizon by deriving separate Riccati equations. Also, Gajic et al. [8] did the same for the case of
finite horizon. Amjadifard et al. [9, 10] addressed the robust disturbance attenuation of a class of
nonlinear singularly perturbed systems and robust regulation of a class of nonlinear singularly
perturbed systems [11], and also position and velocity control of a flexible joint robot manipulator
via fuzzy controller based on singular perturbation analysis [12]. Fridman [13, 14] dealt with the
infinite horizon nonlinear quadratic optimal control problem for a class of non-standard nonlinear
singularly perturbed systems by invariant manifolds of the Hamiltonian system and its
decomposition into linear-algebraic Riccati equations.
In this study, we extend results of [13, 14] to the finite horizon by slow-fast manifolds of the
Hamiltonian system and its decomposition into SDREs. Our contribution is that, we used the
singular perturbation theory and SDRE method together. In the proposed method, first, the state-
dependent Hamiltonian matrix is derived for the system under study. Then, this matrix is
separated into the reduced-order slow and fast subsystems. Using the singular perturbation
theory, the state equations and SDREs are converted into outer layer, initial layer correction and
final layer correction equations, which are then solved to obtain the optimal control law. The block
diagram of the proposed method is shown in Figure 1.
FIGURE 1: The design procedure stages in the proposed method.
The remainder of this study is organized as follows. Section 2 explains the structure of the
singularly perturbed system for optimization. Section 3 involves in the description of steps of the
design procedure in the proposed method. Section 4 presents the simulation results of the
system used in the proposed method. Finally, the study culminates with indication of remarks in
section 5.
2. PROBLEM FORMULATION
The following nonlinear singularly perturbed system is assumed:
,0)0(,)()( xtxuxBxfxE (1)
where 1,2=iRx
x
x
tx in
i ,,)(
2
1
are the states of system, and x=0n is the equilibrium point of the
system (n=n1+n2). This system is full state observable, autonomous, nonlinear in the states, and
affine in the input. Moreover, 1,2=iRB
xxB
xxB
xBRf
xxf
xxf
xf ii n
i
n
i ,,
),(
),(
)(,,
),(
),(
)(
212
211
212
211
are differentiable with respect to x1, x2 for a sufficient number of times. Furthermore, f(0n)=0n,
Optimal
control
law
Description
of the
system as
SDCs
State
dependent
Hamiltonian
matrix
Slow
SDREs
Slow state
equationsSlow
Hamiltonian
matrix
Fast state
equations
Fast
SDREs
Fast
Hamiltonian
matrix
3. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 3
B(x)0nm, xR
n
and
2212
2111
0
0
nnnn
nnnn
I
I
E
that >0 is a small parameter. Provided these, it
is desired to obtain the optimal control law u(x)R
m
such that for k(x)R
n
, k(0n)=0n and
pointwise positive definite matrix R(x)R
n
R
mm
, the following performance index 𝒥 is minimized.
𝒥
Ft
t
TT
F dtuxRuxkxktxh
0
(2)
Suppose that k(x), R(x) are differentiable with respect to x1, x2 for a sufficient number of times.
Moreover, tF is chosen such that it is sufficiently large with respect to the dominant time constant
of the slow subsystem, and x(tF) is free.
3. THE PROPOSED METHOD
The singularly perturbed system (1) with performance index (2) is assumed. Defining the co-state
vector 1,2,=iR
xx
xx
x in
i ,,
),(
),(
)(
212
211
the Hamiltonian function is obtained as (3):
).),(),(()),(),(()(
2
1
)()(
2
1
),,( 21221222112111 uxxBxxfuxxBxxfuxRuxkxkux TTTT
(3)
According to the optimal control theory, the necessary conditions for optimization would be as
follow [2]:
),(,),(),()( 01211211
1
1 txuxxBxxf
H
x T
(4a)
),(,),(),()( 02212212
2
2 txuxxBxxf
H
x T
(4b)
,|
2
1
)(,
)(
2
1)()()(
1
1
11111
1
Ft
T
F
T
T
T
TTT
x
h
tx
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H
(4c)
,|
2
1
)(,
)(
2
1)()()(
2
2
22222
2
Ft
T
F
T
T
T
TTT
x
h
tx
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H
(4d)
.),(),()(0 22121211 xxBxxBuxR
u
H TT
(4e)
3.1 Description of The System As SDCs (The first step)
A continuous nonlinear matrix-valued function A(x) always exists such that
f(x)=A(x)x (5)
Where A(x):R
n
R
nn
is found by mathematical factorization and is, clearly, non-unique when
n>1. A suitable choice for matrix A(x) is ,
1
0 |
d
x
f
xA
xx
where is a dummy variable that
was introduced in the integration [1]. Then, the relations (4) can be written as:
4. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 4
)(,)()( 0txuxBxxAxE (6a)
ft
T
f
T
T
T
TT
T
x
h
txE
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H
E |
2
1
)(,
)(
2
1)()()(
)(
(6b)
)(1
xBxRu T
(6c)
Considering that B(x) and R(x) are nonzero, the optimal control law is proportional to vector .
3.2 Description of The Hamiltonian Matrix As SDCs (The second step)
Assuming that
1
0 |
d
x
k
xK
xx
is available from k(x)=K(x)x and that Q(x)=K
T
(x)K(x) and
S(x)=B(x)R
-1
(x)B
T
(x), the relations (6) can be rewritten as follow:
,)(,)()( 00 xtxxSxxAxE (7a)
xk
x
xK
xxAxxQE
T
i
n
i
i
T
)(
[)(
1
,|
2
1
)(
,]}
)()(
2
1)(
{
1
1
1
11
1
Ft
T
F
m
i
T
i
i
T
m
i
Ti
i
T
T
i
n
i
i
x
h
txE
x
xB
xRxBxBxR
x
xR
xRxB
x
xA
x
(7b)
Where,
,
)()(
)()(
)(
1
1
1
1
n
nini
n
ii
i
x
xA
x
xA
x
xA
x
xA
x
xA
(8a)
,
)()(
)()(
)(
1
1
1
1
n
nini
n
ii
i
x
xK
x
xK
x
xK
x
xK
x
xK
(8b)
,
)()(
)()(
)(
1
1
1
1
n
nini
n
ii
i
x
xB
x
xB
x
xB
x
xB
x
xB
(8c)
5. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 5
.
)()(
)()(
)(
1
1
1
1
n
mimi
n
ii
i
x
xR
x
xR
x
xR
x
xR
x
xR
(8d)
Assumption 1: A(x), B(x), Q(x), R(x),
x
xK
x
xB
x
xA
)(
,
)(
,
)(
and
x
xR
)(
are bounded in a
neighborhood of about the region. Then, the expression in the bracket will be ignored because
of being small. This approximation is asymptotically optimal, in that it converges to the optimal
control close to the origin as [2]. Thus, the relations (7) can be written as:
x
xAxQ
xSxA
E
xE
T
)(
)()(
(9)
Remark 1: Suppose that Tsi, TsF are dominant time constants of the slow subsystem for initial and
final layer correction, respectively. In other words,
islow
si
Jeigreal
T
1
max and
Fslow
sF
Jeigreal
T
1
max
where, Ji and JF are the Jacobian matrices of Hamiltonian system in
initial and final layer correction and,
n
n
F
n
x
ttTF
xx
ttTi
xAxQ
xSxA
J
xAxQ
xSxA
J
0
0
0
)(
)()(
,
)(
)()(
0
0
.
Note that (Tsi+TsF)/2 is the average time constant of the Hamiltonian system and the setting time
is fourfold of one, then a proper selection for tF is
tF > t0+2(Tsi+TsF) (10)
3.3 The Singularly Perturbed SDRE in Finite Horizon
In the proposed method, co-sate vector , can be described as =P(x)x using the sweep method
[3], where,
ji nn
ij
T
RP
xxPxxP
xxPxxP
xP
,
),(),(
),(),(
)(
21222121
21212111
[7] is the unique, non-symmetric,
positive-definite solution of the Riccati matrix equation. By differentiating with respect to time,
we can write:
xxPxxP )()( (11)
By substituting (11) in (9) and with rearrangement of one, we have:
1
0
1
|
2
)(,0)()()()()()()()()( d
x
h
x
E
txPxQxPxSxPxPxAxAxPxPE xx
T
Fnn
TTT (12)
The relation (12) is called a SDRE for nonlinear singularly perturbed system in finite horizon. It
should be noted that the optimal control law is obtained by computing these Riccati matrices.
The solution conditions for SDRE are that {A(x),B(x)} be stabilizable and {A(x),(Q(x))
1/2
} be
detectable for xR
n
. A sufficient test for the stabilizability condition of {A(x),B(x)} is to check that
the controllability matrix Mc= [B(x),A(x)B(x),…,A
n-1
(x)B(x)] has rank(Mc)=n,x. Similarly, a
sufficient test for detectability of {A(x), (Q(x))
1/2
} is that the observability matrix Mo=[(Q(x))
1/2
,
(Q(x))
1/2
A(x),…, (Q(x))
1/2
A
n-1
(x)] has rank(Mo)=n, x [2]. Furthermore, the closed-loop matrix
A(x)-S(x)P(x) should be pointwise Hurwitz for x. Here, is any region such that the
Lyapunov function xdxPxxV T
1
0
)()( is locally Lipschitz around the origin [2]. The SDRE in
6. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 6
(12) consist
2
)1)(( 2121 nnnn
differential equations that number of these equations is reduced
by using singular perturbation theory.
3.4 The Separated Hamiltonian Matrices
In the proposed method, by separating the slow and fast variables as
,
, 211
1
xx
x
Xs
,
, 212
2
xx
x
X f
we can describe the optimization relations (9) in the form of the following
singularly perturbed state-dependent Hamiltonian matrix:
,
),(),(
),(),(
21222121
21122111
f
s
f
s
X
X
xxHxxH
xxHxxH
X
X
(13)
Where,
T
jiij
ijij
ij
xxAxxQ
xxSxxA
xxH
),(),(
),(),(
,
2121
2121
21 and I, j=1,2. Thus, we assume that the 2n1
eigenvalues of the system (13) are pointwise small and the remaining 2n2 eigenvalues are
pointwise large, corresponding to the slow and fast responses, respectively. The state and co-
state equations (13) constitute a singularly perturbed, two point boundary value problem
(TPBVP). Hence, the asymptotic solution is obtained as an outer solution in terms of the original
independent variable t, initial layer correction in terms of an initial stretched variable
0tt
,
and final layer correction in terms of a final stretched variable
ttF
[5]. Thus, the
composite solutions can be written as follow:
),(),(),(),(
),(),(),(),(
),(),(),(),(
),(),(),(),(
2222
1111
fFfifof
sFsisos
Fio
Fio
PPtPtP
PPtPtP
xxtxtx
xxtxtx
(14)
where
0
2
0
10 0,0,
tt
t
tt
tttt FF
F
. The first terms on the right hand sides of
the above relations represent the outer solution. The second and third terms represent boundary-
layer corrections to the slow manifold near the initial and final times, respectively. Indices o, i and
F correspond to the outer layer, initial, and final correction layers. For any boundary condition on
the slow manifold, states and co-states are given by outer solution. For any boundary condition
out of the slow manifold, the trajectory rapidly approaches the slow manifold according to the fast
manifolds.
We now perform the slow-fast decomposition of the singularly perturbed state-dependent
Hamiltonian matrix, in which H22(x1,x2) must be non-singular for all x1, x2 (in what follows,
dependence upon x1, x2 is not represented, for convenience):
22
2111
12
21
2222
11
2221
1
22
2222
22
22
2221
1
221211
2222
1
221222
2221
1211
0
0
0
0
nn
nnnn
nn
nn
nnnn
nn
IHH
I
H
HHHH
I
HHI
HH
HH
(15)
7. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 7
Stated differently:
f
s
nn
nnnn
nn
nn
f
s
nnnn
nn
X
X
IHH
I
H
HHHH
X
X
I
HHI
22
2111
12
21
2212
11
2221
1
22
2222
22
22
2221
1
221211
2222
1
221222
0
0
0
0
(16)
New co-sate vector can be described as new=Pnew(xnew)xnew, where ,
f
s
new
x
x
x
,
,
,
fsf
fss
new
xx
xx
,, 21 n
f
n
s RR and
fsffsb
fsafss
newnew
xxPxxP
xxPxxP
xP
,,
,,
)(
. Then, the
new slow-fast variables are defined as follow:
,
, s
fss
s
s X
xx
x
(17a)
,
, 21
1
22 fs
fsf
f
f XXHH
xx
x
(17b)
Thus, (13) is converted to a new form:
ff
sfs
HX
HHHHXHHX
22
21
1
221211
1
2212
(18)
Finally, the optimization equations in a singular perturbation model framework with the new
variables are obtained as:
fssf
fss
HHHHHHHHHHHHHHHH
HHHHH
1221
1
222221
1
2221
1
2222
1
2221
1
22121121
1
22
1221
1
221211
(19)
Moreover, the separated state-dependent Hamiltonian matrices Hs(xs,xf) and H22(x1,x2) are
described in the form of the following:
,)(
),(),(
),(),(
)(,,,,,
11
1111
1111
11
22
2121
2121
22212121
1
2221122111
nn
nn
T
snns
nnsnns
nnfss
O
xxAxxQ
xxSxxA
OxxHxxHxxHxxHxxH
(20a)
.)(
),(),(
),(),(
)(,, 22
2222
2222
22 22
21222122
21222122
222122 nn
nn
T
nn
nnnn
nnfsf O
xxAxxQ
xxSxxA
OxxHxxH
(20b)
8. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 8
3.5 The slow-fast SDREs (The third step)
In the proposed method, using the singular perturbation theory, the subsystems (19) are
converted into outer-layer and boundary-layer correction subsystems. The separated SDRE
relations are, then derived and solved for obtaining the optimal control law.
Theorem 1: The singularly perturbed system (1) with performance index (2) is assumed. The
slow- fast state equations in the initial layer correction are obtained as follow:
),(|,),(),( 011122
*
122
*
11 0
txxxPxxxSxxxAx toosoioosiooso
(21a)
),()(|,),(),(),(
),(),(
02
*
022121
*
22
*
12222
*
12122
*
121
22
*
22
*
22
*
12222
*
122
2
0
txtxxxPxxxSPxxxSxxxA
xxPxxxSxxxA
d
dx
otiooioosoiooioo
iooiooioo
i
(21b)
Also, the slow- fast SDREs in the final layer correction are obtained as follow:
),(|,0),(
),(),(),(
112
*
1
2
*
12
*
12
*
1
11 Ftsonnooso
sooosososooo
T
sooosososo
tPPxxQ
PxxSPPxxAxxAPP
F
(22a)
).()(|, 22
*
22222222
*
2222
*
2222 FoFtfFfFofFfFoo
T
oooofF
fF
tPtPPPSPPSPAPSAP
d
dP
F
(22b)
where,
1
0
1
2221
2111
|
2))(())((
))(())((
d
x
h
x
E
txPtxP
txPtxP
xx
T
FF
F
T
F
. Furthermore, the optimal control
law is as follows:
,)),(),(),( 22
*
22
*
122
*
12122
*
1122
*
1
1
iofFoocioo
T
osoioo
T
ioo xxPPxPxxxBxPxxxBxxxRu (23)
where, Pso and PfF are the unique, symmetric, positive-definite solutions of (22), and
so
nn
iooioonnfFoc
P
I
xxxHxxxHIPPP 11
22
),(),( 22
*
12122
*
1
1
2222
*
. The solution necessary
conditions of relations (21) and (22) are as follow:
{Aso(x1o,x
*
2o), Bso(x1o,x
*
2o)} and {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o)} should be stabilizable for
., 21
2
*
1
nn
oo RRxx
{Aso(x1o,x
*
2o),(Qso(x1o,x
*
2o))
1/2
} and {A22o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} should be detectable for
., 21
2
*
1
nn
oo RRxx
The outer equations (24) should have solutions (the slow manifolds) as x
*
2o(x1o,P11o),
P
*
21o(x1o,P11o) and P
*
22o(x1o,P11o)
,0 2222222212122112121 noooooooooo xPSAxPSPSA (24a)
,0 212121221121221221222222 nnoooooo
T
oooo
T
o QAPPSPAPSPA (24b)
,0 222222222222222222 nnooooo
T
ooo QPSPPAAP (24c)
It should be noted that in the above relations, all the elements of the state and Riccati matrices
are dependent on state variables, and have not been represented for simplicity.
Proofs of the theorems are given in appendix.
Remark 2: SDREs in (22) have n1n2 the less differential equations respect to (12).
9. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 9
4. EXAMPLE
Consider a single link flexible joint robot manipulator as it has been introduced in [11]. This link is
directly actuated by a D.C. electrical motor whose rotor is elastically coupled to the link. In this
example, the mathematical model of system is as follows:
uqqkqqI
qqkqmglqI
)(
0)()sin(
2122
2111
(25)
FIGURE 2: Single link flexible joint robot manipulator
In Table 1 there is a complete list of notations of the mathematical model of a single link flexible
joint robot manipulator.
TABLE 1: Notations the mathematical model of a single link flexible joint robot manipulator.
Moreover, parameter values are given in Table 2.
TABLE 2: Parameter values of the single link flexible joint robot manipulator.
Defining
2
1
22
1
2
1
13
12
11
1 ,,
x
x
xqx
q
q
q
x
x
x
x
and =J, state equations are as follow:
s
s
xu
xkxkx
x
I
k
x
I
k
x
I
mgl
x
x
x
x
x
x
/0
/0
0
0
0
21211
121111
2
13
2
13
12
11
0
0
3
10
,
1
0
0
0
)sin(
(26)
Notation Description
q1 angular positions of the link
q2 angular positions of the motor
u actuator force (motor torque)
I the arm inertia
J the motor inertia
the motor viscous friction
mgl the nominal load in the rotor link
K the stiffness coefficient of flexible joint
parameter Value of parameter
I 0.031(Kg.m
2
)
J 0.004(Kg.m
2
)
0.007
k 7.13
mgl 0.8 (N.m)
10. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 10
It is desired to obtain the optimal control law such that the following performance index 𝒥 is
minimized.
𝒥
5
0
22
2
2
13
2
12
2
11 dtuxxxx
(27)
In this example, ,1)(,)(,
1
0
0
0
)(,
)sin(
)(
2
13
12
11
21211
121111
2
13
xR
x
x
x
x
xkxB
xkxkx
x
I
k
x
I
k
x
I
mgl
x
x
xf
and
h(x(tF))=0. Moreover, f(x), k(x) are differentiable with respect to x for a sufficient number of times
and x=04 is the equilibrium point of the system. Furthermore, t0=0, tF=5, P(x(tF))=044.
Step 1 (Description of the system as SDCs):
To solve the optimization problem, the nonlinear functions f(x), k(x) must first be represented as
SDCs. A suitable choice, considering [1], is as follows:
0
00
)sin(
1000
0100
)(
11
11
1
0
|
kk
I
k
I
k
Ix
xmgld
x
f
xA
xx (28a)
1000
0100
0010
0001
)(
1
0
|
d
x
k
xK
xx
(28b)
Step 2 (Description of the Hamiltonian matrix as SDCs):
The separated Hamiltonian matrices can be derived:
001100
1
00
1
1
1
)sin(
1
00
11
1
0000
)sin(
0
1
1
00
11
000100
),(
22
2
2
2
11
11
22
2
2
2
11
11
222
21
I
kkkk
I
k
Ix
xmglkkk
I
k
I
k
Ix
xmgl
kk
xxHs
(29a)
1
1
),( 2122 xxH (29b)
Step 3.1 (the outer equations):
The relations (24) have solutions as:
1
)(
2
1323122211121211
2
*
osoosoosooo
o
xPxPxPxxk
x (30a)
11. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 11
1
1
1
2
23
2
22
2
12
21
*
so
so
so
o
P
kP
k
kP
k
P (30b)
12
22
*
oP
(30c)
Moreover,
2
1
2
*
1
2
2
*
1
11
11
222
*
1 ),(,
0
1
1
0
),(,
0
)sin(
0
11
100
),({ oosoooso
o
o
ooso xxQxxB
I
k
I
k
Ix
xmgl
kk
xxA
}
100
0
12
21
2
1
12
21
2
1
0
12
21
2
1
12
21
2
1
2
22
2
22
2
22
2
22
kk
kk
is stabilizable and detectable.
,),({ 2
*
122 ooo xxA
}1),(,1),( 2
1
2
*
1222
*
12 oooooo xxQxxB is also stabilizable and detectable.
Step 3.2 (the state equations):
According to (21), state variables relations in the initial layer correction are as follow:
s
ooo
osoosoosooo
o
o tx
x
I
k
x
I
k
x
I
mgl
xPxPxPxxk
x
x
/0
0
0
01
121111
2
1323122211121211
13
1
0
3
10
)(,
)sin(
1
)(
(31a)
0
2
022012
02
2
2
2
1
)(3)(107
)(,1
tPtPk
txx
d
dx soso
ii
i
(31b)
Step 3.3 (the slow-fast SDREs):
The slow- fast SDREs in (22) have 3 the less equations respect to the original SDRE.
Considering (22), the SDRE relations in the final layer correction are as follow:
33
33_23_13_
23_22_12_
13_12_11_
0)(,
Fso
os
T
os
T
os
osos
T
os
ososos
so tP
PPP
PPP
PPP
P
12. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 12
132
23
2
33
122
232322
2
22
22
2223
33
112
232312
11
1133
2
2
2212
2
2121323
11
1123
2
12
22
1213
11
1113
33_
23_
22_
13_
12_
11_
21
1
1
1
2
12
1
)sin(
11
)sin(
1
22
1
)sin(2
so
so
so
so
sososo
sososo
so
so
sososo
o
oso
sosososososo
o
oso
sososo
o
oso
os
os
os
os
os
os
P
P
I
P
kP
kPPP
PkkP
I
P
k
I
P
kP
kPPP
Ix
xmglP
kPPPP
k
I
PP
k
Ix
xmglP
PkkP
I
kP
Ix
xmglP
P
P
P
P
P
P
(32a)
1)(,12 222
FfFfFfF
fF
tPPP
d
dP (32b)
Step 3.4 (the optimal control law):
Moreover, the optimal control law is as follow:
ifFosoosoosooo xPxPxPxPxx
k
u 2
2
132312221112212112
)1()(
1
)(
1
(33)
The state equations and SDREs are two-point boundary value problem (TPBVP) and dependent
on state variables, but we have no state values in the whole interval [0,5]. To overcome this
problem we solve the above equations by an iterative procedure [3, 4]. Now, running the
simulation programs, Figures 3, 4 show the angular positions and velocities.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-70
-60
-50
-40
-30
-20
-10
0
10
20
Time(sec)
The angular positions(deg) and first angular velocity(deg/s)
q1
q2
dq1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-60
-50
-40
-30
-20
-10
0
10
Time(sec)
The angular positions(deg) and first angular velocity(deg/s)
q1
q2
dq1
FIGURE 3: The slow state variables (The angular positions of q1, q2 and angular velocity of 1q ).
13. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 13
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-40
-20
0
20
40
60
80
100
Time(sec)
Second angular velocity(deg/s)
dq2
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
0
10
20
30
40
50
60
70
80
90
Time(sec)
Second angular velocity(deg/s)
dq2
FIGURE 4: The fast state variable (angular velocity of 2q ).
Also, Figures 5 and 6 show the Riccati gains.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-20
-15
-10
-5
0
5
10
15
20
Time(sec)
The Riccati gains of P
s
FIGURE 5: The Riccati gains of Ps.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time(sec)
The Riccati gains of P
f
4.985 4.99 4.995 5
0.05
0.1
0.15
0.2
0.25
Time(sec)
The Riccati gains of P
f
FIGURE 6: The Riccati gains of Pf.
From Figures 3 and 5, it can be seen that for any initial and final conditions on the slow
manifold, for different values of , states are given by outer solution. On the other hand,
14. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 14
Figures 4 and 6 show that for any initial and final conditions out of the slow manifold, the
trajectories rapidly approach the slow manifold according to the fast manifolds. Moreover,
Figure 7 shows the optimal control law.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time(sec)
The optimal control law
FIGURE 7: The optimal control law u.
5. CONCLUSION
With the proposed method in this study, it is seen that the finite-horizon optimization problem of a
class of nonlinear singularly perturbed systems leads to SDREs for slow and fast state variables.
One of the advantages of SDRE method is that knowledge of the Jacobian of the nonlinearity in
the states, similar to HJB equation, is not necessary. Thus, the proposed method has not only
simplicity of the LQ method but also higher flexibility, due to adjustable changes in the Riccati
gains. On the other hand, one of the advantages of the singular perturbation theory is that it
reduces high-order systems into two lower-order subsystems due to the interaction between slow
and fast variables. Note that SDREs in the proposed method have n1n2 the less differential
equations respect to the original SDRE. Thus, the slow-fast SDREs have the simpler computing
than original SDRE and provide good approximations of one.
6. References
[1] S.P Banks and K.J. Mhana. “Optimal Control and Stabilization for Nonlinear Systems.”
IMA Journal of Mathematical Control and Information, vol. 9, pp. 179-196, 1992.
[2] T. Çimen. ”State-Dependent Riccati Equation (SDRE) Control: A Survey,” in Proc. 17th
World Congress; the International Federation of Automatic Control Seoul, Korea, 2008,
pp. 3761-3775.
[3] H. Khaloozadeh and A. Abdolahi. “A New Iterative Procedure for Optimal Nonlinear
Regulation Problem,” in Proc. III International Conference on System Identification and
Control Problems, 2004, pp. 1256-1266.
[4] H. Khaloozadeh and A. Abdolahi. “An Iterative Procedure for Optimal Nonlinear Tracking
Problem,” in Proc. Seventh International Conference on Control, Automation, Robotics
and Vision, 2002, pp. 1508-1512.
[5] D.S. Naidu and A.J. Calise. “Singular Perturbations and Time Scales in Guidance and
Control of Aerospace Systems: A Survey.” Journal of Guidance, Control and Dynamics,
vol. 24, no.6, pp. 1057-1078, Nov.-Dec. 2001.
15. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 15
[6] W.C. Su, Z. Gajic and X. Shen. “The Exact Slow-Fast Decomposition of the Algebraic
Riccati Equation of Singularly Perturbed Systems.” IEEE Transactions on Automatic
Control, vol. 37, no. 9, pp. 1456-1459, Sep. 1992.
[7] Z. Gajic, X. Shen and M. Lim. ”High Accuracy Techniques for Singularly Perturbed
Control Systems-an Overview.” PINSA, vol. 65, no. 2, pp. 117-127, March 1999.
[8] Z. Gajic, S. Koskie and C. Coumarbatch. “On the Singularly Perturbed Matrix Differential
Riccati Equation,” in Proc. CDC-ECC'05, 44th IEEE Conference on Decision and Control
and European Control Conference ECC 2005, Seville, Spain,2005, pp. 14–17.
[9] R. Amjadifard and M.T. H. Beheshti. "Robust disturbance attenuation of a class of
nonlinear singularly perturbed systems." International Journal of Innovative Computing,
Information and Control (IJICIC), vol. 6, pp. 1349-4198, 2010.
[10] R. Amjadifard and M.T.H Beheshti. "Robust stabilization for a class of nonlinear
singularly perturbed systems." Journal of Dynamic Systems, Measurement and Control
(ASME), In Press, 2011.
[11] R. Amjadifard, M.J. Yazdanpanah and M.T.H. Beheshti. "Robust regulation of a class of
nonlinear singularly perturbed systems," IFAC, 2005.
[12] R. Amjadifard, S.E. Khadem and H. Khaloozadeh. "Position and velocity control of a
flexible Joint robot manipulator via fuzzy controller based on singular perturbation
analysis," IEEE International Fuzzy Systems Conference, 2001, pp. 348-351.
[13]E. Fridman. “Exact Slow-Fast Decomposition of the Nonlinear Singularly Perturbed
Optimal Control Problem.” System and Control Letters, vol. 40, pp. 121-131, Jun. 2000.
[14]E. Fridman. “A Descriptor System Approach to Nonlinear Singularly Perturbed Optimal
Control Problem.” Automatica, vol. 37, pp. 543-549, 2001.
Appendix A: The relation between the P(x) and Pnew(xnew)
In order to compute the optimal control law, the relations between the Riccati matrices
),(),(
),(),(
)(
21222121
21212111
xxPxxP
xxPxxP
xP
T
and
fsffsb
fsafss
newnew
xxPxxP
xxPxxP
xP
,,
,,
)(
must be determined.
Suppose that
1212
1212
2221
1211
21
1
22
nnnn
nnnn
ll
ll
HH , according to (17), we have:
,
,
,)()()(
,)()(
,
1
2112212222
21
1
2112
2
11121121
1
221222
2
11121121
1
211211
221121111211
21
11
TT
f
TT
b
nn
T
a
nn
TT
s
T
f
PlIPlPp
PPlIp
OPllPllPp
OPllPPlIPp
xPlIxPllx
(A1)
Then, for =0, one can write:
1
2111 ,,
1111
x
xxP
I
x
xxP
I nn
s
fss
nn
(A2a)
16. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 16
2
2122
1
2121
1
2111
21
1
22
,,
0
,,
22121122
x
xxP
I
x
xxP
x
xxP
I
HHx
xxP
I nnnnnn
f
fsf
nn
(A2b)
Now, multiplying (A2b) by 22
, nnfsf IxxP , the following relation is obtained.
2
11
22
0,,,
,
, 2212212121
2111
21
1
22 nfsf
nn
nnfsf xxxPxxPxxxP
xxP
I
HHIxxP
(A3)
In other words, we have:
1
)(1 ns Oxx (A4a)
11
)(),(),( 2111 nnfss OxxPxxP (A4b)
22
)(),(),( 2122 nnfsf OxxPxxP (A4c)
12
)(,),( 2121 nnfsc OxxPxxP (A4d)
Where,
.
,
,,
2111
21
1
22
11
22
xxP
I
HHIxxPxxP nn
nnfsffsc Also, for =0, we have:
soo xx 1
(A5a)
),(),( 2111 fososoooo xxPxxP (A5b)
),(),( 2122 fosofoooo xxPxxP (A5c)
fosocoooo xxPxxP ,),( 2121 (A5d)
Appendix B: Proof of Theorem 1
a) The optimal control law
According to =P(x)x [3] and (A4), substituting Riccati matrices in (6c), the optimal control law
would result as in (23).
b) The slow manifolds in boundary-layer correction
According to the singular perturbation theory, for =0, the fast variable should be derived with
respect to the slow variable. Substituting =0 in (19), the outer-layer equations are obtained as
follows:
,120| foososso HH (B1a)
.0 222 2 foon H (B1b)
Substituting (17b) in (B1b), the following relation is derived:
.0 222212 nfoosoo XHXH (B2)
In other words, considering (14), we have:
,0 2222222212122112121 noooooooooo xPSAxPSPSA (B3a)
,0 212121221121221221222222 nnoooooo
T
oooo
T
o QAPPSPAPSPA
(B3b)
17. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 17
,0 222222222222222222 nnooooo
T
ooo QPSPPAAP
(B3c)
For which, x
*
2o(x1o,P11o), P
*
21o(x1o,P11o) and P
*
22o(x1o,P11o) are the solutions. The necessary
conditions for (B3) to be solvable, {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} should be
pointwise stabilizable and detectable for 21
2
*
1 , nn
oo RRxx [2].
In (B1a), 0| sH for inside and out of the fast manifold, is separated as follows:
0|212121
1
22211221110| ,,,, xxHxxHxxHxxHHs
,,
),(),(
),(),(
100
2121
2121
tttt
xxAxxQ
xxSxxA
T
osos
osos
(B4a)
.,
),(),(
),(),(
10
2
*
12
*
1
2
*
12
*
1
FT
oosoooso
oosoooso
tttt
xxAxxQ
xxSxxA
(B4b)
Substituting (B4) in (B1a), we have:
,),(|,),(),(),( 10001112121211 0
tttttxxxxxPxxSxxAx tooosoososo (B5a)
.),(|,
),(),(),(
),(),(),(
1011
12
*
12
*
112
*
1
12
*
12
*
112
*
1
11
1
FFtso
ooosooo
T
sooooso
ooosooosooooso
osooso
o
tttttPP
xxxPxxAxxxQ
xxxPxxSxxxA
xPxP
x
F
(B5b)
Thus, assuming that {Aso(x1o,x
*
2o), Bso(x1o,x
*
2o), (Qso(x1o,x
*
2o))
1/2
} is pointwise stabilizable-
detectable for 21
2
*
1 , nn
oo RRxx [2], with rearrangement of (B5b), the SDRE of the slow
variable is obtained as (22a).
Remark 3: Note that under assumption of above, Pso is unique, symmetric, positive definite
solution of the SDRE (22a) that produces a locally asymptotically stable closed loop solution [2].
Thus the closed-loop matrix As(x1o,x2)-Ss(x1o,x2)Pso is pointwise Hurwitz for (x1o,x2)12.
Here, 12 is any region such that the Lyapunov function is locally Lipschitz around the origin.
c) The fast manifold in initial layer correction
Since the time scale will be changed as
0tt
in the initial layer correction, the time derivative
in this scale will be changed as
dt
d
d
(.)(.)
in forward time. Considering (4b), we have:
)(|,),(),(),( 0222121212122122
2
0
txxuxxBxxxAxxxA
d
dx
toooo
(B6)
Substituting (23) in (B6), according to (A4) and (14), the fast state equation in initial layer is
obtained as (21b).
d) The fast manifold in final layer correction
Since the time scale will be changed as
ttF
in the final layer correction, the time derivative
in this scale will be changed as
dt
d
d
d (.)(.)
in backward time:
fss
f
fs
s
HHHHHHHHHHHHHHHH
d
d
HHHHH
d
d
1221
1
222221
1
2221
1
2222
1
2221
1
22121121
1
22
1221
1
221211
(B7)
18. Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 18
Substituting =0 in (B7), we have 120 ns . Therefore, the final layer correction equation is
obtained as:
.)(),(|,),( 2102
*
122 FFfffoo
f
txtxxxH
d
d
(B8)
Now, substituting (20b) and (17b) in (B8), we have:
.),(|,
),(),(
),(),(
222
2
*
1222
*
122
2
*
1222
*
122
FFFtf
ffoo
T
foo
ffoofoo
f
ff
f
f
tttttPP
xPxxAxxxQ
xPxxSxxxA
x
d
dP
d
dx
P
d
dx
F
(B9)
Thus, assuming that {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} is stabilizable-detectable for
21
2
*
1 , nn
oo RRxx [2], according to (A5) and (14), the SDRE of the fast variable is obtained as
(22b).
Remark 4: Note that under assumption of above, Pf is unique, symmetric, positive definite
solution of the SDRE (22b) that produces a locally asymptotically stable closed loop solution [2].
Thus, the closed-loop matrix A22(x1o,x2)-S22(x1o,x2)P
*
22o is pointwise Hurwitz for (x1o,x2)12.
Here, 12 is any region such that the Lyapunov function is locally Lipschitz around the origin.