Double Inverted Pendulum System Research Articles

LQR and Fuzzy-PID Control Design on Double Inverted Pendulum

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

Online Virtual Reality Networked Control Laboratory Applied in Control Engineering Education.

Online laboratories play an important role in engineering education. This work discusses a WebVR-based virtual laboratory system. The user enters the simulated laboratory environment through a virtual reality (VR) device and interacts with the experimental equipment, similar to hands-on experiments in a physical laboratory. In addition, the proposed system allows users to design their own control algorithms and observe the effects of different control parameters to enhance their understanding of the experiment. To illustrate the features of the proposed virtual laboratory, an example is provided in this paper, which is an experiment on a double inverted pendulum system. The experimental results show that the proposed system allows users to conduct experiments in an immersive and interactive manner and provides users with a complete experimental process from principal design to experimental operation. A solution is also provided to change any virtual laboratory into a WebVR-based virtual laboratory for education and training.

Swing-up control of double-inverted pendulum systems

Swing-up control of double-inverted pendulum systems

Switched Energy and Neural Network-Based Approach for Swing-Up and Tracking Control of Double Inverted Pendulum

The double inverted pendulum (DIP) system is a benchmark underactuated mechanical system. The control problem of the DIP system is challenging since it not only has high nonlinearity, but also has underactuation degree equal to two. In this article, a switched energy-based swing-up and neural network (NN) controller is proposed to solve the swing-up and tracking control problem of the DIP system when the desired position is time varying. The implementation of the proposed controller can be divided into three steps. In step one, the energy-based control method is adopted to drive the first pendulum from the downward position to the neighborhood of the upright position. In step two, the sliding mode control method is adopted to stabilize the first pendulum. Meanwhile, a method combining energy-based control and “equivalent cart” is adopted to swing up the second pendulum. In step three, on the basis of approximate nonlinear output regulation theory, the NN controller is used for achieving satisfactory position tracking performance. Finally, the effectiveness of our design is verified by experimental results.

Optimal LQR Controller Methods for Double Inverted Pendulum System on a Cart

Most of the systems in our lives are inherently nonlinear and unstable. In control problems in the field of engineering, the aim is to define the control laws that maximize the operating efficiency of these systems under diverse security coefficients, and constraints and minimize error rates. This study aimed to model and optimally control a Double-Inverted Pendulum System on a Cart (DIPSC). A DIPSC was modeled using the Lagrange-Euler method, and classical and optimal Linear Quadratic Regulator (LQR) control methods were designed for the control of the system. The purpose of the designed controllers is to keep the arms of the double inverted pendulum on the moving cart vertically in balance and to bring the cart to the determined balance position. The critically important Q and R parameters of the LQR control technique that is one of the optimal control techniques were obtained using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Grey Wolf Optimization (GWO) algorithms. The DIPSC system was checked using classical LQR and optimal LQR methods. All obtained results are given graphically. The proposed methods are presented and analyzed in tabular form using Settling time and Mean-Square-Error (MSE) performance criteria.

Real Time Swinging Up and Stabilizing a Double Inverted Pendulum Using PID-LQR

Abstract This study describes a method for swinging up and stabilizing a double inverted pendulum (DIP) in real-time utilizing a PID-LQR combined control system. Firstly, a dynamic model of the double inverted pendulum system is made up and the equations of motion are constructed. The pendulum then is moved from its unstable position to a stable one using a PID-LQR controller. A comparison of the PID-LQR controller’s output and suggestions for improving system stability is presented and is suggested combined control system.

LQR controller design for stabilization of non-linear DIP system based on ABC algorithm

Inverted pendulum systems, such as double or single, rotational or translational inverted pendulums are non-linear and unstable, which have been the most dominant approaches for control systems. The double inverted pendulum is one kind of a non-linear, unstable system, multivariable, and strong coupling with a wide range of control methods. To model these types of systems, many techniques have been proposed so that motivating researchers to come up with new innovative solutions. The Linear Quadratic Regulator (LQR) controller has been a common controller used in this field. Meanwhile, the Artificial Bee Colony (ABC) technique has become an alternative solution for employing Bee Swarm Intelligence algorithms. The research solutions of the artificial bee colony algorithm in the literature can be beneficial, however, the utilization of discovered sources of food is ineffective. Thus, in this paper, we aim to provide a double inverted pendulum system for stabilization by selecting linear quadratic regulator parameters using a bio-inspired optimization methodology of artificial bee colony and weight matrices Q and R. The results show that when the artificial bee colony algorithm is applied to a linear quadratic regulator controller, it gains the capacity to autonomously tune itself in an online process. To further demonstrate the efficiency and viability of the suggested methodology, simulations have been performed and compared to conventional linear quadratic regulator controllers. The obtained results demonstrate that employing artificial intelligence (AI) together with the proposed controller outperforms the conventional linear quadratic regulator controllers by more than 50 % in transient response and improved time response and stability performance

Advanced Control Strategies for Rotary Double Inverted Pendulum

This paper presents the stabilization control of the rotational double inverted pendulum system. It is generally a non-linear, non-minimum phase, unstable, and underactuated system of high order. It has some significant real-life applications such as position control, aerospace vehicle control, and robotics. The objective is to determine the control law to the motor’s output torque such that the inverted pendulum motion will be stabilized about a vertical axis and position the rotary arm to a commanded angular position. In this study, multi-PID controllers, Linear-Quadratic Regulator (LQR), and Internal Model Control (IMC) controllers are designed and MATLAB-based simulations are performed to understand and compare the performance of the three control schemes.

Input-to-State Stability of the Nonlinear Fuzzy Systems via Small-Gain Theorem and Decentralized Sliding-Mode Control

This article presents the input-to-state stability (ISS) and decentralized sliding-mode control of polynomial fuzzy interconnected systems by using the small-gain theorem. The contribution of this article is threefold. First, based on the boundedness of the solution of fuzzy interconnected subsystems, the sufficient conditions for the boundedness fuzzy interconnected system are provided. Second, the ISS small-gain theorem is extended to the polynomial fuzzy interconnected system for the number of membership functions being the same and different, and the trajectory-based ISS small-gain theorem is proposed. Third, the sliding surface based on state feedback is designed, and the stability of sliding-mode motion is analyzed for the fuzzy interconnected system. Finally, the effectiveness and superiority of the proposed method are illustrated by using a numerical example, a spring-connected double inverted pendulum system, and comparison with the existing results.

Optimal adaptive state feedback control for a three degree-of-freedom series-type double inverted pendulum system by using particle swarm optimisation

ABSTRACT This research tries to stabilise a three degree-of-freedom series-type double inverted pendulum system by employing an adaptive state feedback control approach optimised by a particle swarm optimisation (PSO) algorithm. To this end, the dynamical equations are derived via Lagrangian method and linearised by utilising approximation approaches. Then, the state feedback scheme is implemented to control the system states, that is, the angle and angular velocity of the first and second poles as well as the position and velocity of the cart. In order to timely regulate the control gains, the adaptation rules based on the gradient descent method and sliding surface relations are applied. The particle swarm optimisation algorithm is implemented to optimally tune the design parameters of the controller based on a proper objective function defined as summation of integrals of absolute values of the system errors. The time responses of the angles of the poles and the position of the cart clearly depict the feasibility and efficiency of the introduced control strategy to stabilise the system from different initial conditions to the final desired values.

Power-based modelling and control: experimental results on a cart-pole double inverted pendulum

This paper is concerned with the modeling framework based on power and control for a mechanical system that has nonlinear, unstable, and under-actuated characteristic features, based on an analogy, which is developed by using the Brayton and Moser's (BM) equations between mechanical and electrical systems. The analogy is based on a mixed-potential function generalized for BM. The mixed-potential function for a cart - pole double inverted pendulum (CPDIP) system is used as a new building block for modeling, analysis, and controller design. The analogy allows for the exact transfer of results from electrical circuit synthesis and analysis to the mechanical domain. This paper focuses mainly on the development of the electrical equivalent circuit of CPDIP inspired by the power-based modeling framework. In this brief, a real time CPDIP experimental setup was modeled by using this framework and a linear quadratic regulator (LQR) controller was designed for the stabilization of the system. Experimental results validated that this framework can be used as a new and advantageous modeling method and is a convenient and practical alternative to the Lagrangian framework.

Modeling and Optimization of Security Robot Gait Balance Based on the Principle of Inverted Pendulum

Robots have been increasingly applied to security protection. The modeling and optimal control of robot gait balance were discussed in this study based on a double inverted pendulum system. First, a trolley-type double inverted pendulum system was obtained by abstracting human gaits, a mathematical model corresponding to this system was constructed through the analytical dynamics method. Then a double inverted pendulum system controller was designed using the linear quadratic type control algorithm, a gain matrix K with the optimal performance was obtained by combining the experience piece-try method with regional dichotomy, and the effectiveness of modeling and optimization was demonstrated by comparing the typical value with optimized value in the end. This study is expected to provide a reference for enhancing the gait balance of security robots.

Control of rotary double inverted pendulum system using LQR sliding surface based sliding mode controller

This paper presents LQR sliding surface-based Sliding Mode Controller (LQR–SMC) for balancing control of a Rotary Double Inverted Pendulum (RDIP) system. It is a challenging research topic in control engineering due to its nonlinearity and instability. The RDIP system uses only a motor to control two serially connected pendulums to stand at the upright position. The sliding surface is designed based on the LQR optimal gain. Nonsingular gain matrix is obtained by using the left inverse of the input matrix in the state space form of the system dynamics. The Lyapunov stability theory is used to determine the stability of the controller. To evaluate the performance of LQR–SMC, some performance indices, including the Integral Absolute Error (IAE), Integral Time Absolute Error (ITAE), and the Integrated Square Error (ISE), are used. System stability can be maintained by LQR–SMC under external disturbances as well as model and parameter uncertainties.

T-S Fuzzy System Controller for Stabilizing the Double Inverted Pendulum

This article provides a representation of the double inverted pendulum system that is shaped and regulated in response to torque application at the top rather than the bottom of the pendulum, given that most researchers have controlled the double inverted pendulum based on the lower part or the base. To achieve this objective, we designed a dynamic Lagrangian conceptualization of the double inverted pendulum and a state feedback representation based on the simple convex polytypic transformation. Finally, we used the fuzzy state feedback approach to linearize the mathematical nonlinear model and to develop a fuzzy controller H ∞ , given its great ability to simplify nonlinear systems in order to reduce the error rate and to increase precision. In our virtual conceptualization of the inverted pendulum, we used MATLAB software to simulate the movement of the system before applying a command on the upper part of the system to check its stability. Concerning the nonlinearities of the system, we have found a state feedback fuzzy control approach. Overall, the simulation results have shown that the fuzzy state feedback model is very efficient and flexible as it can be modified in different positions.

Control of interconnected systems with sensor delay based on decentralized adaptive neural dynamic surface method

In this article, an adaptive neural network is proposed for the tracking control problem of unknown nonlinear interconnected systems with inaccessible states and sensor delays based on dynamic surface strategy. The system has unknown nonlinearities and immeasurable states. Thus, a neural network state observer based on delayed outputs of subsystems is applied. The main difficulty in obtaining local observers’ gains is that undelayed outputs are not available. As a result, by utilizing proper Lyapunov–Krasovskii functionals in dynamic surface design procedures, the gains of local observers are given in terms of linear matrix inequalities. Then, appropriate changes in coordinates are defined using delayed outputs, observed states, and filtered virtual controls for the purpose of designing dynamic surface controllers. Subsequently, proper Lyapunov–Krasovskii functionals are introduced to deal with sensor delays and obtain control laws and stability criteria. Furthermore, the proposed decentralized control scheme can suitably conquer the decentralized tracking problem of unknown large-scale systems with sensor delays and guarantee that all the signals in the closed-loop interconnected systems be uniformly ultimately bounded. Finally, to show the effectiveness and efficiency of the proposed approach, the theoretic achievements are employed to design a controller for a double-inverted pendulum system and a cascade chemical reactor system.

LQR-Based Stabilization and Position Control of a Mobile Double Inverted Pendulum

The double inverted pendulum is a system that can be used efficiently to test and validate the performance of control algorithms because of its instability and the many degrees of freedom system. The paper presents the design and development of a unique mobile double inverted pendulum system. LQR is applied for the balancing control of both pendulums and position control of the mobile platform of the system. Both simulations and experiments are conducted to evaluate the system performance.

Resilient decentralized sampled-data [formula omitted] filter design for linear interconnected systems subject to denial-of-service attacks

This paper investigates the design problem of the resilient decentralized sampled-data H∞ filter for linear interconnected systems subject to the denial-of-service attacks. A piecewise Lyapunov function-based method is first developed to determine the filter parameters and the sampled-data communication schemes. Then the obtained results are utilized to facilitate the design of the resilient decentralized sampled-data H∞ filter against the denial-of-service attacks. Compared with the existing results, the communication channels from the subsystems of interconnected systems to the filters are allowed to be independent and be compromised by different adversaries. Finally, numerical simulations on the double-inverted pendulums system and the three-machine power system are provided to illustrate the efficacy of the proposed method.

Calculation of the critical delay for the double inverted pendulum

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristic equation and its derivatives.

Three degrees of freedom rotary double inverted pendulum stabilization by using robust generalized dynamic inversion control: Design and experiments

The aim of this article was to determine control strategy for balance control of rotary double inverted pendulum system, which is highly nonlinear and unstable under-actuated system. The complexities involved in rotary double inverted pendulum dynamics make this system a useful engineering test bed to test and verify newly designed controllers. In this article, a constraint-based control approach titled robust generalized dynamic inversion is designed and implemented for robust stabilization of rotary double inverted pendulum system. The robust generalized dynamic inversion control is designed in two stages; in the first stage, constraint differential equations of the controlled state variables are prescribed, which encompasses the control objectives. To enforce the constraint dynamics, the equivalent control is realized by means of Moore–Penrose generalized inversion. To enhance robustness, the switching (discontinuous) control is introduced in second stage, whose design principle is based on classical sliding mode control theory. Finally, the controllers obtained in two stages are augmented to form the resultant robust generalized dynamic inversion control law. The proposed controller ensures robustness along with improved time domain performance regardless of system nonlinearities, uncertainties, and unwanted disturbances. The stability analysis is presented for guaranteeing semi-global asymptotically stable closed loop performance via Lyapunov stability criteria. Numerical simulation and experimental investigations are carried out along with comparative analysis, to demonstrate the effectiveness of robust generalized dynamic inversion control algorithm over other conventional control methods.

Backstepping controller design with a quadratic error for a double inverted pendulum

This study aims at presenting a conceptualisation of a double inverted pendulum system based on modelling and controlling this model by applying only a single torque in the upper part instead of the lower part of the double inverted pendulum. Nonlinear dynamic equations were analysed using the Lagrangian dynamic formulation, and the graphical presentation of the system has been achieved through applying the bond graphs approach. The double inverted pendulum representation was incarnated using the 20-Sim software. To achieve the system's stability, only one command was applied on the upper part. Most important, a control approach combining the backstepping method with the quadratic error was designed. Indeed, the simulation results affirmed the effectiveness of the backstepping controller with the quadratic error, and good response of the system's flexibility was ensured in the sense that it can be adjusted from the initial position to the equilibrium position.