Nonlinear Inverted Pendulum System Research Articles

Event-triggered neural network control using quadratic constraints for perturbed systems

This paper investigates the event-triggered control problem for perturbed systems under neural network controllers. We propose a novel event-triggering mechanism, based on local sector conditions related to the activation functions, to reduce the computational cost associated with the neural network evaluation. It avoids redundant computations by updating only a portion of the layers instead of evaluating periodically the whole neural network. Sufficient conditions in terms of matrix inequalities are established to design the parameters of the event-triggering mechanism and compute an inner-approximation of the region of attraction for the perturbed feedback system. The theoretical conditions are obtained by using a quadratic Lyapunov function and an abstraction of the activation functions via quadratic constraints to decide whether the outputs of the layers should be transmitted through the network or not. Such conditions allow us to reduce the computational activity on the neural network while preserving the stability and performance level of the perturbed feedback system. To illustrate the efficacy of our approach, we consider the nonlinear inverted pendulum system stabilized by a trained neural network.

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.

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.

A class of general type-2 fuzzy controller based on adaptive alpha-plane for nonlinear systems

This study introduces a general type-2 Takagi–Sugeno–Kang fuzzy controller (GT2-TSKFC) for controlling uncertain systems. The proposed GT2-TSKFC uses equidistant type-2 triangular membership functions (MFs) for the antecedents, Larsen’s implication method, type-1 fuzzy sets for the consequent parameters, and a direct defuzzification method. The analytical structure of the proposed controller indicates that the alpha-plane and apex of the secondary MFs have a noticeable effect on calculating the control signal. Adaptation of the alpha-plane and apex of the secondary MFs is performed using the Lyapunov function to achieve the stability of the controlled system. The proposed controller is applied to an uncertain nonlinear inverted pendulum system. The results of the proposed control algorithm are compared with those of a general type-2 fuzzy controller with a specific number of alpha-planes, a quasi type-2 fuzzy controller, an interval type-2 fuzzy controller, and a type-1 fuzzy controller to demonstrate the robustness and effectiveness of the proposed scheme.

Adaptive delay dependent sliding mode fault-tolerant controller design for nonlinear systems with unknown time-varying input and state delays

This paper presents a novel adaptive delay-dependent fault-tolerant sliding mode control (SMC) strategy for the class of nonlinear Lipschitz systems with time-varying unknown delays in system states and inputs. A modified integral sliding surface (ISS) based on which the SMC approach is developed using a linear matrix inequality. Lyapunov-Krasovskii stability theory is employed to guarantee asymptotic stability of the closed-loop system such that system states starting from any arbitrary initial conditions with time-varying delays reach the predefined sliding surface in a finite time. It is also proved that states stay on the surface for all subsequent time, and the effects of the actuator’s faults are simultaneously attenuated. Adaptive tuning laws are used to adjust controller parameters and estimate actuators’ faults. The controllers’ structures are more straightforward than the most existing recent fault-tolerant control methods. Simulation results of practical nonlinear inverted pendulum system verified the outstanding merits and capabilities of the proposed scheme in the presence of actuators’ faults and multiple delays in the system states and inputs.

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.

Design of optimized interval type-2 fuzzy logic controller based on the continuity, monotonicity, and smoothness properties for a cart-pole inverted pendulum system

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.

Fuzzy logic and gradient descent-based optimal adaptive robust controller with inverted pendulum verification

This paper develops an adaptive robust combination of feedback linearization (FL) and sliding mode controller (SMC) based on fuzzy rules and gradient descent laws. The new suggested control algorithm is tested to stabilize a fourth-order under-actuated nonlinear inverted pendulum system. More precisely, the reliable feedback linearization approach and the robust SMC controller are combined to design a stable control effort. In order to enhance the performance of the suggested controller, an adaptation technique as long as fuzzy rules are applied to update the control gains and the boundary layer parameter. Then, a novel evolutionary algorithm termed multi-objective ant lion optimizer (MOALO) is implemented to determine the control coefficients. The analysis and results conducted on the inverted pendulum system demonstrate the desired performance of the proposed control scheme by providing an optimal smooth control input, suitable tracking performance, and proper time responses.

Orders & Contracts

This paper develops an adaptive robust combination of feedback linearization (FL) and sliding mode controller (SMC) based on fuzzy rules and gradient descent laws. The new suggested control algorithm is tested to stabilize a fourth-order under-actuated nonlinear inverted pendulum system. More precisely, the reliable feedback linearization approach and the robust SMC controller are combined to design a stable control effort. In order to enhance the performance of the suggested controller, an adaptation technique as long as fuzzy rules are applied to update the control gains and the boundary layer parameter. Then, a novel evolutionary algorithm termed multi-objective ant lion optimizer (MOALO) is implemented to determine the control coefficients. The analysis and results conducted on the inverted pendulum system demonstrate the desired performance of the proposed control scheme by providing an optimal smooth control input, suitable tracking performance, and proper time responses.

Robust Passivity Cascade Technique-Based Control Using RBFN Approximators for the Stabilization of a Cart Inverted Pendulum

This paper proposes a novel passivity cascade technique (PCT)-based control for nonlinear inverted pendulum systems. Its main objective is to stabilize the pendulum’s upward states despite uncertainties and exogenous disturbances. The proposed framework combines the estimation properties of radial basis function neural networks (RBFNs) with the passivity attributes of the cascade control framework. The unknown terms of the nonlinear system are estimated using an RBFN approximator. The performance of the closed-loop system is further enhanced by using the integral of angular position as a virtual state variable. The lumped uncertainties (NN—Neural Network approximation, external disturbances and parametric uncertainty) are compensated for by adding a robustifying adaptive rule-based signal to the PCT-based control. The boundedness of the states is confirmed using the passivity theorem. The performance of the proposed approach was assessed using a nonlinear inverted pendulum system under both nominal and disturbed conditions.

How A PID Controlling A Nonlinear Plant

This paper investigates the weakness of feedback controller to find the best parameter of PID controller of a nonlinear inverted pendulum system (NL-IPS). Stability and performance analysis of a NL-IPS are carried out with combination of feedback and PID controllers. It is found that the NL-IPS which was stabilized by feedback controller provides some weakness. By addition of PID controller on closed loop NL-IPS will provide better tracking and response on zero steady state error.

Tuning of parameters of PID controller using Grey Wolf Optimizer

In this paper, the Grey Wolf Optimizer (GWO) is used for tuning the parameters of PID (Proportional, Integral and Derivative) controller. The GWO is a meta-heuristic approach inspired by grey wolves (Canis lupus). The GWO algorithm consists of four types of grey wolves known as alpha (∝) as a leader or decision maker, beta (𝛽) as to assist alpha in decision making, delta (𝛿) as a subordinate wolf and omegas (𝜔) the followers. The hunting behavior of GWO continues with the three main following steps and they are searching for prey, encircling the prey and attacking the prey. Firstly, the parameters of the PID controller are tuned using the Ziegler-Nichols (Z-N) technique. Then these tuned parameters along with some randomly assumed coefficients are used as the first iteration step for the GWO algorithm. The tuning principle of GWO ensures the minimization of performance index. The performance index used in this paper is Integral Square error (ISE). The GWO algorithm for tuning the parameters of PID controller is validated by controlling the highly nonlinear in-verted pendulum system. Simulation results showed that the GWO algorithm minimizes the ISE of the system and stabilized the system within the stipulated time.

An optimal adaptive hybrid controller for a fourth-order under-actuated nonlinear inverted pendulum system

In this paper, a combination of approximate feedback linearization and sliding mode control approaches is applied to stabilize a class of fourth-order nonlinear systems. In order to improve the performance of the proposed controller, design parameters of the sliding surface are adjusted using the adaptation laws, based on the gradient descent technique. Eventually, a new version of particle swarm optimization is implemented to optimize the control gains with respect to the integral of time-weighted absolute errors as the objective function, and a constraint on the maximum control effort value. The introduced idea is put into practice for the under-actuated inverted pendulum system, and the procedure is described with details step by step. The time responses of cart position, pendulum angle, and control effort of the proposed method is then compared with those of several recently published references so as to evaluate the effectiveness of this scenario.

Development of an adaptive radial basis function neural network estimator-based continuous sliding mode control for uncertain nonlinear systems

In this paper an adaptive neural network (NN)-based nonlinear controller is proposed for trajectory tracking of uncertain nonlinear systems. The adopted control algorithm combines a continuous second-order sliding mode control (CSOSMC), the radial basis function neural network (RBFNN) and the adaptive control methodology. First, a second-order sliding mode control scheme (SOSMC), which is published recently in literature for linear uncertain systems, is extended for nonlinear uncertain systems. Second, an adaptive radial basis function neural network estimator-based continuous second order sliding mode control algorithm (CSOSMC-ANNE) is adopted. In CSOSMC-ANNE control methodology, a radial basis function neural network with adaptive parameters is exploited to approximate the unknown system parameters and improve performance against perturbations. Also, the discontinuous switching control of SOSMC is supplanted with a smooth continuous control action to completely eliminate the chattering phenomenon. The convergence and global stability of the closed-loop system are proved using Lyapunov stability method. Numerical computer simulations, with dynamical model of the nonlinear inverted pendulum system, are presented to demonstrate the effectiveness and advantages of the presented control scheme.

Nonlinear SPKF-Based Time-Varying LQG for Inverted Pendulum System

This paper deals with the holing issue of nonlinear inverted pendulum (IP) system. Close to the equilibrium point, IP is considered as a linear system without disruption and linear control is sufficient. However, when the IP swings over a wide range, its nonlinear dynamics becomes significant and the stabilization of IP becomes challenging task due to the inconsistency between its nonlinear dynamics and the controllers designed based on linearized models. Hence, the need for sophisticated control becomes highly demanding. This paper proposes an optimal time-varying linear quadratic Gaussian controller (TV-LQG) that is able to overcome this inconsistency problem. The proposed TV-LQG utilizes a sigma-point Kalman filter (SPKF) and a linear quadratic regulator with a prescribed degree of stability. SPKF is a highly accurate nonlinear state estimator since it does not use any linearization for calculating the state prediction covariance and Kalman gains. This leads, instantaneously, to a more exact nonlinear state estimation. The estimated states are fed to the Jacobian method, which updates, at once, the system dynamics accordingly. Thus, the parameters of the proposed TV-LQG are optimized based on the updated system dynamics. The proposed controller is intensively tested using simulation experiments and compared to the traditional LQG, LQR self-adjusting control and LQR-fuzzy control. The results proved the robustness and competitiveness of the proposed scheme to enhance the performance of the nonlinear IP system.

Adaptive Neuro-Fuzzy Control Approach for a Single Inverted Pendulum System

The inverted pendulum is an under-actuated and nonlinear system, which is also unstable. It is a single-input double-output system, where only one output is directly actuated. This paper investigates a single intelligent control system using an adaptive neuro-fuzzy inference system (ANFIS) to stabilize the inverted pendulum system while tracking the desired position. The non-linear inverted pendulum system was modelled and built using MATLAB Simulink. An adaptive neuro-fuzzy logic controller was implemented and its performance was compared with a Sugeno-fuzzy inference system in both simulation and real experiment. The ANFIS controller could reach its desired new destination in 1.5 s and could stabilize the entire system in 2.2 s in the simulation, while in the experiment it took 1.7 s to reach stability. Results from the simulation and experiment showed that ANFIS had better performance compared to the Sugeno-fuzzy controller as it provided faster and smoother response and much less steady-state error.

Event-triggered fuzzy control of nonlinear systems with its application to inverted pendulum systems

This paper is concerned with the event-triggered fuzzy control design and its application to an inverted pendulum system. First, all of the introduced time-varying delay information from the inverted pendulum system and event-triggered mechanism is considered in the stability analysis. The interval of the time-varying delay is then divided into l nonuniform subintervals by utilizing an improved delay-partitioning technique. Information on each subinterval is processed by employing a reciprocally convex method. The proposed delay-dependent stability condition of the inverted pendulum system is determined to be considerably less conservative than the existing results in the literature, and the reduction in the conservativeness becomes progressively noticeable as the delay partitioning number l becomes progressively thinner. Moreover, by employing the parallel distributed compensation law, conditions sufficient for the resulting fuzzy controller and the event-triggering fuzzy controller are presented for the nonlinear inverted pendulum system. Finally, the advantages and effectiveness of the proposed design schemes are demonstrated by three simulation examples.

Networked fuzzy output feedback control for discrete-time Takagi–Sugeno fuzzy systems with sensor saturation and measurement noise

This paper deals with the problem of networked fuzzy static output feedback control for discrete-time Takagi–Sugeno fuzzy systems with sensor saturation and measurement noise. First, by taking network-induced delays and packet dropouts into account, the resulting closed-loop fuzzy system is modeled as an asynchronous discrete-time T–S fuzzy system subject to both interval-like delays and sector nonlinearity. Second, an augmented fuzzy-basis-dependent Lyapunov functional is introduced. By employing a matrix-based quadratic convex method, an interval-delay-dependent bounded real lemma is derived such that the closed-loop system has a prescribed H∞ performance level. Third, a fuzzy controller design method is presented by using a cone complementarity linearization algorithm. Finally, a nonlinear inverted pendulum system via networked fuzzy control is provided to show that the proposed results are of less conservatism and exhibit satisfactory adaptability to sensor saturation.

Intelligent control of nonlinear inverted pendulum system using Mamdani and TSK fuzzy inference systems: a performance analysis without and with disturbance input

Fuzzy control, an intelligent control technique uses the human expert knowledge to make control decisions. The inverted pendulum, a highly nonlinear unstable system is used as a benchmark for implementing the control methods. In this paper the modelling and control design of nonlinear inverted pendulum-cart dynamic system with and without disturbance input have been presented. Here PID control and fuzzy control methods using Mamdani type and Takagi-Sugeno-Kang (TSK) type fuzzy inference systems (FIS) have been implemented to control the cart position and to stabilise the inverted pendulum in vertically upright position. The MATLAB-Simulink models have been developed for simulation of the control schemes. The simulation results and performance analysis justify the comparative advantages of fuzzy control methods.

Intelligent control of nonlinear inverted pendulum system using Mamdani and TSK fuzzy inference systems: a performance analysis without and with disturbance input

Intelligent control of nonlinear inverted pendulum system using Mamdani and TSK fuzzy inference systems: a performance analysis without and with disturbance input