## 3 Air Pressure Compensation Algorithm
### A-ILC Algorithm  
For a specific control system, the gain matrix of the PD-ILC algorithm cannot be solved and needs to be subjectively selected. This relies more on the designer's experience, and it is easy to cause the system to fail to converge due to improper selection. Therefore, for the air pressure compensation system, an A-ILC algorithm is used to solve the problem to improve the accuracy of the air pressure compensation and control. Figure 7: PD-ILC controller of the air pressure compensation system. The dynamic equation of the nonlinear discrete system at the \(k\)-th iteration is:  
\[y_{k}(t+1)=f(y_{k}(t),y_{k}(t-1),\ldots,y_{k}(t-n),u_{k}(t),u_{k}(t-1),\ldots,u_ {k}(t-n_{u})), \tag{32}\]  
where \(u_{k}(t)\) and \(y_{k}(t)\) are the input and output of the system at the \(k\)-th iteration; \(f(\cdot)\) is the non-linear function. Therefore, relative to the \((k-1)\)-th iteration, the input and output deviations of the \(k\)-th iteration are:  
\[ \begin{array}{l}\Delta y_{k}(t)=y_{k}(t)-y_{k-1}(t)\\ \Delta u_{k}(t)=u_{k}(t)-u_{k-1}(t)\end{array}  \tag{33}\]  
Suppose that the system satisfies the following conditions:  
1. The partial derivative of \(f(\cdot)\) with respect to \(u(t)\) exists and is continuous;
2. When \(\Delta u_{k}(t)\
eq 0\), the system satisfies the generalized Lipschitz continuity. That is: \[\Delta y_{k}(t+1)\leq b|\Delta u_{k}(t)|,\ \forall t\in\{0,\ 1,\ \ldots,T\}, \forall k=0,1,\ldots\] (34) where \(b\) is a constant. It can be proven that the above-mentioned nonlinear system has a \"quasi pseudo partial derivative\" \(\theta_{k}(t)\) to meet the following equation:  
\[\Delta y_{k}(t+1)=\theta_{k}(t)\Delta u_{k}(t), \tag{35}\]  
where \(|\theta_{k}(t)|\leq b\). In the iteration process, Equation (35) can be rewritten as  
\[y_{k}(t+1)=y_{k-1}(t+1)+\theta_{k}(t)\Delta u_{k}(t) \tag{36}\]  
The performance index function of the input of the system is defined as:  
\[J(u_{k}(t))=|e_{k}(t+1)|^{2}+\lambda|u_{k}(t)-u_{k-1}(t)|^{2} \tag{37}\]  
Using the optimal condition \(\frac{\partial f}{2\partial u_{k}(t)}=0\), the following can be obtained:  
\[u_{k}(t)=u_{k-1}(t)+\lambda\frac{\rho\theta_{k}(t)}{\lambda+|\theta_{k}(t)| ^{2}}e_{k-1}(t+1) \tag{38}\]  
Equation (38) is the parameter adaptive iterative learning law, where \(\theta_{k}(t)\) can be obtained by its estimated value \(\hat{\theta}_{k}(t)\) in the iteration process. Combining Equation (37), \(\hat{\theta}_{k}(t)\) can be solved and calculated by the following formula:  
\[\hat{\theta}_{k}(t)=\hat{\theta}_{k-1}(t)+\frac{\eta\Delta u_{k-1}(t)}{\mu+ \left|\Delta u_{k-1}(t)\right|^{2}}\big{(}\Delta y_{k-1}(t+1)-\hat{\theta}_{k -1}(t)\Delta u_{k-1}(t)\big{)}, \tag{39}\]  
where \(\mu>0\) is the weight factor; and \(\eta>0\) is the step sequence. Combining Equations (38) and (39), it can be proven that the system converges when parameters \(\eta\), \(\mu\), \(\rho\) and \(\lambda\) meet the conditions of Equation (40). \[\left\{ \begin{array}{l}0<\left|1-\frac{\eta\Delta u_{k-1}(t)^{2}}{\mu+\left| \Delta u_{k-1}(t)\right|^{2}}\right|<1\\ 0<\rho<\frac{2\left(\lambda+\left|\hat{\theta}_{k}(t)\right|^{2}\right)}{\theta _{k}(t)\hat{\theta}_{k}(t)}\end{array} \right.\right. \tag{40}\]  
Aiming at the presence of unknown external disturbances during the normal operation of high-speed elevators, based on the above-mentioned principle of the parametric adaptive iterative learning control algorithm, the adaptive iterative learning controller of the air pressure compensation system is designed, as shown in Figure 8.  
Before designing the adaptive iterative learning control algorithm for the air pressure compensation system, the following assumptions are given:  
Assumption 1: The initial state of the system is the same at every operation of the high-speed elevator. \[x_{i}(0)=x_{d}(0),\ \forall i\in N \tag{41}\]  
where \(x_{i}(0)\) is the initial state information of the air pressure compensation system in the \(i\)-th iteration; and \(x_{d}(0)\) is the initial state information of the IAPC in the high-speed elevator car.