## 2 Methods
In real systems, the electromechanical time constant Tm is much larger than the current loop time constant \(T_{a}\). The dynamic process of the output value \(I_{a}(t)\) changes rapidly, but the change process of anti-electromotive force of the motor \(E(s)\) reflecting the rotation speed is actually slow. Therefore, in the process of designing the current loop, we can ignore the influence of \(E(s)\) on the current loop, which simplifies the computational complexity of the design. The motor dynamic structure block can be further simplified as a double closed-loop system of current and speed. \(G_{a}(s)\) is the current loop controller; \(G_{v}(s)\) is the speed loop controller; and \(G_{0}(s)\) is the mechanical resonance element, which can approximate to a second-order oscillation element for simulating. In Figure 2, \(U_{a}(s)\) is the current loop input signal and \(U_{v}(s)\) is the speed loop input signal. Current loops reflect the electrical characteristics of the control system. The control object characteristic of the current loop between point A and B is generally a first-order inertial element. Set the current loop controller as shown in Equation (10):  
\[E=C_{e}\cdot n \tag{2}\]0  
\[E=C_{e}\cdot n \tag{2}\]1 Motor dynamic structure block. \[E=C_{e}\cdot n \tag{2}\]2 Current speed double closed-loop system control block. Then, the closed-loop characteristic of the current loop is shown in Equation (11):  
\[E=C_{e}\cdot n \tag{2}\]3  
\({{T^{\prime}}_{a}}={{T_{a}}}/{(1+{{k_{1}}}/{{R_{0}}})}=\) new electrical time constant. \({{T^{\prime}}_{a}}\) is \(1/(1+{{k_{1}}}/{{R_{0}}})\) times less than the electrical time constant \({{T_{a}}}\) without the current feedback loop. Then, the object property of the speed loop is shown in Equation (12):  
\[E=C_{e}\cdot n \tag{2}\]4  
To minimize the occurrence of self-excitation oscillation, the bandwidth of the current loop is designed to be more than 5 times the bandwidth of the speed loop. Then, in the range of mechanical resonance frequency, the control object of the speed loop can be approximated as a pure integral element, as shown in Equation (13). \[E=C_{e}\cdot n \tag{2}\]5  
Without considering the sticky friction, after adding the current loop, the controlled object of the speed loop changes from the second-order inertial element to the pure integral element, which makes the speed loop change from a type 0 system to a type 1 system. This method improves the type of the control loop and reduces the parameter uncertainty of the controlled object of the speed loop. It is beneficial to adopt advanced control methods such as modern model-based control methods. On this basis, this paper continues to construct the friction mathematical model and the mechanical resonant mathematical model. \({{G_{0}}(s)}\) is the mechanical resonance model, and \({{G_{f}}(s)}\) is the friction model. Both of the above two models have mature research results, and this paper is directly borrowed from other research results [27; 28]. Then, Then we can obtain the photoelectric pod speed loop control structure block diagram as shown in \[E=C_{e}\cdot n \tag{2}\]6  
\[E=C_{e}\cdot n \tag{2}\]7  
\[a_{1},a_{2},\cdots,a_{n_{a}},b_{1},b_{2},\cdots,b_{n_{b}}=\text{ parameters to be identified. Taking a continuous value of }k\text{, the system of linear equations can be rewritten into a matrix form, as shown in Equation (17):  
\[ \begin{bmatrix}z(1)\\ z(2)\\ \vdots\\ z(L)\end{bmatrix} = \begin{bmatrix}-z(0)&\cdots&-z(1-n_{a})&u(0)&\cdots&-u(1-n_{ b})\\ -z(1)&\cdots&-z(2-n_{a})&u(1)&\cdots&-u(2-n_{b})\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ -z(L-1)&\cdots&-z(L-n_{a})&u(L-1)&\cdots&-u(L-n_{b})\end{bmatrix} \cdot \begin{bmatrix}a_{1}\\ \vdots\\ a_{n_{a}}\\ b_{1}\\ \vdots\\ b_{n_{b}}\end{bmatrix} + \begin{bmatrix}v(1)\\ v(2)\\ \vdots\\ v(L)\end{bmatrix}  \tag{17}\]  
\(L\) = the total number of data. \(z_{L}\), \(\theta\), \(v_{L}\), \(H_{L}\) are rewritten as Equation (18):  
\[H_{L}= \begin{bmatrix}h^{T}(1)\\ h^{T}(2)\\ \vdots\\ h^{T}(L)\end{bmatrix} =