Inverted Pendulum System Research Articles

Optimizing inverted pendulum control: Integrating neural network adaptability

This study explores the implementation and efficacy of a neural network controller for an inverted pendulum system, contrasting it with traditional state feedback control. Initially, state feedback control exhibited limitations in managing complex system dynamics. Subsequently, a neural network controller was developed, trained using datasets from both uncontrolled and refined state space models. The refined model yielded lower training loss and superior control performance. This research demonstrates the neural network controllers enhanced adaptability and precision, offering significant improvements over traditional methods in controlling dynamic systems like inverted pendulums.

H∞ state feedback design based on LQR robust control for inverted pendulum system

Abstract This paper proposes a robust H∞ design method for the disturbance suppression of an inverted pendulum. Based on the LQR method, the H∞ state feedback design, which has good robustness and disturbance rejection, is provided. The H∞ state feedback problem is ultimately converted into the resolution of the Riccati equation. The key to solving the Riccati equation is how to determine the weighted matrix in the H∞ controller. This paper takes the effective bandwidth of the system, the requirements of high properties at low-frequency ranges, and the coordination between variables and other aspects into consideration, which successfully solves the many parameters’ selection problem. And the physical significance is clear. The outcomes indicate that the advanced H∞ state feedback control scheme, rooted in LQR robustness, significantly enhances the system’s capability to mitigate external interference.

An Application of DC Motor Modelled Classical Sliding Mode Control with Moving Sliding Surface to Rotary Inverted Pendulum System

The rotary inverted pendulum system (RIP), which is highly favored in control applications, is examined in this work. By determining the coordinates of the Rip elements' centers of gravity, the system's total kinetic and potential energies were determined. The kinetic and potential energy expressions were used to generate the Lagrangian function. Expressions providing the system's equations of motion were discovered by taking the Lagrangian approach into consideration. The motor's equations, which will initiate the system, have also been considered. Through the use of state variables and a Matlab program, the system's pendulum angle was managed by using a moving sliding surface and the traditional sliding mode control technique. Based on the dynamics, the slip surface's slope was computed. The sliding surface's slope was computed based on the system's dynamics. The genetic algorithm was utilized to determine the ideal values for the coefficients employed in the control structure. The findings showed that the inaccuracy was roughly zero and that the pendulum angle took around 1.5 seconds to reach the intended reference value. Furthermore, it noted that the motor torque and current values are 12 Nm and 2.5 amps, respectively. The findings show that the motor values are reasonably similar to the values seen in real-world applications. Control in real-time applications won't be an issue if the motor is chosen based on these values.

Online adaptive critic designs with tensor product B-splines and incremental model techniques

As an effective optimal control scheme in the field of reinforcement learning, adaptive dynamic programming (ADP) has attracted extensive attention in recent decades. Neural networks (NNs) are commonly employed in ADP algorithms to realize nonlinear function approximation. However, the learning process with NN approximation typically requires a substantial amount of training data and places a heavy computational burden. To improve learning efficiency and alleviate computational load, this paper develops a novel model-free ADP approach based on multivariate splines and incremental control techniques, aimed at achieving online learning of nonlinear control. By utilizing input and output data, an incremental linear approximation model is identified online without any prior knowledge of system dynamics. To improve nonlinear approximation capabilities and lessen computational demands, tensor product B-splines, instead of NNs, are integrated into the critic module to approximate the value function, and the recursive least squares temporal difference (RLS-TD) algorithm is employed to update the weights of spline bases. Convergence analysis of the proposed control scheme is conducted based on the dual-timescale stochastic approximation theory. To illustrate the effectiveness and feasibility of the proposed control scheme, numerical simulations are performed in solving the online nonlinear control problem within the context of the inverted pendulum system.

A reference model-based backstepping anti-interference control method and its application to inverted pendulum

A reference model-based backstepping anti-interference control method is proposed, which performs well in the application of Inverted Pendulum system. Firstly, the backstepping controller of the single-stage straight-line inverted pendulum is designed and the reference model of the system is obtained. Input the actual state variables into the reference model. If the single-stage straight-line inverted pendulum is disturbed, it will deviate from the ideal value of the reference model. In order to improve the anti-interference performance of single-stage straight-line inverted pendulum system, the generalized state error is introduced, that is, the error between the output of the reference model and the actual output of the system. Then the controller is designed by Lyapunov stability theory based on generalized state error and backstepping control method. Finally, the reference model-based backstepping anti-interference controller is obtained. MATLAB/Simulation results show that the controller can effectively suppress the influence of external interference on the single-stage straight-line inverted pendulum. It has strong robustness and anti-interference ability.

Neuro-modelling and fuzzy logic control of a two-wheeled wheelchair system

Two-wheeled wheelchairs have been used as alternatives for the elderly and disabled people to perform physical activities due to their restriction of movement. Significant challenges posed by two-wheeled wheelchairs control due to their inherent instability, resembling that inverted pendulum system. This research addresses these challenges by developing a dynamic non-linear model and stability control using computational algorithms. A neural network-based Nonlinear Autoregressive model with Exogenous Inputs (NARX) was developed, capturing and behaving similar to the complex dynamics of the wheelchair system with the identification on the experimental input–output data. Ensuring stable and responsive control, design of optimized PD-type and PID-type fuzzy logic controllers using Particle Swarm Optimization (PSO) were established and were tested under a simulation environment. The performance was evaluated across various metrics, including Integral Squared Error (ISE), Integral Absolute Error (IAE), Mean Squared (MSE), and Integral Time Absolute Error (ITAE). The result demonstrates that the PSO Optimized PID-type fuzzy logic controller with scaling factor from MSE index performance come out as best overall, significantly outperforms PD-type fuzzy logic controller, reducing its settling time by 12.5% to 35 s, minimizing overshoot to 0.81°, and achieving a negligible steady-state error of 0.046%. These results highlight the significant of integrating fuzzy logic control and PSO to the neural network model in enhancing the stability and performance of two-wheeled wheelchair systems, offering user safety and comfort.

Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

Eksik Tahrikli Döner Ters Sarkaç Sisteminin Yukarı Yükseltilmesi için Enerji Tabanlı Doğrusal Olmayan Kontrol Algoritması ve Deneysel Doğrulaması

Bu çalışmada, döner ters sarkaç sistemi hareketli koluyla sınırlı bir bölge içinde, salınımı kontrol edilerek ve sarkaç enerjisi artırılarak, kararlı denge noktasından kararsız denge noktasına çıkarılması hedeflenmektedir. Döner ters sarkaç sisteminin doğrusal olmayan modeli ve denge noktalarındaki doğrusal modelleri verildikten sonra sarkaç yukarı yükseltme algoritması tanıtılmaktadır. Yukarı yükseltme algoritması, sarkaç salınırken belli noktalarda kol yardımıyla sarkaca enerji yüklemesine dayanmaktadır. Burada kol ivmelendirilmesi sabit bir ivme ile hareket ettirilmektedir. Yukarı yükseltme işlemindeki algoritma, sarkacın hızının en fazla olduğu yere yani kararlı denge noktasına ilerlerken ve periyodunu tamamlarken devreye girmektedir. Bu analizler hesaplanmış sınırlar dikkate alınarak yapılmaktadır. Sarkacın yukarı yükselmesi gerçekleştiğinde doğrusal model üzerinden elde edilmiş temel bir tam durum geri beslemeli kontrolcü devreye girmektedir ve sarkacı kararsız denge noktasında tutmaktadır. Böylece, anahtarlamalı bir kontrol yapısı elde edilmektedir. Önerilen kontrol algoritmasının doğrulaması için önerilen algoritma bir döner sarkacı üzerinde gerçek zamanlı olarak test edilmiştir. Farklı kol ivmelendirme değerleri ile elde edilen karşılaştırmalı sonuçlar sunulmuştur. Kol ivmesi ±32 rad/s2’den ±64 rad/s2’ye çıkarıldığında sarkacın yukarı yükselme süresi yaklaşık 9 s’den 5 s’ye düşmektedir. Bu sonuçlar, önerilen algoritmanın deneysel başarısını göstermektedir.

Mixed H2/H-infinity Controller Design for Triple Inverted Pendulum System

The steady increase in reliance on robotic systems in industrial applications arises the need to ensure their reliability despite the presence of external disturbances in the environment they work in and their parameters variations due to several factors like: change of temperature, wear and tear effect, and payload variation. Traditional controllers usually fail to maintain the system's stability and performance in the presence of uncertainty, while robust controllers aim to guarantee stability and desired performance across a range of possible uncertainties in the system through incorporating these uncertainties into their design process. This paper investigates the robustness of three controllers namely, H2 controller, H-infinity controller, and mixed H2/H-infinity controller, applied to the triple inverted pendulum system which translates to robustly control legged robots and robotic arms. The results show that among the three controllers, only the mixed H2/H-infinity control system is robustly stable and maintains its performance with stability margin 1.1864 and performance margin 1.1662. Index Terms— H2, H-infinity, Mixed H2/H-infinity, Robust Control, triple inverted pendulum.

Balance Controller Design for Inverted Pendulum Considering Detail Reward Function and Two-Phase Learning Protocol

As a complex nonlinear system, the inverted pendulum (IP) system has the characteristics of asymmetry and instability. In this paper, the IP system is controlled by a learned deep neural network (DNN) that directly maps the system states to control commands in an end-to-end style. On the basis of deep reinforcement learning (DRL), the detail reward function (DRF) is designed to guide the DNN learning control strategy, which greatly enhances the pertinence and flexibility of the control. Moreover, a two-phase learning protocol (offline learning phase and online learning phase) is proposed to solve the “real gap” problem of the IP system. Firstly, the DNN learns the offline control strategy based on a simplified IP dynamic model and DRF. Then, a security controller is designed and used on the IP platform to optimize the DNN online. The experimental results demonstrate that the DNN has good robustness to model errors after secondary learning on the platform. When the length of the pendulum is reduced by 25% or increased by 25%, the steady-state error of the pendulum angle is less than 0.05 rad. The error is within the allowable range. The DNN is robust to changes in the length of the pendulum. The DRF and the two-phase learning protocol improve the adaptability of the controller to the complex and variable characteristics of the real platform and provide reference for other learning-based robot control problems.

Design and Implementation of a Discrete-PDC Controller for Stabilization of an Inverted Pendulum on a Self-Balancing Car Using a Convex Approach

This paper presents a trajectory-tracking controller of an inverted pendulum system on a self-balancing differential drive platform. First, the system modeling is described by considering approximations of the swing angles. Subsequently, a discrete convex representation of the system via the nonlinear sector technique is obtained, which considers the nonlinearities associated with the nonholonomic constraint. The design of a discrete parallel distributed compensation controller is achieved through an alternative method due to the presence of uncontrollable points that avoid finding a solution for the entire polytope. Finally, simulations and experimental results using a prototype illustrate the effectiveness of the proposal.

Integrating of LMIs with Buckingham Π theorem for control of uncertain oscillating systems

This paper introduces a novel framework for robust controller synthesis in oscillating systems by synergistically combining generalized linear matrix inequalities (GLMIs) with the Buckingham Π theorem. The primary goal is to enhance the design of robust controllers for systems with uncertain parameters through the exploitation of dimensionless groups derived from the Buckingham Π theorem. By establishing a dimensionless state-space representation, the proposed approach demonstrates adaptability to a broad spectrum of similar systems. The LMI-based formulation facilitates the systematic design of controllers capable of effectively mitigating the impact of parametric uncertainties. Simulation results for an uncertain inverted pendulum and a vehicle active suspension system validate the superior performance of the proposed method compared to conventional control techniques, such as MPC, LQG, SMC, PID, and LQR. For the uncertain inverted pendulum system, the proposed robust control scheme achieves a cart position stabilization in 5.35 s and pendulum stabilization in 6.3 s, which is faster than standard control algorithms. In the context of the vehicle active suspension system, the performance indices for sprung mass acceleration, suspension deflection, and tire deflection are 0.19, 0.47, and 0.07, respectively. These values indicate significant reductions when compared to conventional passive suspension systems and standard robust control methodologies.

Finite time prescribed performance control for stochastic systems with asymmetric error constraint and actuator faults

This paper investigates the problem of finite time prescribed performance control (PPC) for a number of nonlinear stochastic systems with asymmetric error constraint, unknown control directions, and actuator faults. Firstly, instead of introducing the performance constraint function in the Lyapunov function, a new asymmetric error conversion function (AECF) is presented, which can successfully constrain the tracking errors into the specified asymmetric boundaries and eliminate the feasibility condition of requiring tracking errors to be bounded. Then, in iterative process, the unknown intermediate virtual controller is approximated by the fuzzy logic system (FLS), which makes each actual virtual controller to be reduced to only one item. Furthermore, the investigated finite time PPC strategy can fully compensate the faults impact on the systems and have only one adaptive parameter. In the end, the efficacy of investigated strategy is verified by inverted pendulum systems with stochastic disturbances.

Control and stochastic dynamic behavior of Fractional Gaussian noise-excited time-delayed inverted pendulum system

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H∈(1/2,1), we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic Itoˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

Optimal robust adaptive fuzzy controllers based on sliding surfaces for inverted pendulum systems

Optimal robust adaptive fuzzy controllers based on sliding surfaces for inverted pendulum systems

Fuzzy PID control based on genetic algorithm optimization inverted pendulum system

Abstract For the first-order inverted pendulum control system, a fuzzy PID control system based on the optimization of the genetic algorithm is proposed. The traditional genetic algorithm has the problem that the difference in the fuzzy subset parameter leads to a decrease in the interpretative ability of the fuzzy system. The main problem of the current genetic algorithm is the complexity of the computation and the low efficiency. Based on this problem, this paper proposes an improved genetic algorithm, i.e., it adopts the variance operator and adaptive change of the variance index and elite retention strategy, which solves the premature and local convergence problems of the standard genetic algorithm, in order to optimize the fuzzy system. The experimental results show that the optimized genetic algorithm gives full play to the advantages of fuzzy control in terms of interpretability and robustness, and at the same time guarantees the prediction accuracy, which provides a new research idea in the field of artificial intelligence control.

Optimized Hierarchical Sliding Mode Control for the Swing-up and Stabilization of a Rotary Inverted Pendulum

This paper presents a study on controlling a rotary inverted pendulum (RIP) system using a hierarchical sliding mode control (HSMC) approach. The objective is to swing up and stabilize the pendulum at a desired position. The proposed HSMC controller addresses the underactuation challenge through a hierarchical structure of sliding surfaces. The particle swarm optimization (PSO) algorithm is used to optimize the controller parameters. Simulations were performed to evaluate the performance of the HSMC controller at different initial pendulum angles, demonstrating its effectiveness in achieving swing-up and stabilization. The integration of the PSO algorithm enhances the controller’s adaptability and robustness, emphasizing the benefits of combining optimization algorithms with controller parameter tuning for underactuated systems like the RIP.

Adaptive Multi-Switching-Based Global Tracking Control for Switched Nonlinear Systems With Prescribed Performance

This paper is concerned with the tracking control for a class of switched nonlinear systems subject to prescribed performance. Firstly, a barrier function and a normalized function are introduced to achieve prescribed performance control, which ensures that the tracking error evolves within prescribed boundary. Then, multi-dimensional Taylor network provides a new approach for how to estimate the nonlinearity arising from the backstepping control process. Significantly, the proposed multi-switching-based adaptive controller realizes that all signals in the closed-loop system are globally uniformly ultimately bounded while ensuring asymptotic tracking. Different from most existing network-approximation-based control strategies, the developed method in this paper is not only independent of the initial state, but also can achieve global stability. Finally, it is easy to verify the effectiveness of the proposed control method through two simulations. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This research is motivated by the fact that the control ideas of many practical engineering systems can be provided by making a profound study on switched nonlinear systems. However, most of the existing results focused on semi-global stability of systems. Therefore, this study devotes to develop a novel adaptive prescribed performance control strategy, which can not only ensure global stability of closed-loop system but also realize the asymptotic tracking of the switched nonlinear systems. It is worth noting that the proposed control approach has great significance for many practical systems, such as circuit systems and single-link inverted pendulum systems.

LQR and Fuzzy-PID Control Design on Double Inverted Pendulum

Double inverted pendulum is a non-linear and unstable system. Double inverted pendulum can be stabilized in the upright position by providing control to the system. In this research we compare two types of controllers namely Linear Quadratic Regulator (LQR) and Fuzzy-PID. The objective is to determine the control strategy that provides better performance on the position of the cart and pendulum angle. We modelled the system which is then linearized and given control. From the simulation results, it is proven that LQR and Fuzzy-PID controllers have been successfully designed to stabilize the double inverted pendulum. However, when given a disturbance in the form of noise step, the LQR controller has not been able to achieve the desired reference for up to 20 seconds. In another hand, the Fuzzy-PID controller is able to achieve the desired reference after 8 seconds. Therefore, it can be concluded that the Fuzzy-PID controller when applied to the Double Inverted pendulum system has better performance than the LQR controller.

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.