一级倒立摆的控制算法

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In this paper, the balance control of a single inverted pendulum is studied. Firstly, the mathematical model of the inverted pendulum system is established by using Newtonian mechanics method, and two parallel fuzzy controllers are used to control the position of the car and the angle of the pole respectively, so that the inverted pendulum system can reach a balanced position as soon as possible. Then the control effect of the controller is verified by SIUMLINK simulation and compared with the PID controller. Finally, the 3D model was established by V-Realm Builder and co-simulated with SIMULINK. The results show that the fuzzy controller designed in this paper has a good control effect on the inverted pendulum.

In this paper, a novel recurrent adaptive backstepping optimal control strategy for a single inverted pendulum system is studied. By this method, an inverted pendulum is stabilized using projection recurrent neural network-based adaptive backstepping control (PRNN-ABC). The inverted pendulum is a popular nonlinear system that is used in both industry and academic and is applied various control approaches since it has many applications. Here, first of all, the backstepping control laws are investigated based on the nonlinear dynamic model of the system. Second, by considering control constrains and performance index, the constrained optimization problem is formulated. Later, the optimization problem will be converted to a constrained quadratic problem (QP). To study the recurrent neural network (RNN) according to the Karush- Kuhn-Tucker (KKT) optimization conditions and the variational inequality, the dynamic model of the RNN will be derived. At last, the stability analysis of the system is studied using Lyapunov function.

In this paper, the problem of precise control of single-stage inverted pendulum system is deeply discussed and practiced. The dynamic modeling and simulation of the inverted pendulum system were carried out in the course of the experiment, and the whole process of the inverted pendulum control was vividly reproduced by visual means, which enhanced the intuitiveness and effectiveness of the model design and control strategy analysis. In the design of control strategy, two advanced methods of neural network control and cascade fuzzy control are combined. The neural network controller uses its powerful nonlinear mapping ability and self-learning characteristics to approximate the complex dynamic behavior of the inverted pendulum system in real time and accurately. The cascade fuzzy controller realizes the hierarchical fine control of the Angle and angular velocity of the inverted pendulum through the upper and lower two-level control system, which improves the stability and robustness of the control system. After a series of rigorous experimental testing and optimization adjustment, the control output curve and the response curve of inverted pendulum state under different control methods are obtained successfully. Through the comparative analysis of these results, the advantages of the comprehensive scheme based on cascade fuzzy control and neural network control in improving the stable equilibrium time of inverted pendulum, reducing the overthrow amplitude and enhancing the anti-disturbance ability are revealed, and the effectiveness and superiority of the control strategy are verified, providing a new idea and reference for the theoretical research and practical application of inverted pendulum control.

For a single inverted pendulum control system, the design method of a class of state feedback $H_{\infty}$ controller and the performance of the robust control are studied, then the H∞controller design is studied for the control system mentioned above based on LMI optimization, and a sufficient condition is derived in terms of linear matrix inequalities. The designed controller guarantees the stability of the single inverted pendulum control system, and has good anti-interference ability. Numerical example is given to illustrate the effectiveness of the proposed method.

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This work will present an algorithmic approach for robust control focusing on hydraulic–mechanical systems. The approach is applied to a hydraulic actuator driving a cart with an inverted pendulum. The algorithmic approach aims to satisfy two robust control requirements for single input single output (SISO) linear systems with nonlinear uncertain structure. The first control requirement is robust stabilization, and the second is robust asymptotic command following for arbitrary reference signals. The approach is analyzed in two stages. In the first stage, the stability regions of the controller parameters are identified. In the second stage, a Particle Swarm Optimization Algorithm (PSO) is applied to find suboptimal solutions for the controller parameters in these regions, with respect to a suitable performance cost function. The application of the approach to a hydraulic actuator, driving a cart with an inverted pendulum, satisfies the goal of achieving precise control of the pendulum angle, despite the system’s inherent physical uncertainties.

The discipline of automatic control is making increased use of concepts that originate from the domain of machine learning. Herein, reinforcement learning (RL) takes an elevated role, as it is inherently designed for sequential decision making, and can be applied to optimal control problems without the need for a plant system model. To advance education of control engineers and operators in this field, this contribution targets an RL framework that can be applied to educational hardware provided by the Lucas-N\"ulle company. Specifically, the goal of inverted pendulum control is pursued by means of RL, including both, swing-up and stabilization within a single holistic design approach. Herein, the actual learning is enabled by separating corresponding computations from the real-time control computer and outsourcing them to a different hardware. This distributed architecture, however, necessitates communication of the involved components, which is realized via CAN bus. The experimental proof of concept is presented with an applied safeguarding algorithm that prevents the plant from being operated harmfully during the trial-and-error training phase.

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.

Single input type-2 fuzzy controllers gain increasing popularity for different control problems. In this study, a single input type-2 (SIT2) fuzzy control system is introduced for the control of an inverted pendulum system. The control system consists of two SIT2 fuzzy PD (SIT2-FPD) controllers for the control of the cart position and the pendulum angle. The SIT2-FPD controller is constructed by cascading a single input fuzzy controller with a classical PD controller. In the design of SIT2-FPD controllers, three triangular interval type-2 membership functions are assigned for the input variable error, and three singleton membership functions are used to define the output variable. In order to show the control and robustness performance of the SIT2-FPD control system, simulation studies are performed for different operating conditions and the performance results are compared with the performance of a classical PD control system. The performance comparison results show that the SIT2 fuzzy control system exhibits better control and robustness performance than the classical PD control system.

In order to solve the problem that students cannot fully grasp the analysis and design methods of the engineering physical control system without the automatic control comprehensive experimental device in the laboratory, the virtual simulation experiment teaching design of the linear inverted pendulum control system is carried out by taking the linear first-stage inverted pendulum and the linear second-stage inverted pendulum as examples. Three-dimensional animation technology, visualization technology, computer programming technology and database technology are used to provide students with virtual simulation experiment training on the structure and disassembly of linear inverted pendulum system, as well as photoelectric encoder principle, modeling of linear inverted pendulum system, single loop PID control, double closed-loop PID control, state feedback control and LQR control. Adopting a student-centered teaching approach, and through the self-interactive virtual simulation experiment, the students master the classical nonlinear system modeling, analysis and controller design methods, and the teaching effect of “With the virtual to fill the real, deep interaction, integration of theory and practice ” is achieved.

This paper is primarily focused on the robust control of an inverted pendulum system based on policy iteration in reinforcement learning. First, a mathematical model of the single inverted pendulum system is established through a force analysis of the pendulum and trolley. Second, based on the theory of robust optimal control, the robust control of the uncertain linear inverted pendulum system is transformed into an optimal control problem with an appropriate performance index. Moreover, for the uncertain linear and nonlinear systems, two reinforcement-learning control algorithms are proposed using the policy iteration method. Finally, two numerical examples are provided to validate the reinforcement learning algorithms for the robust control of the inverted pendulum systems.

The cart–inverted pendulum system (CIPS) is a typical example of underactuated mechanical systems. For the CIPS with friction and disturbances, a gain-scheduled model predictive control method is proposed to achieve the upright stabilization objective of the single inverted pendulum (SIP) while controlling the cart to reach a desired new position. To this end, first, a dynamic equation of the CIPS with friction and disturbances is formulated based on the Newton–Euler equation. On the basis of the dynamic equation of the CIPS, its motion characteristics and control process are analyzed. Next, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models at seven different pendulum angles. Then, seven model predictive controllers (MPCs) are designed based on the above-linearized models, respectively. Introducing the idea of the gain-schedule, a gain-scheduled MPC (GSMPC) is designed to switch one of these seven MPCs to realize the control objective of the CIPS, according to the actual pendulum angle of the SIP during the control process. Finally, multi-group simulations that consider the friction and disturbances of the CIPS are implemented to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method.

In this paper we propose a new swing-up strategy for a single inverted pendulum. The proposed method has a feature that can handle the limitation of the pendulum-rail length and actuator constraints using both feedforward and feedback control. The feedforward trajectories are generated by solving an optimal control problem having two-point boundary conditions. The limitation of the rail length and the actuator constraints are taken into account in the problem formulation. Feedback control is combined with the feedforward control to compensate the deviation between the desired trajectories and actual trajectories. The experimental results of the proposed strategy show that it has a good swing-up performance while satisfying all the imposed constraints.

Stabilization of the inverted pendulum by fractional-order proportional-derivative (PD) feedback with two delays is investigated. This feedback law is obtained as a combination of PD feedback with two delays and fractional-order PD feedback with a single delay. Different types of stabilizability boundaries and the corresponding geometric and multiplicity conditions are determined using the D-subdivision method. The stabilizable region is depicted in the plane of the delay parameters for given fractional derivative orders. Several special cases and the concept of delay detuning are also discussed. It is shown that the admissible delay can be slightly increased compared to the integer-order PD feedback by introducing a fractional-order feedback term.

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The present work represents the implementation of the various fuzzy controller with robust sliding mode control (SMC) technique on a nonlinear system considering various external disturbances and model uncertainties. The nonlinear system considered here is a single link inverted pendulum. The proposed work combines the advantages of the sliding mode controlling technique and fuzzy logic controller. A set of linguistic rules are designed in fuzzy logic control, which causes the system to be chattering free. Parameters of the nonlinear system are adjusted according to fuzzy adaptive laws, while the uncertainties of the nonlinear system have been approximated using a fuzzy system. Various types of controller based on fuzzy sliding mode, like approximation based sliding mode control technique; equivalent control based fuzzy sliding mode technique, and switch-gain regulation based sliding mode control methods have been implemented here. A comparative analysis of various methods is also have been discussed.

This study focused on applying and enhancing the Deep Deterministic Policy Gradient (DDPG) algorithm to effectively control a Single Inverted Pendulum (SIP) system. The primary objective was to improve the algorithm's performance by addressing common challenges such as overestimation of Q-values and convergence to local optima. The system's behaviour was analyzed through simulation and real-world experiments, showcasing the algorithm's ability to offer faster responses, enhanced stability, and reduced pendulum displacement. The research introduced key modifications to the experience replay mechanism and the Critic network, which played a significant role in improving the efficiency of the learning process and the robustness of the control strategy. By combining Reinforcement Learning with traditional control methods, this approach successfully managed the nonlinear dynamics of the SIP system. Nevertheless, certain challenges persist, particularly in terms of the efficiency of deep reinforcement learning algorithms and their stability in real-world environments. These findings suggest that future research should focus on further refining DRL algorithms to increase their practical application in physical control systems. In conclusion, the research highlights the potential of combining DRL techniques with conventional control strategies for tackling complex control problems. The success achieved in controlling the SIP system indicates a promising direction for further exploration and development in this field.

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This paper introduces a new continuous integral sliding mode control algorithm, where the discontinuous function of the super-twisting control law is replaced with a continuous disturbance observer for the substantial chattering attenuation. In the present integral sliding mode control, the discontinuous function generates chattering that is undesirable for several real-time applications. The proposed control strategy decreases the amplitude of the controller gain compared to the existing integral sliding mode controls, and as a consequence of this, the attenuation of chattering is achieved to a great extent. The efficacy of the proposed control algorithm is validated successfully on the single-input single-output Inverted Pendulum and 2-DOF Helicopter nonlinear coupled multi-input multi-output systems. The simulation and experimental results demonstrate the successful application of the proposed control approach to follow reference inputs and acquire robustness and stabilization of the system in the presence of limited matched perturbations and nonlinearities.

Abstract Inverted pendulum systems (IPSs) are mostly used to demonstrate the control rules for keeping the pendulum at a balanced upright position due to a slight force applied to the cart system. This paper presents an application for nonlinear control of an x-z type IPS by using a proportional-integral-derivative (PID) controller with newly established evolutionary tuning method Lightning Search Algorithm (LSA). A single, double and triple PID controller system is tested with the conventional and the self-tuning controllers to get a better understanding of the performance of the given system. The mathematical modelling of the x-z type IPS, the theoretical explanation of the LSA and the simulation analysis of the x-z type IPS is put forward entirely. The LSA algorithm solves the optimization problem of PID controller in an evolutionary way. The most effective way of making comparisons is evaluating the performance results with a well-known optimization technique or with the previous accepted results. We have compared the system’s performance with particle swarm optimization and with a classical control study in the literature. According to the simulation results, LSA-tuned PID controller has the ability to decrease the overshoot better than the conventional-tuned controllers. Finally, it can be concluded that the LSA-supported PID can better stabilize the pendulum angle and track the reference better for non-disturbed and the slightly disturbed systems.

Model-Free Control (MFC) is a Data Driven Control (DDC) method which is designed for complex or nonlinear systems. Such a system is the Inverted Pendulum System which is known as unstable in open loop and has a high degree of nonlinearity and limitations. This complexity of the system makes it difficult to be controlled using the classical control methods. Advanced methods as Model Predictive Control (MPC), Linear Quadratic Regulator (LQR) or Sliding Mode Control (SMC) have been proposed over time to control this type of plant. Since it has a single input and two outputs which have a strong correlation, the inverted pendulum system represents a challenge for every control algorithm. There are applications for this type of plant to control the position of the cart or/and the angle of the pendulum. In this paper, a mobile inverted pendulum is considered which is a particular case of the classical inverted pendulum. For this plant, two MFC algorithms are proposed to control the position of the cart and the results are used in a comparative analysis. The MFC method uses intelligent PID (iPID) controllers which represent an improvement of the classical PID algorithm, based on the ultralocal model concept. The first algorithm is represented by an iPD controller while the second one is a conjunction between MFC and SMC which is materialized in a MF-SMC controller that compensates the estimation errors from the iPD design using an augmented control signal. The simulation results present the efficiency of MFC strategies of a positioning application for a mobile inverted pendulum.

In the field of control, inverted pendulum can be used as an experimental device to test various intelligent control theories or optimal control methods, and build a bridge between theory and practice, so it is widely concerned. From the inverted pendulum experiment, we can sum up the effective control experience, and lay the foundation for the stable control of spacecraft, robot and servo system. At the same time, as a high-order nonlinear unstable system, it is difficult to model and control it. In this paper, the mathematical model of the first-order inverted pendulum is deduced firstly, and then different methods are used to control it, and the simulation is carried out by MATLAB [2]; Finally, through the comparative study, it is found that the optimal control correction is better than the traditional control method in terms of the stability performance of the system, and the optimal control correction is easier to achieve. [3]

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Abstract This article proposes a control method for underactuated cartpole systems using semi-implicit cascaded proportional-derivative (PD) controller. The proposed controller is composed of two conventional PD controllers, which stabilizes the pole and the cart second-order dynamics respectively. The first PD controller is realized by transforming the pole dynamics into a virtual PD controller, with the coupling term exploited as the internal tracking target for the cart dynamics. Then, the second PD controller manipulates the cart dynamics to track that internal target. The solution to the internal tracking target relies on an equation set and features a semi-implicit process, which exploits the internal dynamics of the system. Besides, the design of second PD controller relies on the parameters of the first PD controller in a cascaded manner. A stability analysis approach based on Jacobian matrix is proposed and implemented for this fourth-order system. The proposed method is simple in design and intuitive to comprehend. The simulation results illustrate the superiority of proposed method compared with conventional double-loop PD controller in terms of convergence, with the theoretical conclusion of at least locally asymptotic stability.

The inverted pendulum on a cart is a challenging and widely studied control problem in the fields of control systems and robotics. To address the need for more effective control strategies, this paper presents a novel approach that combines the strengths of two powerful control techniques: Linear-Quadratic Regulator (LQR) and Takagi-Sugeno (T-S) fuzzy control. LQR control is well-established for stabilizing linear systems, but it faces limitations with nonlinear systems like the inverted pendulum. On the other hand, T-S fuzzy control excels in handling nonlinearities by approximating the system's behavior with local linear models. Our proposed approach leverages T-S fuzzy systems to approximate complex nonlinearities, while using LQR control for each local linear subsystem. The combined method's efficacy is substantiated by simulation results, specifically considering criteria such as stability under disturbance conditions, variations in the initial angle of the pendulum, and a comparative analysis between stability-only scenarios and those involving both swing-up and stability control.

Stability analysis and control of the inverted pendulum on cart system is an important problem that has been investigated by many researchers in recent years. In this study, nonlinear modeling of the inverted pendulum on cart system is derived and free body diagram is explained. Then, the nonlinear model of the system is created in MATLAB program. In order to keep the pendulum on cart in balance, different types of controllers were designed, and stability analysis was performed by drawing root-locus curves for different controllers. The optimum controller design was obtained to keep the pendulum in balance. The impulse response of the system has been simulated and it has been proven that the designed optimum controller keeps the pendulum in balance.

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This paper proposes a resilient sliding mode control (SMC) strategy for the stabilization of a cart-pendulum system, tackling significant issues in nonlinear control, including parametric uncertainties and external disturbances. The suggested solution uses a two-step process: first, an open-loop energy-based swing-up to lift the pendulum, and then a closed-loop SMC phase to keep it stable. The designed controller uses a saturation function to reduce chattering, which is different from methods that depend on linearized models or complicated gain tuning. The simulation results show that the accuracy is very high, with settling times of about 5 seconds for the pendulum angle and 7 seconds for the cart position. The controller works well even when the system mass and disturbances change by 10%, as long as the cart can only move ±0.5 m and the control forces can only be ±10 N. Stability is reached from the most unfavorable initial condition, the pendulum's downward-hanging position, with a steady-state error of under 1% in essential state variables. This work offers a computationally efficient and adaptive solution, appropriate for real-time applications in robotics and aerospace where resilience to nonlinear dynamics and uncertainty is essential.

In this paper, a pole-independent, single-input, multi-output explicit linear MPC controller is proposed to stabilize the fourth-order cart–inverted-pendulum system around the desired equilibrium points. To circumvent an obvious stability problem, a generalized prediction model is proposed that yields an MPC controller with four tuning parameters. The first two parameters, namely the horizon time and the relative cart–pendulum weight factor, are automatically adjusted to ensure a priori prescribed system gain margin and fast pendulum response while the remaining two parameters, namely the pendulum and cart velocity weight factors, are maintained as free tuning parameters. The comparison of the proposed method with some optimal control methods in the absence of disturbance input shows an obvious advantage in the average peak efficiency in favor of the proposed SIMO MPC controller at the price of slightly reduced speed efficiency. Additionally, none of the compared controllers can achieve a system gain margin greater than 1.63, while the proposed one can go beyond that limit at the price of additional degradation in the speed efficiency.

The inverted cart-pendulum system (ICPS) consists in having a pendulum mounted on a sliding cart, with the pivot point fixed. This real time experiment indeed looks like a rocket and its functionality is akin to a rocket. These are the launchers and the missile guidance and control as well as construction anti-seismic measures also. The control aim in these systems is to maintain the inverted pendulum vertically stable. The system is causal but unstable and, therefore, has no minimum phase. Therefore, the right half plane pole and zero are close to each other. Therefore, the stability of the system can be considered as problematic at some points. Unfortunately, linear time- invariant (LTI) classical controllers are incapable of offering suffient loop robustness for such systems. This paper aims to project a two-loop fractional order controller (2-LFOC) design to stabilize a higher-order nonlinear inverted cart-pendulum system (ICPS). The modeling, linearization, and control of ICPS are demonstrated in this work. The control target is adjusted so that the inverted pendulum stabilizes in its upright state when the cart reaches the required point. To fulfill the control objective, two-loop FOPID-FOPI controllers are proposed, and the Levenberg Marquardt algorithm (LMA) is utilized to tune the controllers. A novel nonlinear integral of time-associated absolute-error (ITAE) based fitness formula considering the settling time and rise time is used to fit the controller parameters for 2-LFOC. A performance comparison with the PID controller in terms of different time domain parameters such as rise_time (T R ), peak_time (T P ), settling_time (T S ), maximum overshoot (OS M ), maximum undershoot (US M ) and steady-state error (E SS ) are investigated. Stability analysis using Riemann surface observation of the system compensated with the proposed controller is presented in this work. The robust behavior of the two-loop FOPID-FOPI controller is verified by the application of disturbances in the system and the Reimann surface observation.

The Inverted Pendulum On a Cart (IPOC) system poses a challenge in control engineering due to its inherent instability, nonlinearity, and underactuation. This addresses the fundamental issues arising from its underactuated nature and introduces an approach that combines Takagi-Sugeno (T-S) fuzzy control with an awareness of real-world constraints to create a control system ensuring both stability and practicality. By aligning theoretical insights with extended considerations, the Linear Matrix Inequality (LMI)-based control design is demonstrated in a comprehensive framework. Theorems are introduced and validated, leading to the derivation of LMI conditions. The simulation results are assessed with accompanying comments to demonstrate the effectiveness of the theorems. Through this integration of T-S fuzzy control with additional considerations, the paper aims to bridge the gap between theory and practical applications, advancing the field of control engineering.

The topic of this paper is the design of two fractional order schemes, based on a state feedback for linear integer order system. In the first one of the state feedback is associated with a fractional order integral () controller. In the second structure the state feedback is associated with a fractional order proportional integral () controller. With such controllers, the closed loop system with state feedback described by the state equations splits in n‐subsystems with different fractional orders derivatives of the state variable. In order to find the optimal parameters value of both controllers () and (), a multi‐objective particle swarm optimization algorithm is used, with the integral of absolute error, the overshoot , the Buslowicz stability criterion are considered as objective functions. The multi‐objective integral fractional order controller and the multi‐objective proportional integral fractional order controller are applied to stabilize the inverted pendulum‐cart system (IP‐C), and their performance is compared to the fractional order controller. The simulation results of these innovative controllers are also compared with those obtained by conventional proportional–integral–derivative and fractional order proportional–integral–derivative controllers. The robustness of the proposed controllers against disturbances is investigated through simulation runs, considering the non‐linear model of the IP‐C system. The obtained results demonstrate that our approach not only leads to high effectiveness but also showcases remarkable robustness, supported by both simulation and experimental results.

This paper presents an innovative control strategy, the Adaptive Fast Terminal Sliding Mode Control (AFTSMC), designed for the stability control of an inverted pendulum on a cart. The proposed controller aims to stabilize the system within a finite time, leveraging the advantages of fast terminal sliding mode techniques. The system’s dynamic model is employed to derive the controller, utilizing an adaptive approach to accommodate uncertainties and disturbances. Simulations are conducted to validate the proposed AFTSMC, comparing its performance with an Adaptive Sliding Mode Controller under various scenarios. The results demonstrate the efficacy of the AFTSMC in achieving stable and precise control, making it a promising solution for the challenging dynamics of inverted pendulum systems.

The scope of inverted pendulum has been widely studied as one of the notable research with respect to standing in balance. The concept of this pendulum is similar to missile guidance, meaning that the center of drag is ahead that of gravity. Mathematical model of inverted pendulum on a cart is moreover presented in this paper. Various rewarding parameters are proposed from the displacement of the pivot, angular rotation, to external force exerted on the carriage so as to gain its equilibrium points and the linearized systems. Due to the severe risk of instability, a reliable closed-loop state feedback controller is designed to stabilize in upright position, even with large deviations. The specific concept proposed is to apply the canonical form of computing the determinant of gain $K$ leading to $K_d$. The results show that the constructed design can maintain the stability of the system by applying three sorts of initial condition and choosing sampling time $T$ under $0.2$ with small possible degrading performance.

This paper discusses the implementation of the $L_{1}$ adaptive control for the inverted pendulum system. The proposed design has two merits. First, it is able to reject the system disturbance and uncertainties. Second, it ensures the overall system stability. Both the continuous-time $L_{1}$ adaptive control and the discrete-time $L_{1}$ adaptive control are designed. Simulation results show that the discrete-time $L_{1}$ adaptive control maintains the system stability in implementation with a guaranteed performance.

The concepts of stability and balance represent many critical problems faced by engineering today. The inverted pendulum on a cart is one such non-linear, unstable, multivariate system whose goal is to determine a suitable control action given to the cart such that it stabilizes the pendulum in an upright vertical position. This paper therefore, aims to design and study a highly robust MISO control structure using Linear Quadratic Regulation, Fuzzy logic and Neural Networks called Two-Stage LQR-based-ANFIS (referred to as TS-LA) for the stabilization of Inverted Pendulums. The proposed controller is implemented on a Simulink model of the Inverted Pendulum constructed through relevant mathematical and state space modelling using Newtonian and Lagrangian mechanics. Applying external disturbances, transient parameters are obtained and are benchmarked against standard conventional controllers to perform comparative analysis and showcase its disturbance rejection capabilities.

This paper discusses the implementation of the explicit model predictive control for the inverted pendulum system. The proposed controller has three merits. First, it is able to produce the optimal control action with constraints satisfaction. Second, it reduces the online computational time by obtaining the offline solution of the optimization problem. Third, it ensures the overall system stability. The simulation results show that the proposed controller achieves an excellent performance.

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Controlling nonlinear systems, such as the inverted pendulum on a moving cart, presents a well-known challenge due to the system’s nonlinearities and highly coupled states. This paper explores the control methodology of the system by linearizing the dynamics around the pendulum’s upright position. The primary objective of this review article is to develop control strategies that can not only stabilize the system but also respond effectively to external disturbances. Although control techniques like Proportional-Integral-Derivative (PID), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC) are commonly used, this research places particular emphasis on the Linear Quadratic Gaussian (LQG) control method. LQG, known for its capacity to handle uncertainties and system noise, is analyzed in detail. MATLAB simulations are conducted to compare the performance of various control strategies, with a specific focus on LQG’s ability to ensure stability and performance under disturbance. The findings highlight LQG’s robustness in managing system uncertainties and its adaptability to changing conditions, making it a strong candidate for practical nonlinear control applications.

This research presents three major contributions to the nonlinear control of underactuated systems. First, the identification and characterization of a 2.5 Hz oscillatory phenomenon in the low-cost inverted pendulum system, addressing challenges from mechanical elasticity and electrical delays, is reported. Second, the Hierarchical Sliding Mode Control (HSMC) framework is developed to control this underactuated system considering its unwanted oscillatory phenomenon. This HSMC control system is used to achieve superior disturbance rejection, maintaining cart position within ± 0.05 m compared to PD-LQR's ± 0.35 m under pure oscillatory disturbances, while reducing energy consumption by 32%. Third, a comprehensive Hardware-in-the-Loop (HIL) implementation using the F28379D microcontroller, which provides real-time parameter adjustment capabilities, is established. The stability of the system is theoretically validated through Lyapunov analysis and homoclinic orbit characterization. Experimental results demonstrate the effectiveness of the HSMC controller in maintaining pendulum angular oscillations within ± 2.5°, significantly outperforming PD-LQR's ± 5° range under combined disturbances.

The inverted pendulum, a classical mechanical system, often serves as a platform for studying stability and control algorithms. Modelling human standing balance as an inverted pendulum controlled by the time-delayed proportional-derivative (PD) feedback controller can be used effectively to study the related biomechanical mechanisms. In this study, to investigate the human balance control strategy, an adjoint sensitivity analysis method and a corresponding optimizer are implemented to directly determine system parameters, control gains and the time delay in the human balancing model. This study validates the accuracy of the optimizer through numerical simulations and experimental verification based on the physical model of the inverted pendulum on a cart. The experimental results confirm the performance of the identification algorithm for systems involving non-smooth dynamics and inherent time delays. Moreover, the identification based on human balance data indicates that the time-delayed PD feedback controller effectively represents the human balance control strategy. Additionally, the identification reveals a tendency in the control strategy: the control gains are located in the lower-left region of the stability diagram, indicating that the human body tends to adopt an optimal control strategy that minimizes energy consumption.

The cart-type inverted pendulum is a typical fourth-order underactuated system that has been widely used in diverse industrial applications. This paper presents a new control method to handle the stability problem of a cart-type inverted pendulum. The system is divided into two second-order subsystems: an inner-layer dynamic system of the rod and an outer-layer dynamic system of the cart. Then a simultaneous planning and executing (SPAE) control method based on double-layer polynomial planning is proposed to make the inner-layer and outer-layer systems asynchronously converge in different time intervals. To achieve this asynchronous convergence, the outer-layer system adjusts the cart displacement. Its goal is to provide a smooth reference angle. The inner-layer system should track this angle quickly and precisely. In contrast to existing control approaches, the proposed method does not require parameter optimization or an accurate nonlinear model. Simulated and experimental results have verified that the proposed control method can achieve the double-layer asynchronous convergence in different time intervals, indicating its superiority over other feedback control approaches.

Over the decades, machines have evolved capabilities to make decisions on the basis of changing situations and environments. Many green technologies have come to light in fighting climate change and other. All these are due to the boom seen in intelligent and adaptive technologies. In this paper, efforts are made to study the various types of intelligent & adaptive controllers that are being employed in controlling the non-linear inverted pendulum cart mechanism. The various characters attributed by this dynamical system enables a control engineer to assess the quality of control algorithms that are used to analyze the stability of this dynamic system. Here a contrast is made between various types of intelligent & adaptive control techniques based on computational time, system parameters & robustness. This system enables the control design for various types of controllers that are presently used in many real time applications such as aerospace vehicles, submarines, ships and many more.

The inverted pendulum system (IPS) is considered the milestone of many robotic-based industries. In this paper, a new variant of variable structure adaptive fuzzy (VSAF) is used with new reduced linear quadratic regulator (RLQR) and feedforward gain for enhancing the stability of IPS. The optimal determining of VSAF parameters as well as Q and R matrices of RLQR are obtained by using a modified grey wolf optimizer with adaptive constants property via particle swarm optimization technique (GWO/PSO-AC). A comparison between the hybrid GWO/PSO-AC and classical GWO/PSO based on multi-objective function is provided to justify the superiority of the proposed technique. The IPS equipped with the hybrid GWO/PSO-AC-based controllers has minimum settling time, rise time, undershoot, and overshoot results for the two system outputs (cart position and pendulum angle). The system is subjected to robustness tests to ensure that the system can cope with small as well as significant disturbances.

The Inverted Pendulum Cart (IPC) system is a significant challenge in control theory, is used as a benchmark for evaluating advanced actuator control techniques, and has critical applications in robotics and autonomous systems. This paper proposes a new control strategy based on a Hierarchical Non-Singular Fast Terminal Sliding Mode (HNFTSM) controller technique enhanced by an Extreme Learning Machine (ELM) neural network to achieve system stability. HNFTSM provides finite time convergence and resistance to disturbances and uncertainty, while the ELM contributes to estimating these disturbances to improve performance. The stability of this strategy is proven using the Lyapunov stability theory, which ensures that all system states reach the desired equilibrium in finite time. Furthermore, the proposed hierarchical control scheme guarantees finite-time convergence of all closed loop IPC states under bounded uncertainties. A comprehensive comparative analysis is conducted against other advanced control techniques, including HSMC, HNTSM, ELM-HNTSM, and conventional NFTSM controllers. Simulation results show that the proposed approach outperforms other methods in tracking accuracy, convergence speed, singularity avoidance, and chattering reduction, which enhances the effectiveness of system control and makes it promising for practical applications.

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In this paper, a PID and PID-liked fuzzy logic controller is designed for optimal control of inverted pendulum. An inverted pendulum is highly a nonlinear and unstable system. Thus, in this paper, the system is modeled, linearized and controlled. The control objective is to keep the pendulum in an upright position despite external disturbances. The stability analysis of the system is performed, which proves the instability of the system. Two separate controllers were developed to cater for the nonlinear behavior of the system. In the PID controller, the PID gains were fine-tuned through trial and error in MATLAB to obtain optimal control of the system. On the other hand, fuzzy logic is used to tune the PID in the PID-like fuzzy logic controller. Control performances of the system by PID and PID-like fuzzy control method were compared. The result indicated PID-like fuzzy control method performs better in respect of rise time, settling time, peak overshoot and steady-state error.

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.

This paper deals with the stabilization of an inverted pendulum on cart; the latter is pneumatically actuated by a double acting cylinder controlled by low cost proportional valves. In particular, a numerical model of whole system is developed in order to find the ability of the pneumatic actuation in stabilizing the pendulum and evaluate its bandwidth. A cascade of two control loops (the inner one for the pendulum angle, the outer one for the cart displacement) are analyzed and proper compensators are defined. The possibility of introducing an additional loop to control the force exerted by the actuator on the cart is evaluated.

Controlling of an Inverted Pendulum is a classical control problem and widely referred in the literature. The system is inherently an unstable nonlinear system. The aim of this project is to design a linear controller for stabilizing the inverted pendulum at its unstable equilibrium point. This work includes detailed mathematical modeling and analysis of this physical system and proposes an efficient design for ensuring stabilizing control. The model with and without controller has been be analyzed theoretically and validated using simulations. A complete developed physical system is implemented using the proposed design to ensure stability of the system. PID based controller has been used in this work with all PID parameter being obtained via proper design process. The PID controllers is tested and implemented on the actual physical hardware as well. National Instruments' MyRIO is used to interface the hardware with LabView.

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Stabilization and tracking problems for cart inverted pendulums under disturbances and uncertainties have posed significant challenges for control engineers. While various controllers have been designed for an inverted pendulum, they often overlook the calibration error of the pendulum angle in practical implementations, which degrades the control performance. Incorrect calibration of the pendulum angle in upright equilibrium position generates an offset of cart position errors. To solve this problem, an augmented model comprising integral cart position errors was first constructed. Afterwards, a sliding mode control was designed for this system based on a linear quadratic controller, to facilitate implementation. Additionally, a stepper motor was employed in the inverted pendulum to enhance the control performance and widen applicability in industrial settings. The effectiveness and performance of the proposed controller were validated by means of experimental studies, focusing on stabilization control and tracking control of a cart inverted pendulum actuated by a stepper motor.

An inverted pendulum is a challenging underactuated system characterized by nonlinear behavior. Defining an effective control strategy for such a system is challenging. This paper presents an overview of the IP control system augmented by a comparative analysis of multiple control strategies. Linear techniques such as linear quadratic regulators (LQR) and progressing to nonlinear methods such as Sliding Mode Control (SMC) and back-stepping (BS), as well as artificial intelligence (AI) methods such as Fuzzy Logic Controllers (FLC) and SMC based Neural Networks (SMCNN). These strategies are studied and analyzed based on multiple parameters. Nonlinear techniques and AI-based approaches play key roles in mitigating IP nonlinearity and stabilizing its unbalanced form. The aforementioned algorithms are simulated and compared by conducting a comprehensive literature study. The results demonstrate that the SMCNN controller outperforms the LQR, SMC, FLC, and BS in terms of settling time, overshoot, and steady-state error. Furthermore, SMCNN exhibit superior performance for IP systems, albeit with a complexity trade-off compared to other techniques. This comparative analysis sheds light on the complexity involved in controlling the IP while also providing insights into the optimal performance achieved by the SMCNN controller and the potential of neural network for inverted pendulum stabilization.

In this paper an integral linear quadratic regulator (ILQR) is proposed for balancing the inverted pendulum (IP) system. Dynamic model of the IP system, using the acceleration of the cart as control input is derived. Subsequently, the dynamic model is linearized at operation region to obtain a linear model for the controller design. This model is further enhanced by incorporating an integral term of the cart position to formulate augmented model for the ILQR control. The control input of the ILQR, essential for balancing the IP and keeping the cart at the desired position, is determined by solving the Riccati equation. Simulation and experimental results are presented to evaluate the effectiveness of the proposed control in balancing the IP while successfully reaching the desired cart position.

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The inverted pendulum cart system (IPCS) be similar to number of real-world systems such as transport vehicles, spacecraft and medical fields equipment’s robots and many more, however it is a traditional problem for the control engineers as it is highly non-linear with more than one equilibrium points. Therefore, the paper investigates a fuzzy logic controller to control cart position while maintaining the balance of an inverted pendulum. Furthermore, the length of inverted pendulum is varied and the performance is evaluated under both controllers’ application simultaneously. The performances of both controllers are compared with number of error indices such as Root mean square error (RMSE), Integral square error (ISE), Integral time square error (ITSE), Integral absolute error (IAE) and Integral time absolute error (ITAE). The fuzzy controlled system shows better results as compared with the classical control approach with PID controller in terms of settling time, rise time, overshoot and undershoot under MATLAB/Simulink environment.

Linear Quadratic Regulator control is one of the most prolific methods used to control a linear system and has found widespread applications. The LQR employs a quadratic cost function, incorporating quadratic terms for state and control variables. The behavior of the controller is shaped by weighting matrices. Despite Linear Quadratic Regulator's (LQR) strong performance and solid resilience, developing these controllers have been challenging, largely because there is no reliable way for choosing the Q and R weighting matrices. A deterministic method is used for choosing them in this paper, providing the designers precise control over performance variables. The method is based on the Newton-Raphson method. The stabilization of inverted pendulum system is used to validate this approach. The inverted pendulum is a classic control engineering problem that has been studied for many years. It is a challenging task to stabilize the pendulum in the upright position, as it is an under-actuated system with non-linear dynamics. The deterministic method is then compared with an Artificial Bee Colony (ABC) optimized LQR controller. Based on MATLAB simulations, the proposed method demonstrates the ability to stabilize the inverted pendulum in the upright position with high accuracy and is found to have better stabilizing characteristics over the ABC optimized controller.

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The paper proposes a modern controller i.e. combination of Linear quadratic regulator (LQR) and Proportional, integral and derivative (PID) to control a highly non-linear inverted pendulum cart system. The detailed mathematical modeling of the inverted pendulum cart system as well as the modern controller are discussed in this paper. The performance of the proposed controller is compared with the conventional PID controller. The comparative results are shown using several simulated waveforms as well as in terms of time response specifications such as rise time, settling time, and peak overshoot and undershoot under MATLAB/Simulated environment.

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Inverted Pendulum (IP) is an inherently unstable and nonlinear system which requires a control system to remain balanced in the upright position. In this study the stabilization in an inverted position of a pendulum setup is accomplished through multiple PID controllers and the setup was extended to introduce PID based Disturbance Observer (DOB) control approach. Mathematical model of the IP is developed using force analysis followed by a state space approach to linearize the model. There are three PIDs introduced in order to control the pendulum angle, cart linear position and current control of the DC motor. PID based DOB control algorithm is implemented in a single board computer and the feedback from the systems are monitored through the data acquisition from encoders along with current sensing module where data will be allowing estimation of external disturbance and compensating by feed-forward to the system. Simulations were carried out under multiple designs and results were verified experimentally using the developed platform.

In the mechanical system control, under-actuation, meaning the situation when the number of actuators is less than the number of position variables, is commonly seen; whereas full state feedback or full position feedback is often utilized, implying that the number of sensors used is at least equal to the number of position variables. What if less sensors are to be used? In such a situation, we say the system is under-sensed. To control an under-sensed system, we need to know where to place the limited number of sensors for the ease of control and how to design the controller effectively under incomplete position information. This paper gives a case study on controlling an under-sensed and under-actuated system. An under-sensed and under-actuated linear (USUAL) inverted pendulum, which has only one position sensor and one force actuator, is investigated. For this USUAL inverted pendulum, we first address the sensor placement problem by analyzing the dependence of system zeros on the sensor location. After a reasonable sensor location is chosen, we then study the issue of controller design. The optimally robust stabilizing controller, which minimizes $H$∞ norm of the Gang of Four transfer matrix, is shown to be effective.

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To research the tendencies of immediate change in the scheduled parameters, a linear parameter–varying control strategy is made in real time for lab-based inverted pendulum system. An adjusted and observing virtual instrument is designed in LabVIEW to construct this mechanism. To obtain the transient response data, a step response is transmitted by virtual instrument to the inverted pendulum by which model of inverted pendulum is identified. System model depicts its characteristics, so this model is employed in designing controller. To improve the performance and robustness, linear parameter–varying controller is created to deal with parameter distinctions. Furthermore, the effectiveness of this designed robust controller is checked by integrating on hardware system. The discovered model is checked by evaluating the presented approaches in real time and simulation outcomes with traditional controller as proportional–integral–derivative controller and linear–quadratic regulator.

This work addresses the finite-time control problem for a class of nonlinear inverted pendulum systems. A series of homogeneous controllers, which are capable of guaranteeing the locally finite-time stability for the closed-loop systems, are first developed using the adding one power integrator method and backstespping technique. Subsequently, the nested saturation control approach has been further proposed in order to achieve global finite-time stability after appropriately adjusting the saturation level. Furthermore, the simulation results are given to validate the effectiveness of the proposed strategy.

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Using Hybrid optimization algorithms for nonlinear systems analysis is a novel approach. It is a powerful technique that uses the exploitation ability of one algorithm and the exploration ability of another algorithm, to find the best solution. Literature survey reveals that hybrid algorithms not only show quality response but also give faster convergence of error for nonlinear systems. In this paper, hybrid optimization techniques based proportional integral derivative (PID) controller is used for benchmark problems: Continuous stirred tank reactor (CSTR), Inverted pendulum and blood glucose system. Two recent hybrid algorithms: Particle swarm optimization‐Gravitational search algorithm (PSO‐GSA) and Particle swarm optimization‐Grey wolf algorithm (PSO‐GWO) are implemented to control the temperature and concentration of CSTR, pendulum angle of inverted pendulum, glucose concentration and insulin level of blood glucose system. In PID and PSOGWO algorithms, the exploration abilities of GSA and GWO combined with the exploitation ability of PSO have been used. The performance of these algorithms is then compared with individual PSO, GSA, and GWO algorithms proving their superiority. Stability is ensured using the Lyapunov approach while the robustness of the systems is checked using the parameter perturbation technique. Simulation results show substantial improvement in the performance of these systems by using these meta‐heuristic hybrid optimization techniques. A comparative analysis of these algorithms has also been done.

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We present the computational stability analysis and the domain of attraction estimation of a non-polynomial system using dynamical embedding. The Lyapunov function is searched in a general quadratic form of rational terms in the embedding state space. To ensure that the Lyapunov conditions are satisfied, sufficient polytopic linear matrix inequalities are formulated using Finsler’s lemma and affine annihilators. A compact forward invariant region is finally given as an appropriate level set of the Lyapunov function in the original system of coordinates. The concepts are demonstrated through a dynamically extended rational fourth-order model of the one degree-of-freedom inverted pendulum system.

In this paper, two mathematical models of the primary inverted pendulum are established based on the Simulink and the Simscape toolboxes of the MATLAB respectively. The inverted pendulum changes from the unstable state to the stable state after performing the state feedback control and the PID control method. Under these two control methods, the overshoot of the Simulink model is 57% and 80% smaller, and the adjustment time is 25% and 20% larger than that of the Simscape model. The control effect of the PID method in the two models is better than that of the state feedback method. The adjustment time of the pendulum rod is 40% and 37.5% smaller, and the overshoot is 82% and 61% smaller than that of state feedback control, respectively.

Controlling of a cart inverted pendulum often becomes learning material for a control system. Several researchers have carried out about stabilization of inverted pendulum using numerous methodologies. However, the stabilization of the pendulum has just been conducted in the inverted area. The aim of this research was designing a control system to raise the pendulum from the lower area to the inverted area using Linear Quadratic Regulator (LQR) and Partial Feedback Linearization control in the Simulink MATLAB. Furthermore, the design was given a noise so that Linear Quadratic Gaussian (LQG) control was needed to make this system more robust. The model of cart pendulum with LQR control alone was unable to raise the pendulum from 0 degree to reach the angle at 180 degrees in an inverted area although the gain K and the voltage of DC motor were increased. For this reason, the Partial Feedback Linearization method was added to raise the pendulum by controlling the energy and slide mode on the cart. The strengthening of the system with the LQG design was carried out to reduce the noise signal that occurs in the DC Motor and sensor. The design was implemented in the simulation using Simulink MATLAB. It shows that the design of LQR and Partial Feedback Linearization control can make the pendulum stable at 180 degrees in 5.2 seconds and the cart is stable at 0 cm in 4.5 seconds. Whereas, the design of LQG control can reduce the noise well in the system.

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.

This paper presents the methodology for implementing the Exact Feedback Linearization (EFL) technique in an inverted pendulum coupled to a DC motor. This technique enables the control of a nonlinear system while preserving the original dynamics, in contrast to conventional linearization methods. A nonlinear feedback control is proposed that allows a smooth switch, which would protect the DC motor from mechanical damage caused by sudden rotation changes. A comparison is made with the LQR control to evaluate the performance of the proposed control. The steady state RMS errors are calculated. System dynamics are simulated using Matlab software.

To control the nonlinear inverted pendulum system, PID and LQR control techniques have been proposed. This paper described the method for problem of swing up control and stabilize the pendulum system. Controlling of angle of pendulum has been done with PID and LQR is implemented to stabilize the pendulum at upright(vertical) position. The optimal matrices of LQR obtained using GA. These matrices helped to find the feedback gain of the system with performance index (PI). In this paper time response specification has been noted down considering with these proposed controllers. And showed that these controllers stabilize this nonlinear system very quickly and smoothly. Some time response parameters have been improved in this paper. The simulation results are done with MATLAB Simulink.

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An inverted pendulum on cart is an object which is a nonlinear, unstable system, is used as a standard for designing the control methods and finds most versatile application in the field of control theory. To achieve the stabilization of a inverted pendulum system pole placement method is used to design a state feedback controller and then optimal linear quadratic regulator has been applied to the system. The results shown that are the comparison between the performance of two controllers, first the state feedback controller, which is designed by pole placement method and the optimal LQR controller. Both controller stabilizes the inverted pendulum system but deviates in their performance. Simulation has been carried out to show two different approaches for comparative study of their performance.

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The paper presents swinging up and stabilization of inverted pendulum cart system (IPCS) and analysis of applied controller via root locus method. The IPCS is highly non-linear system which is analysed by the response after application of impulse input to the system. The controller which has combined feature of linear quadratic regulator (LQR) and conventional PID is proposed to handle the system. The root locus are drawn with and without application of the conventional PID and proposed controllers. It reveals that after application of controller, the roots shift towards left hand side (LHS) which are shown via root locus method. Initially, the system is merely controlled with conventional controller and then after LQR is incorporated in the feedback to control the system. The time specifications result are calculated and compared. The results reveal that the controller which is composed with PID and LQR shows better results as compared to the merely application of PID controller. The disturbance effect is also presented to show performance of the proposed controller. A number of simulated waveforms e.g. cart position and pendulum angle response are presented to show controller’s performance.

The aim of the paper is to present an inverted pendulum system that operates in two different modes. Firstly, it operates in a static balancing mode, during which the controller tries to keep the pendulum balanced during which the controller tries to keep the pendulum balanced and maintain the current position of the pendulum carriage. Secondly, it also operates in a velocity control mode, so that the carriage can move at a selective velocity while simultaneously maintaining pendulum balance. In order to realize these two modes of control we implement a state feedback controller and schedule gain depending on the selected mode of operation of the system. We first describe the design and construction of the system. We then perform state space analysis, build state feedback controllers designed as linear quadratic regulators (LQR), and run tests to examine the operation of the system whilst subjecting the pendulum to impulsive disturbances. In particular, we investigate differences in control behavior in static position mode and in velocity-controlled mode. We present the experimental results and discuss their implications.

Inverted Pendulum is used as benchmark test in Control System Engineering. As a result, controller which shows robust performance on Inverted Pendulum should be robust for most non-linear systems. This paper presents a study of control schemes like PID, FLC and ANFIS for Inverted Pendulum. Comparative analysis of these modern control schemes on Full State Feedback (FSF) model of Inverted Pendulum is carried out to first stabilize the unstable Inverted Pendulum using LQR and then choose the satisfactory results of controllers on the basis of parameters like overshoot, rise time, settling time, control input required, disturbance rejection and reference tracking etc. MATLAB Simulation of Inverted Pendulum has been presented to compare the behavior and suitability of these control schemes.

In this paper, a problem of inverted pendulum stabilization considering constant input delay has been investigated. A state feedback controller using a back-stepping method has been designed, it was recently created for parabolic PDEs, moreover the unstable PDE is characterized by integral transformation and boundary feedback. A time-delay system which is unstable is morphed into a stable system which further converges to zero in a finite period of time. In back-stepping controller it needs all the past input, and using this method the infinite dimensional delayed system can be represented by a cascade ODE-PDE equation, where the boundary of the PDE is defined by the constant delay. These ideas makes the input delay problem more interesting to handle. Moreover, we compare the optimal state feedback based stabilizing controller (LQR) with and without back-stepping approach for stabilization of inverted pendulum on cart, and finally, the results are depicted in a MATLAB environment

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This paper presents a study on the design of arbitrary-order controllers for controlling an inverted pendulum system in two distinct targets: the angle of rotation of the pendulum and the position of the cart. A systematic and methodological approach, using cascade control, is used to tackle the non-linearity of the system. The behavior of the arbitrary-order integral and derivative parts of each controller is approximated through the Curve-Fitting approximation method. Numerical simulations were conducted within a Simulink/Matlab environment to demonstrate the improved performance of the transient response of the nonlinear system when it is compared to the traditional integer-order controllers.

The education of control theory requires experimental verification and interesting examples to keep the students focused during classes. In this paper a tiny two-wheeled inverted pendulum robot is presented for this purpose. The prototyped robot can be used for education related to microcontroller programming, discrete control systems, signal filtering, PID controller, wind-up phenomena, and cascade control structure. The paper presents the simplicity of the construction, firmware, and control application of a two-wheeled inverted pendulum. The possibilities are presented using the well-known cascade control structure with PID controllers. Moreover, the Madgwick's algorithm has been implemented to provide satisfactory tilt angle measurement using an IMU sensor. The results presented show that the prototyped construction of a tiny two-wheeled inverted pendulum robot may be used successfully for education purposes. The project has been published using the CC BY-NC license on GitHub repository.

Inverted pendulum systems are used in the academic world due to their numerous applications in industrial and control engineering that are used as a benchmark, e.g., modern robots and others. These systems investigate several challenges, such as stability analysis and control design, focusing on robust stability and performance when considering polytopic uncertainties. This paper proposes a robust control methodology considering polytopic uncertainties in an inverted pendulum tracking problem. The proposed method combines the cascade control structure and the LMI approach with polynomial uncertainty theory. The proposed methodology consists of the inner loop design based on LMI theory. Then, the outer control loop is based on Kharitonov’s theorem to ensure robust stability and performance under parametric uncertainties. The proposed methodology is compared with the other three classic control techniques: LQR, PID+LQR, and robust tracking and disturbance rejection (RTDR)). Many simulations and experimental tests showed that the proposed methodology outperforms the other classic approaches when the system is subject to parametric variations (i.e., rod length and mass variations), ratifying the proposed method’s robustness, effectiveness, and accuracy.

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This paper explores the theoretical and experimental implementation of proportional-integral-derivative (PID) and sliding mode control (SMC) on an inverted pendulum system, a well-known problem in control engineering that is inherently unstable and highly nonlinear. The primary objective of this study is to evaluate and compare the effectiveness of these two control strategies in achieving system stabilization and robustness against disturbances. The PID controller, widely utilized due to its straightforward design and implementation, is developed based on the linearized model of the inverted pendulum. On the other hand, the SMC technique, known for its robustness to system uncertainties and external disturbances, is employed to tackle the nonlinear nature of the system. The controllers are tested in both simulation and real-time experimental environments to ensure the reliability of the findings. The results from the experiments indicate that while the PID controller performs adequately under nominal conditions, it struggles to maintain stability when faced with parameter variations and external perturbations. In contrast, the SMC exhibits superior performance by consistently stabilizing the pendulum even under adverse conditions, demonstrating its robustness and effectiveness in managing nonlinear systems. This comparative analysis provides valuable insights into the practical applications of PID and SMC, highlighting the trade-offs between simplicity and robustness in control system design.

Stabilizing inverted pendulum systems remains a challenging and open control problem due to their inherent instability and relevance in a wide range of real-world applications, including robotics and aerospace systems. While PID and fractional-order PID (FOPID) controllers offer distinct advantages, they individually suffer from trade-offs between performance and control energy. This paper presents the design, implementation, and experimental validation of a switched SW FOPID-PID controller for the stabilization of an inverted pendulum (InvP) system, aiming to achieve an improved balance between system performance and control energy used. The controller was tuned offline using particle swarm optimization (PSO) and a mathematical model of the system for simulation. Additional PID and FOPID controllers were also designed, tuned and validated for comparison purposes. Their performance was assessed through key indicators, including ITAE, ISI, settling time, peak values, and variance and compared against a manufacturer-provided PID controller. The experimental results demonstrated that all three designed controllers outperformed the manufacturer’s PID under nominal conditions. The SW FOPID-PID controller achieved the best overall performance, balancing control energy efficiency and response quality. Under external disturbances, the FOPID and SW FOPID-PID controllers exhibited superior robustness, with the switched controller being the most effective, responding quickly to disturbances while minimizing positional and angular errors. Still, this research is limited to a specific plant and switching strategy; thus, further validation on other systems and switching criteria is necessary to generalize these findings.

This paper proposes a PID controller optimized by the HO algorithm for stabilizing the cart displacement and pendulum angle of a first-stage linear inverted pendulum. The controller combines the HO algorithm with a cyclic perturbation strategy to enhance the ability of global search. Unlike the traditional PID control that relies on expert experience and experimental data to find the optimal parameters, HO can adjust the PID controller gain to find the optimal parameters, which effectively solves the problem of local convergence and lack of efficiency of the traditional optimization algorithm. Simulation experiments are completed in a MATLAB/Simulink environment, including the design of a multi-objective fitness function that comprehensively evaluates the overshooting amount, stabilization time, in which the HO-PID controller is compared with the traditional PID controller. The simulation results show that the parameters found in HO are effective in smoothing the cart displacement and pendulum angle.

Manual tuning of Proportional-Integral-Derivative (PID) controller is time consuming, tedious and generally lead to poor performance. PID controller is a simple and intuitive feedback-based control mechanism being useful to track set points and to reject disturbances. This paper presents a computational Intelligence (CI) technique of Whale Optimization Algorithm (WOA) for tuning the PID controller parameters for inverted pendulum. The inverted pendulum is a benchmark problem in control system design because of its high level of instability and non-linearity. It can mimic the behaviour of open-loop unstable systems such as robot manipulator and missile launcher. The metaheuristic algorithm achieved rise time (0.0913), settling time (1.57), maximum overshot (7.9%) and steady-state error (0.0294). Simulation results demonstrated that the WOA-optimized PID controller can provide an improved closed-loop performance over the Ziegler- Nichols tuned PID controller Parameters. The computational intelligence approach also has superior features, such as easy implementation, stable convergence characteristic and good computational efficiency over the conventional methods of Ziegler- Nichols.

The inverted pendulum system is a typical control problem for comparing advanced control approaches because of its unstable and nonlinear characteristics. Balancing the kind of system described is fundamental to many branches of engineering such as robotics, aeronautical systems, and automobile dynamics. In this work, the comparative ability of Self-Tuning Fuzzy PID and the traditional PID to stabilize an inverted pendulum system is investigated. The main issue resolved is the performance constraint of fixed-gain PID controllers in dynamic operating conditions. For this purpose, the Self-Tuning Fuzzy PID controller uses a fuzzy inference system to adaptively vary PID gains in real time, which supports better adaptability and disturbance rejection. In contrast, the traditional PID controller with fixed gain parameters renders suboptimal output when the operating conditions change. Simulations carried out in the Simulink platform compare the two control schemes in terms of stability parameters such as overshoot, settling time, rise time, and steady-state error. Outcomes show that the traditional PID controller has a rise time of 1.2s, an overshoot of 20%, and a settling time of 5s, resulting in slower stabilization and more oscillations. The Self-Tuning Fuzzy PID controller greatly enhances transient response, minimizing rise time to 0.8s, overshoot to 5%, and settling time to 2.5s, thereby stabilizing faster and with better robustness.

Inverted pendulum is a classic control problem where it often used a benchmark for research in control theory and robotics. Tuning linear controllers such as the Proportional-Integral-Derivative (PID) and Linear Quadratic Regulator (LQR) for this system is challenging. While this is commonly addressed with computationally intensive metaheuristic algorithms, this work explores the effectiveness of simpler, classical optimization techniques as a viable alternative. This paper presents a systematic comparison of three classical methods: gradient descent, simplex search, and pattern search for tuning both PID and LQR controllers. A model of an inverted pendulum system, consisting of a cart, pendulum, and DC motor, is developed and simulated in Simulink. The system is then simulated in Simulink using two common control strategies: a PID/PD controller and an LQR controller. The parameters for both controllers are subsequently optimized using three classical methods: gradient descent, simplex search, and pattern search and their performances are compared. The simulation results demonstrate that the LQR controller tuned with the pattern search method provides the best overall performance, achieving a maximum peak of 0.0125 radians and settling time of 0.7572s. To validate these findings, the optimal LQR controller was implemented on the physical inverted pendulum hardware. The hardware implementation demonstrates a performance that closely matches the simulation results, confirming the accuracy of the system model and the effectiveness of the automated tuning process.

The inverted pendulum system is a widely studied benchmark problem in control engineering due to its inherent nonlinearity and instability. In this work, a nonlinear multibody model of the inverted pendulum is developed and analyzed to capture the complex dynamics of the system. A cascade Proportional-Derivative (PD) control scheme is implemented, with the dual objectives of controlling the pendulum angle and the cart position through nested control loops. The optimal tuning of PD gains, a critical factor for robust performance, is performed using Genetic Algorithm optimization technique. The proposed approach is implemented and simulated in MATLAB Simulink with Simscape to evaluate its effectiveness. Simulation results demonstrate the successful control of both pendulum angle and the cart position. The study highlights the success of cascade PD control and superior performance of the GA tuning approach over conventional methods, emphasizing its efficacy in handling nonlinear and dynamic control challenges.

The operation of an inverted pendulum and its respective type of control are affected by the change of the values of its internal parameters. Changes with high uncertainty result in responses with undesirable outputs. In this work, a comparison is presented for the control of an inverted pendulum to determine the operation and characteristics of three types of control systems: Neuro-Fuzzy Control (NFC), Indirect Adaptive Control (IAC) and a Proportional Integral Derivative control (PID). The study considers several indices such as stabilization time, rise time, mean square error, overshoots, convergence, computational load, error, mathematical requirements, and performance indices for control systems. To demonstrate its operation, the controls are implemented in hardware, one for the NFC and another for the IAC under an Arduino UNO platform. The results indicate that the NFC and IAC controls do not generate a transient or impulse response, only a small delay and the rise and stabilization time are minimal. While PID presents a transient response and overshoot, as well as a stabilization time to reach the steady state response. •The characteristics and operation of an NFC, IAN, and a PID were analyzed using an inverted pendulum. •The prototype was built to carry out experimental tests with the controllers. •For the controllers, the initial requirements, implementation results (hardware), and efficiency are described. •The characteristics and operation of an NFC, IAN, and a PID were analyzed using an inverted pendulum. •The prototype was built to carry out experimental tests with the controllers. •For the controllers, the initial requirements, implementation results (hardware), and efficiency are described.

In electromechanical systems, backlash in gear trains can lead to a degradation in control performance. We propose a drive–anti-drive mechanism to address this issue. It consists of two DC motors that operate in opposite directions. One motor acts as the drive, while the other serves as the anti-drive to compensate for the backlash. This work focuses on switching between the drive and anti-drive motors, controlled by a switched-mode PID controller. Simulation results on an inverted pendulum demonstrate that the proposed scheme effectively compensates for backlash, improving position accuracy and control. This switched controller approach enhances the performance of electromechanical systems, particularly where gear backlash poses challenges to closed-loop performance.

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.

The inverted pendulum is a classic nonlinear unstable system used in control theory for testing control methods. In this article a nonlinear and a linearized model is obtained to design a PID and LQR controller. The MATLAB®/Simulink® software is used for linear and nonlinear model simulation and controller design. WEBOTS software is used for implementation and for comparative analysis between the controllers performance using the standard deviation of the tilt angle and cart position, and the RMS value of the force applied to the cart as performance indices. To validate the robustness of each controller the system is tested against disturbances and reference changes in the cart position. The results of this study show that the LQR controller has minimum steady-state error and a smooth control signal. Nonetheless, according to the performance indices, the PID controller performs better than the LQR, with a position standard deviation of 0.0903m, angle standard deviation of 0.0120 rad and RMS value of the control signal of 1.1007N.

In this work, a simple cascade control structure (CCS) with filter is used to control the cart inverted pendulum system. As the CCS comprises of Secondary loop and primary loop, the secondary loop is designed to counteract disturbances affecting the pendulum and maintain its upright position, while the primary loop aims to achieve precise setpoint tracking by adjusting the cart’s position. The controllers of both the loops are designed based on model matching techniques in the low-frequency range and positioning a pole within the characteristic equation. The only tuning parameter is chosen based on maximum sensitivity of the system. Performance is assessed through simulations and comparisons with recent literature. Results indicate a significant improvement in servo tracking and regulatory performance for the cart-inverted pendulum system under both nominal and noisy conditions when compared with the available literature.

In midst of new development advance of controller in control theory studies, classical proportional-integral-derivative (PID) controller and it combinations are considered the most applicable and widely used controller. PID controller is a simple yet effective controller to applied in linear systems. It is what makes it used in 95% of control loops process in industries. Although its linearity is what makes it harder to implement when the system becomes more complex, non-linear application of several variable to consider. Other types of control system can handle it with completely different ways using state space model of the system. But a more quick method is by develop a second loop of PID control. This paper focus on performance comparison for stabilize gravitational pendulum on a cart using bang-bang control, single PID Control, and cascade PID control. The system parameters such as rise time, steady time, overshoot and distance traveled will also measured. The result shows the significance of PID control for sistem stablization over simple on/off control or bang-bang control. While conventional single PID controller does the job more efficiently, in real application cascade PID controller shown to be more realistic to consider for system limit and capability.

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The Proportional, Integral, Derivative (PID) controller finds widespread use in various control applications, yet accurately tuning it can prove challenging and time-consuming, especially for intricate systems characterized by non-linearities and uncertainties. To address these challenges, an approach rooted in swarm intelligence-inspired optimization techniques, drawn from honeybee foraging behavior, is proposed for PID tuning, specifically employing the Artificial Bee Colony (ABC) algorithm. This algorithm is tailored for the control of an inverted pendulum and cart system. The ABC algorithm offers an efficient and automated avenue to navigate the parameter space and dynamically adjust PID gains according to performance criteria. Within this study, the inverted pendulum and cart model serves as a testing ground to evaluate the effectiveness of the proposed ABC-based PID tuning algorithm. The intricate presence of non-linearities and uncertainties in the system renders achieving precise control a formidable task. By harnessing the potential of the ABC algorithm, the objective is to uncover optimal PID gains that augment control performance, stability, and robustness amidst these complexities. The outcomes underscore the algorithm’s adeptness in exploring the parameter space, tailoring gains based on performance feedback, and adeptly mitigating the system’s non-linearities and uncertainties. The proposed approach bears multiple advantages, including diminished reliance on manual trial-and-error tuning, heightened automation, and improved control performance. By capitalizing on the strengths inherent to the ABC algorithm, engineers can streamline the PID tuning process and realize enhanced control outcomes for intricate systems.

This paper considers the problem of inverted pendulum stabilization and performance improvement by using PID regulation. The dynamical system model is derived in the framework of the Lagrange - Euler equations and the model parameters are evaluated by using specific identification procedures. The parameter tuning of the PID controller is based on the method of dominant pole selection. Different simulations are performed, showing the influence of the third pole placement on the closed-loop system performance.

This presentation is on studies of application of combined constant rate reaching law and proportional-integral-derivative (PID) control law (EPID) for the control of inverted pendulum system. The inverted pendulum system is similar to a typical attitude control of booster rocket undergoing the takeoff process. Recent studies indicated records of higher rates of accidents in aircrafts. The records show that about half results due to malfunctions of aircraft systems and close to one-third from propulsion system malfunctions. Others are higher complexities of modern aircraft systems and in trying to reduce cost of maintenance. Therefore, the need enhanced automation, fault detection, faults isolation, faults tolerance, faults diagnosis and faults correction. Linear control techniques may not yield the desired performances in aircraft systems due to high levels of system nonlinearities. Applications of the intelligent control counterparts may not guarantee the generation of mathematical model for in-depth analysis. The major demerit of nonlinear control methods is higher requirement of computational burden making practical implementation difficult. The model of the inverted pendulum system is a linearized analytical model. The system performance of the system was observed with the EPID and PID controllers. Furthermore the effect of sudden changes such as wind, gust or other related variations on the system was also studied using step disturbance. It was a simulation studies using MATLAB/SIMULINK software. Results showed that with the EPID a near zero deviation was achieved. Whereas with PID controller was able to only maintain a deviation of about 40. Results also indicated a near zero disturbance rejection ability with the EPID, while the PID was able to only suppress the disturbance to some extent. It implies a more robust control with the EPID was achieved for aircraft/inverted pendulum control. Hence, it implies enhanced performance and with further improvement it can be used for application in this class of systems as well as similar systems. The work would help to for basic researches in aircraft/inverted pendulum control for beginners as well as experienced researchers in the field.

This study explores the application of deep reinforcement learning (DRL) to solve the control problem of a single swing-up inverted pendulum. The primary focus is on investigating the impact of discount factor parameterization within the DRL framework. Specifically, the Deep Deterministic Policy Gradient (DDPG) algorithm is employed due to its effectiveness in handling continuous action spaces. A range of discount factor values is tested to evaluate their influence on training performance and stability. The results indicate that a discount factor of 0.99 yields the best overall performance, enabling the DDPG agent to successfully learn a stable swing-up strategy and maximize cumulative rewards. These findings highlight the critical role of the discount factor in DRL-based control systems and offer insights for optimizing learning performance in similar nonlinear control problems.

Abstract This paper presents a new design and implementation of an adaptive swing-up control algorithm for a real inverted pendulum system. The main core of the control algorithm is a sliding mode technique with the Lyapunov stability method. The goal of the proposed adaptive swing-up controller is to get the optimal force control action for the inverted pendulum in the real-time in order to precisely and quickly swing the pendulum up to the inverted position. An on-line auto-tuning hybrid intelligent algorithm based on Culture-Bees algorithms is carried out as a stable and robust algorithm to obtain and adjust the control parameters for the proposed controller. To eliminate the chattering effect of the fast switching surface, the sigmoid function is used as a Signum function for reducing the amplitude of the output function. The numerical simulation results in MATLAB and the experimental work in LabVIEW illustrate the improved performance of the adaptive swing-up controller in terms of robust performance and adaptation effectiveness that minimized the pendulum angle error to a zero value and obtained the best force control action for the pendulum cart, in addition to reducing the fitness evaluation number. These results were confirmed by a comparative study with different nonlinear controller types.

The inverted pendulum, is a classical experiment widely used as a benchmark for research in control systems, due to its challenging dynamics. In this paper, Deep Reinforcement Learning is used to control a real inverted pendulum on a cart. The Soft Actor Critic algorithm with automatic entropy tuning is used to train an agent capable of acting as a controller. The agent is trained on real data collected on an episodic basis and learns to carry out the swing up control task successfully.

An inverted pendulum is a high non-linear, chaotic and dynamically complex system, which presents problems for traditional controllers that require feedback loops and a precise dynamic model of the system. Reinforcement learning is an promising approach, since it does not need the dynamic model and generates autonomous actions based on experience. However, solving a control problem with reinforcement learning is challenging, because every dynamic system has a continuous state space. In this paper, an algorithm that uses Q-learning with function approximation is proposed to control an inverted pendulum. The algorithm consists of two stages, one for swing up, and another for the control at upright position. Results show that the proposed approach reaches the control objectives.

Electrohydraulic actuators (EHAs) are critical in applications demanding compact designs, high power density, and precise motion control. However, their nonlinear dynamics and inherent uncertainties pose considerable control challenges. This study develops a reinforcement learning (RL)-based control framework, integrating an actor–critic algorithm [deep deterministic policy gradient (DDPG)] and a novel reward-shaping method to address these challenges without requiring prior expert knowledge. The proposed approach is validated on a heavy-duty EHA-driven inverted pendulum testbench, as a benchmark system for nonlinear and unstable dynamics. Experimental results demonstrate the RL controller’s effectiveness in achieving the dual-control goal: swing-up and balancing of the EHA-driven pendulum. Furthermore, a comparative analysis with the classical linear quadratic regulator (LQR) highlights the strengths and limitations between RL-based control and model-based control. This study serves as fundamental research, offering practical and theoretical contributions to the application of RL in advanced motion control for EHAs and other complex systems.

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This paper presents the modeling, simulation, and experimental implementation of an inverted pendulum system mounted on a cart—a classical nonlinear system in control engineering. The system dynamics are first derived and serve as the basis for designing three control strategies: cascaded proportional–integral–derivative (PID) control, linear quadratic regulator (LQR), and an energy-based swing-up algorithm. In the cascaded PID structure, the outer loop controls the cart position while the inner loop stabilizes the pendulum angle. The proposed control methods are validated through both numerical simulation and real-time implementation using an STM32F103-based embedded platform. Experimental results confirm that the system can stabilize the pendulum in the upright position and drive the cart to the desired positions with acceptable accuracy. Among the tested strategies, LQR demonstrates superior performance under external disturbances. The results from this study can be used to examine and train algorithms for learners in control laboratories.

SUMMARY In Part I a technique for the swing-up control of single inverted pendulum system is presented. The requirement is to swing-up a carriage mounted pendulum system from its natural pendent position to its inverted position. It works for all carriage balancing single inverted pendulum systems as the swing-up control algorithm does not require knowledge of the system parameters. Comparison with previous swing-up controls shows that the proposed swing-up control is simpler, eaiser. more efficient, and more robust. In Part II the technique is extended to the case of the swing-up control of double inverted pendulum systems. Use is made of a novel selective partial-state feedback control law. The nonlinear, open-loop unstable, nonminimum-phase. and interactive MIMO pendulum system is actively linearised and decoupled about a neutrally stable equilibrium by the partial-state feedback control. This technique for swing-up control is not at all sensitive to uncertainties such as modelling error and sensor noise, and is both reliable and robust.

Aiming at the control planning of inverted pendulum task, the planning strategy based on Soft Actor-Critic (SAC) algorithm was studied. An agent based on Actor-Critic framework is designed, which takes inverted pendulum state as input and planned target position as output as controller. The agent has a total of 5 neural networks, namely, action network actor, 2 current evaluation networks and 2 target evaluation networks. Actor network outputs the next planned position according to the current inverted pendulum state, and the current Critic network and target Critic network respectively output the value evaluation of the current action and the next action according to the reward feedback from the environment. The PD control method is used to convert the planned target position of the mobile network output into the control voltage value, and actually control the inverted swing. Simulation results show that the proposed method is effective and efficient.

This paper proposes a modified swing control for a nonlinear inverted pendulum system by utilizing the sliding mode controller based on the on-line tuning Bees algorithm as speed of optimization and accuracy of results. The goal of the proposed nonlinear controller is to obtain the optimal force control action for the pendulum cart in order to stabilize the pendulum in the inverted position precisely and quickly. The optimal parameters of the nonlinear controller are on-line tuned by Bees algorithm and guided by Lyapunov stability criterion to reduce the amplitude of the sliding mode signum function in order to eliminate the chattering phenomena and make the smoothness control action. Matlab simulation results confirm the validity of the proposed controller algorithm in terms of fast dynamic response, minimizing the pendulum’s angle tracking error to the zero radian at 2.5 second and obtaining the optimal and smooth force control action without saturation state, with the minimum number of fitness evaluation of the algorithm.

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In this study, an interval type-2 fuzzy logic controller design method is proposed and validated on the real-time under-actuated nonlinear inverted pendulum system for swing-up and stabilization control problems. The swing-up algorithm is designed to give a faster response and the stabilization controller is designed based on continuity, monotonicity, and smoothness theorems for worst and best cases using the Mamdani fuzzy inference system in order to show the effectiveness of the proposed method. Outstanding type reduction methods are analyzed for the stabilization controller based on the design approach to determine the best type reduction algorithm for the effective and robust control performance. Real-time experiments are conducted to investigate the capability of the proposed controller in terms of adaptation performance and robustness ability. The controller also is able to handle structured and unstructured uncertainties such as measurement noise, external payload, undesirable internal/external disturbances, and parameter uncertainties. The results show that the proposed method clearly improves the control performance of the system in six experimental tasks conducted.

Inverted Pendulum System is a typical high-order nonlinear system and always used to verify the effectiveness of the control algorithm. This article uses the PID tracking control to predict the pendulum swing angle, and builds a real environment to verify the algorithm. Firstly, the inverted pendulum is analyzed and simulated to get its system characteristics. Then build a real physical model of the whole control system. Finally, based on the real system, this paper uses the double closed-loop PID control algorithm and PID tracking control algorithm individually to verify and compare control effect. The experimental result shows that the PID tracking control has good robustness and can avoid the logic contradiction of speed loop’s parameter adjustment in the typical double closed-loop algorithm and this algorithm has higher efficiency of the parameter adjustment and practicability.

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This paper presents a novel approach to designing a Hedge Algebra Controller named Hedge Algebra Controller with Recursive Semantic Values (RS-HAC). This approach incorporates several newly introduced concepts, including Semantically Quantifying Simplified Mapping (SQSM) featuring a recursive algorithm, Infinite General Semantization (IGS), and Infinite General De-semantization (IGDS). These innovations aim to enhance the optimizability, scalability, and flexibility of hedge algebra theory, allowing the design of a hedge algebra-based controller to be carried out more efficiently and straightforward. An application of stabilizing an inverted pendulum on a cart is conducted to illustrate the superiority of the proposed approach. Comparisons are made between RS-HAC and a fuzzy controller of Takagi-Sugeno type (FC), as well as a linear quadratic regulator (LQR). The results indicate that the RS-HAC surpasses the FC by up to 400\% in control efficiency and is marginally better than the LQR regarding transient time in balancing an inverted pendulum on a cart.

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Reinforcement learning can handle what traditional control theory can not cope with. In the field of visual reinforcement learning given only a partially observed high-dimensional image state, the agent still can learn a policy to achieve certain tasks. In this paper, we show the potential and limitations of visual reinforcement learning with an experiment on cart-pole balancing tasks both in simulation and semi-real environment systems.

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In this paper, we provide the details of implementing various reinforcement learning (RL) algorithms for controlling a Cart-Pole system. In particular, we describe various RL concepts such as Q-learning, Deep Q Networks (DQN), Double DQN, Dueling networks, (prioritized) experience replay and show their effect on the learning performance. In the process, the readers will be introduced to OpenAI/Gym and Keras utilities used for implementing the above concepts. It is observed that DQN with PER provides best performance among all other architectures being able to solve the problem within 150 episodes.

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In recent trend stabilization of the system is a very basic classical control problem in the field of the control system. Moreover, the techniques don't just stabilize the system, but improve the system response is having its significant impact. In this paper, a nonlinear system like an inverted pendulum has been utilized. The dynamics of an inverted pendulum on the cart are analogous to various real word applications like robot arm, missile launching system, balancing systems. The purpose of the control mechanism is to balance the pendulum. It has been done with optimally controlled Linear Quadratic Regulator (LQR). Stability analysis has been done with the classical pole placement method and the result has been compared with Linear Quadratic Regulator.

The inverted pendulum, typically realized as a cartpole system, is a benchmark for studying control of underactuated and nonlinear dynamics. In this system, a pole is hinged to a cart that moves horizontally, and the control objective is to balance the pole upright by applying forces to the cart. Unlike the standard Gymnasium implementation, our cart-pole environment incorporates high-fidelity Unified Robot Description Format (URDF) modeling to better emulate real-world conditions. We compare three control strategies for balancing: classical proportional-integral-derivative (PID) control, reinforcement learning (RL), and deep reinforcement learning (DRL). Our results demonstrate that proximal policy optimization (PPO) achieves superior robustness, outperforming PID and Q-learning in both reward accumulation and episode longevity. PID remains viable for short-term stabilization, while Q-learning suits energyconstrained scenarios.

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An adaptive proportional–integral–derivative (PID) controller based on Q-learning algorithm is proposed to balance the cart–pole system in simulation environment. This controller was trained using Q-learning algorithm and implemented the learned Q-tables to change the gains of linear PID controllers according to the state of the system during the control process. The adaptive PID controller based on Q-learning algorithm was trained from a set of fixed initial positions and was able to balance the system starting from a series of initial positions that are different from the ones used in the training session, which achieved equivalent or even better performances in comparison with the conventional PID controller and the controller only uses Q-learning algorithm. This indicates the advantage of the adaptive PID controller based on Q-learning algorithm both in the generality of balancing the cart–pole system from a relatively wide range of initial positions and in the stabilisability of achieving smaller steady-state error.

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We present an inverted pendulum design using readily available V-slot rail components and 3D printing to construct custom parts. To enable the examination of different pendulum characteristics, we constructed three pendulum poles of different lengths. We implemented a brake mechanism to modify sliding friction resistance and built a paddle that can be attached to the ends of the pendulum poles. A testing rig was also developed to consistently apply disturbances by tapping the pendulum pole, characterizing balancing performance. We perform a comprehensive analysis of the behavior and control of the pendulum. This begins by considering its dynamics, including the nonlinear differential equation that describes the system, its linearization, and its representation in the s-domain. The primary focus of this work is the development of two distinct control modes for the pendulum: a velocity control mode, designed to balance the pendulum while the cart is in motion, and a position control mode, aimed at maintaining the pendulum cart at a specific location. For this, we derived two different state space models: one for implementing the velocity control mode and another for the position control mode. In the position control mode, integral action applied to the cart position ensures that the inverted pendulum remains balanced and maintains its desired position on the rail. For both models, linear observer-based state feedback controllers were implemented. The control laws are designed as linear quadratic regulators (LQR), and the systems are simulated in MATLAB. To actuate the physical pendulum system, a stepper motor was used, and its controller was assembled in a DIN rail panel to simplify the integration of all necessary components. We examined how the optimized performance, achieved with the medium-length pendulum pole, translates to poles of other lengths. Our findings reveal distinct behavioral differences between the control modes.

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This article proposes a fuzzy cerebellar model articulation controller with reinforcement-strategy-based modified bacterial foraging optimization for solving the cart-pole balancing control problem. The proposed reinforcement-strategy-based modified bacterial foraging optimization is used to adjust the parameters of fuzzy receptive field functions and fuzzy weights for improving the accuracy of the fuzzy cerebellar model articulation controller output. An efficient strategic approach is applied in the chemotaxis step in the traditional bacterial foraging optimization algorithm. In the approach, each virtual bacterium swims for different run lengths and increases the bacterial diversity. Experimental results are presented to show the performance and effectiveness of the proposed reinforcement-strategy-based modified bacterial foraging optimization method.

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This paper presents ModelicaGym toolbox that was developed to employ Reinforcement Learning (RL) for solving optimization and control tasks in Modelica models. The developed tool allows connecting models using Functional Mock-up Interface (FMI) to OpenAI Gym toolkit in order to exploit Modelica equation-based modeling and co-simulation together with RL algorithms as a functionality of the tools correspondingly. Thus, ModelicaGym facilitates fast and convenient development of RL algorithms and their comparison when solving optimal control problem for Modelica dynamic models. Inheritance structure of ModelicaGym toolbox's classes and the implemented methods are discussed in details. The toolbox functionality validation is performed on Cart-Pole balancing problem. This includes physical system model description and its integration using the toolbox, experiments on selection and influence of the model parameters (i.e. force magnitude, Cart-pole mass ratio, reward ratio, and simulation time step) on the learning process of Q-learning algorithm supported with the discussion of the simulation results.

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This paper introduces a new method for learning to refine a rule-based fuzzy logic controller. A reinforcement learning technique is used in conjunction with a multilayer neural network model of a fuzzy controller. The approximate reasoning based intelligent control (ARIC) architecture proposed here learns by updating its prediction of the physical system's behavior and fine tunes a control knowledge base. Its theory is related to Sutton's temporal difference (TD) method. Because ARIC has the advantage of using the control knowledge of an experienced operator and fine tuning it through the process of learning, it learns faster than systems that train networks from scratch. The approach is applied to a cart-pole balancing system.

Numerically computing global policies to optimal control problems for complex dynamical systems is mostly intractable. In consequence, a number of approximation methods have been developed. However, none of the current methods can quantify how much the resulting control underperforms the elusive globally optimal solution. Here we propose policy decomposition, an approximation method with explicit suboptimality estimates. Our method decomposes the optimal control problem into lower-dimensional subproblems, whose optimal solutions are recombined to build a control policy for the entire system. Many such combinations exist, and we introduce the value error and its LQR and DDP estimates to predict the suboptimality of possible combinations and prioritize the ones that minimize it. Using a cart-pole, a 3-link balancing biped and N-link planar manipulators as example systems, we find that the estimates correctly identify the best combinations, yielding control policies in a fraction of the time it takes to compute the optimal control without a notable sacrifice in closed-loop performance. While more research will be needed to find ways of dealing with the combinatorics of policy decomposition, the results suggest this method could be an effective alternative for approximating optimal control in intractable systems.

This paper presents a new technique for the design of approximate reasoning based controllers for dynamic physical systems with interacting goals. In this approach, goals are achieved based on a hierarchy defined by a control knowledge base and remain highly interactive during the execution of the control task. The approach has been implemented in a rule-based computer program which is used in conjunction with a prototype hardware system to solve the cart-pole balancing problem in real-time. It provides a complementary approach to the conventional analytical control methodology, and is of substantial use where a precise mathematical model of the process being controlled is not available.

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Despite the numerous advances, reinforcement learning remains away from widespread acceptance for autonomous controller design as compared to classical methods due to lack of ability to effectively tackle uncertainty. The reliance on absolute or deterministic reward as a metric for optimization process renders reinforcement learning highly susceptible to changes in problem dynamics. We introduce a novel framework that effectively quantify the uncertainty in the design space and induces robustness in controllers by switching to a reliability-based optimization routine. A model-based approach is used to improve the data efficiency of the method while predicting the system dynamics. We prove the stability of learned neuro-controllers in both static and dynamic environments on classical reinforcement learning tasks such as Cart Pole balancing and Inverted Pendulum.

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