## 3 A Fractiona-Order Sliding Mode Controller (FOSMC)  
Fractional calculus is the extension of classical calculus to derivation and integration operations using non-integer orders. In the time domain, fractional (non-integer) derivatives and integrals are defined by the convolution operation. Hence, they are especially suited to representing memory phenomena, with several applications in science and engineering [35]. In this paper, a factional-order sliding mode controller (FOSMC) is proposed in order to track a predefined operating path of a photobioreactor (eutrphying microalgae photobioreactor). The Riemann-Liouville and Caputo definitions of fractional operators are the most used. In this work, the Caputo definition is used. According to Reference [36], the Caputo fractional derivative of order \(\beta\) of function \(\phi(t)\) on the half-positive real axis is defined as  
\[D^{\beta}\phi(t)=\frac{1}{\Gamma(n-\beta)}\int_{0}^{t}\frac{\phi^{(n)}(\tau) }{(t-\tau)^{\beta+n-1}}d\tau, \tag{8}\]  
where \(\phi(t)\) is a function assumed to be sufficiently smooth and locally integrable; \(\Gamma\) is is the so-called gamma function defined as \(\Gamma(n)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}dt\); and \(n-1<\beta<n\), with \(n\) being and integer. This Caputo definition is used the most in engineering applications since this definition incorporates initial conditions for \(\phi(t)\) and its integer-order derivatives; this is initial conditions that are physically appealing in the traditional way. Notice that \(\phi^{(n)}(t)\) stands for the integer derivative of order \(n\) of function \(\phi(t)\) in Equation (8). Also, to simplify the notation, we will denote the Caputo fractional derivative of order \(\beta\) of function \(\phi(t)\), \(D^{\beta}\phi(t)\), as \(\phi^{(\beta)}\). In addition, the fractional integral of order \(\beta\) of function \(\phi(t)\) on the half-positive real axis is defined, also in accordance with [36], as  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]  
where \(\beta\) is the fractional order defined above. It is important to notice that the notation \(D^{-\beta}\phi(t)\) is used as well as to denote the fractional integral of order \(\beta\) of function \(\phi(t)\), more precisely \(D^{-\beta}\phi(t)\equiv l^{\beta}\phi(t)\). The definitions of fractional derivative and fractional integral, as stated above, cannot be used in practice; thus, numeric methods such as the one based on the Grunwald-Letnikov approach are commonly used [37]. We consider the following first-order nonlinear perturbed system:  
\[\dot{x}(t)=f(x)+g(x)u(t)+\delta(x), \tag{10}\]  
where \(f(x)\) and \(g(x)\) are bounded, smooth, and locally integrable functions; \(u(t)\) is a scalar input variable; and \(\delta(x)\in\mathbb{R}\) is a disturbance term that satisfies the bound in Equation (4) in Assumption 1. To design a reference tracking controller, the reference signal \(x(t)_{ref}=x_{r}\) is proposed together with the tracking error:\[e=x_{r}-x. \tag{11}\]  
Based on the sliding mode control methodology [38], the following switching function \(s\) is then proposed:  
\[s(t)=e(t)+k_{1}\underset{0}{\overset{t}{\int}}e(\tau)d\tau, \tag{12}\]  
where \(k_{1}\) is a nonzero positive real parameter. The switching function in Equation (12) defines the sliding surface  
\[s=0=e(t)+k_{1}\underset{0}{\overset{t}{\int}}e(\tau)d\tau, \tag{13}\]  
for which \(k_{1}\) is selected in such a way that the corresponding first-order differential equation \(0=\dot{e}+k_{1}e\) has a solution that exponentially converges to zero; as a consequence, \(x\) converges to \(x_{r}\) exponentially when the system dynamics is constrained to the sliding surface in Equation (13).