# Robust Model-Free Control for MMC Inverters in Cold Ironing Systems
## 3 Proposed Control Strategy  
Most advanced control strategies require at least partial knowledge of the physical model of the system to be controlled. This is a major drawback, as obtaining these physical equations can be complex. What's more, the nature of these models is often difficult to exploit, especially if they are partial differential equations [41].  
In this study, we use model-free control (MFC). In [42], [PERSON] and [PERSON] introduced the concept of combining an ultra-local model with the notion of model-free control, thus eliminating the need for mathematical modeling. This control method is based on an ultra-local model that represents only the input-output behavior of the system, as shown in Figure 5. The intelligent proportional-integral (iPI) controller continuously updates its parameters to adapt to the dynamic evolution of the system. It is a key component of this approach as it disregards the model of both Single Input Single Output (SISO) and Multi-Input Multi-Output (MIMO) systems because it uses an ultra-local model expressed in an affine form given by Equation (13), using a specific order, \(v\), as follows:  
\[y^{(v)}(t)=F(t)+\alpha u(t) \tag{13}\]  
where \(y^{(v)}(t)\) is the \(v\) order derivative of the system output (typically, \(v\) is set to 1 or 2), \(u(t)\) is the control signal, \(\alpha\in\mathbb{R}\) is a constant input so that the magnitude orders of \(au(t)\) and \(y^{(v)}(t)\) close, and \(F(t)\) is a corrective coefficient, estimated at each computation step.  
Closing the loop using an intelligent PI controller gives the control law expressed by Equation (14):  
\[u(t)\ =\ \frac{y^{(v)}_{d}(t)\ -\ F_{e}(t)\ +\ k_{p}\varepsilon(t)\ +\ k_{i}\int \varepsilon(t)dt}{\alpha} \tag{14}\]  
where \(y^{(v)}_{d}(t)\) is the reference, \(\varepsilon(t)\) is the tracking error, \(\ k_{p}\) and \(\ k_{i}\) are the tuning gains.  
The control law (14) requires the estimation of \(F(t)\), noted \(F_{e}(t)\). In [43], [PERSON] proposed two methods of calculating \(F_{e}(t)\). In this study, \(F_{e}(t)\) is calculated by Equation (15):  
\[F_{e}(t)=y^{(v)}(t)-\alpha u(t-T_{e}) \tag{15}\]  
where \(T_{e}\) represents the sampling period.  
Equation (14) of the ultra-local model of model-free control shows that there are two parameters that we can adjust: the derivative order \(v\) and the parameter \(\alpha\). In this study, we focus on adjusting \(\alpha\) to improve the performance of model-free control. The parameter \(\alpha\) represents a scaling factor between the time derivative of the output and the input. Until now, its adjustment has been based on an empirical approach. However, in systems subject to large variations in input or output amplitude (whether known or unknown), an initial fixed choice of \(\alpha\) may become inadequate.  
In our case study, where the maritime loads fluctuate significantly, the dynamic adaptation \(\Delta\alpha\) of \(\alpha\) would allow us to analyze its response to these fluctuations and optimize the robustness of the system, as represented by Equations (16)-(18):  
\[y^{(v)}(t)=F(t)+(\alpha_{0}+\Delta\alpha)u(t) \tag{16}\]  
\[u(t)\ =\ \frac{y^{(v)}_{d}(t)\ -\ F_{e}(t)\ +\ k_{p}\varepsilon(t)\ +\ k_{i}\int \varepsilon(t)\mathrm{dt}}{\alpha_{0}+\Delta\alpha} \tag{17}\]  
\[F_{e}(t)=y^{(v)}(t)-(\alpha_{0}+\Delta\alpha)u(t-T_{e}) \tag{18}\]  
Figure 5: Block diagram of the model-free control.