scholarly journals Stabilizing a Rotary Inverted Pendulum Based on Logarithmic Lyapunov Function

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Jie Wen ◽  
Yuanhao Shi ◽  
Xiaonong Lu
Keyword(s):  
Lyapunov Function ◽  
Inverted Pendulum ◽  
Control Method ◽  
Quadratic Function ◽  
Control Law ◽  
Logarithmic Function ◽  
Control Laws ◽  
Lyapunov Control ◽  
Comparative Results ◽  
Rotary Inverted Pendulum

The stabilization of a Rotary Inverted Pendulum based on Lyapunov stability theorem is investigated in this paper. The key of designing control laws by Lyapunov control method is the construction of Lyapunov function. A logarithmic function is constructed as the Lyapunov function and is compared with the usual quadratic function theoretically. The comparative results show that the constructed logarithmic function has higher numerical accuracy and faster convergence speed than the usual quadratic function. On this basis, the control law of stabilizing Rotary Inverted Pendulum is designed based on the constructed logarithmic function by Lyapunov control method. The effectiveness of the designed control law is verified by experiments and is compared with LQR controller and the control law designed based on the quadratic function. Moreover, the system robustness is analyzed when the system parameters contain uncertainties under the designed control law.

scholarly journals Swinging up the Furuta Pendulum and its Stabilization Via Model Predictive Control

2013 ◽  
Vol 64 (3) ◽  
pp. 152-158 ◽  
Author(s):  
Pavol Seman ◽  
Boris Rohal’-Ilkiv ◽  
Martin Juh´as ◽  
Michal Salaj
Keyword(s):  
Predictive Control ◽  
Inverted Pendulum ◽  
Control Law ◽  
Classical Approach ◽  
Second Stage ◽  
Total Potential ◽  
Predictive Controller ◽  
Power Law Function ◽  
Two Stages ◽  
Rotary Inverted Pendulum

This paper deals with certain options on controlling an inverted rotary pendulum also known as the Furuta pendulum. Controlling an inverted pendulum involves two stages. The first stage is the swing up of the pendulum and the second stage is its balancing in the up-right position. The paper describes two possibilities on swinging up the pendulum. First one is the classical approach based on comparing the current total (potential and kinetic) energy of the system with the energy in its stabilized up-right position. The second option uses an exponentiation operation over the pendulum position since the trend of power law function is very convenient for determining the amount of required energy to be delivered to the system. For the purposes of balancing the pendulum in the up-right position a predictive controller based on optimal control law with perturbation was proposed, which is an LQ controller with control signal corrections when constraints are exceeded. The results are illustrated by real-time experiments on a laboratory rotary inverted pendulum setup.

scholarly journals Velocity control of a two-wheeled inverted pendulum mobile robot: a fuzzy model-based approach

2019 ◽  
Vol 8 (3) ◽  
pp. 808-817
Author(s):  
Mustapha Muhammad ◽  
Amir A. Bature ◽  
Umar Zangina ◽  
Salinda Buyamin ◽  
Anita Ahmad ◽  
...  
Keyword(s):  
Lyapunov Function ◽  
Mobile Robot ◽  
Inverted Pendulum ◽  
Fuzzy Model ◽  
Control Law ◽  
Velocity Control ◽  
Wheeled Inverted Pendulum ◽  
Fuzzy Lyapunov Function ◽  
Tracking Controller ◽  
Velocity Tracking

This paper presents the design of a fuzzy tracking controller for balancing and velocity control of a Two-Wheeled Inverted Pendulum (TWIP) mobile robot based on its Takagi-Sugino (T-S) fuzzy model, fuzzy Lyapunov function and non-parallel distributed compensation (non-PDC) control law. The T-S fuzzy model of the TWIP mobile robot was developed from its nonlinear dynamical equations of motion. Stabilization conditions in a form of linear matrix inequalities (LMIs) were derived based on the T-S fuzzy model of the TWIP mobile robot, a fuzzy Lyapunov function and a non-PDC control law. Based on the derived stabilization conditions and the T-S fuzzy model of the TWIP mobile robot, a state feedback velocity tracking controller was then proposed for the TWIP mobile robot. The balancing and velocity tracking performance of the proposed controller was investigated via simulations. The simulation result shows the effectiveness of the proposed control scheme.

Continuous time-varying feedback control of a robotic manipulator with base vibration and load uncertainty

2020 ◽  
pp. 107754632092759
Author(s):  
Xi Wang ◽  
Baolin Hou
Keyword(s):  
Feedback Control ◽  
Lyapunov Function ◽  
Continuous Time ◽  
Position Control ◽  
Control Method ◽  
Robotic Manipulator ◽  
Control Law ◽  
Time Varying ◽  
Load Uncertainty ◽  
Feedback Control Method

To solve precise and fast position control of a robotic manipulator with base vibration and load uncertainty, a continuous time-varying feedback control method based on the implicit Lyapunov function is studied. This method is proportional–derivative-like in the form of control law, but its proportional and differential coefficients depend on the system Lyapunov function, which are differentiable functions of system error variables. In the motion process of the robotic manipulator, the system performance is influenced by three main nonlinear factors: system friction, balance torque, and base vibration. As the former two factors are available to be modeled and identified through experiments, compensation of the two terms is added to the proposed control law to reduce the effects of system nonlinearities to a certain extent. Experimental results show that the proposed control strategy is robust to base vibration and load uncertainty. Besides, the compensation of system friction and balance torque can shorten the positioning time by 27.3%, from 1.32 s to 0.96 s. Meanwhile, the positioning precision is guaranteed, which verifies the effectiveness of the proposed control scheme.

Continuum of Motion Equations and Control Laws for the Inverted Pendulum Cart and Rotary Pendulum

2020 ◽  
Author(s):  
Constance Lare ◽  
Warren N. White
Keyword(s):  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Control Law ◽  
Control Laws ◽  
Motion Equations ◽  
Base Radius ◽  
Limiting Condition ◽  
And Control ◽  
The Given ◽  
Insight Into

Abstract This paper questions whether the controller properties for a given rigid body mechanical system still apply as the given system is changed. As a first attempt in this investigation, the controller for the underactuated rotary pendulum is investigated as the system morphs into an underactuated inverted pendulum cart. As the limiting condition of the inverted pendulum cart is approached, the investigation allows the controller to also morph. The authors show that, as the pendulum base radius grows, the rotary pendulum equations of motion morph into the inverted pendulum cart dynamics. The paper presents necessary conditions for the successful morphing of the dynamic equations. The morphing process for the controller tests the idea whether the control law also satisfies the same continuum basis as the motion equations. The paper presents a framework for the class of controllers investigated for providing insight into when the controller morphing may be successful. This paper presents dimensionless quantities that render the equations of motion and controller for the inverted pendulum cart and rotary pendulum into dimensionless form. These dimensionless quantities allow comparison of controllers and systems that are not possible through simple inspection. This comparison ability is especially useful for quantifying the nonlinearities of a given system and controller compared to another system and controller having different parameter sizes, a comparison rarely seen in the control literature.

Sampled-data Control of a Class of Nonlinear Flat Systems With Application to Unicycle Trajectory Tracking

2005 ◽  
Vol 128 (3) ◽  
pp. 722-728 ◽  
Author(s):  
N. Léchevin ◽  
C. A. Rabbath
Keyword(s):  
Trajectory Tracking ◽  
Control Method ◽  
Digital Design ◽  
Control Law ◽  
Sampled Data ◽  
Control Laws ◽  
Control Approach ◽  
Flat Systems ◽  
Data Control ◽  
Sampled Data Control

In this paper we propose a flatness-based nonlinear sampled-data control approach for the trajectory tracking of nonlinear differentially flat systems that can be expressed in cascade form. The nonlinear sampled-data control method relies on the flatness property for the generation of appropriate trajectories, with the design of one-step predictive control laws, and on controller discretization by means of an averaging-like method. In the paper we demonstrate that the causality problem that might arise in the implementation is avoided by using an estimator based on numerical integration techniques of sufficiently high order. Stability-like properties are proved. Numerical simulations show that the proposed sampled-data control law offers the best closed-loop performance when compared with nonlinear direct digital design for the trajectory tracking of a rotorcraft-like UAV modeled as the unicycle. The synthesis of the nonlinear sampled-data control law takes advantage of the feedback linearizability property of the unicycle model. Furthermore, the proposed nonlinear sampled-data control does not rely on approximated discretization techniques and is computed from exponentially convergent steering trajectories that result from the stabilization of the linearized unicycle model.

scholarly journals Luenberger-Sliding Mode Observer Based Backstepping Control for the SCR System in a Diesel Engine

Energies ◽  
2019 ◽  
Vol 12 (22) ◽  
pp. 4270
Author(s):  
Zheng ◽  
Yang ◽  
Li ◽  
Ma
Keyword(s):  
Lyapunov Function ◽  
Control Method ◽  
Sliding Mode ◽  
Setting Time ◽  
Tracking Error ◽  
Backstepping Control ◽  
Sliding Mode Observer ◽  
Control Law ◽  
Virtual Control ◽  
Scr System

In order to keep the ammonia (NH3) slip of the downstream selective catalytic reduction (SCR) system at a low level and simultaneously achieve a high nitrogen oxide (NOX) conversion rate, a Luenberger-sliding mode observer based backstepping control method is proposed. Considering that the internal working condition of the catalyst cannot be measured by commercial sensors directly, a Luenberger-sliding mode observer is designed to estimate the ammonia concentration at the middle of the catalyst. In addition, based on the stepped distributed characteristic of the surface ammonia coverage ratio along the SCR axial direction, a backstepping control method is utilized for the SCR system, in which the SCR system is decomposed into two subsystems. Firstly, the Lyapunov function is designed to ensure the convergence of the downstream subsystem, and then the virtual control law is obtained. After that, taking the virtual control law as the tracking target of the upstream subsystem, the Lyapunov function of virtual control law is given. Finally, the actual control law of the whole closed loop system is acquired. Simulations under different conditions are conducted to investigate the effect of the proposed control method. In addition, comparisons with the traditional PID (Proportion Integration Differentiation) control are presented. Results show that the proposed method is much better than the PID control method in overshoot, setting time, and tracking error.

scholarly journals Preparation of Quantum Gates for Open Quantum Systems by Lyapunov Control Method

2016 ◽  
Vol 23 (01) ◽  
pp. 1650005 ◽  
Author(s):  
Jie Wen ◽  
Shuang Cong
Keyword(s):  
Lyapunov Function ◽  
Control Method ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Quantum Gates ◽  
Control Laws ◽  
Open Quantum ◽  
Matrix Logarithm ◽  
The Matrix ◽  
And Control

In this paper the control laws of preparing quantum gates are designed based on Lyapunov stability theorem for two level open quantum systems. We propose a novel Lyapunov function according to the matrix logarithm function, which has higher accuracy and faster convergence speed by comparing them with those of the Lyapunov function of distance. Based on the proposed function, we design two types of control laws to prepare quantum gates for different systems including Markovian quantum systems with phase damping and amplitude damping, non-Markovian quantum systems and closed quantum systems. Furthermore, the system robustness when the Hamiltonian contains uncertainty is further investigated. In order to verify the superiorities of proposed function and control method, NOT gates are prepared by the designed control laws for different systems in the numerical experiments, and the results are comparatively analyzed.

Robust Adaptive Super-twisting Sliding Mode Stability Control for Underactuated Rotary Inverted Pendulum with Experimental Validation

2021 ◽  
Author(s):  
Omid Mofid ◽  
Khalid A Alattas ◽  
Saleh Mobayen
Keyword(s):  
Inverted Pendulum ◽  
Control Method ◽  
Sliding Mode ◽  
Lyapunov Stability Theory ◽  
Stability Control ◽  
Control Technique ◽  
Control Procedure ◽  
Pendulum System ◽  
Inverted Pendulum System ◽  
Rotary Inverted Pendulum

Abstract In this paper, an adaptive proportional-integral-derivative (PID) sliding mode control method combined with super-twisting algorithm is designed for the stabilization control of rotary inverted pendulum system in the appearance of exterior perturbation. The state-space model of rotary inverted pendulum in the presence of exterior disturbance is obtained. Then, the super-twisting PID sliding mode controller is designed for finite time stability control of this underactuated control system. The upper bounds of perturbation are presumed to be unknown; accordingly, the adaptive control procedure is taken to approximate the uncertain bound of the external disturbances. The stability control of rotary inverted pendulum system is proved by means of the Lyapunov stability theory. In order to validate accuracy and efficiency of the recommended control technique, some simulation outcomes are prepared and compared with other existing method. Moreover, experimental results are implemented to show the success of the proposed method.

Passivity Based Control of a Cart with Inverted Pendulum

2013 ◽  
Vol 332 ◽  
pp. 339-344 ◽  
Author(s):  
Matthias Jörgl ◽  
Kurt Schlacher ◽  
Hubert Gattringer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lyapunov Function ◽  
Mechanical System ◽  
Simple Proof ◽  
Control Strategy ◽  
Inverted Pendulum ◽  
Equation Of Motion ◽  
Control Law ◽  
Passivity Based Control

This paper deals with passivity based control of a cart , which can moves on a skew plain. On the cart a non actuated pendulum is assembled. Therefore, the mechanical system is non-linear, one degree under actuated and instable. The equation of motion is derived by the method of Lagrange. The goal of the control strategy is to stabilize the cart and the upper position of the pendulum. The applied control strategy is called Interconnection Damping Assignment Passivity Based Control (IDA-PBC). This method allows a physical approach with a simple proof of stability having appropriate Lyapunov function on hand. The simple proof of stability is an advantage of this theory. However, the proof of stability is fulfilled, challenging partial differential equation have to be solved to get the control law. Measurements results are presented.

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