Reduced-Order Observer-Based LQR Controller Design for Rotary Inverted Pendulum

The Rotary Inverted Pendulum(RIP)is a widely used underactuated mechanical system in various applications such as bipedal robots and skyscraper stabilization where attitude control presents a significant challenge.Despite the implementation of various control strategies to maintain equilibrium,optimally tuning control gains to effectively mitigate uncertain nonlinearities in system dynamics remains elusive.Existing methods frequently rely on extensive experimental data or the designer’s expertise,presenting a notable drawback.This paper proposes a novel tracking control approach for RIP,utilizing a Linear Quadratic Regulator(LQR)in combination with a reduced-order observer.Initially,the RIP system is mathematically modeled using the Newton-Euler-Lagrange method.Subsequently,a composite controller is devised that integrates an LQR for generating nominal control signals and a reduced-order observer for reconstructing unmeasured states.This approach enhances the controller’s robustness by eliminating differential terms from the observer,thereby attenuating unknown disturbances.Thorough numerical simulations and experimental evaluations demonstrate the system’s capability to maintain balance below50Hz and achieve precise tracking below1.4 rad,validating the effectiveness of the proposed control scheme.

Project-Based Teaching in Control Theory Education Based on V-REP: A Cart Inverted Pendulum Case

This paper provides a teaching concept for control theory education based on Virtual Robot Experimentation Platform(V-REP).A cart inverted pendulum virtual physical model is developed on V-REP.Students must analyze,design,and implement a suitable controller for the cart inverted pendulum system using their knowledge of the control theory.Different from traditional experiment and numerical simulation,virtual experiment is safe and less constrained.Moreover,the experiment results are more intuitive and obvious.This study can improve students’interest in learning the control theory and help students understand the relevant content better.

Numerical Study of a Novel Procedure for Installing the Tower and Rotor Nacelle Assembly of Offshore Wind Turbines Based on the Inverted Pendulum Principle

Current installation costs of offshore wind turbines(OWTs) are high and profit margins in the offshore wind energy sector are low, it is thus necessary to develop installation methods that are more efficient and practical. This paper presents a numerical study(based on a global response analysis of marine operations) of a novel procedure for installing the tower and Rotor Nacelle Assemblies(RNAs) on bottom-fixed foundations of OWTs. The installation procedure is based on the inverted pendulum principle. A cargo barge is used to transport the OWT assembly in a horizontal position to the site, and a medium-size Heavy Lift Vessel(HLV) is then employed to lift and up-end the OWT assembly using a special upending frame. The main advantage of this novel procedure is that the need for a huge HLV(in terms of lifting height and capacity) is eliminated. This novel method requires that the cargo barge is in the leeward side of the HLV(which can be positioned with the best heading) during the entire installation. This is to benefit from shielding effects of the HLV on the motions of the cargo barge, so the foundations need to be installed with a specific heading based on wave direction statistics of the site and a typical installation season. Following a systematic approach based on numerical simulations of actual operations, potential critical installation activities, corresponding critical events, and limiting(response) parameters are identified. In addition, operational limits for some of the limiting parameters are established in terms of allowable limits of sea states. Following a preliminary assessment of these operational limits, the duration of the entire operation, the equipment used, and weather-and water depth-sensitivity, this novel procedure is demonstrated to be viable.

Optimal feedback gains of a delayed proportional-derivative(PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a controlled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a systematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost characteristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial values around the origin are fixed. The main results are checked by numerical simulation.

Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations (ODEs) .Complete rigidity is the approximation of practical models ; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Because of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.

Novel active fault-tolerant control scheme and its application to a double inverted pendulum system

On the basis of the gain-scheduled H∞ design strategy, a novel active fault-tolerant control scheme is proposed. Under the assumption that the effects of faults on the state-space matrices of systems can be of affine parameter dependence, a reconfigurable robust H∞ linear parameter varying controller is developed. The designed controller is a function of the fault effect factors that can be derived online by using a well-trained neural network. To demonstrate the effectiveness of the proposed method, a double inverted pendulum system, with a fault in the motor tachometer loop, is considered.

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration(PDA)feedback,which can be used to understand human balance in quiet standing.The closed-loop system is described by a neutral delay differential equation(NDDE).The optimal feedback gains(OFGs)that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4.Such a property is called multiplicity-induced dominancy of time-delay systems,and has been discussed intensively by many authors for retarded delay differential equations(RDDEs).This paper shows that multiplicity-induced dominancy can be achieved in NDDEs.In addition,the OFGs are delay-dependent,and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays.Thus,the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains.The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

Designing Discrete Predictor-Based Controllers for Networked Control Systems with Time-varying Delays:Application to A Visual Servo Inverted Pendulum System

A discrete predictor-based control method is developed for a class of linear time-invariant networked control systems with a sensor-to-controller time-varying delay and a controller-to-actuator uncertain constant delay,which can be potentially applied to vision-based control systems.The control scheme is composed of a state prediction and a discrete predictor-based controller.The state prediction is used to compensate for the effect of the sensor-to-controller delay,and the system can be stabilized by the discrete predictor-based controller.Moreover,it is shown that the control scheme is also robust with respect to slight message rejections.Finally,the main theoretical results are illustrated by simulation results and experimental results based on a networked visual servo inverted pendulum system.

Using Adaptive Gain Scheduling LQR Method Control of Arm Driven Inverted Pendulum System Based on PIC18F4431

The arm driven inverted pendulum system is a highly nonlinear model, muhivariable and absolutely unstable dynamic system so it is very difficult to obtain exact mathematical model and balance the inverted pendulum with variable position of the ann. To solve this problem, this paper presents a mathematical model for arm driven inverted pendulum in mid-position configuration and an adaptive gain scheduling linear quadratic regulator control method for the stabilizing the inverted pendulum. The proposed controllers for arm driven inverted pendulum are simulated using MATLAB-SIMULINK and implemented on an experiment system using PIC 18F4431 mieroeontroller. The result of experiment system shows the control performance to be very good in a wide range stabilization of the arm position.

Balance control of a 12-DOF mobile manipulator based on two-wheel inverted pendulum robot

Humanoid mobile manipulator which is based on two-wheel inverted pendulum robot has been studied.Balance control is a key problem for this kind of centroid-variable robot.Due to the principle of two wheel inverted pendulum,a timely angle compensation is necessary to make the system keep balance when the centroid changes.In this paper,a method based on coordinate transformation is introduced to get the compensatory angle and a 12-DOF mobile manipulator is also used to check the method.Simulation and experimental results show the effectiveness of the method.

Swinging up a Real-Time Inverted Pendulum System

In this study, a real-time control of the cart inverted pendulum system was developed using Mamdani type Fuzzy Logic Controller. Swing-up and stabilization of the inverted pendulum were implemented directly in a Fuzzy Logic Controller. The fuzzy logic controller was designed in the Matlab-Simulink environment and applied into in a Quasar controller board. Swing-up algorithm brings the pendulum near to its inverted position in 5 seconds from downward position. External forces were applied on the inverted pendulum to test the robustness of the fuzzy logic controller under internal as well as external disturbances. The inverted pendulum system showed an acceptable robustness to the external and internal disturbances.

LQG Control Design for Balancing an Inverted Pendulum Mobile Robot

The objective of this paper is to design linear quadratic controllers for a system with an inverted pendulum on a mobile robot. To this goal, it has to be determined which control strategy delivers better performance with respect to pendulum’s angle and the robot’s position. The inverted pendulum represents a challenging control problem, since it continually moves toward an uncontrolled state. Simulation study has been done in MATLAB Simulink environment shows that both LQR and LQG are capable to control this system successfully. The result shows, however, that LQR produced better response compared to a LQG strategy.

Design and implementation of 2DOF spherical inverted pendulum

The dynamics of 2DOF spherical inverted pendulum system is analyzed. The motion of the pendulum may be projected onto the orthogonal planes in the Cartesian Space. In this way the system can be decoupled into two classical cart-pendulum systems and the design of controllers aimed at each subsystem separately are proposed. The linear quadratic optimal control strategy is applied in order to balance the pendulum system at the 'inverted' status. The method proposed is verified by the simulation and actual system experiments and the performance of the controller is discussed.

Steering performance of an inverted pendulum vehicle with pedals as a personal mobility vehicle

From the viewpoints of environmental protection, support for the aged and ensuring the right to mobility, there is a need to develop a new type of mobility vehicle that provides more effective transportation. The authors propose an inverted pendulum vehicle with pedals as one of the forms of personal mobility vehicles （PMVs）. In this paper, the steering performance of the inverted pendulum vehicle with pedals is discussed based on experiments on a prototype. From the experimental results, it was confirmed that the errors from the five subjects for the target trajectory and the five-grade evaluation of the maneuverability were similar. Finally, we created an inverted pendulum vehicle with pedals to which was added a reaction actuator for the steering system. From the experimental results, it was found that setting appropriate feedback gains for the handle steering angle and its rate of rotation, which control the right and left wheel driving torques, resulted in greatly improved maneuverability. C 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi：10.1063/2.1301309]

Simulation of a Triple Inverted Pendulum Based on Fuzzy Control

The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. This paper studies the use of fuzzy control method to study the stability control problem of a triple inverted pendulum system. By the linear model of the system, the feedback weight matrix of the LQR optimal control and the feedback parameters of the linear optimal control are designed to determine the parameters of the fuzzy controller. The simulation results show that the proposed method can achieve the stability control of the three stage inverted pendulum, and has good dynamic performance with simple parameter selection.

Modified grey wolf optimised adaptive super-twisting sliding mode control of rotary inverted pendulum system

This paper proposes a modified grey wolf optimiser-based adaptive super-twisting sliding mode control algorithm for the trajectory tracking and balancing of the rotary inverted pendulum system.The super-twisting sliding mode algorithm severely alleviates the chattering present in the classical sliding mode control.It provides robustness against model uncertainties and external disturbances with the knowledge of the upper bounds of the uncertainties and disturbances.The gains of the super-twisting sliding mode algorithm are selected through adaptive law.Parameters of the adaption law are tuned using a modified grey wolf optimisation algorithm,a meta-heuristic optimisation technique.Lyapunov stability analysis is carried out to analyse the overall control system stability.The performance of the proposed control algorithm is compared with two other sliding mode control strategies present in the literature,therein showing better performance of the proposed control scheme.

Spring-Loaded Inverted Pendulum Hopping via Hybrid Averaging and Control Lyapunov Function

The Spring-Loaded Inverted Pendulum(SLIP)has been regarded as a canonical model for hopping and running dynamics of legged robots.This paper presents a novel control of the actuated-SLIP hopping on unknown terrains.We propose that in the neighborhood of the desired stable hybrid limit cycle,the local dynamical behavior of a hybrid system can be expressed by a set of phase coordinates and transverse coordinates.Under some acceptable assumptions,the hybrid averaging theorem is applied on the SLIP non-integrable dynamics to simplify the controller design.Using the inherent symmetry of SLIP dynamics,a control Lyapunov function-based hybrid averaging controller is developed to ensure the exponential stability of the desired gait orbit.This results in a set of linear constraints on the control signal,which can be readily solved by a quadratic programming optimization.Furthermore,a novel method is introduced to improve the robustness against unknown disturbances through the online constraint adjustment.The proposed controller is evaluated in various simulations,demonstrating the SLIP hopping on diverse terrains,including flat,sin-wave,and unregular terrains.The performance of the approach is also validated on a quadruped robot SCIT Dog for generating dynamic gaits such as pronking.

Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input

Linear quadratic regulator(LQR) and proportional-integral-derivative(PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions.The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

Variable universe adaptive fuzzy control on the quadruple inverted pendulum

This paper focuses on the control problem of the quadruple inverted pendulum by variable universe adaptive fuzzy control.First,the mathematical model on the quadruple inverted pendulum is described and its controllability is versified.Then,an efficient controller on the quadruple inverted pendulum is designed by using variable universe adaptive fuzzy control theory.Finally the simulation of the quadruple inverted pendulum is shown in detail.Besides,the experimental results on the hardware systems,i.e.real object systems,on a single inverted pendulum,a double inverted pendulum and a triple inverted pendulum are briefly introduced.

STABILIZATION OF THE QUADRUPLE INVERTED PENDULUM BY VARIABLE UNIVERSE ADAPTIVE FUZZY CONTROLLER BASED ON VARIABLE GAIN H_∞ REGULATOR

In this paper, the variable universe adaptive fuzzy controller based on variable gain Ha regulator （VGH∞R） is designed to stabilize a quadruple inverted pendulum. The VGH∞R is a novel robust gain-scheduling approach. By utilizing VGH∞R technique, a more precise real-time feedback gain matrix, which is changing with states, is obtained, Via the variable gain matrix 10 state variables of quadruple inverted pendulum are transformed into a kind of synthesis error （E） and synthesis rate of change of error （EC） at sampling time. Therefore, the dimension of the multivariable system is reduced and the variable universe adaptive fuzzy controller is built. Experiments illustrate the effectiveness of the proposed control scheme.