LMI-Based Robust Stabilization of a Class of Input-Constrained Uncertain Nonlinear Systems with Application to a Helicopter Model

Abstract

This paper is concerned with the robust stabilization of a class of continuous-time nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, via an affine state-feedback control law using the linear matrix inequality (LMI) approach. The nonlinear dynamics is subject to norm-bounded parametric uncertainties and disturbances. In addition, the problem of actuator nonlinearity is addressed by considering the saturation effect of the control law. We demonstrate first that the synthesis problem of the saturated controller is expressed in terms of bilinear matrix inequalities (BMIs). Thanks to the Schur complement lemma and the matrix inversion lemma, we convert these BMIs into LMIs allowing the simultaneous computation of the two gains of the affine controller. Furthermore, we address in this work the estimation problem of the domain of attraction using the invariant set concept. This is solved by computing the largest attractive invariant ellipsoid. Compared with previous works, the research procedure of such ellipsoidal set is achieved in a single step with a reduced number of LMI constraints and then with less conservative conditions. A portfolio of numerical results is presented. The effectiveness and robustness of the proposed saturated controller in the stabilization of the adopted helicopter pitch model toward parametric uncertainties and disturbances are illustrated through simulation results.

A Hierarchical Optimal Control of Uncertain and Time-Varying Knowledge Dissemination Model in Complex Network

“…Turki et al [32] adopted a state-feedback controller to robustly control the motion of a 1-DoF double-side impact mechanical oscillator subject to norm-bounded parametric uncertainties. Gritli [33] studied the robust calming problem of a class of continuous-time nonlinear systems and applied it to a simple helicopter model. He also addressed the problem of static output feedback (SOF) stabilization for continuoustime linear systems subject to norm-bounded parameter uncertainties using the LMI approach [34].…”

Robust integral controller design based on composite nonlinear feedback in uncertain systems with constrained inputs and external disturbances

“…In uncertain control systems, most studies have focused on LMI‐based controller synthesis. Recently, based on LMI representation, robust stabilization is accomplished in constrained uncertain systems 34 . Then, a robust controller is synthesized for an underactuated system with saturation and motion constraints 35 …”

“…Recently, based on LMI representation, robust stabilization is accomplished in constrained uncertain systems. 34 Then, a robust controller is synthesized for an underactuated system with saturation and motion constraints. 35 Several LMI-based methods have been addressed for integral control.…”

Comparison Between Some Nonlinear Controllers for the Position Control of Lagrangian-type Robotic Systems

“…The first class being the underactuated robotic systems Choukchou-Braham et al (2014); Liu and Yu (2013); Zilong Zhang (2022). This type has less of actuators than the degrees of freedom (DoF), such as the acrobot, the pendubot and the inverted pendulum on a cart, and the inertia wheel inverted pendulum, just to mention a few (Choukchou-Braham et al 2014;Gritli andBelghith 2018, 2021;Krafes et al 2018;Liu and Yu 2013;Parulski et al 2021, 11;Zilong Zhang 2022). The second class is called the fully-actuated robotic systems.…”