## 1 Introduction
Our objective is to develop and put into practice a robust state feedback control law with an affine structure to proficiently oversee and stabilize the knee exoskeleton robot at an intended position. The proposed controller carefully considers the nonlinear dynamics of the exoskeleton robot, encompassing crucial variables like viscous and solid frictions, as well as handling uncertain parameters and external disturbances. These factors could exert a substantial impact on the effectiveness of the control process. To accomplish this goal, we employ an affine state-feedback controller coupled with the Lyapunov approach. Through four distinct approaches, we utilize the Linear Matrix Inequality (LMI) techniques [33; 34] to formulate a series of LMI criteria that enable the computation of the gain for the proposed affine state feedback control law. These conditions are established based on various technical lemmas, ensuring the stabilization and resilience of the rehabilitation exoskeleton robot at the knee-joint level. To handle the nonlinearities present in the exoskeleton robot, we introduce constraints aimed at minimizing it, satisfying specific second-order polynomial conditions. Our initial development yields a nonlinear matrix inequality (NLMI), which can be challenging to solve numerically. To address this challenge, we employ theoretical techniques, including the matrix inversion lemma and the Schur complement in conjunction with the Young inequality and the S-procedure lemma, to convert the NLMI into a more tractable LMI. Adhering to the derived LMI conditions enables us to develop an affine state-feedback controller that effectively mitigates uncertainties and external disturbances, thereby enhancing control performance. Alongside the theoretical analysis and design, we present simulation results that validate the application of the established LMI requirements and demonstrate the efficacy of the adopted affine state-feedback controller. These simulations underscore the system's robustness and stability across diverse scenarios and in the presence of perturbations. The main contributions of this paper are summarized as follows:  
1. An affine state-feedback controller is developed for stabilizing a knee exoskeleton robot with nonlinear dynamics, accounting for critical factors such as solid and viscous friction, uncertainties, and external disturbances. These uncertainties and disturbances are combined into a single term, the lumped disturbance. 2. Stabilization conditions are derived using the Lyapunov approach, Young's inequality, and the S-procedure lemma, initially formulated as nonlinear matrix inequalities (NMIs). 3. The derived NMIs are converted into linear matrix inequalities (LMIs) using the Schur complement, matrix inversion lemma, and congruence transformations. 4. The stability criteria in LMI form are refined by introducing optimization problems to maximize or minimize critical parameters, including the magnitude of the lumped disturbance and a constraint parameter on the controller. 5. Numerical simulations validate the theoretical results, demonstrating the controller's ability to mitigate the effects of the lumped disturbance effectively. The structure of this paper is outlined as follows in the sections that follow: Section 2 provides a literature review on the applications of LMIs in rehabilitation robotics. Section 3 presents the problem formulation and describes the nonlinear dynamics used to control the knee joint of the rehabilitation exoskeleton system. In Section 4, we explore the specifics of the linearized dynamics and elaborate on the implementation of the affine state-feedback control law. Section 5 focuses on elaborating the design of LMI stability conditions through four distinct approaches. A comparison with the classical PD and PID controllers is discussed in Section 6. The simulation results, shedding light on practical implications, are unveiled in Section 7.