This document describes the modeling and simulation of an inverted pendulum system using different dynamic analysis techniques. It first applies a recursive Newton-Euler analysis to derive the equations of motion, identifying two minor discrepancies with a previous analysis. It then uses Lagrangian mechanics to independently derive the equations of motion. Finally, it numerically simulates the highly nonlinear system using MATLAB's ode45 solver, requiring the second-order equations of motion to first be converted to first-order form.
A Self-Tuned Simulated Annealing Algorithm using Hidden Markov ModeIJECEIAES
Simulated Annealing algorithm (SA) is a well-known probabilistic heuristic. It mimics the annealing process in metallurgy to approximate the global minimum of an optimization problem. The SA has many parameters which need to be tuned manually when applied to a specific problem. The tuning may be difficult and time-consuming. This paper aims to overcome this difficulty by using a self-tuning approach based on a machine learning algorithm called Hidden Markov Model (HMM). The main idea is allowing the SA to adapt his own cooling law at each iteration, according to the search history. An experiment was performed on many benchmark functions to show the efficiency of this approach compared to the classical one.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
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.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
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.
Comparison of a triple inverted pendulum stabilization using optimal control ...Mustefa Jibril
In this paper, modelling design and analysis of a triple inverted pendulum have been done using
Matlab/Script toolbox. Since a triple inverted pendulum is highly nonlinear, strongly unstable
without using feedback control system. In this paper an optimal control method means a linear
quadratic regulator and pole placement controllers are used to stabilize the triple inverted
pendulum upside. The impulse response simulation of the open loop system shows us that the
pendulum is unstable. The comparison of the closed loop impulse response simulation of the
pendulum with LQR and pole placement controllers results that both controllers have stabilized
the system but the pendulum with LQR controllers have a high overshoot with long settling time
than the pendulum with pole placement controller. Finally the comparison results prove that the
pendulum with pole placement controller improve the stability of the system.
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.
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
A Self-Tuned Simulated Annealing Algorithm using Hidden Markov ModeIJECEIAES
Simulated Annealing algorithm (SA) is a well-known probabilistic heuristic. It mimics the annealing process in metallurgy to approximate the global minimum of an optimization problem. The SA has many parameters which need to be tuned manually when applied to a specific problem. The tuning may be difficult and time-consuming. This paper aims to overcome this difficulty by using a self-tuning approach based on a machine learning algorithm called Hidden Markov Model (HMM). The main idea is allowing the SA to adapt his own cooling law at each iteration, according to the search history. An experiment was performed on many benchmark functions to show the efficiency of this approach compared to the classical one.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
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.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
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.
Comparison of a triple inverted pendulum stabilization using optimal control ...Mustefa Jibril
In this paper, modelling design and analysis of a triple inverted pendulum have been done using
Matlab/Script toolbox. Since a triple inverted pendulum is highly nonlinear, strongly unstable
without using feedback control system. In this paper an optimal control method means a linear
quadratic regulator and pole placement controllers are used to stabilize the triple inverted
pendulum upside. The impulse response simulation of the open loop system shows us that the
pendulum is unstable. The comparison of the closed loop impulse response simulation of the
pendulum with LQR and pole placement controllers results that both controllers have stabilized
the system but the pendulum with LQR controllers have a high overshoot with long settling time
than the pendulum with pole placement controller. Finally the comparison results prove that the
pendulum with pole placement controller improve the stability of the system.
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.
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
Hierarchical algorithms of quasi linear ARX Neural Networks for Identificatio...Yuyun Wabula
This document summarizes a research paper on hierarchical algorithms for training a quasi-linear ARX neural network model for identification of nonlinear systems. The key points are:
1) A hierarchical algorithm is proposed that first estimates the system using a linear sub-model and least squares estimation to obtain linear parameters. It then trains a neural network nonlinear sub-model to refine the errors of the linear sub-model.
2) The linear parameter estimates are fixed and used as biases for the neural network, which is trained to minimize the residual errors of the linear sub-model.
3) This hierarchical approach separates the identification into linear and nonlinear parts, allowing analysis of the system linearly while also capturing nonlinearities. The neural
Secure Communication and Implementation for a Chaotic Autonomous SystemNooria Sukmaningtyas
In this paper, a three-dimensional chaotic autonomous system is presented. The stability of the
equilibrium and the conditions of the Hopf bifurcation are studied by means of nonlinear dynamics theory.
Then, the circuit of chaotic system is structured out in Multisim platform by the unit circuit. The chaotic
system is applied to secure communications by linear feedback synchronization control. All simulations
results performed on three-dimensional chaotic autonomous system are verified the applicable of secure
communication.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
Chapter 3 mathematical modeling of dynamic systemLenchoDuguma
The document discusses mathematical modeling of dynamic systems, including obtaining differential equations to represent system dynamics, different representations like transfer functions and impulse response functions, using block diagrams to visualize system components and signal flows, modeling various physical systems like mechanical, electrical, and thermal systems, and representing systems using signal flow graphs. It provides examples of obtaining transfer functions for different system types and using block diagram reduction techniques to find overall transfer functions.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This document summarizes research on synchronization of a four-dimensional hyperchaotic system. It presents a four-wing hyperchaotic system and derives an active control law to synchronize two identical systems based on Lyapunov stability theory. Numerical simulations demonstrate effective synchronization of the states and attractors between the two systems using the proposed active control method. The synchronization errors converge to zero, indicating synchronization is achieved.
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.
This formula book gives simple and useful formulas related to control system. It helps students in solving numerical problems, in their competitive examinations
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
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.
The document discusses several topics related to analog control systems including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they impact system response.
4. Defining characteristics of second order systems and analyzing their step response.
5. Calculating steady state error for different input types.
6. Analyzing stability by examining the location of poles in the complex s-plane.
This document discusses model reference adaptive control (MRAC). It provides an overview of the concept, the MIT rule for updating controller parameters, and an example of applying MRAC to control the position of a pendulum. Simulation and experimental results show the controller requires proportional-derivative feedback and tuning to stabilize the unstable pendulum system. More advanced control methods could provide better practical performance than the basic MRAC approach presented.
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.
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.
An optimal control for complete synchronization of 4D Rabinovich hyperchaotic...TELKOMNIKA JOURNAL
This paper derives new results for the complete synchronization of 4D
identical Rabinovich hyperchaotic systems by using two strategies: active
and nonlinear control. Nonlinear control strategy is considered as one
of the powerful tool for controlling the dynamical systems. The stabilization
results of error dynamics systems are established based on Lyapunov second
method. Control is designed via the relevant variables of drive and response
systems. In comparison with previous strategies, the current controller
(nonlinear control) focuses on convergence speed and the minimum limits
of relevant variables. Better performance is to achieve full synchronization
by designing the control with fewer terms. The proposed control has
certain significance for reducing the time and complexity for strategy
implementation.
1. The document discusses the state estimation, locomotion, kinematics, dynamics, and control of quadruped robots. It focuses on the ANYmal robot from ETH Zurich as a key example.
2. Key topics covered include the use of an extended Kalman filter for state estimation, the inverse kinematics and dynamics challenges of quadruped robots, and approaches for support consistent inverse kinematics and dynamics that respect contact constraints.
3. The document provides an overview of concepts for quadruped control, including kinematic control, impedance control, inverse dynamics control, and whole-body control through simultaneous optimization of posture, contact forces and joint torques.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
This document discusses system modeling and properties of linearity and time invariance. It provides examples of modeling a resistor, square-law system, delay operator, and time compressor to illustrate these properties. A model is a set of mathematical equations relating the output and input signals of a physical system. A system is linear if the response to a sum of inputs is the sum of the responses. It is time invariant if its behavior does not change over time. Developing an accurate but simplified model is important for understanding system behavior and designing controllers.
Detecting Assignable Signals via Decomposition of MEWMA Statisticinventionjournals
The MEWMA Chart is used in Process monitoring when a quick detection of small or moderate shifts in the mean vector is desired. When there are shifts in a multivariate Control Charts, it is clearly shows that special cause variation is present in the Process, but the major drawback of MEWMA is inability to identify which variable(s) is/are the source of the signals. Hence effort must be made to identify which variable(s) is/are responsible for the out- of-Control situation. In this article, we employ Mason, Young and Tracy (MYT) approach in identifying the variables for the signals.
This document discusses modeling mechanical systems using three basic elements: springs, dampers, and masses. It describes the properties and dynamic responses of ideal spring and damper elements and provides examples of real-world springs and dampers. The document also discusses modeling nonlinear springs and damping effects in mechanical systems.
This document discusses using cascade compensation to improve control system performance. Cascade compensation involves adding additional poles and zeros to the open-loop transfer function. This can improve the transient response by placing poles farther out in the s-plane, and improve steady-state error by increasing the system type. An example shows designing a PI controller to reduce steady-state error to zero without affecting the 57.4% overshoot transient response. Pole-zero cancellation is used to maintain the original transient response while increasing the system type.
ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.
This paper proposes a hybrid approach combining Particle Swarm Optimization (PSO) and Simulated Annealing (SA) to solve the n-queen problem. PSO is first used to find a good local maximum solution. SA is then applied to the best solution from PSO to escape local maxima and converge to the global maximum faster than SA alone. The representation of problem states and parameters of the hybrid algorithm such as temperature stages and Markov chain lengths are described. Experimental results show the hybrid method converges faster than SA for high-dimension n-queen problems.
Hierarchical algorithms of quasi linear ARX Neural Networks for Identificatio...Yuyun Wabula
This document summarizes a research paper on hierarchical algorithms for training a quasi-linear ARX neural network model for identification of nonlinear systems. The key points are:
1) A hierarchical algorithm is proposed that first estimates the system using a linear sub-model and least squares estimation to obtain linear parameters. It then trains a neural network nonlinear sub-model to refine the errors of the linear sub-model.
2) The linear parameter estimates are fixed and used as biases for the neural network, which is trained to minimize the residual errors of the linear sub-model.
3) This hierarchical approach separates the identification into linear and nonlinear parts, allowing analysis of the system linearly while also capturing nonlinearities. The neural
Secure Communication and Implementation for a Chaotic Autonomous SystemNooria Sukmaningtyas
In this paper, a three-dimensional chaotic autonomous system is presented. The stability of the
equilibrium and the conditions of the Hopf bifurcation are studied by means of nonlinear dynamics theory.
Then, the circuit of chaotic system is structured out in Multisim platform by the unit circuit. The chaotic
system is applied to secure communications by linear feedback synchronization control. All simulations
results performed on three-dimensional chaotic autonomous system are verified the applicable of secure
communication.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
Chapter 3 mathematical modeling of dynamic systemLenchoDuguma
The document discusses mathematical modeling of dynamic systems, including obtaining differential equations to represent system dynamics, different representations like transfer functions and impulse response functions, using block diagrams to visualize system components and signal flows, modeling various physical systems like mechanical, electrical, and thermal systems, and representing systems using signal flow graphs. It provides examples of obtaining transfer functions for different system types and using block diagram reduction techniques to find overall transfer functions.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This document summarizes research on synchronization of a four-dimensional hyperchaotic system. It presents a four-wing hyperchaotic system and derives an active control law to synchronize two identical systems based on Lyapunov stability theory. Numerical simulations demonstrate effective synchronization of the states and attractors between the two systems using the proposed active control method. The synchronization errors converge to zero, indicating synchronization is achieved.
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.
This formula book gives simple and useful formulas related to control system. It helps students in solving numerical problems, in their competitive examinations
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
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.
The document discusses several topics related to analog control systems including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they impact system response.
4. Defining characteristics of second order systems and analyzing their step response.
5. Calculating steady state error for different input types.
6. Analyzing stability by examining the location of poles in the complex s-plane.
This document discusses model reference adaptive control (MRAC). It provides an overview of the concept, the MIT rule for updating controller parameters, and an example of applying MRAC to control the position of a pendulum. Simulation and experimental results show the controller requires proportional-derivative feedback and tuning to stabilize the unstable pendulum system. More advanced control methods could provide better practical performance than the basic MRAC approach presented.
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.
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.
An optimal control for complete synchronization of 4D Rabinovich hyperchaotic...TELKOMNIKA JOURNAL
This paper derives new results for the complete synchronization of 4D
identical Rabinovich hyperchaotic systems by using two strategies: active
and nonlinear control. Nonlinear control strategy is considered as one
of the powerful tool for controlling the dynamical systems. The stabilization
results of error dynamics systems are established based on Lyapunov second
method. Control is designed via the relevant variables of drive and response
systems. In comparison with previous strategies, the current controller
(nonlinear control) focuses on convergence speed and the minimum limits
of relevant variables. Better performance is to achieve full synchronization
by designing the control with fewer terms. The proposed control has
certain significance for reducing the time and complexity for strategy
implementation.
1. The document discusses the state estimation, locomotion, kinematics, dynamics, and control of quadruped robots. It focuses on the ANYmal robot from ETH Zurich as a key example.
2. Key topics covered include the use of an extended Kalman filter for state estimation, the inverse kinematics and dynamics challenges of quadruped robots, and approaches for support consistent inverse kinematics and dynamics that respect contact constraints.
3. The document provides an overview of concepts for quadruped control, including kinematic control, impedance control, inverse dynamics control, and whole-body control through simultaneous optimization of posture, contact forces and joint torques.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
This document discusses system modeling and properties of linearity and time invariance. It provides examples of modeling a resistor, square-law system, delay operator, and time compressor to illustrate these properties. A model is a set of mathematical equations relating the output and input signals of a physical system. A system is linear if the response to a sum of inputs is the sum of the responses. It is time invariant if its behavior does not change over time. Developing an accurate but simplified model is important for understanding system behavior and designing controllers.
Detecting Assignable Signals via Decomposition of MEWMA Statisticinventionjournals
The MEWMA Chart is used in Process monitoring when a quick detection of small or moderate shifts in the mean vector is desired. When there are shifts in a multivariate Control Charts, it is clearly shows that special cause variation is present in the Process, but the major drawback of MEWMA is inability to identify which variable(s) is/are the source of the signals. Hence effort must be made to identify which variable(s) is/are responsible for the out- of-Control situation. In this article, we employ Mason, Young and Tracy (MYT) approach in identifying the variables for the signals.
This document discusses modeling mechanical systems using three basic elements: springs, dampers, and masses. It describes the properties and dynamic responses of ideal spring and damper elements and provides examples of real-world springs and dampers. The document also discusses modeling nonlinear springs and damping effects in mechanical systems.
This document discusses using cascade compensation to improve control system performance. Cascade compensation involves adding additional poles and zeros to the open-loop transfer function. This can improve the transient response by placing poles farther out in the s-plane, and improve steady-state error by increasing the system type. An example shows designing a PI controller to reduce steady-state error to zero without affecting the 57.4% overshoot transient response. Pole-zero cancellation is used to maintain the original transient response while increasing the system type.
ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.
This paper proposes a hybrid approach combining Particle Swarm Optimization (PSO) and Simulated Annealing (SA) to solve the n-queen problem. PSO is first used to find a good local maximum solution. SA is then applied to the best solution from PSO to escape local maxima and converge to the global maximum faster than SA alone. The representation of problem states and parameters of the hybrid algorithm such as temperature stages and Markov chain lengths are described. Experimental results show the hybrid method converges faster than SA for high-dimension n-queen problems.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems 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 uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
This document analyzes the suspension system of an automobile modeled as a two-degree-of-freedom spring-mass-damper system. Equations of motion were derived using Lagrange's equations and modeled in SIMULINK. Natural frequencies were found to be 5.1 rad/s and 6.5 rad/s. Increasing damping reduced bounce by 3x10-3 m to 1x10-3 m and pitch by 5x10-4 m to less than 1x10-4 m. MATLAB modal analysis verified results and natural frequencies were compared.
Cone Crusher Model Identification Using Block-Oriented Systems with Orthonorm...ijctcm
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
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.
An Unmanned Rotorcraft System with Embedded DesignIOSR Journals
This document describes the design and implementation of an embedded control system for an unmanned rotorcraft. It uses a backstepping control law with state estimation provided by an extended Kalman filter using data from a marker-based vision system and inertial measurement unit. A self-organizing map is used to select local operating regimes for modeling the nonlinear dynamics. The control algorithm was tested in a hardware-in-the-loop simulation and able to track spiral paths in 3D space with the embedded controller implemented on a Freescale MPC555 processor.
Oscar Nieves (11710858) Computational Physics Project - Inverted PendulumOscar Nieves
This document describes a numerical simulation of an inverted pendulum system created in MATLAB using a 4th order Runge-Kutta algorithm. The simulation models an inverted pendulum attached to a horizontally moving cart. Forces like air drag and friction are included, and parameters like mass, pendulum length, and initial conditions can be varied. Small changes to initial conditions can lead to large differences in motion, demonstrating the system's chaotic behavior. The document also outlines the methodology for adapting the Runge-Kutta algorithm to solve systems of coupled differential equations.
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.
Sliding mode control-based system for the two-link robot armIJECEIAES
In this research, the author presents the model of the two-link robot arm and its dynamic equations. Based on these dynamic equations, the author builds the sliding mode controller for each joint of the robot. The tasks of the controllers are controlling the Torque in each Joint of the robot in order that the angle coordinates of each link coincide with the desired values. The proposed algorithm and robot model are built on Matlab-Simulink to investigate the system quality. The results show that the quality of the control system is very high: the response angles of each link quickly reach the desired values, and the static error equal to zero.
r5.pdf
r6.pdf
InertiaOverall.docx
Dynamics of Mechanical Systems
Inertia and Efficiency Laboratory
1 Overview
The objectives of this laboratory are to examine some very common mechanical drive components, and hence to answer the following questions:
· How efficient is a typical geared transmission system?
· How do gearing and efficiency affect the apparent inertia of a geared system as observed at (i.e. referred to) one of the shafts?
The learning objectives are more generic:
· To give experience of the kinematic equations relating displacement, velocity, acceleration and time of travel of a particle.
· To give experience of applying Newton’s second law to linear and rotational systems.
· To introduce the concept of mechanical power and its relationship to torque and angular velocity.
The completed question sheet must be submitted to the laboratory demonstrator at the end of the lab, and is worth 6% of module mark.
Please fill in the sheet neatly (initially in pencil, perhaps, then in ink once correct!) as you will be handing it in with the remainder of your report.
Note: it is a matter of Departmental policy that students do not undertake laboratories unless they are equipped with safety shoes (and laboratory coat). The reasons for this policy are apparent from the present lab, where descending masses are involved, and could cause injury if they run out of control. Safety shoes therefore MUST be worn.
Also, keep fingers clear of rotating parts, whether guarded or not, taking particular care when winding the cord onto the capstans. In particular, do not touch (or try to stop) the flywheel when it is rotating rapidly. Do not move the rig around on the bench – if its position needs changing, please ask the lab supervisor.
1
Inertia and Efficiency Laboratory
2 Mechanical efficiency, inertia and gearing
2.1 Theory
2.1.1 Kinematics: motion in a straight line
The motion of a particle in a straight line under constant acceleration is described by the following equations:
v u at
s (u v) t
2
s ut 12 at 2 s vt 12 at 2 v2 u 2 2as
where s is the distance travelled by the particle during time t, u is the initial velocity of the particle, v is its final velocity, and a is the acceleration of the particle.
To think about: which one of these equations will you need to use to calculate the acceleration of a mass as it accelerates from rest to cover a distance s in time t? (Hint: note that u is zero while v is both unknown and irrelevant. You will need to rearrange one of the above equations to obtain a in terms of s and t).
2.2 Kinematics: gears and similar devices
If two meshing gears1 have numbers of teeth N1 and N2 and are connected to the input and output shafts respectively, then the gear ratio n is said to be the ratio of the input rotational angle to the output rotational angle (and angular velocity and angular acceleration), see Fig. 1:
N
2
1
1
Gear ratio n
...
Design of airfoil using backpropagation training with mixed approachEditor Jacotech
Levenberg-Marquardt back-propagation training method has some limitations associated with over fitting and local optimum problems. Here, we proposed a new algorithm to increase the convergence speed of Backpropagation learning to design the airfoil. The aerodynamic force coefficients corresponding to series of airfoil are stored in a database along with the airfoil coordinates. A feedforward neural network is created with aerodynamic coefficient as input to produce the airfoil coordinates as output. In the proposed algorithm, for output layer, we used the cost function having linear & nonlinear error terms then for the hidden layer, we used steepest descent cost function. Results indicate that this mixed approach greatly enhances the training of artificial neural network and may accurately predict airfoil profile.
Design of airfoil using backpropagation training with mixed approachEditor Jacotech
The document describes a new algorithm for designing airfoils using neural networks. The algorithm uses a mixed training approach: it trains the output layer of the neural network using a cost function with linear and nonlinear error terms for faster convergence, while training the hidden layer using steepest descent. Results show the mixed approach converges much faster than traditional backpropagation or Levenberg-Marquardt algorithms alone. The algorithm more accurately predicts airfoil profiles with fewer training iterations.
This document describes a track reconstruction software using the Hough Transform algorithm. It introduces the theoretical background of the algorithm and how it is applied to reconstruct tracks in the ALICE Time Projection Chamber. It explains the linear and helix Hough Transform algorithms in detail, including how to generate points, fill the Hough space, visualize results, and find peaks. It evaluates the performance by looking at reconstruction efficiency for different numbers of lines/helices, points, and resolutions. The software was able to accurately reconstruct up to 15 straight lines and over 25 helices.
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.
Power system transient stability margin estimation using artificial neural ne...elelijjournal
This paper presents a methodology for estimating the normalized transient stability margin by using the multilayered perceptron (MLP) neural network. The complex relationship between the input variables and output variables is established by using the neural networks. The nonlinear mapping relation between the normalized transient stability margin and the operating conditions of the power system is established by using the MLP neural network. To obtain the training set of the neural network the potential energy boundary surface (PEBS) method along with time domain simulation method is used. The proposed method is applied on IEEE 9 bus system and the results shows that the proposed method provides fast and accurate tool to assess online transient stability.
A New Method of Determining Instability of Linear SystemIDES Editor
This document presents a new method for determining instability in linear systems based on the Gerschgorin theorem and bisection method. The method identifies real eigenvalues in the right half of the s-plane without computing the full characteristic polynomial or eigenvalues. It is more efficient than existing methods as it requires fewer computations. The algorithm involves calculating Gerschgorin bounds and evaluating the determinant of the system matrix at different points on the real axis to check for signs of instability. Several examples are provided to demonstrate the method.
- The document details a state space solver approach for analog mixed-signal simulations using SystemC. It models analog circuits as sets of linear differential equations and solves them using the Runge-Kutta method of numerical integration.
- Two examples are provided: a digital voltage regulator simulation and a digital phase locked loop simulation. Both analog circuits are modeled in state space and simulated alongside a digital design to verify mixed-signal behavior.
- The state space approach allows modeling analog circuits without transistor-level details, improving simulation speed over traditional mixed-mode simulations while still capturing system-level behavior.
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.
This document discusses fuzzy identification of systems and its applications to modeling and control. It begins by introducing fuzzy logic and fuzzy controllers. It then provides details on the format of fuzzy implications and reasoning algorithms using Takagi-Sugeno controllers. An identification algorithm is presented as a mathematical tool to build fuzzy models of systems. The document applies this fuzzy identification method to modeling a human operator's control of a water cleaning process and a converter in a steel-making process. Results show the fuzzy models accurately capture the operators' actions.
1.
I. INTRODUCTION
To
show
the
power
of
dynamics,
an
inverted
pendulum
(shown
below)
that
is
capable
of
driving
the
un-‐actuated
pendulum
to
the
unstable
equilibrium
position
and
holding
it
there
was
modeled,
designed
and
built.
Fig.
1.
Dr.
Craig’s
inverted
pendulum
design
The
purpose
of
this
paper
is
to
re-‐analyze
the
dynamics
of
the
system
using
principles
learned
in
MEEN-‐5220
and
also
simulate
the
dynamics
derived
using
numerical
integration
techniques.
II. PART
1
–
RECURSIVE
NEWTON
EULER
(RNE)
DYNAMIC
ANALYSIS
The
RNE
approach
to
dynamic
modeling
is
a
computer
programmable
way
of
finding
the
dynamics
of
a
system
by
using
an
algorithmic-‐based
set
of
coordinate
frames.
Often
times,
it
is
used
on
multi
DOF
systems
as
the
dynamic
analysis
of
systems
rapidly
becomes
more
complex
as
the
number
of
DOF
increases.
This
method
was
applied
to
the
inverted
pendulum
problem.
First,
find
coordinate
frames
using
the
Denavit-‐Hartenberg
(D-‐H)
Distal
Variant
procedure.
Fig.
2.
Distal
D-‐H
Coordinate
Frames
applied
to
the
given
2
DOF
inverted
pendulum
N
θn+1
dn+1
an+1
αn+1
0
θ
0
0
-‐π/2
1
-‐α
L1
0
0
Using
these,
it
is
possible
to
systematically
compute
each
characteristic
of
the
system
all
the
way
through
to
joint
torques.
The
equation
for
joint
torque
(for
a
revolute
joint
i)
is
shown
below.
See
Appendix
A
for
code
used
to
compute
the
remainder
of
system
characteristics.
𝜏! = 𝑛!!!
!,!!!
!
∗
0
0
1
+ 𝐵! 𝑞! + 𝑇! ∗ 𝑠𝑖𝑔𝑛(𝑞!)
(1)
Note
that
in
Eq.
(1),
𝜏!
will
be
equal
as
the
input
torque
and
𝜏!
will
be
equal
to
zero
as
there
is
no
actuator
on
the
second
joint.
In
addition,
𝑛!!!
!,!!!
!
is
the
moment
on
each
joint
found
using
the
RNE
method.
Completing
the
analysis,
equations
of
motion
(EOM)
were
found
to
be
as
follows:
𝜏! = 𝑇!"#$% = [𝑚! 𝑙!!
!
+ 𝐼!!! + 𝑚! 𝑙!"
!
cos!
𝜙 + 𝐼!!! cos!
𝜙 +
𝐼!!! sin!
(𝜙)]𝜃 + −𝑚! 𝑙! 𝑙!" sin 𝜙 𝜙 + −𝑚! 𝑙! 𝑙!" cos 𝜙 𝜙!
+
−𝑚! 𝑙!"
!
− 𝐼!!! + 𝐼!!! ∗ 2 cos 𝜙 sin 𝜙 𝜃𝜙 − 𝐵! 𝜃 − 𝑇! ∗
𝑠𝑖𝑔𝑛(𝜃)
(2)
𝜏! = 0 = 𝑚! 𝑙!"
!
+ 𝐼!!! 𝜙 + −𝑚! 𝑙! 𝑙!" sin 𝜙 𝜃 + 𝑚! 𝑙!"
!
+
𝐼!!! − 𝐼!!! ∗ cos 𝜙 sin 𝜙 𝜃!
− 𝑚! 𝑔𝑙!" cos 𝜙 + 𝐵! 𝜙 + 𝑇! ∗
𝑠𝑖𝑔𝑛(𝜙)
(3)
Compared
to
the
EOM
derived
by
Dr.
Craig
(given
in
the
project
handout),
the
equations
derived
above
contain
two
discrepancies.
In
Eq.
(3)
the
term
highlighted
in
red
has
the
opposite
sign
from
the
equations
given.
However,
this
may
simply
be
a
discrepancy
in
the
sign
of
g.
In
addition,
Eq.
(2)
is
missing
one
term
(𝑚! 𝑙!
!
)
in
the
𝜃
term
compared
to
equations
given.
However,
this
analysis
did
prove
the
power
of
the
RNE
as
the
script
used
to
derive
the
EOM
was
only
90
lines
long
(approximately)
and
no
free
body
diagrams
were
required.
III. PART
3
–
LAGRANGIAN
DYNAMIC
ANALYSIS
Another
means
by
which
one
can
acquire
the
EOM
of
a
system
is
a
Lagrange
Dynamic
analysis.
This
is
an
energy-‐
based
method
and
therefor,
one
does
not
need
to
consider
the
many
+/-‐
values
found
in
Newton-‐Euler
methods.
However,
it
is
also
not
very
intuitive
and
joint
forces
are
lost
hence
why
RNE
was
completed
as
well.
See
Appendix
D
for
a
complete
derivation
using
Lagrangian
mechanics.
IV. PART
4
–
MODELING
USING
NUMERICAL
INTEGRATION
AND
ODE45
–
A
SIMPLE
CASE
As
the
EOM
of
the
system
are
highly
nonlinear,
numerical
analysis
will
be
required
to
find
the
time
based
position
response
of
the
system.
Using
MATLAB,
two
different
approaches
were
used:
first,
a
proprietary
Euler
algorithm
was
created
and
second,
an
internal
MATLAB
function
(ode45)
was
used.
Each
of
these
codes
can
be
found
in
Appendix
A
of
this
report.
However,
both
of
these
methods
only
work
for
1
st
order
differential
equations.
Since
the
EOM
of
the
system
consists
of
two
2
nd
order
equations,
it
is
required
to
break
each
one
2. up
into
two
1
st
order
equations.
This
can
be
done
by
using
the
following
definition
of
a
system
of
equations
of
motion:
𝑀 𝑞 𝑡 = {𝐹 𝑞, 𝑞, 𝑡 }
(4)
!
!"
𝑥 =
𝑞
𝑀 !! 𝐹
(5)
𝑥 =
𝑞
𝑞
(6)
See
Appendix
C
for
the
values
of
the
mass
matrix
and
force
vector,
which
were
derived
through
the
rearranging
of
the
EOM
(Eq.
(2)
&
(3)).
To
test
the
model,
a
simple
case
that
is
intuitively
understandable
would
be
employed.
The
case
tested
is
one
where
there
is
no
actuator
torque
and
the
pendulum
is
released
from
rest
at
the
near
vertical
position.
In
theory,
the
pendulum
should
oscillate
(with
decaying
amplitude
due
to
damping)
about
the
stable
equilibrium
position.
In
addition,
link
1
should
rotate
due
to
the
momentum
of
the
pendulum.
Results
are
plotted
below.
(note:
plots
found
using
EOM
from
handout):
From
these
plots,
it
is
immediately
obvious
that
the
modeled
data
does
not
perfectly
match
up
with
experimental
data.
In
addition,
the
two
sets
of
modeled
data
do
not
match
up
with
one
another.
However,
this
should
be
expected
as
ODE45
is
far
more
sophisticated
than
the
Euler
integration
used,
which
is
a
0
th
order
integration.
To
improve
the
Euler
method,
a
higher
order
integration
(trapezoidal
rule,
etc.)
should
be
used.
V. PART
6-‐
COMPARISON
TO
KNOWN
MODEL
In
addition,
a
Simulink
model
used
to
determine
control
torque
at
the
actuator
to
drive
the
pendulum
to
the
unstable
equilibrium
position
and
hold
it
there
was
provided.
In
theory,
if
the
model
generated
matches
up
with
the
one
used
by
Dr.
Craig,
the
angle
profiles
of
the
system
will
be
the
same
regardless
of
which
model
is
used.
Fig.
5.
Comparison
of
Θ(t)
values
for
modeled
values
Fig.
6.
Comparison
of
Φ(t)
values
for
modeled
values
From
the
plots,
it
is
immediately
clear
that
the
model
generated
for
this
project
does
not
match
up
with
Dr.
Craig’s
Simulink
model.
The
reasons
for
this
could
be
many
and
will
be
discussed
in
Section
VI,
Conclusion.
VI. CONCLUSION
To
conclude,
it
was
found
that
the
model
used
for
the
purpose
of
this
report
did
not
match
perfectly
with
the
one
used
by
Dr.
Craig.
This
could
be
for
one
of
two
reasons:
first,
the
EOM
used
for
the
purposes
of
this
report
may
not
be
correct.
A
reference
to
a
product
of
inertia
(Ixy2)
was
found
in
the
parameters
given
however,
it
was
not
used
in
the
final
system
model
which
draws
concern.
In
addition,
Euler
integrations
are
characterized
by
their
propensity
for
instability.
This
issue
is
only
exacerbated
by
the
attempt
to
position
the
pendulum
at
the
unstable
equilibrium
position.
This
would
explain
why
Euler
matches
up
with
Dr.
Craig’s
analysis
until
approximately
the
time
at
which
unstable
equilibrium
is
reached.
Finally,
a
great
deal
was
learned
through
this
project
as
it
did
encompass
many
if
not
every
topic
touched
upon
in
the
course.
Specifically,
the
difference
between
intelligent
methods
(ode45)
and
brute
force
methods
(Euler)
was
reinforced
through
this
project.
In
the
end,
it
can
be
said
summarily
as
thus:
if
you
have
an
analytical
solution,
great!
But
if
you
don’t
and
need
to
use
a
numerical
method,
it
is
best
to
work
smarter,
not
harder.
3. APPENDIX
A. Matlab
Code
%Alex Folz
%MEEN-5220 Project
%Recursive Newton Euler Calculation
clc
clear
%initialize symbolic variables
syms L21 t td tdd a ad add L1 ixx2 izz2 m2 g iyy2 L1 L11 ixx1 iyy1
izz1 ixy2 m1
z = [0;0;1];
%omega,alpha,v,a values of the zero frame
w0 = [0;0;0];
al0 = [0;0;0];
v0 = [0;0;0];
a0 = [0;0;0];
%Inertia values of each of the links
I1 = [ixx1 0 0; 0 iyy1 0; 0 0 izz1];
I2 = [ixx2 0 0; 0 iyy2 0; 0 0 izz2];
%Distance to each frame and center of mass
r1 = [0;0;0];
r2 = [0;0;L1];
rc1 = [0;0;L11];
rc2 = [0; -L21; 0];
%Rotation matrices and inverse rotation matrices between each
frame
R01 = [cos(t) 0 -sin(t); sin(t) 0 cos(t); 0 -1 0];
R12 = [cos(-a) -sin(-a) 0; sin(-a) cos(-a) 0; 0 0 1];
R10 = [cos(t) sin(t) 0; 0 0 -1; -sin(t) cos(t) 0];
R21 = [cos(-a) sin(-a) 0; -sin(-a) cos(-a) 0; 0 0 1];
%gravity in the zero frame
g0 = [0;0;g];
%calculation of the angular velocity of each frame
r.w1 = R10*(w0+z*td);
r.w2 = R21*(r.w1+z*ad);
%calculation of the angular acceleration of each frame
r.al1 = R10*(al0+z*tdd+cross(w0,(z*td)));
r.al2 = R21*(r.al1+z*add+cross(r.w1,(z*ad)));
%calculation of the velocity of each frame
r.v1 = R10*v0+cross(r.w1,r1);
r.v2 = R21*r.v1+cross(r.w2,r2);
%calculation of the acceleration of each frame
r.a1 = R10*a0+cross(r.al1,r1)+cross(r.w1,cross(r.w1,r1));
r.a2 = R21*r.a1+cross(r.al2,r2)+cross(r.w2,cross(r.w2,r2));
%calculation of the acceleration of each center of mass
r.ac1 = r.a1+cross(r.al1,rc1)+cross(r.w1,cross(r.w1,rc1));
r.ac2 = r.a2+cross(r.al2,rc2)+cross(r.w2,cross(r.w2,rc2));
%calculation of the gravity vector in each frame
r.g1 = R10*g0;
r.g2 = R21*r.g1;
%forces acting on joint 2, in both frame 1 and 2
r.f221 = [0;0;0]+m2*r.g2+m2*r.ac2;
r.f121 = R12*r.f221;
%forces acting on join 1, in frame 1
r.f110=r.f121+m1*r.g1+m1*r.ac1;
%moments on joint 2, in both frame 1 and 2
r.n221=[0;0;0]+cross((r2+rc2),r.f221)-
cross(rc2,[0;0;0])+I2*r.al2+cross(r.w2,(I2*r.w2));
r.n121=R12*r.n221;
%moments on joint 1, in both frame 1 and 0
r.n110=r.n121+cross((r1+rc1),r.f110)-
cross(rc1,r.f121)+I1*r.al1+cross(r.w1,(I1*r.w1));
r.n010=R01*r.n110;
%actuator torque at each joint
r.T1 = transpose(r.n010)*z;
r.T2 = transpose(r.n121)*z;
%Use matlab to simplify the expressions of torque as much as
possible
r.m1 = collect(r.T1,ad^2);
r.n1 = collect(r.m1,tdd);
r.o1 = collect(r.n1,add);
r.T1f = r.o1;
r.m2 = collect(r.T2,td^2);
r.n2 = collect(r.m2,tdd);
r.o2 = collect(r.n2,add);
r.T2f = r.o2;
%Alex Folz & Dr. Phil Voglewede
%MEEN-5220 Project
% Euler's Method solver
%clear variables generated by this code
clc
clear x1 y1 x2 y2 m f x2d y2d om al theta_eu alpha_eu tout_eu
% Initial conditions for Euler's method
x1(1) = 0;
x2(1) = 0;
y1(1) = -pi()/2;
y2(1) = 0;
x2d(1) = 0;
y2d(1) = 0;
om(:,:,1) = [x2(1); y2(1)];
al(:,:,1) = [x2d(1); y2d(1)];
% Increment for time1
deltat1 = 0.001;
tspan_eu = [0:deltat1:10];
% Euler's method1
for i=1:length(tspan_eu)-1
%Determine actuator torque; 0 for experimental test case and
%control_torque to compare with Dr. Craig's model
Torque(i) = control_torque(i);
%Calculate componenents of the mass Matrix
m1(i) =
M1*L11^2+I_bar_1_z1+M2*L1^2+M2*L21^2*cos(y1(i))^2+I_bar_2_z
2*cos(y1(i))^2+I_bar_2_y2*sin(y1(i))^2;
m2(i) = -(M2*L1*L21)*sin(y1(i));
m3(i) = -(M2*L1*L21)*sin(y1(i));
m4(i) = M2*L21^2+I_bar_2_x2;
%Calculate components of the force vector
4. f1(i) = M2*L1*L21*cos(y1(i))*y2(i)^2-(I_bar_2_y2-M2*L21^2-
I_bar_2_z2)*(2*cos(y1(i))*sin(y1(i)))*y2(i)*x2(i)+Torque(i)-
B_theta*x2(i)-Tf_theta*sign(x2(i));
f2(i) = -(M2*L21^2-
I_bar_2_y2+I_bar_2_z2)*cos(y1(i))*sin(y1(i))*x2(i)^2-
M2*g*L21*cos(y1(i))-B_phi*y2(i)-Tf_phi*sign(y2(i));
%Initialize the mass matrix and force vector
m(:,:,i) = [m1(i) m2(i); m3(i) m4(i)];
F(:,:,i) = [f1(i); f2(i)];
%Find x1 and y1 of the next cycle, where x1 and y1 are theta and
alpha respectively
x1(i+1) = x1(i)+deltat1*x2(i);
y1(i+1) = y1(i)+deltat1*y2(i);
%Find dx2 and dy2 of the current cycle, where dx2 and dy2 are
the angular
%accelerations of theta and alpha,respectively
acc(:,:,i) = m(:,:,i)^(-1)*F(:,:,i);
x2d(i) = acc(1,1,i);
y2d(i) = acc(2,1,i);
%Find x2 and y2 of the next cycle, where x2 and y2 are the
angular
%velocity of theta and alpha, respectively
x2(i+1) = x2(i)+deltat1*x2d(i);
y2(i+1) = y2(i)+deltat1*y2d(i);
%convert x and y values to known variables
alpha_eu(i) = (y1(i)-pi()/2)*360/(2*pi());
theta_eu(i) = (x1(i))*360/(2*pi());
tout_eu(i) = i*deltat1;
end
%plot relevant data
figure(1)
plot(tout_eu,theta_eu)
figure(2)
plot(tout_eu,alpha_eu)
%Alex Folz & Dr. Phil Voglewede
%Meen-5220 Project
%ODE initialization code
% Initial conditions for ODE45
ic = [0 0 pi()/2-0.0061 0];
tspan45 = [0 10];
% Run the solver
[tout45,pout45] = ode45('alex_model',tspan45,ic);
%Convert pout45 values into known variables
theta45 = pout45(:,1)*360/(2*pi());
theta45_dot = pout45(:,2)*360/(2*pi());
alpha45 = pout45(:,3)*360/(2*pi())-90;
alpha45_dot = pout45(:,4)*360/(2*pi());
%Plot Results
figure(1)
plot(tout45,theta45);
figure(2)
plot(tout45,alpha45);
%Alex Folz
%Meen-5220 Project
%System Model to be used with ODE45
%p - [theta;dtheta;alpha;dalpha]
%pdot - derivative of p
function pdot = alex_model(t,p)
%Establish Global Variables
global g M1 M2 L1 L11 L21 I_bar_1_z1 I_bar_2_x2 I_bar_2_y2
I_bar_2_z2 B_phi B_theta Tf_theta Tf_phi
%No accuator torque in test case
Torque = 0;
pdot = zeros(4,1); %initialize column vector
%From the state model definition, p(2) & p(4) are the derivatives of
%p(1) & p(3), respectively
pdot(1) = p(2);
pdot(3) = p(4);
%Calculate components of the Mass matrix
m1 =
M1*L11^2+I_bar_1_z1+M2*L1^2+M2*L21^2*cos(p(3))^2+I_bar_2_z
2*cos(p(3))^2+I_bar_2_y2*sin(p(3))^2;
m2 = -(M2*L1*L21)*sin(p(3));
m3 = -(M2*L1*L21)*sin(p(3));
m4 = M2*L21^2+I_bar_2_x2;
%Caculate components of the Force vector
f1 = M2*L1*L21*cos(p(3))*p(4)^2-(I_bar_2_x2-M2*L21^2-
I_bar_2_z2)*(2*cos(p(3))*sin(p(3)))*p(4)*p(2)+Torque-B_theta*p(2)-
Tf_theta*sign(p(2));
f2 = -(M2*L21^2-
I_bar_2_y2+I_bar_2_z2)*cos(p(3))*sin(p(3))*p(2)^2-
M2*g*L21*cos(p(3))-B_phi*p(4)-Tf_phi*sign(p(4));
%Initialize Mass Matrix and Force vector
m = [m1 m2; m3 m4];
F = [f1; f2];
%Using the Mass Matrix and Force vector, determine pdot(2)
&pdot(4)
%which are the angular acceleration of theta and alpha,
respectively
acc= m^(-1)*F;
pdot(2) = acc(1,1);
pdot(4) = acc(2,1);
end
B. Matlab
Output
for
RNE
T1:
(-‐L1*L21*m2*cos(a))*add
+
(iyy1
+
cos(a)*(m2*cos(a)*L1^2
+
iyy2*cos(a))
-‐
L11*(L1*m2*cos(a)^2
+
L1*m2*sin(a)^2)
+
L11*(L1*m2*cos(a)^2
+
L1*m2*sin(a)^2
+
L11*m1)
+
sin(a)*(m2*sin(a)*L1^2
+
m2*sin(a)*L21^2
+
ixx2*sin(a)))*tdd
-‐
cos(a)*(L1*(L21*m2*cos(a)*sin(a)*td^2
+
g*m2*sin(a))
+
ad*ixx2*td*sin(a)
-‐
ad*iyy2*td*sin(a)
-‐
ad*izz2*td*sin(a))
+
sin(a)*(L1*(g*m2*cos(a)
-‐
L21*m2*td^2*sin(a)^2)
+
L21*m2*(L1*td^2*cos(a)^2
+
L1*td^2*sin(a)^2
-‐
2*L21*ad*td*cos(a))
-‐
ad*ixx2*td*cos(a)
+
ad*iyy2*td*cos(a)
-‐
ad*izz2*td*cos(a))
+
(-‐
L1*L21*m2*sin(a))*ad^2