This paper discusses the design of observer-based reduced order controllers for the stabilization of large
scale linear discrete-time control systems. This design is carried out via deriving a reduced-order model
for the given linear plant using the dominant state of the linear plant. Using this reduced-order linear
model, sufficient conditions are derived for the design of observer-based reduced order controllers. A
separation principle has been established in this paper which demonstrates that the observer poles and
controller poles can be separated and hence the pole-placement problem and observer design are
independent of each other.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
MATHEMATICAL MODELING OF COMPLEX REDUNDANT SYSTEM UNDER HEAD-OF-LINE REPAIREditor IJMTER
Suppose a composite system consisting of two subsystems designated as ‘P’ and
‘Q’ connected in series. Subsystem ‘P’ consists of N non-identical units in series, while the
subsystem ‘Q’ consists of three identical components in parallel redundancy.
A New Approach to Design a Reduced Order ObserverIJERD Editor
This document proposes a new method for designing reduced order observers for linear time-invariant systems. The approach is based on inverting matrices of proper dimensions. It reduces the arbitrariness of previous methods by using pole-placement techniques. The method is applied to design a reduced order observer for a 3rd order system. Simulation results show the observer estimates converge to the true system states.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
Measures of different reliability parameters for a complex redundant system u...Alexander Decker
This document summarizes a mathematical model of a complex redundant system consisting of two subsystems (A and B) connected in series. Subsystem A has N non-identical units in series, while subsystem B has 3 identical components in parallel. The model analyzes the system's reliability under a "head-of-line" repair policy where failures follow exponential and repair times follow general distributions. Differential equations are formulated and solved using Laplace transforms to obtain state probabilities and an expression for the expected total cost of the system over time.
Observer design for descriptor linear systemsMahendra Gupta
In this presentation, A method is given to design the observer for descriptor (singular) systems. This talk was delivered at International Conference of Mathematical Sciences, Sathyama University, Chennai, in July 2014
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
MATHEMATICAL MODELING OF COMPLEX REDUNDANT SYSTEM UNDER HEAD-OF-LINE REPAIREditor IJMTER
Suppose a composite system consisting of two subsystems designated as ‘P’ and
‘Q’ connected in series. Subsystem ‘P’ consists of N non-identical units in series, while the
subsystem ‘Q’ consists of three identical components in parallel redundancy.
A New Approach to Design a Reduced Order ObserverIJERD Editor
This document proposes a new method for designing reduced order observers for linear time-invariant systems. The approach is based on inverting matrices of proper dimensions. It reduces the arbitrariness of previous methods by using pole-placement techniques. The method is applied to design a reduced order observer for a 3rd order system. Simulation results show the observer estimates converge to the true system states.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
Measures of different reliability parameters for a complex redundant system u...Alexander Decker
This document summarizes a mathematical model of a complex redundant system consisting of two subsystems (A and B) connected in series. Subsystem A has N non-identical units in series, while subsystem B has 3 identical components in parallel. The model analyzes the system's reliability under a "head-of-line" repair policy where failures follow exponential and repair times follow general distributions. Differential equations are formulated and solved using Laplace transforms to obtain state probabilities and an expression for the expected total cost of the system over time.
Observer design for descriptor linear systemsMahendra Gupta
In this presentation, A method is given to design the observer for descriptor (singular) systems. This talk was delivered at International Conference of Mathematical Sciences, Sathyama University, Chennai, in July 2014
The aim of this report is to use eigenvectors, eigenvalues, and orthogonality to understand the concept of Principal Component Analysis (PCA) and to show why PCA is useful.
Evaluation the affects of mimo based rayleigh network cascaded with unstable ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document contains 10 problems related to analyzing and working with discrete-time linear systems. Problem 1 involves block diagram realization, finding the pulse transfer function, and state space transformations. Problem 2 involves partial fraction expansion and writing state equations in diagonal form for a given transfer function. Problem 3 involves determining equivalence between state space representations and finding transformation matrices.
This document presents a study on using dynamic surface control (DSC) to achieve synchronization and anti-synchronization of identical Arneodo chaotic systems. DSC is proposed as a method to address issues with conventional backstepping control, such as the "explosion of terms". The paper describes the Arneodo chaotic system, then designs DSC controllers to synchronize and anti-synchronize a master and slave Arneodo system. Simulation results demonstrate the controllers drive the errors to zero, achieving synchronization and anti-synchronization. In conclusion, DSC is an effective method for controlling synchronization in identical chaotic systems.
Principal component analysis (PCA) is a technique used to simplify complex datasets. It works by converting a set of observations of possibly correlated variables into a set of linearly uncorrelated variables called principal components. PCA identifies patterns in data and expresses the data in such a way as to highlight their similarities and differences. The main implementations of PCA are eigenvalue decomposition and singular value decomposition. PCA is useful for data compression, reducing dimensionality for visualization and building predictive models. However, it works best for data that follows a multidimensional normal distribution.
Principal component analysis (PCA) is a technique used to reduce the dimensionality of data by transforming correlated variables into a smaller number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA involves computing the covariance matrix of the data and then determining the eigenvectors with the highest eigenvalues, which become the principal components.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
"Principal Component Analysis - the original paper" presentation @ Papers We ...Adrian Florea
Principal Component Analysis (PCA) is a technique for dimensionality reduction that was introduced by Karl Pearson in 1901. PCA finds a set of orthogonal vectors (principal components) along which the variance in the data is maximized. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA is performed by calculating the eigenvalues and eigenvectors of the covariance matrix of the data, and projecting the data onto only the top r principal components that capture most of the variance, where r is typically much less than the original number of dimensions.
Availability of a Redundant System with Two Parallel Active Componentstheijes
This paper considers a redundant system which consists of two parallel active components. The time-to-failure
and the time-to-repair of the components follow an exponential and a general distribution, respectively. The
repairs of failed components are randomly interrupted. The time-to-interrupt is taken from an exponentially
distributed random variable and the interrupt times are generally distributed. We obtain the availability for the
system
Optimal PID Controller Design for Speed Control of a Separately Excited DC Mo...ijscmcj
This paper presents a new approach to determine the optimal proportional-integral-derivative controller parameters for the speed control of a separately excited DC motor using firefly optimization technique. Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in nature. The firefly optimization technique is successfully implemented using MATLAB software. A comparison is drawn from the results obtained between the linear quadratic regulator and firefly optimization techniques. Simulation results are presented to illustrate the performance and validity of the design method.
Generalized synchronization of nonlinear oscillators via opcl couplingIJCI JOURNAL
In this communication, open-plus-closed-loop (OPCL) coupling method is applied to make generalized
synchronization between two non-linear chaotic dynamical systems. For this reason, a transformation
matrix is considered which can be chosen arbitrarily. We have used five different cases to establish our
claim. Chaotic behaviours and the efficiency of the generalized synchronization using OPCL method are
verified by numerical simulations.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document discusses the union find algorithm and its time complexity. It defines the union find problem and three operations: MAKE-SET, FIND, and UNION. It describes optimizations like union by rank and path compression that achieve near-linear time complexity of O(m log* n) for m operations on n elements. It proves several lemmas about ranks and buckets to establish this time complexity through an analysis of the costs T1, T2, and T3.
The document discusses queue data structures. A queue is a linear data structure where additions are made at the end/tail and removals are made from the front/head, following a First-In First-Out (FIFO) approach. Common queue operations include add, remove, check if empty, check if full. A queue can be stored using either a static array or dynamic linked nodes. The key aspects are maintaining references to the head and tail of the queue.
This document contains 8 problems related to systems engineering. The problems cover various topics like determining states and outputs of linear systems given state space representations, controllability/observability analysis, designing state feedback controllers and observers, and computing step responses of closed loop systems.
1) The Group Steiner Problem involves finding a minimum-cost tree connecting at least one node from each of k disjoint groups in a graph. It has applications in VLSI circuit routing.
2) The authors propose a CUDA-aware MPI-based approach to solve large instances of the Group Steiner Problem using a heuristic that constructs 2-star trees in parallel.
3) Experimental results on a supercomputer show the parallel implementation achieves up to 302x speedup over sequential algorithms and finds accurate solutions to industry benchmark instances.
This document contains a homework assignment for a 4th year biomedical engineering systems engineering course. It involves solving several problems related to modeling and analyzing dynamic systems using MATLAB. Students are asked to write individual reports for each problem and not copy solutions from classmates. The problems cover topics like determining states of linear systems, showing equivalence of different representations, analyzing controllability and observability, designing state feedback gains, and developing an integral control system. MATLAB functions for obtaining state feedback and integral gains are also to be implemented.
Debabrata Pal, Aksum University, College of Engineering and Technology Department of Electrical and Computer Engineering Ethiopia, NE Africa, Email:debuoisi@gmail.com,website:www.ijrd.in
The Design of Reduced Order Controllers for the Stabilization of Large Scale ...sipij
This paper investigates the design of reduced order controllers for the stabilization of large scale linear discrete-time control systems. Sufficient conditions are derived for the design of reduced order controllers by obtaining a reduced order model of the original large scale linear system using the dominant state of the system. The reduced order controllers are assumed to use only the state of the reduced order model of the original plant.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
The aim of this report is to use eigenvectors, eigenvalues, and orthogonality to understand the concept of Principal Component Analysis (PCA) and to show why PCA is useful.
Evaluation the affects of mimo based rayleigh network cascaded with unstable ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
This document contains 10 problems related to analyzing and working with discrete-time linear systems. Problem 1 involves block diagram realization, finding the pulse transfer function, and state space transformations. Problem 2 involves partial fraction expansion and writing state equations in diagonal form for a given transfer function. Problem 3 involves determining equivalence between state space representations and finding transformation matrices.
This document presents a study on using dynamic surface control (DSC) to achieve synchronization and anti-synchronization of identical Arneodo chaotic systems. DSC is proposed as a method to address issues with conventional backstepping control, such as the "explosion of terms". The paper describes the Arneodo chaotic system, then designs DSC controllers to synchronize and anti-synchronize a master and slave Arneodo system. Simulation results demonstrate the controllers drive the errors to zero, achieving synchronization and anti-synchronization. In conclusion, DSC is an effective method for controlling synchronization in identical chaotic systems.
Principal component analysis (PCA) is a technique used to simplify complex datasets. It works by converting a set of observations of possibly correlated variables into a set of linearly uncorrelated variables called principal components. PCA identifies patterns in data and expresses the data in such a way as to highlight their similarities and differences. The main implementations of PCA are eigenvalue decomposition and singular value decomposition. PCA is useful for data compression, reducing dimensionality for visualization and building predictive models. However, it works best for data that follows a multidimensional normal distribution.
Principal component analysis (PCA) is a technique used to reduce the dimensionality of data by transforming correlated variables into a smaller number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA involves computing the covariance matrix of the data and then determining the eigenvectors with the highest eigenvalues, which become the principal components.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
"Principal Component Analysis - the original paper" presentation @ Papers We ...Adrian Florea
Principal Component Analysis (PCA) is a technique for dimensionality reduction that was introduced by Karl Pearson in 1901. PCA finds a set of orthogonal vectors (principal components) along which the variance in the data is maximized. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA is performed by calculating the eigenvalues and eigenvectors of the covariance matrix of the data, and projecting the data onto only the top r principal components that capture most of the variance, where r is typically much less than the original number of dimensions.
Availability of a Redundant System with Two Parallel Active Componentstheijes
This paper considers a redundant system which consists of two parallel active components. The time-to-failure
and the time-to-repair of the components follow an exponential and a general distribution, respectively. The
repairs of failed components are randomly interrupted. The time-to-interrupt is taken from an exponentially
distributed random variable and the interrupt times are generally distributed. We obtain the availability for the
system
Optimal PID Controller Design for Speed Control of a Separately Excited DC Mo...ijscmcj
This paper presents a new approach to determine the optimal proportional-integral-derivative controller parameters for the speed control of a separately excited DC motor using firefly optimization technique. Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in nature. The firefly optimization technique is successfully implemented using MATLAB software. A comparison is drawn from the results obtained between the linear quadratic regulator and firefly optimization techniques. Simulation results are presented to illustrate the performance and validity of the design method.
Generalized synchronization of nonlinear oscillators via opcl couplingIJCI JOURNAL
In this communication, open-plus-closed-loop (OPCL) coupling method is applied to make generalized
synchronization between two non-linear chaotic dynamical systems. For this reason, a transformation
matrix is considered which can be chosen arbitrarily. We have used five different cases to establish our
claim. Chaotic behaviours and the efficiency of the generalized synchronization using OPCL method are
verified by numerical simulations.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document discusses the union find algorithm and its time complexity. It defines the union find problem and three operations: MAKE-SET, FIND, and UNION. It describes optimizations like union by rank and path compression that achieve near-linear time complexity of O(m log* n) for m operations on n elements. It proves several lemmas about ranks and buckets to establish this time complexity through an analysis of the costs T1, T2, and T3.
The document discusses queue data structures. A queue is a linear data structure where additions are made at the end/tail and removals are made from the front/head, following a First-In First-Out (FIFO) approach. Common queue operations include add, remove, check if empty, check if full. A queue can be stored using either a static array or dynamic linked nodes. The key aspects are maintaining references to the head and tail of the queue.
This document contains 8 problems related to systems engineering. The problems cover various topics like determining states and outputs of linear systems given state space representations, controllability/observability analysis, designing state feedback controllers and observers, and computing step responses of closed loop systems.
1) The Group Steiner Problem involves finding a minimum-cost tree connecting at least one node from each of k disjoint groups in a graph. It has applications in VLSI circuit routing.
2) The authors propose a CUDA-aware MPI-based approach to solve large instances of the Group Steiner Problem using a heuristic that constructs 2-star trees in parallel.
3) Experimental results on a supercomputer show the parallel implementation achieves up to 302x speedup over sequential algorithms and finds accurate solutions to industry benchmark instances.
This document contains a homework assignment for a 4th year biomedical engineering systems engineering course. It involves solving several problems related to modeling and analyzing dynamic systems using MATLAB. Students are asked to write individual reports for each problem and not copy solutions from classmates. The problems cover topics like determining states of linear systems, showing equivalence of different representations, analyzing controllability and observability, designing state feedback gains, and developing an integral control system. MATLAB functions for obtaining state feedback and integral gains are also to be implemented.
Debabrata Pal, Aksum University, College of Engineering and Technology Department of Electrical and Computer Engineering Ethiopia, NE Africa, Email:debuoisi@gmail.com,website:www.ijrd.in
The Design of Reduced Order Controllers for the Stabilization of Large Scale ...sipij
This paper investigates the design of reduced order controllers for the stabilization of large scale linear discrete-time control systems. Sufficient conditions are derived for the design of reduced order controllers by obtaining a reduced order model of the original large scale linear system using the dominant state of the system. The reduced order controllers are assumed to use only the state of the reduced order model of the original plant.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
Global stabilization of a class of nonlinear system based on reduced order st...ijcisjournal
The problem of global stabilization for a class of nonlinear system is considered in this paper.The sufficient
condition of the global stabilization of this class of system is obtained by deducing thestabilization of itself
from the stabilization of its subsystems. This paper will come up with a designmethod of state feedback
control law to make this class of nonlinear system stable, and indicate the efficiency of the conclusion of
this paper via a series of examples and simulations at the end. Theresults presented in this paper improve
and generalize the corresponding results of recent works.
T-S Fuzzy Observer and Controller of Doubly-Fed Induction GeneratorIJPEDS-IAES
This paper aims to ensure a stability and observability of doubly fed induction generator DFIG of a wind turbine based on the approach of fuzzy control type T-S PDC (Parallel Distributed Compensation) which determines the control laws by return state and fuzzy observers.First, the fuzzy TS model is used to precisely represent a nonlinear model of DFIG proposed and adopted in this work. Then, the stability analysis is based on the quadratic Lyapunov function to determine the gains that ensure the stability conditions.The fuzzy observer of DFIG is built to estimate non-measurable state vectors and the estimated states converging to the actual statements. The gains of observatory and of stability are obtained by solving a set of linear matrix inequality (LMI).Finally, numerical simulations are performed to verify the theoretical results and demonstrate satisfactory performance.
Adaptive Stabilization and Synchronization of Hyperchaotic QI SystemCSEIJJournal
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of four-
dimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization
of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to
stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control
theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations
are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization
schemes.
On the principle of optimality for linear stochastic dynamic systemijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
This document summarizes research on adaptive chaos control and synchronization of a hyperchaotic Liu system. The system is a 4-dimensional hyperchaotic system with unknown parameters. Adaptive control laws are designed to stabilize the hyperchaotic system to the origin based on Lyapunov stability theory. Adaptive control laws are also derived to achieve synchronization between identical hyperchaotic Liu systems with unknown parameters. Numerical simulations demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM cseij
This document summarizes research on adaptive stabilization and synchronization of hyperchaotic systems. It specifically focuses on applying these techniques to the hyperchaotic Qi system, a 4-dimensional hyperchaotic system. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to equilibrium based on adaptive control theory and Lyapunov stability theory. Then, adaptive control laws are derived to achieve global synchronization between identical hyperchaotic Qi systems with unknown parameters. Numerical simulations demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
1. The document describes the syllabus for the course EE1354 - Modern Control Systems. It includes 5 units that cover topics like state space analysis of continuous and discrete time systems, z-transforms, nonlinear systems, and MIMO systems.
2. Key concepts discussed include state variable representation, eigenvectors and eigenvalues, solution of state equations, controllability and observability, and deriving state space models from transfer functions.
3. Methods like pole placement, state feedback, and observer design for state estimation are also covered in the context of analysis and design of control systems.
Margin Parameter Variation for an Adaptive Observer to a Class of SystemsCSCJournals
In this paper, adaptive observer robustness is studies. The adaptive observer is proposed to a nonlinear system linearized by output injected for variable structure systems. Despites, the adaptive observer is suggesting to supply the parameter variation but it isn't usually verified. Unfortunately, the parameter uncertainty impact state convergence. For that a margin parameter variation is determined which the convergence of the state is guaranty. Simulation results show that the adaptive observer is robust only in the definite parameter margin variation.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Controller design of inverted pendulum using pole placement and lqreSAT Journals
Abstract In this paper modeling of an inverted pendulum is done using Euler – Lagrange energy equation for stabilization of the pendulum. The controller gain is evaluated through state feedback and Linear Quadratic optimal regulator controller techniques and also the results for both the controller are compared. The SFB controller is designed by Pole-Placement technique. An advantage of Quadratic Control method over the pole-placement techniques is that the former provides a systematic way of computing the state feedback control gain matrix.LQR controller is designed by the selection on choosing. The proposed system extends classical inverted pendulum by incorporating two moving masses. The motion of two masses that slide along the horizontal plane is controllable .The results of computer simulation for the system with Linear Quardatic Regulator (LQR) & State Feedback Controllers are shown in section 6. Keyword-Inverted pendulum, Mathematical modeling Linear-quadratic regulator, Response, State Feedback controller, gain formulae.
DYNAMICS, ADAPTIVE CONTROL AND EXTENDED SYNCHRONIZATION OF HYPERCHAOTIC SYSTE...ijccmsjournal
This work studies the dynamics, control and synchronization of hyperchaotic Lorenz-stenflo system and its
application to secure communication. The proposed designed nonlinear feedback controller control and
globally synchronizes two identical Lorenz-stenflo hyperchaotic systems evolving from different initial
conditions with unknown parameters. Adaptive synchronization results were further applied to secure
communication. The numerical simulation results were presented to verify the effectiveness of the designed
nonlinear controller and its success in secure communication application.
This document describes a new method for designing fuzzy controllers based on extending the classical Lyapunov synthesis method. The key points are:
1) The method assumes minimal knowledge about the plant and systematically derives the fuzzy rule base for the controller.
2) The method is demonstrated by designing Mamdani-type controllers for stabilizing and tracking an inverted pendulum system. The rule bases are derived analytically rather than heuristically.
3) Simulation results show the controllers achieve local stability of the closed-loop systems, even in the presence of noise, indicating the effectiveness of the new fuzzy Lyapunov synthesis method.
The Controller Design For Linear System: A State Space ApproachYang Hong
The controllers have been widely used in many industrial processes. The goal of accomplishing a practical control system design is to meet the functional requirements and achieve a satisfactory system performance. We will introduce the design method of the state feedback controller, the state observer and the servo controller with optimal control law for a linear system in this paper.
DESIGN OF OBSERVER BASED QUASI DECENTRALIZED FUZZY LOAD FREQUENCY CONTROLLER ...ijfls
This paper proposes Fuzzy Quasi Decentralized Functional Observers (FQDFO) for Load Frequency Control of inter-connected power systems. From the literature, it is well noticed about the need of
Functional Observers (FO’s) for power system applications. In past, conventional Functional Observers are used. Later, these conventional Functional Observers are replaced with Quasi Decentralized
Functional Observers (QDFO) to improve the system performance. In order to increase the efficacy of the
system, intelligent controllers gained importance. Due to their expertise knowledge, which is adaptive in
nature is applied successfully for FQDFO. For supporting the validity of the proposed observer FQDFO,
it is compared with full order Luenberger observer and QDFO for a two-area inter connected power system by taking parametric uncertainties into consideration. Computational results proved the robustness of the proposed observer.
DESIGN OF OBSERVER BASED QUASI DECENTRALIZED FUZZY LOAD FREQUENCY CONTROLLER ...Wireilla
ABSTRACT
This paper proposes Fuzzy Quasi Decentralized Functional Observers (FQDFO) for Load Frequency Control of inter-connected power systems. From the literature, it is well noticed about the need of Functional Observers (FO’s) for power system applications. In past, conventional Functional Observers are used. Later, these conventional Functional Observers are replaced with Quasi Decentralized Functional Observers (QDFO) to improve the system performance. In order to increase the efficacy of the system, intelligent controllers gained importance. Due to their expertise knowledge, which is adaptive in nature is applied successfully for FQDFO. For supporting the validity of the proposed observer FQDFO, it is compared with full order Luenberger observer and QDFO for a two-area inter connected power system by taking parametric uncertainties into consideration. Computational results proved the robustness of the proposed observer.
The document discusses using clustering models like subtractive fuzzy clustering (SFC) and fuzzy c-means clustering (FCM) to generate an adaptive neuro-fuzzy inference system (ANFIS) for medical diagnoses. Experimental results on medical diagnosis datasets show that ANFIS models using SFC and FCM clustering (ANFIS-SFC and ANFIS-FCM) had better average training and checking errors compared to ANFIS without clustering. Specifically, ANFIS-SFC performed best using backpropagation learning, while ANFIS-FCM performed best using a hybrid learning model. Clustering the datasets without ANFIS was also able to identify different disease clusters.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Control system introduction for different applicationAnoopCadlord1
The document provides an overview of control systems design. It begins by describing the general process for designing a control system, which involves modeling interconnected system components to achieve a desired purpose. Examples of early control systems are discussed to illustrate fundamental feedback principles still used today. Modern applications of control engineering are then briefly mentioned. The document notes that a design gap exists between complex physical systems and their models. An iterative design process is used to effectively address this gap while meeting performance and cost objectives.
Similar to OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS (20)
A Free 200-Page eBook ~ Brain and Mind Exercise.pptxOH TEIK BIN
(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptxCapitolTechU
Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx
OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
1. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
DOI : 10.5121/ijcseit.2011.1508 85
OBSERVER-BASED REDUCED ORDER CONTROLLER
DESIGN FOR THE STABILIZATION OF LARGE SCALE
LINEAR DISCRETE-TIME CONTROL SYSTEMS
Sundarapandian Vaidyanathan1
and Kavitha Madhavan2
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
2
Department of Mathematics, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
kavi78_m@yahoo.com
ABSTRACT
This paper discusses the design of observer-based reduced order controllers for the stabilization of large
scale linear discrete-time control systems. This design is carried out via deriving a reduced-order model
for the given linear plant using the dominant state of the linear plant. Using this reduced-order linear
model, sufficient conditions are derived for the design of observer-based reduced order controllers. A
separation principle has been established in this paper which demonstrates that the observer poles and
controller poles can be separated and hence the pole-placement problem and observer design are
independent of each other.
KEYWORDS
Dominant State, Reduced-Order Model, Stabilization, Observers, Discrete-Time Linear System.
1. INTRODUCTION
The reduced order controller design and observer design are active research problems in the linear
systems literature and there has been a significant attention paid in the literature on these two
problems during the past four decades [1-10]. In the recent decades, there has been a
considerable attention paid to the control problem of large scale linear systems. This is mainly
owing to the practical and technical issues like data acquisition, sensing, computing facilities,
information transfer networks and the cost associated with them, which arise from the use of full
order controller design. Thus, there is a great demand in the industry for the control of large scale
linear systems with the use of reduced-order controllers rather than full-order controllers.
A recent approach for obtaining the reduced-order controllers is via the reduced-order model of a
linear plant preserving the dynamic as well as static properties of the system and then devising
controllers for the reduced-order model thus obtained [4-8]. This approach has practical and
2. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
86
technical benefits for the reduced-order controller design for large-scale linear systems with high
dimension and complexity.
The observer-based reduced-order controller design is motivated by the fact that the dominant
state of the linear plant may not be available for measurement and hence for implementing the
pole placement law, only the reduced order exponential observer can be used in lieu of the
dominant state of the given discrete-time linear system.
In this paper, we derive a reduced-order model for any linear discrete-time control system and our
approach is based on the approach of using the dominant state of the given linear discrete-time
control system, i.e. we derive the reduced-order model for a given discrete-time linear control
system keeping only the dominant state of the given discrete-time linear control system. Using
the reduced-order model obtained, we characterize the existence of a reduced-order exponential
observer that tracks the state of the reduced-order model, i.e. the dominant state of the original
linear plant. We note that the reduced-order observer design detailed in this paper is a discrete-
time analogue of the results of Aldeen and Trinh [8] for the observer design of the dominant state
of continuous-time linear control systems.
Using the reduced-order model of the original linear plant, we also characterize the existence of a
stabilizing feedback control law that uses only the dominant state of the original plant. Also,
when the plant is stabilizable by a state feedback control law, the full information of the dominant
state may not be always available. For this reason, we establish a separation principle so that the
state of the exponential observer may be used in lieu of the dominant state of the original linear
plant, which facilitates the implementation of the stabilizing feedback control law derived. The
design of observer-based reduced order controllers for large scale discrete-time linear systems
derived in this paper has important practical, industrial applications.
This paper is organized as follows. In Section 2, we describe the construction of the reduced-
order plant of a given linear discrete-time control system. In Section 3, we derive necessary and
sufficient conditions for the exponential observer design for the reduced-order linear control
plant. In Section 4, we derive necessary and sufficient conditions for the reduced-order plant to be
stabilizable by a linear feedback control law and we also present a separation principle for the
observer-based reduced order controller design. In Section 5, we describe a numerical example.
In Section 6, we summarize the main results derived in this paper.
2. REDUCED ORDER MODEL FOR DISCRETE-TIME LINEAR SYSTEMS
In this section, we consider a large scale linear discrete-time control system 1S given by
( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
+ = +
=
(1)
where n
x R∈ is the state, m
u R∈ is the control or input and p
y ∈R is the system output. We
assume that ,A B and C are constant matrices with real entries having dimensions
,n n n m× × p n× respectively.
First, we suppose that we have made an identification of the dominant (slow) and non-
dominant (fast) states of the original linear system (1) using the modal approach as
described in [9].
Without loss of generality, we may assume that
3. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
87
,
s
f
x
x
x
=
where r
sx R∈ represents the dominant state and n r
fx R −
∈ represents the non-dominant
state of the system.
Then the linear system (1) becomes
( 1) ( )
( )
( 1) ( )
( )
( )
( )
s ss sf s s
f fs ff f f
s
s f
f
x k A A x k B
u k
x k A A x k B
x k
y k C C
x k
+
= + +
=
(2)
From (2), we can write the plant equations as
( 1) ( ) ( ) ( )
( 1) ( ) ( ) ( )
( ) ( ) ( )
s ss s sf f s
f fs s ff f f
s s f f
x k A x k A x k B u k
x k A x k A x k B u k
y k C x k C x k
+ = + +
+ = + +
= +
(3)
Next, we shall assume that the matrix A has a set of n linearly independent eigenvectors.
In most practical situations, the matrix A has distinct eigenvalues and this condition is
immediately satisfied.
Thus, it follows that A is diagonalizable.
Thus, we can find a non-singular (modal) matrix P consisting of n linearly independent
eigenvectors of A such that
1
,P AP−
= Λ
where Λ is a diagonal matrix consisting of the n eigenvalues of .A
Now, we introduce a new set of coordinates on the state space given by
1
P xξ −
= (4)
In the new coordinates, the plant (1) becomes
1
( 1) ( ) ( )
( ) ( )
k k P Bu k
y k CP k
ξ ξ
ξ
−
+ = Λ +
=
Thus, we have
4. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
88
1
( 1) 0 ( )
( )
( 1) 0 ( )
( )
( )
( )
s s s
f f f
s
f
k k
P Bu k
k k
k
y k CP
k
ξ ξ
ξ ξ
ξ
ξ
−
+ Λ
= + + Λ
=
(5)
where sΛ and fΛ are r r× and ( ) ( )n r n r− × − diagonal matrices respectively, consisting of the
dominant and non-dominant eigenvalues of .A
Define matrices , ,s f sΓ Γ Ψ and fΨ by
1 s
f
P B−
Γ
= Γ
and s fCP = Ψ Ψ (6)
where , ,s f sΓ Γ Ψ and fΨ are , ( ) ,r m n r m p r× − × × and ( )p n r× − matrices respectively.
From (5) and (6), we see that the plant (3) has the following simple form in the new coordinates
(4).
( 1) ( ) ( )
( 1) ( ) ( )
( ) ( ) ( )
s s s s
f f f f
s s f f
k k u k
k k u k
y k k k
ξ ξ
ξ ξ
ξ ξ
+ = Λ + Γ
+ = Λ + Γ
= Ψ + Ψ
(7)
Next, we make the following assumptions:
(H1) As ,k → ∞ ( 1) ( ),f fk kξ ξ+ ≈ i.e. fξ takes a constant value in the steady-state.
(H2) The matrix fI − Λ is invertible.
Then it follows from (7) that for large values of ,k we have
( ) ( ) ( )f f f fk k u kξ ξ≈ Λ + Γ
i.e.
1
( ) ( ) ( )f f fk I u kξ −
≈ − Λ Γ (8)
Substituting (8) into (7), we obtain the reduced-order model of the linear plant (1) in the
ξ coordinates as
1
( 1) ( ) ( )
( ) ( ) ( ) ( )
s s s s
s s f f f
k k u k
y k k I u k
ξ ξ
ξ −
+ = Λ + Γ
= Ψ + Ψ − Λ Γ
(9)
5. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
89
To obtain the reduced-order model of the linear plant (1) in the x coordinates, we proceed as
follows.
By the linear change of coordinates (4), it follows that
1
.P x Qxξ −
= =
Thus, we have
( ) ( ) ( )
( ) ( ) ( )
s s ss sf s
f f fs ff f
k x k Q Q x k
Q
k x k Q Q x k
ξ
ξ
= =
(10)
Using (9) and (10), it follows that
1
( ) ( ) ( ) ( ) ( )f fs s ff f f fk Q k Q k I u kξ ξ ξ −
= + = − Λ Γ
or
1
( ) ( ) ( ) ( )ff f fs s f fQ k Q x k I u kξ −
= − + − Λ Γ (11)
Next, we assume the following:
(H3) The matrix ffQ is invertible.
Using the assumption (H3), the equation (11) becomes
1 1 1
( ) ( ) ( ) ( )f ff fs s ff f fx k Q Q x k Q I u k− − −
= − + − Λ Γ (12)
To simplify the notation, we define the matrices
1
ff fsR Q Q−
= − and 1 1
( )fff fS Q I− −
= − Λ Γ (13)
Using (13), the equation (12) can be simplified as
( ) ( ) ( )f sx k Rx k Su k= + (14)
Substituting (14) into (3), we obtain the reduced-order model 2S of the given linear system 1S as
( 1) ( ) ( )
( ) ( ) ( )
s s s s
s s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(15)
where the matrices , ,s s sA B C∗ ∗ ∗
and sD∗
are defined by
6. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
90
s ss sf
s s sf
s s f
s f
A A A R
B B A S
C C C R
D C S
∗
∗
∗
∗
= +
= +
= +
=
(16)
3. REDUCED ORDER OBSERVER DESIGN
In this section, we state a new result that prescribes a simple procedure for estimating the
dominant state of the given linear control system 1S that satisfies the assumptions (H1)-(H3)
described in Section 2.
Theorem 1. Let 1S be the linear system described by
( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
+ = +
=
(17)
Under the assumptions (H1)-(H3), the reduced-order model 2S of the linear system 1S can be
obtained (see Section 2) as
( 1) ( ) ( )
( ) ( ) ( )
s s s
s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(18)
where , ,s s sA B C∗ ∗ ∗
and sD∗
are defined as in (16).
To estimate the dominant state sx of the system 1,S consider the candidate observer 3S defined
by
( 1) ( ) ( ) ( ) ( ) ( )s s s s s s s sz k A z k B u k K y k C z k D u k∗ ∗ ∗ ∗ ∗
+ = + + − − (19)
Let the estimation error be defined as
s se z x= −
Then ( ) 0e k → exponentially as k → ∞ if and only if the matrix sK∗
is such that
s s sE A K C∗ ∗ ∗
= − is convergent. If ( ),s sC A∗ ∗
is observable, then we can always construct an
exponential observer of the form (19) having any desired speed of convergence.
Proof. From Eq. (2), we have
( 1) ( ) ( ) ( )s ss s sf f sx k A x k A x k B u k+ = + + (20)
Adding and subtracting the term ( ) ( )sf s f sA K C Rx k∗
− in the RHS of Eq. (20), we get
7. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
91
( 1) ( ) ( ) ( ) ( ) ( ) ( )s ss sf s f s sf f sf s f s sx k A A R K C x k A x k A K C Rx k B u k∗ ∗
+ = + − + − − + (21)
Subtracting (21) from (19) and simplifying using the definitions (16), we get
( 1) ( ) ( ) ( )[ ( ) ( ) ( )]s s s sf s f f se k A K C e k A K C x k Rx k Su k∗ ∗ ∗ ∗
+ = − − − − − (22)
As proved in Section 2, the assumptions (H1)-(H3) yield
( ) ( ) ( )f sx k Rx k Su k≈ + (23)
Substituting (23) into (22), we get
( 1) ( ) ( ) ( )s s se k A K C e k Ee k∗ ∗ ∗
+ = − = (24)
which yields
( ) ( ) (0) (0)k k
s s se k A K C e E e∗ ∗ ∗
= − = (25)
From Eq. (25), it follows that ( ) 0e k → as k → ∞ if and only if E is convergent.
If ( ),s sC A∗ ∗
is observable, then it is well-known [10] that we can find an observer gain matrix sK∗
that will arbitrarily assign the eigenvalues of the error matrix s s sE A K C∗ ∗ ∗
= − . In particular, it
follows from Eq. (25) that we can find an exponential observer of the form (19) having any
desired speed of convergence.
4. OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN
In this section, we first state an important result that prescribes a simple procedure for stabilizing
the dominant state of the reduced-order linear plant derived in Section 2.
Theorem 2. Suppose that the assumptions (H1)-(H3) hold. Let 1S and 2S be defined as in
Theorem 1. For the reduced-order model 2 ,S the state feedback control law
( ) ( )s su k F x k∗
= − (26)
stabilizes the dominant state sx of the reduced-order model 2S having any desired speed of
convergence.
In practical applications, the dominant state sx of the reduced-order model 2S may not be directly
available for measurement and hence we cannot implement the state feedback control law (26).
To overcome this practical difficulty, we derive an important theorem, usually called as the
Separation Principle, which first establishes that the observer-based reduced-order controller
indeed stabilizes the dominant state of the given linear control system 1S and also demonstrates
that the observer poles and the closed-loop controller poles can be separated.
8. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
92
Theorem 3 (Separation Principle). Suppose that the assumptions (H1)-(H3) hold. Suppose that
there exist matrices sF∗
and sK∗
such that s s sA B F∗ ∗ ∗
− and s s sA K C∗ ∗ ∗
− are both convergent
matrices. By Theorem 1, we know that the system 3S defined by (19) is an exponential observer
for the dominant state sx of the control system 1.S Then the observer poles and the closed-loop
controller poles are separated and the control law
( ) ( )s su k F z k∗
= − (27)
also stabilizes the dominant state sx of the control system 1.S
Proof. Under the feedback control law (27), the observer dynamics (19) of the system 3S
becomes
( )( 1) ( ) ( ) ( )s s s s s s s s s s s s s f fz k A B F K C K D F z k K C x k C x k∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
+ = − − − + + (28)
By (14), we know that
( ) ( ) ( ) ( ) ( )f s s s sx k Rx k Su k Rx k SF z k∗
≈ + = − (29)
Substituting (29) into (28) and simplifying using the definitions (16), we get
( 1) ( ) ( ) ( )s s s s s s s s s sz k A B F K C z k K C x k∗ ∗ ∗ ∗ ∗ ∗ ∗
+ = − − + (30)
Substituting the control law (27) into (3), we also obtain
( 1) ( ) ( )s s s s s sx k A x k B F z k∗ ∗ ∗
+ = − (31)
In matrix representation, we can write equations (30) and (31) as
( 1) ( )
( 1) ( )
s ss s s
s ss s s s s s s
x k x kA B F
z k z kK C A B F K C
∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗
+ −
= + − −
(32)
Since the estimation error e is defined by ,s se z x= − it is easy to see from equation (32) that the
error e satisfies the equation
( )( 1) ( )s s se k A K C e k∗ ∗ ∗
+ = − (33)
Using the ( , )x e coordinates, the composite system (32) can be simplified as
( 1) ( ) ( )
( 1) ( ) ( )0
s s ss s s s s
s s ss s s
x k x k x kA B F B F
M
e k e k e kA K C
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗
+ − −
= = + −
(34)
where
9. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
93
.
0
s s s s s
s s s
A B F B F
M
A K C
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗
− −
=
−
(35)
Since the matrix M is block-triangular, it is immediate that
( ) ( )eig( ) eig eigs s s s s sM A B F A K C∗ ∗ ∗ ∗ ∗ ∗
= − −U (36)
which establishes the first part of the Separation Principle namely that the observer poles are
separated from the closed-loop controller poles.
To show that the observed-based control law (27) indeed works, we note that the closed-loop
regulator matrix s s sA B F∗ ∗ ∗
− and s s sA K C∗ ∗ ∗
− are both convergent matrices. From Eq. (36), it is
immediate that M is also a convergent matrix. From Eq. (35), it is thus immediate that
( ) 0sx k → and ( ) 0se k → as k → ∞ for all (0)sx and (0).e
5. NUMERICAL EXAMPLE
In this section, we consider a fourth-order linear discrete-time control system described by
( 1) ( ) ( )
( ) ( )
x k A x k B u k
y k Cx k
+ = +
=
(37)
where
2.0 0.6 0.2 0.3 1
0.4 0.4 0.9 0.5 1
,
0.1 0.3 0.5 0.1 1
0.7 0.9 0.8 0.8 1
A B
= =
and [ ]1 2 1 1 .C = (38)
The eigenvalues of the matrix A are
1 2 32.4964, 1.0994, 0.3203λ λ λ= = = and 4 0.2161.λ = − (39)
From (39), we note that 1 2,λ λ are unstable (slow) eigenvalues and 3 4,λ λ are stable (fast)
eigenvalues of the system matrix .A
For this linear system, the dominant and non-dominant states are calculated. A simple calculation
using the procedure in [9] shows that the first two states{ }1 2,x x are the dominant (slow) states,
while the last two states { }3 4,x x are the non-dominant (fast) states for the given system (37).
Using the procedure described in Section 2, the reduced-order linear model for the given linear
system (37) can be obtained as
10. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
94
( 1) ( ) ( )
( ) ( ) ( )
s s s
s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(40)
where
2.0077 1.1534 0.8352
,
0.3848 1.5581 1.1653
s sA B∗ ∗
= =
[ ]1.0092 4.0011sC∗
= and 0.2904sD∗
= −
The step responses of the original plant and the reduced order plant are plotted in Figure 1, which
validates the reduced-order model obtained for the given linear plant.
Figure 1. Step Responses for the Original and Reduced Order Linear Systems
We note also that the reduced-order linear system (40) is completely controllable and completely
observable. Thus, reduced-order observers and observer-based controllers for this plant can be
easily constructed as detailed in Sections 3 and 4.
6. CONCLUSIONS
In this paper, using the dominant state analysis of the given large-scale linear plant, we obtained
the reduced-order model of the linear plant. Then we derived sufficient conditions for the design
of observer-based reduced-order controllers. The observer-based reduced order controllers are
constructed by combining the reduced order controllers for the original linear system which
11. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
95
require the dominant state of the original system and reduced order observers for the original
linear system which provide an exponential estimate of the dominant state of the original linear
system. We also established a separation principle in this paper which shows that the pole
placement problem and observer problem are independent of each other.
REFERENCES
[1] Cumming, S.D. (1969) “Design of observers for reduced dynamics”, Electronics Letters, Vol. 5, pp
213-214.
[2] Fortman, T.E. & Williamson, D. (1971) “Design of low-order observers for linear feedback control
laws”, IEEE Trans. Automat. Control, Vol. 17, pp 301-308.
[3] Litz, L. & Roth, H. (1981) “State decomposition for singular perturbation order reduction – a modal
approach”, Internat. J. Control, Vol. 34, pp. 937-954.
[4] Lastman, G. & Sinha, N. & Rozsa, P. (1984) “On the selection of states to be retained in reduced-
order model”, IEE Proc. Part D, Vol. 131, pp 15-22.
[5] Anderson, B.D.O. & Liu, Y. (1989) “Controller reduction: concepts and approaches”, IEEE Trans.
Automat. Control, Vol. 34, pp 802-812.
[6] Mustafa, D. & Glover, K. (1991) “Controller reduction by H∞ balanced truncation”, IEEE Trans.
Automat. Control, Vol. 36, pp 668-692.
[7] Aldeen, M. (1991) “Interaction modelling approach to distributed control with application to
interconnected dynamical systems”, Internat. J. Control, Vol. 241, pp 303-310.
[8] Aldeen, M. & Trinh, H. (1994) “Observing a subset of the states of linear systems”, IEE Proc.
Control Theory Appl., Vol. 141, pp 137-144.
[9] Sundarapandian, V. (2005) “Distributed control schemes for large scale interconnected discrete-time
linear systems”, Math. Computer Modelling, Vol. 5, pp 313-319.
[10] Ogata, K. (2010) Discrete-Time Control Systems, Prentice Hall, New Jersey.
Authors
Dr. V. Sundarapandian obtained his Doctor of Science degree in Electrical and
Systems Engineering from Washington University, Saint Louis, USA under the guidance
of Late Dr. Christopher I. Byrnes (Dean, School of Engineering and Applied Science) in
1996. He is currently Professor in the Research and Development Centre at Vel Tech Dr.
RR & Dr. SR Technical University, Chennai, Tamil Nadu, India. He has published over
210 refereed international publications. He has published over 100 papers in National
Conferences and over 50 papers in International Conferences. He is the Editor-in-Chief
of International Journal of Mathematics and Scientific Computing, International Journal
of Instrumentation and Control Systems, International Journal of Control Systems and Computer
Modelling, etc. His research interests are Linear and Nonlinear Control Systems, Chaos Theory and
Control, Soft Computing, Optimal Control, Process Control, Operations Research, Mathematical
Modelling, Scientific Computing using MATLAB etc.
Mrs. M. Kavitha received the B.Sc. degree in Mathematics from Bharatidasan
University, Tiruchirappalli, Tamil Nadu in 1998, M.Sc. degree in Mathematics from
Bharatidasan University, Tiruchirappalli, Tamil Nadu in 2000, P.G. Diploma in
Computer Applications from Bharatidasan University, Tiruchirappalli, Tamil Nadu,
M.Phil in Mathematics from Madurai Kamaraj University, Madurai, Tamil Nadu in
2004 and currently pursuing PhD degree in Mathematics from Vel Tech Dr. RR & Dr.
SR Technical University, Chennai, India. She has 9 years of teaching experience and
her thrust areas of research are Difference and Differential Equations and Linear
Control Systems.