## 4 Control Strategies
### The Second Strategy  
This strategy is similar to the previous one but it differs in the derivative action implemented in \(G_{PID}\), that is replaced by the fractional-order derivative (see \(G_{PID}^{\alpha}\)). It is computed in the following way.  
The desired \(\alpha\)-order fractional derivative of the computed signal of error,  
\[e_{l}(t)=\theta_{lr}(t)-\theta_{la}(t),\quad\tilde{e}_{l}(t)=\beta_{1}e_{l}(t), \tag{15}\]  
can be estimated using moments of a function.  
The fractional derivative of order \(\alpha\) will be approximated by a finite sum of \(N\) terms, including derivatives of integer order employing continuous expansion of the left Riemann-Liouville derivative.  
Figure 3: Block diagram of the closed-loop model reference Proportional Integral Derivative (PID) control with the additional time-dependent disturbances of the damping coefficient \(b_{la}(t)\) (case A) and the loading torque \(T_{la}(t)\) (case B).  
According to Lemma 2.12 in Reference [21], let \(\tilde{e}_{l}(t)\in C[a,b]\) and \(0<\alpha<1\). Then, the left Riemann-Liouville fractional derivative \({}_{a}D_{t}^{a}\tilde{e}_{l}(t))\) exists almost everywhere in \([a,b]\). Moreover, \(t\in(a,b)\), \({}_{a}D_{t}^{a}\tilde{e}_{l}(t)\in L_{p}[a,b]\), for \(1\leq p<1/\alpha\), and  
\[{}_{a}D_{t}^{a}\tilde{e}_{l}(t)=\frac{1}{\Gamma(1-\alpha)}\left[\frac{\tilde{ e}_{l}(t)(a)}{(t-a)^{\alpha}}+\int_{a}^{t}(t-\tau)^{-\alpha}\tilde{e}_{l}(\tau)d \tau\right]. \tag{16}\]  
Integration of Equation (16) by parts yields  
\[{}_{a}D_{t}^{a}\tilde{e}_{l}(t)= \frac{\tilde{e}_{l}(a)(t-a)^{-\alpha}}{\Gamma(1-\alpha)}+\frac{( t-a)^{1-\alpha}}{\Gamma(2-\alpha)}\Big{[}\tilde{e}_{l}(a)+\int_{a}^{t}C^{ \prime}(\tau)\tilde{e}_{l}(\tau)d\tau\Big{]}, \tag{17}\]  
where, by definition  
\[C^{\prime}(\tau)=\left(1-\frac{\tau-a}{t-a}\right)^{1-\alpha}. \tag{18}\]  
Since the condition \((\tau-a)\leq(t-a)\) is satisfied, applying the binomial theorem to the function \(C^{\prime}(\tau)\), one gets  
\[C^{\prime}(\tau)=\sum_{p=0}^{\infty}\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha- 1)p!}\left(\frac{\tau-a}{t-a}\right)^{p}. \tag{19}\]  
Integration of Equation (17) by parts, including the definition (19) of the term \(C^{\prime}(\tau)\) under the integral, gives  
\[{}_{a}D_{t}^{a}\tilde{e}_{l}(t)= A(\alpha)(t-a)^{-\alpha}\tilde{e}_{l}(t)+B(\alpha)(t-a)^{1-\alpha} \tilde{e}_{l}(t)-\sum_{p=2}^{\infty}C(\alpha,p)(t-a)^{1-\alpha-p}V_{p}(\tilde {e}_{l}(t)), \tag{20}\]  
where, as it is defined in Reference [22], the moment of a function \(\tilde{e}_{l}(t)\in C^{2}[a,b]\), for \(p\in\mathbb{N}\) where \(p\geq 2\), \(t\geq a\), is given by the formula  
\[V_{p}(\tilde{e}_{l}(t))=(1-p)\int_{a}^{t}(\tau-a)^{p-2}\tilde{e}_{l}(\tau)d\tau, \tag{21}\]  
and  
\[\begin{split} A(\alpha)=\frac{1}{\Gamma(1-\alpha)}\left[1+\sum_ {p=2}^{\infty}\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha)(p-1)!}\right],\quad B( \alpha)=\frac{1}{\Gamma(2-\alpha)}\left[1+\sum_{p=1}^{\infty}\frac{\Gamma(p- 1+\alpha)}{\Gamma(\alpha-1)p!}\right],\\ C(\alpha,p)=\frac{\Gamma(p-1+\alpha)}{\Gamma(2-\alpha)\Gamma( \alpha-1)(p-1)!}.\end{split} \tag{22}\]  
In this work, the above algorithm does not take into account any closed form function of \(\tilde{e}_{l}(t)\). One assumes in the method that a finite discrete series of \(n+1\) samples \(\tilde{e}_{l}(t_{j})\) is available at the time instants \(t_{j}=jh\), for \(j=0,\ldots,n\), and the constant sampling period \(h\). Therefore, in Equations (20) and (22), the time \(t\) is to be replaced with a \(j\)-th measurement sample at \(t_{j}\in[t_{0},t_{n}]\), while \(t_{0}=a\) and \(t_{n}=b\), and accordingly, \(\infty\) is to be replaced with a finite number \(N\) of terms of approximation of the fractional left Riemann-Liouville derivative of order \(\alpha\).  
In the numerical simulation the parameters of the regulator for both considered types of disturbance will be valued as follows: \(K_{p}=300\), \(T_{i}=1\), \(\beta_{1}=1\), \(\alpha=0.7\) (the fractional derivative part), \(t_{\rm hr}=0.2\) (the time horizon reset).