Rotary Inverted Pendulum System Research Articles

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

Adaptive neural network-based sliding mode control of rotary inverted pendulum system

This paper introduces a new adaptive neural network-based sliding mode control of rotary inverted pendulum system (RIPS) subject to model uncertainties and disturbances. First, the sliding mode control is designed using the double power rate reaching law. Second, an adaptive neural network (ANN) is exploited to deal with model uncertainties and disturbances. The weight matrices of the neural network are adaptively tuned. In the proposed control method the exact dynamic model of the RIPS is not required due to the exploitation of the universal property of the neural network. Moreover, the implementation of a double power-reaching law can greatly suppress the undesirable chattering. The convergence and stability analysis is validated using Lyapunov stability theory. Finally, the robustness and excellence of the proposed control strategy are verified by comparing the system performance with the existing SMC method and improved super-twisting SMC method.

Rotary inverted pendulum control using neuro-adaptive robust generalized dynamic inversion

The objective of this research is to design a robust control system for trajectory tracking of the under-actuated Rotary Inverted Pendulum (RIP). The focus is on achieving semi-global asymptotic stability in the absence of knowledge about the RIP geometric and inertia parameters. The control design commences by formulating differential servo constraint dynamics (VCD) that encapsulates the control objectives for the RIP. The baseline generalized dynamic inversion (GDI) control law is derived by inverting the VCD for the control voltage input using the Moore–Penrose generalized inverse. To enhance robustness against modeling uncertainties and exogenous disturbances, a sliding mode control (SMC) element is integrated within the GDI control system, resulting in the robust GDI (RGDI) control system. Furthermore, an adaptive estimator that is based on Radial Basis Function Neural Networks (RBF-NN) is adopted to eliminate the dependency of the RGDI control system on the RIP mathematical model. The weighting matrices of the RBF-NN are updated with the aid of a Lyapunov control function. The proposed control system offers robust solution and guarantees semi-global asymptotic stability to the unstable equilibria of the RIP dynamics in the absence of a mathematical model. The proposed control system’s performance is evaluated through computer simulations and experimental tests on a real RIP system. Comparative analyses with traditional SMC and linear quadratic control methodologies are conducted to show the effectiveness of the proposed approach.

Study of Q-learning and deep Q-network learning control for a rotary inverted pendulum system

The rotary inverted pendulum system (RIPS) is an underactuated mechanical system with highly nonlinear dynamics and it is difficult to control a RIPS using the classic control models. In the last few years, reinforcement learning (RL) has become a popular nonlinear control method. RL has a powerful potential to control systems with high non-linearity and complex dynamics, such as RIPS. Nevertheless, RL control for RIPS has not been well studied and there is limited research on the development and evaluation of this control method. In this paper, RL control algorithms are developed for the swing-up and stabilization control of a single-link rotary inverted pendulum (SLRIP) and compared with classic control methods such as PID and LQR. A physical model of the SLRIP system is created using the MATLAB/Simscape Toolbox, the model is used as a dynamic simulation in MATLAB/Simulink to train the RL agents. An agent trainer system with Q-learning (QL) and deep Q-network learning (DQNL) is proposed for the data training. Furthermore, agent actions are actuating the horizontal arm of the system and states are the angles and velocities of the pendulum and the horizontal arm. The reward is computed according to the angles of the pendulum and horizontal arm. The reward is zero when the pendulum attends the upright position. The RL algorithms are used without a deep understanding of the classical controllers and are used to implement the agent. Finally, the outcome indicates the effectiveness of the QL and DQNL algorithms compared to the conventional PID and LQR controllers.

Effective Control Strategies for Rotary Inverted Pendulum System: Energy-Based, Linear Quadratic Regulator, and Hierarchical Sliding Mode Control Techniques

This work presents a comprehensive study of control problem for the rotary inverted pendulum (RIP), a challenging underactuated system with potential applications in robotics and aerospace. The paper develops effective swing-up energy-based control and stabilization task. It also presents two controllers, linear quadratic regulator (LQR) and hierarchical sliding mode control (HSMC), that effectively handle the stabilization problem. System modelling is based on Lagrangian mechanics, and the control strategies are evaluated using simulations and compared in terms of performance and robustness. The results demonstrate that energy-based control is effective for swing-up, while linear quadratic regulator and hierarchical sliding mode control are effective for stabilization. The proposed controllers show promising results and contribute to the development of robust and efficient control strategies for the rotary inverted pendulum system. The study has implications for the development of control strategies for other underactuated systems and can potentially lead to advancements in the field of robotics and aerospace.

A signal compensation-based balance control for the rotary inverted pendulum system

In this paper, we study the balance control for the rotary inverted pendulum system with two joints both subject to uncertainty. To facilitate the experiment, we apply a classical energy-based swing-up control, eschewing compensation for the uncertainty. In balance control, we employ a novel approach in the form of a modified signal compensation-based balance control, which models the uncertainty as its previously sampled term and its variation. The signal compensation-based balance control involves three steps. First, for the linearized discrete-time state-space model around the upright equilibrium point, we design the main control signal by using the pole-placement technique. Second, we estimate the discrete-time model of the system by using the least squares identification method, thereby unveiling key points in the identification procedure. The discrete-time model presents the uncertainty as a previously sampled term and its variation. Third, we design two discrete-time compensation signals for the previously sampled term and the variation. By combining the main control signal and the two compensation signals, we formulate the balance control law. We further optimize the signal compensation-based balance control by adding an extra parameter, successfully circumventing the critical stability problem, and analyze the performance of the closed-loop system. Numerical simulation and comparative experiments demonstrate its effectiveness and strong robustness against the uncertainty in frictions and modeling errors of the system.

Stabilization of Double Inverted Pendulum Systems Based on Hierarchical Sliding Mode Control Techniques

The double rotary inverted pendulum (DRIP) system belongs to the class of under-actuated mechanical systems, and it is a highly nonlinear, unstable, and benchmark system to test the different control techniques. This paper successfully designed the nonlinear hierarchical sliding mode control (HSMC) techniques to stabilize the DRIP system. We compare the performance of these techniques numerically with each other and with the previously designed control technique. We propose an aggregated HSMC technique as it has a much shorter stabilization time than other designed techniques.

Combining Passivity-Based Control and Linear Quadratic Regulator to Control a Rotary Inverted Pendulum

In this manuscript, new combination methodology is proposed, which named combining Passivity-Based Control and Linear Quadratic Regulator (for short, CPBC-LQR), to support the stabilization process as the system is far from equilibrium point. More precisely, Linear Quadratic Regulator (for short, LQR) is used together with Passivity-Based Control (for short, PBC) controller. Though passivity-based control and linear quadratic regulator are two control methods, it is possible to integrate them together. The combination of passivity-based control and linear quadratic regulator is analyzed, designed and implemented on so-called rotary inverted pendulum system (for short, RIP). In this work, CPBC-LQR is validated and discussed on both MATLAB/Simulink environment and real-time experimental setup. The numerical simulation and experimental results reveal the ability of CPBC-LQR control scheme in stabilization problem and achieve a good and stable performance. Effectiveness and feasibility of proposed controller are confirmed via comparative simulation and experiments.

Implementation of Sliding Surface Moving Anfis Based Sliding Mode Control to Rotary Inverted Pendulum

This study covers the control of the pendulum angle by taking into account the dynamic equations and motor dynamics of the rotary inverted pendulum system, with the help of state variables in the Matlab program, by using the sliding mode control method with sliding surface moving and the adaptive neural fuzzy inference system together. The sliding mode control method with a changing sliding surface is a part of the control structure. The slope of the sliding surface was calculated using the adaptive neural fuzzy inference technique. The optimum values of the coefficients in the adaptive neural-fuzzy inference system structure have been calculated by genetic algorithm. The finding of the coefficients, the sum of the squares of the errors chosen as the objective function. The input of the adaptive neural fuzzy inference system structure consists of the error of the pendulum and the derivative of the error of the pendulum. The gradient of the sliding surface of the sliding mode control structure is the output of the adaptive neural fuzzy inference system structure. According to the findings, the pendulum angle achieved the appropriate reference value after 1.5 seconds, with an error of around zero. It obtained that the engine torque value reaches up to 50 Nm. From here, it is seen that the motor torque values used in practical applications and the motor torque values as a result of this study overlap.

Optimal FOPI-FOPD controller design for rotary inverted pendulum system using grey wolves’ optimization technique

The rotary inverted pendulum (RIP) has been used in various control application areas. This system can be represented as two degree of freedom (2-DOF), consisting of a rotating arm and rotating pendulum rod. RIP is an excellent example of designing a single-input multi-output (SIMO) system. Due to unstable RIP system dynamics, and its nonlinear model, multiple control techniques have been used to control this system. This paper uses integer and fractional order proportional integral-proportional derivative (PI-PD) controllers to stabilize the pendulum in the vertical direction. Constrained optimization approaches, such as the grey wolf optimization (GWO) methodology, are utilized to estimate the parametric values of the controllers. The simulation results showed that the fractional order PI-PD controller outperforms the integer order PI-PD controller with and without disturbance signal existence. A multiple results comparison has illustrated the superiority of fractional order controller over a previous work.

Optimal fractional-order proportional–integral–derivative control enabling full actuation of decomposed rotary inverted pendulum system

Allowing for a “virtual” full actuation of a rotary inverted pendulum (RIP) system with only a single physical actuator has been a challenging problem. In this paper, a hybrid control scheme that involves a pole-placement feedback controller and an optimal proportional–integral–derivative (PID) or fractional-order PID (FOPID) controller is proposed to simultaneously enable the tracking control of the rotary arm and the stabilization of the pendulum arm in an input–output feedback linearized RIP system. The PID controller is optimized first with the particle swarm optimization (PSO) to obtain three optimal gains, and then the other two parameters of the FOPID controller are optimized with the PSO. Compared to the optimized PID controller, the optimized FOPID controller improves the tracking and stabilizing accuracy by 53% and 29%, respectively, and demonstrates better adaptability for tracking different reference signals. Moreover, the hybrid FOPID controller exhibits 74.8% and 53% higher tracking accuracy than previous optimized model reference adaptive control PID (MRAC-PID) and linear–quadratic regulator (LQR) controllers, respectively. The proposed hybrid controllers are also digitized with different rules and sampling times, showing a closer performance between the discrete-time and continuous-time hybrid controllers under smaller sampling times.

Rotary Inverted Pendulum Control using Robust Generalized Dynamic Inversion with Adaptive Neural Estimation

A novel robust generalized dynamic inversion control system with adaptive neural estimation (RGDI-ANE) is designed for trajectory tracking of the under actuated rotary inverted pendulum (RIP). The RGDI control design begins by prescribing a virtual differential constraint dynamics (VCD) that imitates the desired tracking control objectives. The baseline RGDI control law is derived by inverting the VCD using dynamically scaled Generalized Inversion, and then augmenting a sliding mode control (SMC) element to provide robustness against performance deterioration due to dynamic scaling of the Moore-Penrose generalized inverse. Furthermore, an adaptive estimator that is based on radial basis function neural networks (RBF-NN) is adopted to limit the dependency of RGDI control on the RIP mathematical model. The weighting matrices of the RBF-NN are updated using a Lyapunov control function. The closed loop RGDI-ANE control system is shown to guarantee semi-global asymptotic stability. Computer simulations along with experimental tests have been conducted on Quanser's RIP system to validate the performance of RGDI-ANE control, and the results are compared with the results obtained by using the SMC and the linear quadratic control methodologies.

Fuzzy Sliding Mode Control with Moving Sliding Surface of Rotary Inverted Pendulum

In this study, considering the dynamic equations of the rotary inverted pendulum system and the motor dynamics, pendulum angle is controlled with fuzzy logic sliding mode control method which has moving sliding surface using state variables in Matlab program. The sliding surface of sliding mode control method is selected as moving. A fuzzy logic structure is used to calculate the slope of slip surface. The boundary values of the membership functions in the fuzzy logic structure have been optimized using genetic algorithm codes in Matlab program. The sum of the squares of the errors is preferred as the objective function. Inputs of the fuzzy logic structure are the error of the pendulum angle and the derivative of pendulum angle error. In the fuzzy logic structure, the slope of sliding surface of sliding mode control structure is obtained as an output. From the results, it was seen that the pendulum angle reached the desired reference value around the first second and the error was approximately zero. In addition, it has been observed that the motor torque value is at the levels of 20 Nm and the motor current value is at the levels of 3 ampers. It has been obtained from the results that the motor values are at reasonable levels close to the values in practical applications. When the motor is selected according to these obtained values, there won't be a problem with the implementation of this control method in real-time applications of the rotary inverted pendulum system

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.

Robust Adaptive Super-Twisting Sliding Mode Stability Control of Underactuated Rotational Inverted Pendulum With Experimental Validation

In this study, an adaptive proportional-integral-derivative (PID) sliding mode control technique combined with the super-twisting algorithm is planned for the stabilization of rotational inverted pendulum in the appearance of exterior perturbation. The state-space model of rotational inverted pendulum in the existence of exterior disturbance is attained. Then, the super-twisting PID sliding mode controller is designed for finite time stability control of the considered underactuated control system. The upper bounds of perturbation are presumed to be unknown; consequently, the adaptive control procedure is taken into account to approximate the uncertain bounds of external disturbances. The stability control of rotational inverted pendulum system is verified by means of the Lyapunov stability theory. In order to validate the accuracy and efficiency of the recommended control technique, some simulation outcomes are prepared and compared with other existing scheme. Finally, the experimental results are implemented to show the success of the designed method.

Swing up and stabilization control of rotary inverted pendulum based on energy balance, fuzzy logic, and LQR controllers

Rotary Inverted Pendulum (RIP) mimics the behavior of many practical control systems like crane mechanism, segway, unicycle robot, traction control in vehicles, rocket stabilization, and launching. RIP is a fourth-order nonlinear open-loop unstable dynamical system and is widely used for testing the effectiveness of the newly developed control algorithms. In this paper, a Hybrid Control Scheme (HCS) based on energy balance and fuzzy logic controllers is proposed to implement the swing up and stabilization control of RIP. In the proposed control scheme, the fuzzy logic-based state feedback gains are dynamically tuned in real-time by minimizing the absolute error between the desired and actual states to get robust control performance. The proposed HCS is also compared with the conventional Linear Quadratic Controller (LQR) for this application. The comparative results show that the proposed fuzzy logic-based hybrid control scheme gives the optimal control performance in terms of achieving satisfactory transient, steady-state, and robust responses from a given RIP system, as compared to the conventional LQR based control scheme. The proposed control scheme is also relatively less complex with a low computational cost and provides desired response characteristics as compared to the existing ones in the literature.

Lyapunov-based continuous-time nonlinear control using deep neural network applied to underactuated systems

Several learning-based control with computational intelligence strategies handle challenges related to the difficulty of modeling complex systems or the need for control strategies with provably safe. In recent years, learning-based control using machine learning has been successfully demonstrated in robotics applications and applied to deal with nonlinearities. These control methods may lead to better solutions to nonlinear problems, such as the safety-critical industry, which requires strong guarantees about the controller behavior. Learning-based neural network control can comprehend and learn about plants, disturbances, the environment, and operating conditions. In this paper, we presented a Lyapunov-based nonlinear control determined from a deep neural network, which uses the Lyapunov theory to compute a control law for a nonlinear system. For advance stability analysis, an estimation of the region of attraction is presented. A numerical example and experimental simulations using the rotational inverted pendulum system are performed and compared with a conventional control technique. The proposed method calculated a control law that provided the stabilizability of the system and produced better solutions considering different tracking and process disturbance.

Arm Angle Tracking Control with Pole Balancing Using Equivalent Input Disturbance Rejection for a Rotational Inverted Pendulum

This paper proposes a robust tracking control method for swing-up and stabilization of a rotational inverted pendulum system by applying equivalent input disturbance (EID) rejection. The mathematical model of the system was developed by using a Lagrangian equation. Then, the EID, including external disturbances and parameter uncertainties, was defined; and the EID observer was designed to estimate EID using the state observer dynamics and a low-pass filter. For robustness, the linear-quadratic regulator method is used with EID rejection. The closed-loop stability is proven herein using the Lyapunov theory and input-to-state stability. The performance of the proposed method is validated and verified via experimental results.

Data-Driven State Prediction and Sensor Fault Diagnosis for Multi-Agent Systems with Application to a Twin Rotational Inverted Pendulum

When a multi-agent system is subjected to faults, it is necessary to detect and classify the faults in time. This paper is motivated to propose a data-driven state prediction and sensor fault classification technique. Firstly, neural network-based state prediction model is trained through historical input and output data of the system. Then, the trained model is implemented to the real-time system to predict the system state and output in absence of fault. By comparing the predicted healthy output and the measured output, which can be abnormal in case of sensor faults, a residual signal can be generated. When a sensor fault occurs, the residual signal exceeds the threshold, a fault classification technique is triggered to distinguish fault types. Finally, the designed data-driven state prediction and fault classification algorithms are verified through a twin rotational inverted pendulum system with leader-follower mechanism.

Tuning rules: Graphical analysis and experimental validation of a simplified fractional order controller for a class of open‐loop unstable systems

AbstractThe design of non‐integer order controllers and tuning of its fractional parameters have attracted the research community of science and engineering in previous years. However, few attempts already have been done by the researchers to present the excellent results for some limited cases of models using fractional controllers; still, many physical world problems are untouched and waiting to be discovered. The big obstacle which impounds the adaptation of non‐integer controllers is the complexity in the tuning rules for physical systems. The objective of this article is to present a novel graphical tuning approach of mono and dual degree fractional controllers to fulfill the five different design requirements such as iso‐damping, gain crossover frequency, sensitivity, phase margin, and steady‐state error cancellation. To demonstrate the effectiveness of the suggested tuning technique, two test bench setups, rotary inverted pendulum and magnetic levitation system, have been taken as a case study. The experimental results are given to validate the effectiveness of the proposed method.