## 1 Introduction
In [19], the PTOS is applied to the hard disk drive system, and the experimental results at 180 degrees, 360 degrees, and 720 degrees set-point tracking indicate the swiftness, smoothness, and accuracy of the control scheme. However, the response performance to SRSs is not analyzed by this method, and the controller switching time is extremely difficult to determine. It is difficult to apply it to systems that have a strong demand for SRSs tracking. A PTOS control scheme based on damping ratio scheduling is proposed in [20], which uses adaptive velocity gain to achieve variable damping ratio of system, but the description of system stability is also not covered. On the other hand, facing the control input saturation in SRS tracking when the process encounters the model uncertainty, the authors of [21] proposed a new anti-windup compensator for an uncertain linear system, which is the three degrees of freedom (3-DOF) structure composed of the controller, prefilter, and anti-windup compensator. Specifically, it is necessary to consider the Horowitz-Sidi bounds of the 3-DOF at the same time in Nichols chart to obtain the anti-saturation compensator. From the perspective of data-driven methods, the authors of [22] proposed a model-free neural network (NN)-based control scheme in an Adaptive Actor-Critic (AAC) learning framework. The initial controller for AAC designs is obtained from input-state-output data collected of the process in open-loop setting. Then the resulting suboptimal state-feedback controller is next improved under the AAC learning framework by online adaptation of a critic NN and a controller NN. In [23], the traditional two step anti-windup control methodology is applied to multiagent systems and the input constrained consensus tracking problem is solved based on distributed anti-windup compensator design. In this paper, the transition process (TP) is employed by recalculating the reference input signal. The controller can follow the new reference input without entering the saturation region, which ensures that the controller input is always equal to actuator output, thereby eliminating the windup phenomenon. Specifically, the TP is derived using the optimal control theory, and the performance index is the time optimum to maintain its swiftness. The derivation results show that the TP can be designed by the target position signal of any amplitude and the limit performance of actuator, which imply the design method is universal. In the view of above discussions, the control scheme novelty and contributions of this paper is as follows. (1) A proximate time-optimal servomechanism based on transition process (PTSTP) control scheme is proposed to reduce transition time and avoid big chatter simultaneously in set-point tracking processes, which incorporates a TP into the framework of PTOS. Furthermore, the stability region of PTSTP control scheme is proved. Finally, the proposed methodology is verified in an electro-optical set-point tracking system with inertial sensor-based closed loop. (2) Compared with other improved PTOS control schemes, the PTSTP completely separates the design of the closed loop controller and the TP, which reduces much of the complex controller design work. Moreover, PTSTP has no limitations on the amplitude of the target position signal, which fully reflects the effectiveness of the control scheme for set-point tracking in the whole range. (3) On the other hand, the design of TP is also an anti-windup method in essence. Compared with the former methods, it focuses on the processing of input signals rather than a back-calculation structure. It provides the engineers another choice to anti-windup, which can be directly added to the closed-loop outside according to the positioning amplitude and the limit performance of actuator. The rest of this paper is organized as follows. In Section 2, we elaborate on the theoretical derivation of the standard PTOS and the analysis of the control algorithm. The PTSTP control scheme is shown in Section 3.