# An Experience Transfer Approach for the Initial Data of Iterative Learning Control
## 3 ILC of Recombining Transform  
### Problem Description and Process Analysis  
In the mathematic model of control systems, when the damping of the system is changed, the inertial time parameters in the transfer function and the matrix \(A\) of the state equation are usually changed. When ILC is applied to this kind of system again, an ILC of re-combining transform strategy is proposed based on the tests of step response and impulse response. The re-combined experience is transformed as the control data of the initial iteration learning in the new system. Firstly, the inertia time constant of the original and new systems are estimated by open-loop test of unit step response, as shown in Figure 2.  
Secondly, according to the slope of the system's step response curve and damping, the two systems are modeled approximately. The unit impulse responses of two systems are simulated and analyzed respectively, as shown in Figure 3.  
Figure 1: Iterative Learning Control (ILC) with Different Initial Experiences. Figure 2: Step Response of Systems with Different Time Constants. Next, record the impulse response of the new and the original controlled systems as sequence \(y_{a}\) and \(y_{b}\), respectively. The experience data sequence of ILC is regarded as a set of impulse functions that are input into the controlled system in a timely and sequential way, and the following quantitative analysis is carried out. In the original controlled system, the inertia time constant is set as \(T_{a}\). Then, the control value sequence obtained after \(i\) turns of iterative learning is \(U_{a,i}\). The control value \(u_{a,i}(k)\) at a certain time \(k\) on the corresponding time axis can be expressed as:  
\[ \begin{array}{l}u_{a,i}(0)\Leftrightarrow y_{a,i}(0,1)z^{-1}+y_{a,i}(0,2)z^ {-2}+\cdots+y_{a,i}(0,p)z^{-p}\\ u_{a,i}(1)\Leftrightarrow y_{a,i}(1,2)z^{-2}+y_{a,i}(1,3)z^{-3}+\cdots+y_{a,i} (1,p+1)z^{-p-1}\\ \cdots\\ u_{a,i}(k)\Leftrightarrow y_{a,i}(k,k+1)z^{-k-1}+y_{a,i}(k,k+2)z^{-k-2}+ \cdots+y_{a,i}(k,k+p)z^{-k-p}\end{array}  \tag{5}\]  
where \(p\) is 3 times of the system's inertia time constant \(T_{a}\). \(z^{-i}\) denotes the delayed effect of pulse input in the discrete system. The subsequent effect after \(p\) could be considered to decay towards 0 and be ignored. Similarly, in the new controlled system, the inertia time constant is \(T_{b}\). Its initial iterative learning data is supposed as \(U_{b,0}\). The control value sequence \(u_{b,0}\)(\(k\)) at a certain time \(k\) on the corresponding time axis can be expressed as:  
\[ \begin{array}{l}u_{b,0}(0)\Leftrightarrow y_{b,0}(0,1)z^{-1}+y_{b,0}(0,2)z^ {-2}+\cdots+y_{b,0}(0,q)z^{-q}\\ u_{b,0}(1)\Leftrightarrow y_{b,0}(1,2)z^{-2}+y_{b,0}(1,3)z^{-3}+\cdots+y_{b,0 }(1,q+1)z^{-q-1}\\ \cdots\\ u_{b,0}(k)\Leftrightarrow y_{b,0}(k,k+1)z^{-k-1}+y_{b,0}(k,k+2)z^{-k-2}+ \cdots+y_{b,0}(k,k+q)z^{-k-q}\end{array}  \tag{6}\]  
where \(q\) is 3 times of the system's inertia time constant \(T_{b}\). Additionally, the subsequent effect after \(p\) could be considered to decay towards 0 and be ignored. In Equation (5), the output response sequences \(y_{a}\) of control data sequence \(u_{a}\) combine together to form the whole desired trajectory. So do those of \(y_{b}\). The control data sequences in Equation (6) are desired to have the same effect as that of Equation (5). The relation can be obtained via the discrete impulse response of the two systems shown in Figure 3. According to the energy-result balance and superposition principle in the linear system, there is the following relation:  
Figure 3: Impulse Response of Two Systems.