## 2 A DISO Adaptive Slope-Seeking Strategy  
The control strategy is built upon the assumption that the process can be described by a block-oriented model, which is made of the interconnection of nonlinear static building blocks and linear time-invariant dynamic building blocks. A great variety of configurationsare possible, including series and parallel structures. In this study, we restrict ourselves to the classical Hammerstein model, where a nonlinear static block followed by a linear dynamical model represent process and sensor dynamics. Such block-oriented models are often used to model nonlinear systems [22], and in the following, we will consider a class of systems that can be modeled locally using quadratic static building blocks and second-order linear time invariant dynamic blocks:  
**Assumption 1**. : _The process input/output static map can be approximated locally by a quadratic function of the form:_  
\[x(k)=\underline{u}^{T}\,A\,\underline{u}+\underline{b}^{T}\,\underline{u}+c \tag{1}\]  
_where \(\underline{u}\) represents the input vector and \(x\) the output of the nonlinear static block, \(A\in\mathbb{R}^{m\times m}\), \(b\in\mathbb{R}^{m\times 1}\), \(c\in\mathbb{R}\) and \(\underline{u}\in\mathbb{R}^{m\times 1}\), with \(m\) being the number of inputs._  
This first assumption implies that (i) driving the system to a stable steady-state manifold is feasible and that (ii) the system in the steady state is likely to be a convex/concave function of the input. In bioprocess applications, these statements find their justifications in the bifurcation analysis developed in [23], which can be considered as a basis for further investigations of bioprocesses [24]. **Assumption 2**. : _The actuator, process and sensor dynamics can be approximated locally by linear time-invariant dynamic blocks. In particular, they can be described by proper and stable second-order transfer functions with unitary steady-state gains:_  
\[G(z^{-1})=K\,\frac{1+\gamma_{1}\,z^{-1}+\gamma_{2}\,z^{-2}}{1+\beta_{1}\,z^{- 1}+\beta_{2}\,z^{-2}} \tag{2}\]  
_with \(K=\frac{1+\beta_{1}+\beta_{2}}{1+\gamma_{1}+\gamma_{2}}\) and \(z^{-1}\) the back-shift operator such that \(z^{-1}\,x(k)=x(k-1)\)._  
The relatively slow dynamics of (bio)processes allows considering the latter in steady state and to consequently isolate the fast actuator dynamics in a separate block. The low-order approximation of the process nonlinearity and dynamics is justified by the fact that an adaptive control approach will be developed, where the parameters of the block-oriented models are estimated online. The estimation of these parameters requires persistency of excitation of the input signals, which corresponds to the following assumption:  
**Assumption 3**. : _Input signals are persistently exciting and allow closed-loop parameter estimation of the block-oriented models._  
In practice, this is achieved using appropriate dither signals in the gradient estimation algorithm, such as pseudo-random binary sequences [25; 26]. This point will be further elaborated in the subsequent sections. Using the notation of Assumptions 1 and 2, the quadratic DISO Hammerstein model can be described by the following state-space model:  
\[x(k)= \underline{u}^{T}\,A\,\underline{u}+b^{T}\,\underline{u}+c \tag{3a}\] \[J(k)= -\beta_{1}\,J(k-1)-\beta_{2}\,J(k-2)\] \[+K\,x(k)+K\,\gamma_{1}x(k-1)+K\,\gamma_{2}x(k-2) \tag{3b}\]  
where \(J\) is the measured performance index. A last assumption is now required regarding the existence of a unique extremum (maximum) point of the measurable performance index:  
**Assumption 4**. : _The performance index is concave, or in other words, there exists a unique set of inputs \(\underline{u}^{opt}\in\mathbb{R}^{m\times 1}\) such that:_  
\[\
abla J= \begin{bmatrix}\frac{\partial I}{\partial\underline{u}_{ \underline{r}}}\\ \frac{\partial I}{\partial\underline{I}_{\underline{r}}}\end{bmatrix}