Discrete-time backstepping control with nonlinear adaptive disturbance attenuation for the inverted-pendulum system

Abstract

A discrete-time backstepping controller with an active disturbance attenuation property for the Inverted-Pendulum system is constructed in this paper. The main purpose of this study is to show that Immersion and Invariance (I & I) approach can be used to design a nonlinear observer for disturbance estimation and demonstrate its effectiveness considering a nonlinear system with an unstable equilibrium point, namely Inverted-Pendulum system, by utilizing the estimated values in backstepping control design. All designs are directly performed in discrete-time domain to obtain directly implementable observer and controller in discrete processors with superior performance compared to emulators. The Inverted-Pendulum system is not in strict feedback form therefore backstepping procedure cannot be directly applied. In order to enable backstepping construction, firstly a partial feedback linearization is performed and afterwards a novel discrete-time coordinate transformation is proposed. Prior to the construction of partial feedback linearizing and backstepping controller, a nonlinear disturbance estimator design is proposed with Immersion and Invariance approach. The estimated disturbance values used in the partial feedback linearization and construction of the backstepping controller. The global asymptotic stability of the estimator and local asymptotic stability of overall closed loop system are proved in the sense of Lyapunov. Performance of proposed direct discrete-time backstepping control with discrete I & I observer is compared with a backstepping sliding mode controller with another nonlinear disturbance observer (NDO) by simulations.

Backstepping control for a class of underactuated nonlinear mechanical systems with a novel coordinate transformation in the discrete-time setting

“…where k 2 is a controller gain. Similarly, considering (36) and (49), equation (43) can be reorganized as…”

“…35 In the conference paper, 35 we presented a coordinate transformation that enables design of a backstepping controller where the underactuated coordinates are derived to desired operating points but actuated coordinates are assumed to be free, namely, it does not allow to control all the coordinates. On the contrary, in a journal paper, 36 we have mainly addressed the adaptive disturbance attenuation. Nevertheless, we used a special case of the coordinate transform given in Adigu¨zel and Yalc xn 35 to be able to construct a discretetime backstepping controller.…”

Immersion and invariance sliding mode control with time-delay estimation for rigid spacecraft with uncertainties and actuator faults

“…It not only contains the classical “integral action” but also achieves “proportional action,” which is potentially superior to purely integral adaptation based on certainty-equivalence. Therefore, I&I adaptive methodology is applied to various control systems (Adıgüzel and Yalçın, 2021; Gao et al, 2018; Lee and Singh, 2020; Mannarino and Mantegazza, 2014). However, it can only estimate constant parameter or disturbance.…”

“…The exiting I&I methodology (Adıgüzel and Yalçın, 2021; Astolfi, 2008; Gao et al, 2018; Lee and Singh, 2020; Mannarino and Mantegazza, 2014) only estimates constant parameter or disturbance. A novel I&I scheme is developed to estimate time-varying disturbances.…”

A large-stroke rapid motion controller for servo applications with speed and control constraints

“…A cascade control scheme based on extended state observer (ESO) was developed for an omnidirectional quad-rotor (Lu et al, 2021). Adiguzel and Yalcin (2021) designed a discrete-time backstepping control scheme with an immersion & invariance observer for the inverted-pendulum system. A multi-rate feed-forward control design framework was developed by Zundert et al (2020) to overcome the traditional performance/cost tradeoff, and successful application to an industrial dual-stage wafer system was reported.…”