# Dynamic High-Type Interval Type-2 Fuzzy Logic Control for Photoelectric Tracking System
## 2 Problem Formulation and Preliminaries
### Dhtc  
The commonly used photoelectric tracking control method is the classic type-I double closed-loop feedback control, as shown in Figure 1. In the figure, Gv is the mathematical model identified from the FSM experimental platform, and Cv is the speed controller. These two parts form an inner loop called the speed loop; the speed loop and the position controller Cp in the series form an outer loop called the position loop. The open-loop transfer function of the system is  
\[G(s)=\frac{a\!\cdot\!k\!\cdot\!(T_{3}s+1)}{s\!\cdot\!(T_{4}s+1)\!\cdot\!(a+(T_{ 1}s+1)\!\cdot\!(T_{2}s+1))}=\frac{K\prod_{i=1}^{m}(\tau_{i}s+1)}{s^{\
u}\prod_ {j=1}^{m-\
u}(T_{j}s+1)} \tag{1}\]  
where \(a\) and \(k\) are the gains of the inner loop and the outer loop, respectively, and \(T_{1}\), \(T_{2}\), \(T_{3}\), and \(T_{4}\) are the time constants of the transfer function of each controller. According to the definition in the principle of automatic control, since there is an integrator in the system, namely \(v=1\), it is a type-I system. The systematic error \(e\) is defined as  
\[e(s)=r(s)-y(s)=(1-\Phi(s))r(s)=E(s)r(s), \tag{2}\]  
where \(r\) is reference input, \(y\) is systematic output, \(E(s)\) is the error transfer function, \(\Phi(s)\) is the closed-loop transfer function of the system  
\[\Phi(s)=\frac{y(s)}{r(s)}=\frac{b_{m}s^{m}+b_{m-1}s^{m-1}+\ldots+b_{1}s+b_{0}} {a_{n}s^{n}+a_{n-1}s^{n-1}+\ldots+a_{1}s+a_{0}} \tag{3}\]  
Figure 1: The classic type-I double closed-loop feedback system.  
Therefore, \(e(s)\) can be expressed as  
\[e(s)=\frac{a_{n}s^{n}+\ldots+c_{m}s^{m}+c_{m-1}s^{m-1}+\ldots+c_{1}s+c_{0}}{a_{n} s^{n}+a_{n-1}s^{n-1}+\ldots+a_{1}s+a_{0}} \tag{4}\]  
where \(c_{i}=(a_{i}-b_{i}),i=0,\ldots,m\). According to the final value theorem, when the system is in a stable state, the steady-state error \(e(\infty)\) can be expressed as  
\[e(\infty)=\underset{s\to 0}{\lim}\bigg{(}\frac{a_{n}s^{n}+\ldots+c_{m}s^{m} \ldots+c_{1}s+c_{0}}{a_{n}s^{n}+a_{n-1}s^{n-1}+\ldots+a_{1}s+a_{0}}\bigg{)}r(s) =\frac{\underset{s\to 0}{\lim}\big{[}s^{r+1}r(s)\big{]}}{K+\underset{s\to 0}{\lim}s^{r}} \tag{5}\]  
The step signal, velocity signal, and acceleration signal existing in the physical system are expressed as \(r(s)=\frac{R_{1}}{s},r(s)=\frac{R_{2}}{s^{2}},r(s)=\frac{R_{3}}{s^{3}}\). For the superimposed input of these three signals, \(e(\infty)\) will remain in a certain range only when \(v\geq 2\); that is, the system can reach a stable state only if the system type is greater than 2.  
To improve the steady-state accuracy of the system, multiple integrators are added to the system to change the system into a high-type state. However, the increase of the system type will increase the step response overshoot and increase the system oscillation; as the gain of the integrator becomes larger, the oscillation will be greater, and in some extreme cases it will even reach the integral saturation state, causing the system to lose stability completely.  
To solve the above-mentioned problems, the 'dynamic high-type' method was proposed. As shown in Figure 2, multiple integrators are connected in parallel in the forward path, and the on and off of the integrators are judged according to the system error state, and the type of the system is dynamically changed. However, there are two problems to be solved:  
Question 1. The standard for type switching is not defined. When to increase the type and when to decrease it, no theoretical guidance is given in the current literature. Besides, the manual switching method used in most of the literature is inaccurate and not practical.  
Question 2. The jitter caused by type switching is difficult to eliminate. In the existing research, every moment of type switching will bring greater jitter to the system, increase instability, and reduce its robustness.