Abstract: To address the vibration control problem of servo systems, an H-mixed sensitivity design method is introduced based on the established mathematical model of the flexible arm. The selection of each weighting function is discussed. Simulation results show that the H-mixed sensitivity method has certain advantages. It is robust to the uncertainty of the object and also has good tracking performance and anti-interference ability. It is a promising new research direction in this field. Keywords : flexible arm; mixed sensitivity; vibration control This approach has advantages in robustness against uncertainty and abilities in position tracking and dtsturbance rejection, therefore providing a potential research direction in this domain. Key words: flexible arm; mixed sensitivity; vibration restrained control. As is well known, the mechanical transmission connection between the motor and the load is not an absolutely rigid body. During high-speed operation, it will excite unmodeled dynamics, producing the so-called "overflow," causing the entire unit to resonate. Under such circumstances, traditional PI control is difficult to achieve satisfactory results. How to better solve the mechanical resonance problem in the position servo system is what we are facing. The above problem can theoretically be classified into the active research field of flexible arm control. With the increasing demand for lightweight, high-speed, and high-performance machines in space exploration and manufacturing, the research on flexible arm modeling and control has become increasingly in-depth. In recent years, many scholars have carried out some new research attempts, among which the application of H control theory is more eye-catching. It mainly presents two branches: the idea of ​​H linear control theory and the method involving H nonlinear control theory. The former is widely used because it is easy to implement, but due to the existence of nonlinear unmodeled dynamics, the modeling is not accurate enough. It is inevitable that the system's performance will be affected; the latter requires a deep understanding of the system's dynamic model. As for how to effectively put it into practice, we still need to continue our efforts, but this will undoubtedly be a promising new research direction in the field of elastic arm control. This paper applies the H hybrid sensitivity theory to design the anti-vibration control of the position servo system and explores a new research approach. 1 System Model This paper takes a small permanent magnet DC torque motor as the research object. Without considering the elastic arm, the armature circuit voltage balance equation is obtained. The motor shaft torque balance equation is obtained from equations (1) and (2) to obtain the system structure in Figure 1, that is, the nominal model. When the unmodeled dynamic is excited, the action of the elastic arm makes θ on the load shaft not equal to θ on the motor shaft. Equation (2) becomes the load shaft torque balance equation: The structure in Figure 2 is obtained from equations (1), (4), and (5): The object model when the unmodeled dynamic is excited can be obtained. The meaning of each letter in the above equations and their specific data are shown in Table 1. Among them, K[sub]c[/sub]J[sub]L [/sub] will undergo perturbation changes during operation. Obviously, when the unmodeled dynamics are excited, a flexible arm element will be generated, adding two weakly damped points to the system. This results in a significant lag in the open-loop frequency response and a reduction in the phase margin, ultimately affecting the system stability. The feedback system structure with weighted functions is shown in Figure 3, where W is the external input; Y is the system output signal; e is the error signal; U is the control input; K(S) is the controller; and Z1, Z2, and Z3 are the evaluation signals of the weighted functions W1, W2, and W3, respectively. 2 Hybrid Sensitivity Design The hybrid sensitivity problem can be represented by Figure 4, where Y is the controller input. The design requires S(S) and T(S) to be called the sensitivity and complementary sensitivity functions, respectively, and W(S) to be called the sensitivity and complementary sensitivity weighting functions, respectively. Hybrid sensitivity design can suppress the influence of disturbances on control errors and the influence of object model uncertainty on the system. The object of this study, the motor, is subject to disturbances during operation, and as mentioned earlier, unmodeled dynamics with flexible arms are generated during high-speed operation. These two problems are precisely what hybrid sensitivity design can solve. Therefore, it is necessary and meaningful to introduce the mixed sensitivity method in servo system control. The specific selection of weights is the key to solving the problem. The sensitivity function marks the anti-interference ability and tracking performance of the system. The smaller the singular value of the sensitivity function, the stronger the anti-interference and tracking ability. Considering that these factors mostly occur in the low frequency band, ||W1(S)S(S)||＜1 can make the gain of W(S) in the low frequency band as large as possible, and its cutoff frequency is selected in the low frequency band. At the same time, given that the object itself has a pure integral element, this makes S(s) have a pure differential element. It is well known that W(S) represents the norm bound of multiplicative perturbation. Therefore, equation (11) can be used. Since R(s) is the transfer function of the input to the control quantity, w(s) can limit u to be too large. In order to make the controller less complicated, it is advisable to take W(s)=k. In the design process, it is required to find a compromise between bandwidth and interference suppression. In this paper, the design is carried out by changing w1(s) and W2(s). During simulation, it was found that the system cutoff frequency increases with increasing P, but decreases with increasing J. However, excessively large values ​​do not yield satisfactory results, and may even lead to no solution. Finally, k1=0.03 was chosen, and P and k were adjusted to meet certain bandwidth requirements, ultimately resulting in P=80 and k=0.1. 3. Simulation The mathematical model introduced in Part 1 yields the following two equations. The H-mixed sensitivity design method described in the text is applied to this model: Meanwhile, the PI controller (K=0.08, K=0.1) can meet the system's requirements. The two controllers are compared. Figure 5 shows the system step response under the nominal model; Figure 6 shows the system step response when dynamically excited without modeling (and K and J undergo 20% perturbation); Figure 7 shows the nominal model response under external disturbance (M = 0.49 N·m); Figure 8 shows the response under external disturbance and when dynamically excited without modeling. From Figure 8, it can be seen that under the nominal model... Both PI and H controllers can meet good performance requirements, but when dynamically excited without modeling, the system using the PI controller will produce large oscillations, resulting in poor robustness. The system using the H controller, however, still maintains good results. 4. Conclusion This paper introduces the H hybrid sensitivity design into the vibration damping control of servo systems. The singular problem is transformed into a standard H design. The selection of each weighting value is discussed. Simulations show that the controller using the above design does indeed have good tracking performance, robustness, and anti-interference capabilities. The hybrid sensitivity design method is a promising research direction in the vibration damping control of servo systems. Click here to download the original text.