Finite-Time State Estimation for an Inverted Pendulum under Input-Multiplicative Uncertainty

Abstract

A sliding mode observer is presented, which is rigorously proven to achieve finite-time state estimation of a dual-parallel underactuated (i.e., single-input multi-output) cart inverted pendulum system in the presence of parametric uncertainty. A salient feature of the proposed sliding mode observer design is that a rigorous analysis is provided, which proves finite-time estimation of the complete system state in the presence of input-multiplicative parametric uncertainty. The performance of the proposed observer design is demonstrated through numerical case studies using both sliding mode control (SMC)- and linear quadratic regulator (LQR)-based closed-loop control systems. The main contribution presented here is the rigorous analysis of the finite-time state estimator under input-multiplicative parametric uncertainty in addition to a comparative numerical study that quantifies the performance improvement that is achieved by formally incorporating the proposed compensator for input-multiplicative parametric uncertainty in the observer. In summary, our results show performance improvements when applied to both SMC- and LQR-based control systems, with results that include a reduction in the root-mean square error of up to 39% in translational regulation control and a reduction of up to 29% in pendulum angular control.

Disturbance observer-based controller for inverted pendulum with uncertainties: Linear matrix inequality approach

“…− A disturbance observer-based controller is proposed for the inverted pendulum system with uncertainties that has not been found in any previous paper. Unlike previous papers [21]- [27], the uncertainties in this study do not need to fulfill the bounded constraints. Therefore, it is impossible to employ the methods in paper [21]- [27] to design a controller for our case.…”

“…Unlike previous papers [21]- [27], the uncertainties in this study do not need to fulfill the bounded constraints. Therefore, it is impossible to employ the methods in paper [21]- [27] to design a controller for our case. In this paper, first step, the uncertainties are transformed to the input disturbances, and then the disturbance observer-based controller is synthesized in the second step to estimate unknown states, and input disturbances simultaneously.…”

“…Regarding synthesizing for the uncertain inverted pendulum system, a sliding mode technique was employed to design an observer to calculate the unknown states and reject the impacts of the uncertainties. However, it is seen that the uncertainties of the previous articles [21]- [27] have to be bounded. It means that the upper and lower bounds of the uncertainties should be provided in advance, otherwise it is impossible to design the controller and observer for these systems.…”

“…The inverted pendulum is not an exception, hence, stabilizing the inverted pendulum with the impacts of the uncertainties is a pressing and interesting issue. There are many articles paying attention to stabilizing the uncertain inverted pendulum system in recent years [21]- [27]. In paper [21], a fuzzy type-2 PID controller was synthesized for the inverted pendulum to reject the influence of uncertainties and stabilize the inverted pendulum system.…”

Design of an Adaptive Super-Twisting Control for the Cart-Pole Inverted Pendulum System

“…A different approach was proposed in [5], where the adaptive control logic was applied to solve the system problem. In general, there are three types of control problems related to inverted pendulum systems: (i) swinging up the pendulum upright from the downward position [6] [7] [8], (ii) stabilization or balancing of the pendulum upright [7][9] [10], and (iii) tracking control of the inverted pendulum [11] [12] [13] [14]. Since this system is popular for several decades, many control methods have been proposed to solve these three problems.…”