## 4 Proposed Control Strategy for Maximum Power Extraction
### Arbitrary Order Sliding Mode Control Design  
In this section, a NAAOSMC technique is proposed to obtain maximum power from the PV module. The output of the controller \(u\) controls duty cycle of switches of DC-DC converter. For designing the control strategy, the system (14) is converted in canonical form as  
\[\left\{ \begin{array}{l}\dot{y}_{1}=y_{2}\\ \dot{y}_{2}=\frac{\dot{y}_{re}}{C_{1}}-u\frac{\dot{x}_{2}}{C_{1}}-\dot{u}\frac {\dot{x}_{2}}{C_{1}}+\Delta(y_{1},y_{2},t)\end{array} \right. \tag{22}\]  
The system (22) is the controllable canonical form in term of output and its derivative, which is assumed convenient for designing a control strategy. Now, defining tracking error as difference between the reference and maximum PV voltage i.e.,  
\[e=y_{1}-y_{ref} \tag{23}\]  
where \(y_{ref}=V_{ref}\) and  
\[ \begin{array}{l}y_{1}=x_{1}\\ y_{2}=\dot{x_{1}}=\frac{\dot{y}_{re}}{C_{1}}-u\frac{\dot{x}_{2}}{C_{1}}\end{array}  \tag{24}\]  
We can achieve our results by converging the tracking error to zero. Taking derivative of (23) and simplify using (24), we get  
\[\dot{e}=\frac{\dot{i}_{pv}}{C_{1}}-u\frac{\dot{x}_{2}}{C_{1}}-\dot{y}_{ref} \tag{25}\]  
As the reference voltage has a fix value so \(\dot{y}_{ref}=0\). Taking derivative of (25)  
\[\ddot{e}=\frac{\dot{i}_{pv}}{C_{1}}-u\frac{\dot{x}_{2}}{C_{1}}-\frac{\dot{x}_{ 2}}{C_{1}}\dot{u} \tag{26}\]  
Now, the sliding manifold \(\sigma\) in terms of error is characterized as follows  
\[\sigma=\dot{e}+\lambda e+\int f(t)dt \tag{27}\]  
Figure 10: The error histogram with 20 bins. Here, \(\lambda\) has a constant value and \(f(t)\) is a forcing function, which carries following expression for AOSMC  
\[ \begin{split} f(t)=c_{2}\mid\dot{e}\mid^{a_{2}}sign(\dot{e})+b_{2} \mid\dot{e}\mid^{\beta_{2}}sign(\dot{e})+\\ c_{1}\mid\dot{e}\mid^{a_{1}}sign(\dot{e})+b_{1}\mid\dot{e}\mid^{ \beta_{1}}sign(\dot{e})\end{split}  \tag{28}\]  
Now, taking the time derivative of (27)  
\[\dot{v}=\ddot{e}+\lambda\dot{e}+f(t) \tag{29}\]  
Now, integrating the values of \(\dot{e}\), \(\ddot{e}\) and posing \(\dot{v}=0\), we get the following control law equation, which operates dynamics of system on the sliding manifold \(\sigma=0\)  
\[\dot{u}_{equ}=\frac{1}{x_{2}}[i_{pv}-\dot{x}_{2}u+\lambda(i_{pv}-x_{2}u)+C_{1} f(t)] \tag{30}\]  
Since a practical system works during uncertainties, so, equivalent control law given in (30) will not be able to enforce sliding mode. The overall control law which can enforce the sliding manifold is computed as  
\[\dot{u}=\dot{u}_{equ}+\dot{u}_{d} \tag{31}\]  
where \(\dot{u}_{d}\) is given by  
\[\dot{u}_{d}=-k_{1}\sigma-k_{2}sign(\sigma) \tag{32}\]  
Here, \(k_{1}\) and \(k_{2}\) are positive gain constants. Stability Analysis  
The objective here is to prove the zero dynamic stability of the system. AOSMC law has been designed using dynamics of first two equations of the system (14). The dynamics of the third equation of the system (14) is clearly the internal dynamics of the given PV system i.e.,  
\[\dot{x}_{3}=(\frac{1}{C_{2}}-\frac{u}{C_{2}})x_{2}-\frac{1}{R_{L}C_{2}}x_{3} \tag{33}\]  
Since, the control input \(u\) directly affects control driven states like \(x_{1}\) and \(x_{2}\) therefore, zero dynamics of a system can be achieved by putting \(u=x_{1}=x_{2}=0\). So, one can get  
\[\dot{x}_{3}=-\frac{1}{R_{L}C_{2}}x_{3} \tag{34}\]  
Since, the typical parameters \(R_{L}\) and \(C_{2}\) are positive, which ensures that the (34) has its poles in the left half-plane at \(-\frac{1}{R_{L}C_{2}}\). This shows that the system (34) has exponentially stabilized the zero dynamics and validated the PV system's minimum phase nature.