## 2 Related Work
### PID Search Algorithm  
The PID algorithm is a fundamental control strategy widely used in various applications. It can be categorized into two types: incremental PID control and positional PID control. The PID controller, which operates based on proportional, integral, and differential deviation, is one of the most widely utilized automatic control systems. PSA is a meta-heuristic approach designed for analog incremental PID control. The user sets the target value, and the controller continuously adjusts the system deviation to provide an output value to the actuator, thereby converging the entire system to the optimal state [15].  
In a PID control system, the target value of the controlled system is denoted as \(V_{t}\), while the actual value at time \(t\) is represented as \(V(t)\). The deviation in the control system at time \(t\) is defined as \(E(t)=V_{t}-V(t)\)[16]. The proportional control formula can be expressed by Equation (4):  
\[P(t)=K_{p}\cdot E(t)+P_{0} \tag{4}\]where \(P(t)\) represents the output value of the proportional control at time \(t\), and \(K_{p}\) is the proportional coefficient, a factor used to adjust the deviation in the system. Additionally, \(P_{0}\) is a constant introduced to prevent the output value from approaching zero.  
Integral control takes into account the past states of the system being controlled, with the controller's output being proportional to the integral of the input error signal. To eliminate steady-state errors, an integral term in the controller must be introduced. The integral control output value at time \(t\) can be expressed as follows:  
\[I(t)=K_{i}\cdot\int_{0}^{t}E(t)dt+I_{0} \tag{5}\]  
where \(K_{i}\) and \(I_{0}\) represent the integral coefficient and the constant, respectively.  
To mitigate potential instability during error regulation, an early correction signal that predicts the trend in error changes should be introduced into the system. The output value at time \(t\) can be expressed as follows:  
\[D(t)=K_{d}\cdot\frac{dE(t)}{dt}+D_{0} \tag{6}\]  
where \(K_{d}\) represents the differential coefficient. Thus, the classical PID algorithm can be expressed as follows:  
\[O(t)=P(t)+I(t)+D(t)+O_{0} \tag{7}\]  
where \(O(t)\) is the control quantity output by the PID controller at moment \(t\). \(O_{0}\) refers to the output value of the PID controller at the initial time, which is used to compensate for the static error of the system. Its value depends on the static characteristics of the system and the control target.  
In practice, only discrete data returned by the sensors are usually available. Therefore, the discretized form of the conventional PID algorithm is  
\[O(t)=K_{p}E(t)+K_{i}{\sum_{i=0}^{t}E(i)}+K_{d}[E(t)-E(t-1)] \tag{8}\]  
Incremental PID refers to the output of the digital controller as the increment of the control quantity. In this case, the incremental PID control algorithm can be expressed as follows:  
\[\Delta O(t)=K_{p}[E(t)-E(t-1)]+K_{i}E(t)+K_{d}[E(t)-2E(t-1)+E(t-2)] \tag{9}\]  
The above formula shows that the incremental PID control is related to the systematic deviation at times \(t\), \(t-1\), and \(t-2\). Since the deviation of the system will not accumulate, the stability of the control system is improved.