Inverted Pendulum System Research Articles

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.

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

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

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

Dynamic compensator design and [formula omitted] admissibilization for delayed singular jump systems via Moore–Penrose generalized inversion technique

This paper is intended to study output feedback-based H∞ admissibilization for singular Markovian jump time-varying delays systems via Moore–Penrose generalized inversion technique. The main aim is to design singular dynamic compensator based on output feedback control idea and Moore–Penrose generalized inversion technique, such that the time-varying delays singular Markovian jump system realizes not only stochastic stability, but also regularity, impulse-freeness (which are collectively known as stochastic admissibility) and achieves H∞ interference level. Time delay-dependent and mode-dependent L–K candidate functional is constructed, Moore–Penrose generalized inversion technique is utilized, then the improved H∞ admissibilization conditions for singular Markovian jump time-varying delays systems are provided by virtue of linear matrix inequalities. Simulation examples including a direct current motor-controlled inverted pendulum system are employed to confirm the effectiveness and practicability of the addressed singular dynamic compensation technique.

Further Results on Optimal Tracking Control for Nonlinear Systems With Nonzero Equilibrium via Adaptive Dynamic Programming.

This article develops a novel cost function (performance index function) to overcome the obstacles in solving the optimal tracking control problem for a class of nonlinear systems with known system dynamics via adaptive dynamic programming (ADP) technique. For the traditional optimal control problems, the assumption that the controlled system has zero equilibrium is generally required to guarantee the finiteness of an infinite horizon cost function and a unique solution. In order to solve the optimal tracking control problem of nonlinear systems with nonzero equilibrium, a specific cost function related to tracking errors and their derivatives is designed in this article, in which the aforementioned assumption and related obstacles are removed and the controller design process is simplified. Finally, comparative simulations are conducted on an inverted pendulum system to illustrate the effectiveness and advantages of the proposed optimal tracking control strategy.

Structure Analysis of General Type-2 Fuzzy Controller and Its Application

In this paper, a general type-2 fuzzy controller is proposed. The proposed general type-2 fuzzy controller is based on α-plane representation, which converts the centroid of general type-2 fuzzy sets to centroids of several interval type-2 fuzzy sets. Also, the mathematical expression of general type-2 fuzzy controller is derived by fuzzy product operate and NT type reduction. By the mathematical expression, the relation between general type-2 fuzzy controller output with parameter of triangular secondary membership function is discussed. Finally, the control performances of proposed general type-2 fuzzy controller are compared with conventional PID controller, tradition type-1 fuzzy controller, and interval type-2 fuzzy controller. Furthermore, to show the effective and practical of the proposed controller, a nonlinear inverted pendulum system is tested. The simulation results show that general type-2 fuzzy controller achieves better control efforts than other compared controllers.

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems, in this paper, a new iterative adaptive dynamic programming algorithm, which is the discrete-time time-varying policy iteration (DTTV) algorithm, is developed. The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance. The admissibility of the iterative control law is analyzed. The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution. To implement the algorithm, neural networks are employed and a new implementation structure is established, which avoids solving the generalized Bellman equation in each iteration. Finally, the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm, where the mass and pendulum bar length are permitted to be time-varying parameters. The effectiveness of the developed method is illustrated by numerical results and comparisons.

Optimum of fractional order fuzzy logic controller with several evolutionary optimization algorithms for inverted pendulum

AbstractThis paper compared the performance between Integer Order Fuzzy PID (IOFPID) and Fractional Order Fuzzy PID (FOFPID) controllers for inverted pendulum system as a controlling plant. The parameters of each controller were tuned with four evolutionary optimization algorithms (Social Spider Optimization (SSO), Swarm Optimization (PSO), Genetic Algorithm (GA), and Particle Ant Colony Optimization (ACO)). The comparisons were carried out between the two controllers IOFPID and FOFPID, as well as among the four optimization algorithms for the two controllers. The results of comparisons proved that the FOFPID controller with SSO has achieved the best time response characteristics and the least tuning time.

Reset control and ℒ2-gain analysis of piecewise-affine systems

This paper is concerned with reset controller design and L 2 -gain stability of piecewise-affine systems under the framework of hybrid systems. Firstly, stability conditions of piecewise-affine systems under dynamic state-feedback control are established through bilinear matrix inequality conditions. Secondly, a reset controller with reset rules under the hybrid systems framework is proposed in the sense of Lyapunov and sufficient conditions for exponential and L 2 -gain stability of the closed-loop systems are provided. Different from the piecewise-affine systems with the dynamic state-feedback controller, the reset controller is designed such that the L 2 -gain performance of piecewise-affine systems can be enhanced. Thirdly, an LMI approach is proposed to avoid the difficulty of solving the bilinear matrix inequalities. Furthermore, robustness to inflations of the flow and jump sets is established, and robustness to the norm-bounded uncertainties is proposed. Finally, numerical simulations including a robot arm system, an inverted pendulum system and the Chua's circuit system are provided to illustrate the results.

A third‐order sliding mode controller for nonlinear multivariable systems

AbstractThis paper proposes a third‐order sliding mode controller for nonlinear multivariable systems with uncertain parameters and subject to external disturbances. The controller achieves fast convergence rate, high tracking accuracy, and a reduced level of chattering. The stability of the controller and its global ultimately uniform convergence is proved by the Lyapunov stability theory. Simulation results on a single inverted pendulum system are given to illustrate the effectiveness of the proposed control scheme by comparing it with methods such as a second‐order supertwisting controller, a third‐order supertwisting controller, and an integral terminal third‐order sliding mode controller.

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

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

Robust preview control of interconnected continuous-time systems with parametric uncertainties

This paper designs a decentralized robust preview tracking controller for interconnected continuous-time systems with parametric uncertainties. The uncertain error system, which is composed of the derivative equation of the states, the tracking error dynamics, and the available future reference dynamics, is constructed. Then, a cost function is introduced to evaluate the tracking quality. Meanwhile, a performance signal is defined to discuss the influence of disturbances on the tracking performance. Based on the above operations, the considered tracking control is converted to the H-infinity control of the uncertain error system. The sufficient conditions which guarantee that the closed-loop system is asymptotically stable with H-infinity disturbance attenuation are derived. Further, the decentralized preview controller is obtained with the help of the linear matrix inequality (LMI). The result is applied to double-parallel inverted pendulum system and 3-interconnected machine power system. Numerical simulation demonstrate that the suggested preview controller can make the output track the reference signal, and the tracking quality be improved as the preview length increases.

Decentralized Dynamic Event-Triggered Adaptive Fuzzy Control for Switched Nonlinear Systems With Unstable Inverse Dynamics

In this article, the problem of decentralized dynamic event-triggered (ET) adaptive fuzzy control for a class of strong interconnected switched uncertain nonlinear systems with unstable inverse dynamics and unknown control gains is studied. By utilizing the adaptive backstepping technique and new fuzzy logic systems, decentralized ET adaptive fuzzy controllers for subsystems and a new type of switching decentralized dynamic event-triggering mechanisms are constructed. Then, the asynchronous switching problem between candidate controllers and subsystems is addressed, and the upper bound of asynchronous time is not directly restricted. Meanwhile, by ensuring that the interexecution times are lower bounded by a positive constant, Zeno behavior is avoided. Furthermore, it is proved by the multiple Lyapunov functions method that all signals in the switched closed-loop system are bounded under a new class of slow switching signals. The validity of the derived result is illustrated using two examples which include a two inverted pendulum system.

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

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

Simultaneous Multiplicative and Additive Actuator Faults Estimation‐Based Sliding Mode FTC for a Class of Uncertain Nonlinear System

In this article, an adaptive sliding mode fault tolerant control (FTC) is improved in the case of uncertain nonlinear system which is affected by both multiplicative and additive faults in actuator. Especially, when the nonlinear system is modeled by Takagi–Sugeno (T‐S) fuzzy system with local nonlinear model. The main contribution of this paper is developing a model of multiplicative faults, which offers a more realistic dynamic evolution of the actuator degradation. The degradation process is modeled by Wiener process and estimated by the maximum likelihood estimation (MLE). Sliding mode observer (SMO) is conceived to realize the additive actuator faults using convex multiobjective optimization. On these bases, the estimated multiplicative and additive actuator faults are used to design the adaptive sliding mode controller (SMC). Finally, the proposed fault‐tolerant control scheme is demonstrated by the results of inverted pendulum system simulation.

Gradient Descent-Barzilai Borwein-Based Neural Network Tracking Control for Nonlinear Systems With Unknown Dynamics.

In this article, a combined gradient descent-Barzilai Borwein (GD-BB) algorithm and radial basis function neural network (RBFNN) output tracking control strategy was proposed for a family of nonlinear systems with unknown drift function and control input gain function. In such a method, a neural network (NN) is used to approximate the controller directly. The main merits of the proposed strategy are given as follows: first, not only the NN parameters, such as weights, centers, and widths but also the learning rates of NN parameter updating laws are updated online via the proposed learning algorithm based on Barzilai-Borwein technique; and second, the controller design process can be further simplified, the controller parameters that should be tuned can be greatly reduced. Theoretical analysis about the stability of the closed-loop system is manifested. In addition, simulations were conducted on a numerical discrete time system and an inverted pendulum system to validate the presented control strategy.

Attack Detection for Networked Control Systems Using Event-Triggered Dynamic Watermarking

Dynamic watermarking schemes can enhance the cyber attack detection capability of networked control systems (NCSs). This paper presents a linear event-triggered solution to conventional dynamic watermarking (CDW) schemes. Firstly, the limitations of CDW schemes for event-triggered state estimation based NCSs are investigated. Secondly, a new event-triggered dynamic watermarking (ETDW) scheme is designed by treating watermarking as symmetric key encryption, based on the limit convergence theorem in probability. Its security property against the generalized replay attacks (GRAs) is also discussed in the form of bounded asymptotic attack power. Thirdly, finite sample ETDW tests are designed with matrix concentration inequalities. Finally, experimental results of a networked inverted pendulum system demonstrate the validity of our proposed scheme.

Neural-Network based Swing-up and Stabilization Control of Rotary Inverted Pendulum Systems

We propose the neural-network based control (NNC) approach for rotary inverted pendulum (RIP). The control structure for RIP consists of two phases: (i) swing-up phase, which drives the pendulum up towards the desired upright position; and (ii) stabilization phase, which enables the pendulum to keep the desired upright position. In our paper, the swing-up controller is designed based on energy control approach with feedback linearization technique, where the corresponding control gain Kec is determined by the proposed NNC. Next, the stabilization controller is designed based on modified super-twisting sliding-mode control (MSTSMC) algorithm, which requires to design a new sliding surface in order to achieve the asymptotic stability of RIP. Similar to the swing-up phase, the control gain KSS of the MSTSMC is determined intelligently by the proposed NNC. In fact, the proposed NNC allows to select Kec and KSS adaptively based on the operation of RIP. Finally, we provide the experimental comparisons between the proposed NNC and the control system of RIP without the neural-network framework. Through the experimental results, we show that by the NNC, the proposed control system of RIP provides the better swing-up and stabilization performances.

A Stackelberg Game-Theoretic Exploration Rendering Robustness and Optimality for Performance Improvement of Fuzzy Mechanical Systems.

We consider mechanical systems with uncertainty. The uncertainty may be time varying. The bound of the uncertainty is described by its fuzzy characteristics. To design a feasible control, we start with a robust phase, which renders a control scheme that guarantees the system performance regardless of the actual value of the uncertainty. This robust phase is then followed by an optimal phase. There are design parameters in the control, which can be fine-tuned. We proposed multiple performance objectives. The goal of the choice of the control design parameters is to minimize the performance objectives. However, since these objectives are nonconciliating (meaning one's minimum is not the other one's minimum), we invoke the Stackelberg strategy for the optimal parameters. The game strategy mimics two players: one is the leader and one is the follower. Through the interplay between the two players, we show how to select the design parameters. The design procedure in both robust and optimal phases is demonstrated by a coupled inverted pendulum system.

Neuro-Based Higher Order Sliding Mode Control for Perturbed Nonlinear Systems

One of the great concerns when tackling nonlinear systems is how to design a robust controller that is able to deal with uncertainty. Many researchers have been working on developing such type of controllers. One of the most efficient techniques employed to develop such controllers is sliding mode control (SMC). However, the low order SMC suffers from chattering problem which harm the actuators of the control system and thus unsuitable to be used in many practical applications. In this paper, the drawbacks of low order traditional sliding mode control (FOTSMC) are resolved by presenting a novel adaptive radial basis function neural network–based generalized rth order sliding mode control strategy for nth order uncertain nonlinear systems. The proposed solution adopts neural networks for their excellent capability in function approximation and thus used to approximate the nonlinearities and uncertainties for systems under consideration. The approximation errors are completely considered in the developed approach. The proposed approach can be used with any order of sliding mode and thus can be generally used with various types of applications. The global stability of the proposed control approach is proved through Lyapunov stability criterion. The proposed approach is validated and assessed through simulations on the nonlinear inverted pendulum system with severe modeling uncertainties. The simulations results show that the proposed approach provide superior performance compared with other approaches in the literature.