## 2 Process Modelling  
A photobioreactor is a special type of bioreactor where photosensitive living organisms are cultivated [3]. Despite the complexity of the process dynamics of this system, it is possible to derive a treatable mathematical model that represents well the main features of the photobioreactor behavior [4]. Applying mass balances, the proposed model considers biomass concentration (represented by \(x_{1}(t)\)) and substrate concentration (\(x_{2}(t)\)) in the bioreactor [4]. The specific biomass growth rate mostly defines the bioreactor dynamics; hence, an accurate representation of this term is crucial (see, for example, Reference [3]). Our growth rate representation includes the substrate concentration and a complex global condition \(\Sigma(\cdot)\), i.e., \(\mu(x_{2}(t),\Sigma(\cdot))=\mu(\cdot)\). This second component incorporates all those non-modelled effects that have a significant impact on the growth dynamics, such as sudden changes in pH, luminous intensity, metabolism, temperature, and substrate concentration gradients. Biomass and substrate concentration dynamics are given by the corresponding mass balances in the fed-batch bioreactor:  
\[\textit{Accumulation = Inputs-Outputs + Generation-Consumption}.\]  
Therefore,  
\[\frac{d}{dt}(x_{3}(t)x_{1}(t)) = x_{3}(t)\mu(\cdot)x_{1}(t),\] \[\frac{d}{dt}(x_{3}(t)x_{2}(t)) = -\alpha_{1}x_{3}(t)\mu(\cdot)x_{1}(t)+\alpha_{2}u(t),\] \[\frac{d}{dt}x_{3}(t) = u(t), \tag{1}\]  
where \(x_{3}(t)\) is the bioreactor volume, \(\alpha_{1}\) is the substrate/biomass yield coefficient, \(\alpha_{2}\) is the substrate concentration in the inlet stream, and \(u(t)\) is the control variable which in this case is the inlet stream flow-rate. Usually, the volume varies until a desirable steady state is reached (fed-batch operation), after which the bioreactor is operated in continuous mode (constant volume and equal inlet and outlet flow-rates). In this paper, we are only concerned with the fed-batch stage. Equation (1) can be written as follows:  
\[\frac{dx_{3}}{dt}x_{1}(t)+\frac{dx_{1}(t)}{dt}x_{3}(t) = x_{3}(t)\mu(\cdot)x_{1}(t),\] \[\frac{dx_{3}}{dt}x_{2}(t)+\frac{dx_{2}(t)}{dt}x_{3}(t) = -\alpha_{1}x_{3}(t)\mu(\cdot)x_{1}(t)+\alpha_{2}u(t),\] \[\frac{dx_{3}}{dt} = u(t), \tag{2}\]  
or, equivalently,  
\[\dot{x}_{1} = \mu(\cdot)x_{1}(t)-\frac{u(t)}{x_{3}(t)}x_{1}(t),\] \[\dot{x}_{2} = -\alpha_{1}\mu(\cdot)x_{1}(t)+\frac{u(t)}{x_{3}(t)}(\alpha_{2}-x _{2}(t)),\] \[\dot{x}_{3} = u(t). \tag{3}\]  
An unknown additive disturbance function \(\Sigma(\cdot)=f(pH,T,P,\
abla x_{2}(t),l\_av(t))\) is assumed to be present in the dynamics of \(x\_1\) and \(x\_2\), where \(pH=pH(t)\), \(T=T(t)\) is the temperature, \(P=P(t)\) is the pressure, \(\
abla x_{2}(t)\) is the substrate concentration gradient, and \(\
abla l\_av(t)\) is the average luminous intensity. We consider the following assumption for function \(\Sigma(\cdot)\). **Assumption 1**. : _The disturbance function \(\Sigma(\cdot)\) is bounded by a strictly nonnegative smooth function \(\delta(t)\), which is bounded by a positive real number \(\delta_{\max}\):_  
\[\frac{d}{dt}(x_{3}(t)x_{1}(t)) = x_{3}(t)\mu(\cdot)x_{1}(t),\]0  
In addition, we consider that the effect of substrate concentration on biomass growth rate is well represented by the Monod model. This is one of the most used biomass growth kinetic models, since it is a good compromise between accuracy and simplicity. Moreover, this kinetic model will allow us to develop analytically appropriate feedback control laws that will compensate the effects of non-modeled disturbances. More precisely, we consider the following model:  
\[\frac{d}{dt}(x_{3}(t)x_{1}(t)) = x_{3}(t)\mu(\cdot)x_{1}(t),\]1  
with \(\mu_{m}=\mu_{mond}(x_{2}(t))\). Thus, it is proposed in this paper that the dynamic complex denoted by \(\mu(\cdot)\) is the arithmetic sum of two terms, the Monod kinetics, and the bounded non-modeled disturbance function:  
\[\frac{d}{dt}(x_{3}(t)x_{1}(t)) = x_{3}(t)\mu(\cdot)x_{1}(t),\]2  
**Assumption 2**.