## 1 Introduction  
One of the eminent problems in modern control theory is the precise modeling of systems for robust control design. The second major challenge in robust control design is tackling unmodeled internal nonlinearities or unmeasurable external disturbances that may reduce the efficiency of the closed-loop system [1; 2; 3; 4]. Implementing disturbance estimation techniques, which can eliminate these uncertainties and modeling errors, is always helpful in controller design due to the increase in the importance of precise control. Various disturbance observer-based control (DOBC) techniques [5; 6; 7; 8; 9; 10; 11] have been developed in the past few years. The survey paper [10] will provide a complete overview of the DOBC study. As per [10], disturbance observer (DOB) [5], unknown input observer (UIO) [11], and extended state observer (ESO) [8] are the most widely researched and applied among these disturbance estimation approaches. Although many researchers have utilized DOBC on various systems, satisfying matching conditions, very few studies addressed systems with mismatched uncertainties [12; 13; 14; 15]. ESO based control (ESOBC) only requires the relative order of system and approximate model information [16; 17; 18]. Originally ESOBC design was in the form of a typical chain of integrators with a matching condition considered [19], which restricts the ESOBC applicability to the mismatched system with non-integral chain form of systems [16]. Recent studies [13; 14] suggested the use of GESO for such systems. The above survey motivates us to augment states observer quality of EGESO [14] with robust sliding mode controller to improve the performance of systems with mismatched uncertainty. This technique reduces the number of sensors required for plant states to be measured and makes the plant cost-effective. An SMC is one of the famous, robust, nonlinear control techniques for uncertain systems because of its simple design procedure [20]. SMC was previously used for aircraft, robotic manipulators, batteries, and photo-voltaic systems [21], marine vehicles, and process control applications [22; 23]. The designing of sliding mode control requires a sliding function to reach the desired behavior of a sliding surface and a discontinuous controller that pushes the system trajectories towards the sliding (switching) manifold in a finite time and stays there for the future [24; 25; 26]. An SMC is sensitive to lumped uncertainty before it reaches the sliding surface, so it is necessary to eliminate this reaching phase to make the system insensitive to uncertainties [12; 27]. The ESO-based SMC [28; 29; 30; 31; 32] approach has been utilized previously but lacked in addressing issues like chattering in SMC, reaching phase elimination, non-integral chain form, mismatched system, and proper disturbance estimation. To address some of these issues, many researchers [33; 34; 35; 36] used a single-link flexible manipulator (SLFM) model as an application, but it fails to address all the issues and to achieve the desired performance. Due to the intrinsic underactuated nature of SLFM, designing a controller for such systems is a challenging task. In this paper, the EGESO based SMC, along with reaching phase elimination, is proposed for SLFM along with the mismatched uncertainties and succeeded in reducing the tip chattering (\(\alpha\)) of the flexible arm in real-time experimentation. The proposed controller is found to be effective in set-point tracking, handling mismatched uncertainties, and reducing the chattering effect in comparison to [13; 14; 30]. The remainder of the paper is structured as follows. Section 2 reviews the requisite preliminaries for problem statement identification. Section 3 defines the proposed control law along with the design of an EGESO and a modified stable sliding surface, with the elimination of reaching phase property, and EGESO stability is also indicated here. The proposed closed-loop system control law's stability is described in Section 4.