## 1 Introduction  
Flexible bars establish an important role in many engineering applications such as space tethers, catheters, robotic links, cantilever cranes, astronomical telescopes, etc. [1; 2; 3; 4]. They are widely used in industries due to their higher speed operation and lower energy consumption as a result of their lower overall mass. Moreover, light-weight links require smaller actuators which are beneficial for cost reduction of space robot transportation. Hence, flexible bars are superior to rigid ones in the above respects. However, it is well known that these systems could give rise to oscillations which make it difficult to obtain a precise tracking response [5]. Additionally, the mathematical model of the system becomes more complicated because of the flexibility. These are the two major challenges for the usage of flexible bars. Therefore, many works have been aimed at dynamic modeling and control of such systems and counsel into new channels to design controlled structures with high-performance characteristics. The dynamic modeling of the flexible bar is essential for understanding the system behavior and consequently designing the controller. One way to describe the mathematical model of these systems is through discretization. The discretization approach can be realized either mathematically or physically [6]. In the mathematical discretization, the finite difference method, as well as the Ritz-Galerkin procedure, can be applied. Although this is an efficient method, it becomes inconvenient for more than a few modes and can lead to numerical difficulties for high numbers of modes [7]. On the other hand, physical discretization models the system as finite elements or point mass elements. This method has been frequently adopted for describing different physical systems like rope [8; 9], chain [10], whip [11], space tethers [12], and has provided reasonably accurate results. The stringent behavior requirements imposed on flexible structures, on the other hand, demands precise control of these structures. In order to control the system efficiently both the controller and the actuator should be selected carefully. As for the actuator, smart materials such as shape memory alloys (SMA), have been frequently adopted in controllable structures including flexible bars [12]. SMAs are able to return to a predefined shape upon the supply of heat. The significant advantages of SMA wires in comparison to other smart materials such as piezoelectrics are the large displacement, complete recovery deformation, and high stiffness [13]. Furthermore, SMA can be heated easily by passing an electrical current through it [14]. This makes the shape control of structures integrated with SMA actuators more feasible. During the heating and cooling process, a phase transformation occurs in the crystalline structure of the SMA. This phenomenon introduces a hysteresis effect, which is highly nonlinear. Due to the hysteresis, the control of flexible structures actuated by shape memory alloys has turned to a challenging task [15]. Various linear control approaches including proportional-integral (PI) [16] and proportional-integral-derivative (PID) [17; 18] as well as nonlinear optimal and fuzzy logic controls [19], and variable structure/sliding mode control has been applied on SMA actuated devices. Among these methods, the sliding mode algorithms have shown promising results thanks to their less sensitivity to system parameter variations and noise disturbances [20; 21; 22]. The sliding-mode control (SMC), is a nonlinear feedback control whose structure is intentionally changed to achieve the desired performance. However, this change of structure gives rise to an undesirable phenomenon of oscillations having finite frequency and amplitude, which is known as chattering. Although some important general results for controlling flexible structures have been obtained using these controllers, most of these methods are application-specific and/or lack rigorous stability proof.