Inverted Pendulum System Research Articles

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

An Optimization-based Adaptive Sliding Mode Control for an Inverted Pendulum

The work proposes an optimization-based robust adaptive gain sliding mode control technique to handle bounded behavior of uncertainties/perturbations without having the exact knowledge of the bound a priori. The constant of the sliding surface follows dynamical adaptive nature using three optimization techniques, i.e. Harmony Search Algorithm, Bacterial Foraging Optimization Algorithm and Gradient Search Algorithm to extenuate the extent of chattering. The gain of the switching control law, which is a part of the sliding mode control design, is calculated using the gain adaption technique. Lyapunov Stability Criterion is applied to assure the stability of the closed-loop system with adaptable gain and the optimized value of the sliding constant. An inverted Pendulum system has been taken to validate the proposition both in the simulation environment and the hardware platform.

Designing Discrete Predictor-Based Controllers for Networked Control Systems with Time-varying Delays: Application to A Visual Servo Inverted Pendulum System

A discrete predictor-based control method is developed for a class of linear time-invariant networked control systems with a sensor-to-controller time-varying delay and a controller-to-actuator uncertain constant delay, which can be potentially applied to vision-based control systems. The control scheme is composed of a state prediction and a discrete predictor-based controller. The state prediction is used to compensate for the effect of the sensor-to-controller delay, and the system can be stabilized by the discrete predictor-based controller. Moreover, it is shown that the control scheme is also robust with respect to slight message rejections. Finally, the main theoretical results are illustrated by simulation results and experimental results based on a networked visual servo inverted pendulum system.

Super-opposition spiral dynamic-based fuzzy control for an inverted pendulum system

This paper presents a hybrid spiral dynamic algorithm with a super-opposition spiral dynamic algorithm (SOSDA) strategy. An improvement on the spiral dynamic algorithm (SDA), this method uses a concept centered on opposition-based learning, which is used to evaluate the fitness of agents at the opposite location to the current solution. The SDA is a simple-structured and deterministic type of algorithm, which also performs competitively in terms of solution accuracy. However, its deterministic characteristic means the SDA suffers premature convergence caused by the unbalanced diversification and intensification during its search procedure. Thus, the algorithm fails to achieve highly accurate solutions. It is proposed that adopting super-opposition into the SDA would enable the deterministic and random techniques to complement one another. The SOSDA was tested on four benchmark functions and compared to the original SDA. To analyze the result statistically, the Friedman and Wilcoxon tests were conducted. Furthermore, the SOSDA was applied to optimize the parameters of an interval type-2 fuzzy logic control (IT2FLC) for an inverted pendulum (IP). The statistical results produced by the SOSDA for both benchmark functions and the IP show that the proposed algorithm significantly outperformed the SDA. The SOSDA-based IT2FLC scheme also produced better IP responses than the SDA-based IT2FLC.

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

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

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.

Exponential stabilization of chaotic systems based on fuzzy time-triggered intermittent control

The exponential stabilization of chaotic systems is studied via fuzzy time-triggered intermittent control (FTIC). For the Takagi-Sugeno (T-S) fuzzy model representing a chaotic system, the mathematical description of FTIC is presented initially. Compared with fuzzy intermittent control (FIC), FTIC just needs the information at sampling instants on control time intervals. Compared with fuzzy sampled-data control (FSC), FTIC only transmits partial sampling data. Then, for the deduced FTIC system, a novel mixed Lyapunov functional is constructed to establish an exponential stabilization theorem. Based on it, FTIC can be designed. Further, the amount of transmitted data and the cost function are considered as two performance indexes. Finally, the inverted pendulum system and the chaotic Lorenz system are taken as examples to show the effectiveness and superiority of FTIC.

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.

Distributed Adaptive Fault-Tolerant Control for Heterogeneous Multiagent Systems With Time-Varying Communication Delays

This article considers the distributed adaptive fault-tolerant control problem for heterogeneous linear multiagent systems with actuator faults and nonuniform time-varying communication delays. First, novel distributed switching observers are proposed to estimate the system matrix and the state of the exosystem. The observers allow each agent to share its information with its underlying neighbors only at sampled instants of time, thus making a distinct difference from the existing results that require the exosystem matrix to be accessible to all agents at every continuous instant of time. Second, a sufficient condition is derived such that the observer error systems are exponentially stable under the influence of communication delays among interacting agents. Third, new distributed adaptive fault-tolerant controllers, which promise fewer adaptive parameters, are presented to compensate for the actuator faults. It is shown that the developed controllers are capable to effectively and efficiently solve the cooperative fault-tolerant output regulation problem with reduced computational complexity. Finally, three illustrative examples, including an inverted pendulum system, an electronic double-integrator circuit system, and an AC microgrid system, are presented to show the validity and efficiency of the developed method.

Intelligent and Robust Controller Tuned with WOA: Applied for the Inverted Pendulum

Inverted pendulum is a well-known problem in the control theory because several systems such as robot balancing, Segway, hover board riding and operation of a rocket propeller are inherently based on Inverted Pendulum, furthermore it possesses a height non-linear and unstable dynamics. The main objective of our study is to introduce a comparative analysis of fuzzy logic (FLC), radial basis function neural network (RBF) and integral sliding mode control (ISMC) tuned with whale optimizer algorithm (WOA) for the control of the angle position and velocity of the inverted pendulum system. The implemented controller schemas can adequately reflect and approximate a certain type of uncertainties, nevertheless their parameters should be fine-tuned in order to get height and efficient performance, therefore all the antecedents and consequences of those controllers were tuned with WOA. This later provide height accuracy and fast convergence with height dimensional cost function. Comparative results shows that ISMC-WOA outperforms other techniques in term of settling time and overshoot.

Optimization of Interval Type-2 Fuzzy Logic Controller Using Real-Coded Quantum Clonal Selection Algorithm

In recent years, quantum computing has gained immense popularity with the production of real quantum computers. Researchers have developed quantum-inspired evolutionary algorithms (QIEAs) to solve combinatorial optimization problems and have obtained successful results. As a special case of QIEAs, real-coded quantum evolutionary algorithm (RCQEA) is used in the optimization of high-dimensional complex problems. In this study, a novel mechanism of the quantum rotation gate (QRG) that is used to determine the rotation angle of the qubit in the RCQEA is introduced and implemented to accelerate the evolutionary process and increase the possibility of finding the optimal solution. Moreover, the skeleton of RCQEA is modified by using a clonal selection mechanism, and the real-coded quantum clonal selection algorithm (RCQCSA) is developed. Our proposed QRG accelerates the convergence speed of the algorithm. The main purpose of this study is to present a more effective algorithm that inspires quantum computing principles for optimizing the interval type-2 fuzzy logic controller (IT2FLC) membership functions (MFs). In this study, four different comparisons are made with these two different algorithms that have the original version of QRG and our proposed QRG. Optimized IT2FLC provides stabilization of the inverted pendulum system. The results show that the RCQCSA having our proposed QRG outperforms RCQEA in stabilizing the inverted pendulum system by optimizing the IT2FLC parameters.

Fast nonlinear model predictive controller using parallel PSO based on divide and conquer approach

The high computationally expensive nature of nonlinear optimisation algorithms leads to limitations of its use in a real-time era. So, this is the main challenge against researchers to develop a fast algorithm that is used in real-time computations. This paper proposes a fast nonlinear model predictive control algorithm that utilises a parallel particle swarm optimisation with synchronous and asynchronous methods inclusion, that handles nonlinear optimisation problems with constraints. The additional divide and conquer approach of the proposed algorithm improves the speed of computation and disturbance rejection capability which proves its efficacy in real-time applications. A highly nonlinear fast dynamic real-time inverted pendulum system with hybrid embedded hardware platform (ARM + FPGA) is used to validate the performance of this algorithm under constraints. The solution presented in the paper is computationally feasible for smaller sampling times i.e. in miliseconds and it gives promising results with synchronous and asynchonous parallel PSO compared to the state-of-art PSO algorithm.

Reinforcement Learning-Aided Performance-Driven Fault-Tolerant Control of Feedback Control Systems

This article is concerned with a fault-tolerant control (FTC) scheme for feedback control systems with multiplicative faults by optimizing system performance with the aid of a reinforcement learning (RL) approach. To be specific, initially, based on the Youla–Kučera (YK) and dual YK parameterizations, a new performance-driven FTC method is proposed and its capability in dealing with multiplicative faults is proven. Then, data-driven implementation of this method using RL is elaborated. This implementation shows that RL can be applied efficiently by utilizing both plant model and data to recover the fault-induced system performance degradation. Finally, a benchmark study on an inverted pendulum system demonstrates the application of the proposed performance-driven FTC method.

Adaptive Stabilization of Uncertain Nonlinear Systems Under Output Constraint

In this article, we extend the adaptive tracking control to more general nonlinear systems with multiple uncertainties, including output constraint, input delay, unknown parameter, and external disturbances for the first time. Without any growth assumptions, the adaptive backstepping technique is combined with the parameter separation technique to solve the parametric nonlinearities while the related results need to apply restrictive assumptions or use an approximation-based scheme to deal with them. To alleviate the serious uncertainties caused by output constraint and input delay, the barrier Lyapunov function (BLF) and the Pade approximation method are employed in a unified framework when the control coefficient is known. Under the case of the unknown control coefficient, the Nussbaum gain technique is further combined to compensate it. Then, under both cases, universal adaptive state-feedback control strategies merged with rigorous stability analysis are proposed, respectively, which guarantees all the signals are uniformly ultimately bounded. In addition, the reference signal tracks the system output into a compact set of the origin and the constraint of output is not violated. Finally, the proposed controllers are applied to an inverted pendulum system, which demonstrates the designed controller is effective.

Fixed‐time non‐singular terminal sliding mode control with globally fast convergence

This paper investigates a fixed-time terminal sliding mode (TSM) control scheme of second-order non-linear uncertain systems. A globally fast fixed-time stable system is proposed, and the convergence time is established with the Lyapunov method. Then, a novel non-singular TSM utilising the proposed fixed-time stable system and an auxiliary polynomial function is constructed. Meanwhile, a fixed-time disturbance observer technique is employed to estimate the matched lumped perturbation. Globally fast fixed-time convergence of the closed-loop system is guaranteed with the phase plane analysis and Lyapunov tools, and the upper bound of the settling time independent of the initial conditions can be predefined by the controller parameters. Finally, the simulation results of a single inverted pendulum system are included to confirm the validity of the proposed methodology while comparing with some popular TSM control strategies.

Modified grey wolf optimised adaptive super-twisting sliding mode control of rotary inverted pendulum system

This paper proposes a modified grey wolf optimiser-based adaptive super-twisting sliding mode control algorithm for the trajectory tracking and balancing of the rotary inverted pendulum system. The super-twisting sliding mode algorithm severely alleviates the chattering present in the classical sliding mode control. It provides robustness against model uncertainties and external disturbances with the knowledge of the upper bounds of the uncertainties and disturbances. The gains of the super-twisting sliding mode algorithm are selected through adaptive law. Parameters of the adaption law are tuned using a modified grey wolf optimisation algorithm, a meta-heuristic optimisation technique. Lyapunov stability analysis is carried out to analyse the overall control system stability. The performance of the proposed control algorithm is compared with two other sliding mode control strategies present in the literature, therein showing better performance of the proposed control scheme.

Robust output-feedback SOSM control subject to unmatched disturbances and its application: A fixed-time observer-based method

This paper investigates the output-feedback control for a general class of multi-input multi-output (MIMO) linear systems in the presence of unmatched disturbances. Firstly, a new observer composed of a Luenberger observer and a novel hierarchical high-order sliding mode (HOSM) observer is proposed to identify the system states and disturbances, simultaneously. As one of the most remarkable properties, the convergence time of the proposed observer is bounded by a positive constant which is free of the system initial error conditions. Secondly, based on the proposed observer, a new second-order sliding mode (SOSM) controller is constructed by using a novel sliding surface with unmatched disturbances compensation. The proposed control law is a simply continuous function of time and thus can certainly reduce numerical chattering. Finally, to show the effectiveness of the theoretical results, an application to inverted pendulum system is used to make simulation comparison.

Networked‐based H∞ control for discrete‐time slow sampling singularly perturbed systems via an improved event‐triggered approach

AbstractThis paper investigates the H∞ control problem for a class of slow sampling singularly perturbed systems (S3PSs) with an improved event‐triggered method (ETM). Compared with the conventional static ETM, an improved one with time‐varying threshold is exploited to enhance the dynamic performance for the S3PSs with certain communication frequency reduction. By adjusting the threshold with the triggering error, a faster converge speed can be achieved. Sufficient conditions are derived by constructing a singular perturbation parameter (SPP) dependent Lyapunov function, with which both asymptotic stability can be guaranteed for the closed‐loop S3PSs and the ill‐conditioned numerical issues can be avoided. By resorting to the matrix inequality techniques, an ETM‐based controller can be developed within the upper bound of the SPP. Further demonstration for the feasibility of the proposed algorithm is presented by designing an ETM‐based controller for an inverted pendulum system.

H ∞ Fault Estimation and Fault-Tolerant Control for T-S Fuzzy Systems with Actuator and Sensor Faults Using Sliding Mode Observer

This paper investigates the problems of the robust fault estimation (FE) and fault-tolerant control (FTC) for the Takagi-Sugeno (T-S) fuzzy systems with unmeasurable premise variables (PVs) subject to external disturbances, actuator, and sensor faults. An adaptive fuzzy sliding mode observer (SMO) with estimated PVs is designed to reconstruct the state, actuator, and sensor faults simultaneously. Compared with the existing results, the proposed observer is with a wider application range since it does not require the knowledge of the upper bound of faults that some FE methods demand. Based on the FE information, a dynamic output-feedback fault-tolerant controller (DOFFTC) is designed to compensate the effect of faults by stabilizing the closed-loop systems. By using the H ∞ filtering method, sufficient conditions for the existence of the proposed SMO and DOFFTC are derived in terms of linear matrix inequalities (LMIs) optimization. Finally, a nonlinear inverted pendulum system is given to validate the proposed methods.