## 1 Introduction  
Set-point tracking systems, such as hard disk drive reading systems [1], surface mount device placement systems [2], electro-optical tracking systems [3; 4], as well as mechanical arm control systems, play an significant role in the field of modern industrial process and are widely applied in energy exploration [5], celestial target observation, space beam communication [6], and other fields [7; 8; 9]. These processes require certain fastness and smoothness in target pointing with different amplitudes. In consideration of fastness, the most natural choice is time-optimal control (TOC) under the framework of stability [10; 11], which is bang-bang control, deriving from the minimum principle. The TOC control scheme uses the maximum control capability to achieve the fastest acceleration and deceleration process. However, the uncertainty of the controlled object and the measurement noise will cause the system to generate chattering between the maximum and minimum values [12]. Moreover, the system may diverge because of the low sampling frequency of sensors when using the maximum control capability. Therefore, TOC is not advisable in practical systems. To overcome the shortcomings of TOC and improve the smoothness of the control system, a proximate time-optimal servomechanism (PTOS) is proposed [12; 13]. PTOS indicates that when the system tracking error is large, the control law adopts bang-bang control. When the system tracking error becomes small, the control law switches to linear control instead of bang-bang control in TOC. This operation can reduce the system chattering to a certain extent, but the system performance of swiftness will be deteriorated in exchange for partial smoothness improvement. The set-point tracking control performance is quite different as the amplitude of the target position signal changed. Therefore, the input signal is usually divided into a wide-range signal (WRS) and a small-range signal (SRS). The standard of division is determined by the amplitude of the target position signal and the linear controller parameters in PTOS. It is related to drive capability, natural frequency, and damping ratio in an actual system. The input signal belongs to the SRSs when only the linear part of PTOS applied as the control scheme. If the complete PTOS control scheme is directly used, the input signal is WRS. In this paper, WRSs refer to the input signal with an amplitude greater than \(10^{\circ}\) in an electro-optical set-optical tracking system. Correspondingly, SRSs are the input signal with an amplitude less than or equal to \(10^{\circ}\). In fact, the research of PTOS is still a topic of interest in academic publications, and its advancement in tracking signals at WSRs has been confirmed [14; 15; 16; 17; 18]. The linear control employed by the PTOS during the tracking of signals in SRSs requires a compromise between swiftness and smoothness. Therefore, when using PTOS to track a SRS, there is still much room for improvement in system performance. The authors of [15] proposed a dynamically damped PTOS to nearly eliminate the conservatism of the original PTOS, which is combined with composite nonlinear feedback (CNF). When the input signal is a SRS, the linear feedback part of CNF is used to achieve the swiftness of response, and the nonlinear feedback part makes the system damping ratio in a changing state to get a better dynamic performance. However, the range of the input signal is limited by the parameters of the controller, and the controller parameter values need to be adjusted for different input signals to achieve the expected performance. When applying the control scheme into an actual system, the cumbersome parameter tuning cannot be ignored. To improve the set-point tracking performance of the PTOS control scheme in a SRS, [PERSON] et al. [18] proposed proximate time-optimal sliding-mode control scheme, which uses time-varying sliding surface and a third-order velocity profile to reduce the settling time; however, the paper does not analyze the system stability.