## 1 Introduction  
The magnetic resonance imaging (MRI) technique is widely used in clinical diagnosis due to its ability to image without the use of ionizing x-rays and superior soft tissue contrast as compared to computed tomography (CT) scanning. Recently, pneumatically actuated MRI-compatible robots, which enable real-time magnetic resonance (MR) image-guided needle placement, are designed for brachytherapy and biopsy by several researchers [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. To fulfill the requirements for MR compatibility of the robotic systems, pneumatic valves are commonly placed outside the scanner room in the aforementioned works. Therefore, long transmission lines between the actuators and valves are used. Since long transmission lines have a significant influence on the pressure dynamics of the pneumatic system and MRI-compatible pressure sensors are not commercially available, precise position control is one of the main technical challenges in robot development. Indeed, considerable research effort has been devoted to addressing the issue related to long transmission lines. [PERSON] et al. [16] suggested a formula for the time delay and amplitude attenuation between the mass flows at the outlet and inlet of the line, as well as a formula for the pressure drop along the tube. Although this approach was later employed in many other studies ([PERSON] et al. [14], [PERSON] et al. [17]), it has been proven improper for long transmission lines (\(>\)2 m) [18]. [PERSON] et al. [8] adopted a first order transfer function with time delay to describe the 9-m transmission line dynamics. However, for some unknown reason, the authors omitted the transmission line dynamics in the subsequent controller design. Based on the discretization of a general set of equations from fluid mechanics, two similar distributed models of long pneumatic transmission lines were derived by [PERSON] et al. [19] and [PERSON] et al. [20]. However, this type of model is not suitable for controller design for their high order and low computational speed. [PERSON] et al. [21] developed a simplified distributed model to characterize the additional dynamics brought by the long transmission lines. As previously mentioned, direct pressure measurement is unavailable for the lack of MRI-compatible pressure sensors. However, pressure states are commonly used in controller design for precise position control of a pneumatic actuator. To solve this problem, pressure observers were used in place of a pressure sensor in many studies. [PERSON] et al. [22] investigated two design methods of an observer to estimate the chamber pressures in a pneumatic cylinder. While a continuous gain observer was used to estimate one of the chamber pressures with the assumption that the other one is measured in the first method, a sliding mode observer was utilized to estimate both chamber pressures in the second method. In this study, the mass flow rate was assumed to be exactly known, which is apparently too restrictive in practice. [PERSON] et al. [23] conducted an analysis on observability and concluded that the pressure states in the pneumatic servo system are not locally observable from the measurement of the output motion within several regions in the state space. Based on the actuator pressure dynamics, various observers are developed, such as the sliding-mode observer ([PERSON] [24]), energy-based Lyapunov observer ([PERSON] et al. [25]), and adaptive nonlinear observer ([PERSON] et al. [26]). It should be noted that the performance of these observers relies on the accuracy of the valve model. [PERSON] et al. [27] developed a pressure estimation algorithm by utilizing the measured actuation force and the hypothesized average air pressure in the actuator. [PERSON] et al. [18] presented a pressure observation method by using the measured force and piston displacement.