The behavior of phytoplankton cells in a continuous reactor is usually described by the Droop model. Cell growth is limited by a nutrient with concentration $S$. The biomass has a concentration $N$ and $Q$ represents the cell quota of assimilated nutrient, expressed as the amount of intracellular nutrient per biomass unit. The dilution rate $D$ corresponds to the flow rate of renewal medium over the volume of the reactor, and $D$ is the input of the system.

We denote $D = D_0 + u$, and the system fits

$$\sum_D \begin{cases} \dot{x}_i = f(x) + ug(x)\\ y=h(x_1) \end{cases}$$

with

$f(x)=\begin{pmatrix} a_2\left(1-\dfrac{1}{x_2}\right)x_1 - D_0x_1\\ a_3\dfrac{x_3}{a_1+x_3} - a_2(x_2 - 1)\\ D_0(1-x_3)-\dfrac{x_1x_3}{a_1+x_3} \end{pmatrix}$

$g(x)=\begin{pmatrix} -x_1\\ 0\\ 1-x_3 \end{pmatrix}$, and $h(x_1)=x_1$, where

$ x_1 = (\rho_m N/S_i);\\ x_2 = (Q/K_Q);\\ x_3 = (S/S_i);\\ a_1 = (K_{\rho}/S_i);\\ a_2 = \mu_m;\\ a_3 = (\rho_m/K_Q). $

The transfer function of the three-mass-system is much more complex than it is for one dominant elasticity (two-mass-system).

$${G_{\rm mech}}(s) = \underbrace{{\dfrac{1} {T_{ \Sigma} \cdot s}}}_{G_{\rm rs}(s)} \cdot \underbrace{ \dfrac{ a_{7} \cdot s^{4} + a_{6} \cdot s^{3}+a_{5} \cdot s^{2} + a_{4} \cdot s + 1}{a_{3} \cdot s^{4} + a_{2} \cdot s^{3}+ a_{1} \cdot s^{2} + a_{4} \cdot s + 1}} _{G_{\rm nrs}(s)}$$

with

$T_{\Sigma} = T_{\rm M} + T_{{\rm L}1} + T_{{\rm L}2}$

and

$\begin{align*} a_{1}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}\left(T_{\rm M}+T_{{ \rm L}1}\right) \cdot \frac{T_{C2}}{T_{\Sigma }}+T_{\rm M}\left(T_{{\rm L}1}+T_{{\rm L}2}\right) \cdot \frac{T_{{\rm C}1}}{T_{\Sigma}} \\ a_{2}&=\frac{T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}}\cdot\left(d_{1}T_{{\rm L}2}\left(T_{\rm M}+T_{{\rm L}1}\right)+d_{2}T_{\rm M} \left(T_{{\rm L}1}+T_{{\rm L}2}\right)\right) \\ a_{3}&=\frac{T_{\rm M}T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}} \\ a_{4}&=d_{1}T_{{\rm C}1}+d_{2}T_{{\rm C}2} \\ a_{5}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}T_{{\rm C}2}+\left(T_{{\rm L}1}+T_{{\rm L}2}\right)\cdot T_{{\rm C}1} \\ a_{6}&=\left(\left(d_{1}+d_{2}\right)T_{{\rm L}2}+d_{2}T_{{\rm L}1}\right)\cdot T_{{\rm C}1}T_{{\rm C}2} \\ a_{7}&=T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}. \end{align*}$

The transfer function of a nonrigid mechanical system with two concentrated masses is given by

$$G_{\rm mech}(s) = \underbrace{ \dfrac{1}{s \cdot \left(T_{\rm M} + T_{\rm L}\right)} }_{G_{\rm rs}(s)} \cdot \underbrace{ \dfrac{T_{ \rm L} \cdot T_{\rm C} \cdot s^{2} + d \cdot T_{\rm C} s + 1} {\dfrac{ T_{\rm L} \cdot T_{\rm C} \cdot T_{\rm M}}{T_{\rm M} + T_{\rm L}} \cdot s^{2} + d \cdot T_{\rm C} \cdot s + 1}}_{G_{\rm nrs}(s)}.$$

$T_M$ and $T_L$ are the run-up times of the motor and the load. The nonrigid shaft of the two-mass-configuration is modeled as a damper-spring-system. $T_C$ is the normalized spring-constant and $d$ is the related damping of the spring.

Simulation studies are carried out for the following MIMO discrete-time system with triangular form inputs

$$\begin{align*} x_{1,1}(k+1) &= f_{1,1}({\bar x}_{1,1}(k))+g_{1,1}({\bar x}_{1,1}(k))x_{1,2}(k) \\ x_{1,2}(k+1) &= f_{1,2}({\bar x}_{1,2}(k))+g_{1,2}({\bar x}_{1,2}(k))u_{1} (k)+d_{1}(k) \\ x_{2,1}(k+1) &= f_{2,1}({\bar x}_{2,1}(k)) +g_{2,1}({\bar x}_{2,1}(k))x_{2,2}(k) \\ x_{2,2}(k+1) &= f_{2,2}({\bar x}_{2,2}(k),u_{1}(k)) +g_{2,2}({\bar x}_{2,2}(k))u_{2}(k)+d_{2}(k) \\ y_{1}(k) &= x_{1,1}(k) \\ y_{2}(k) &= x_{2,1}(k), \end{align*}$$

where

$\begin{cases} f_{1,1}({\bar x}_{1,1}(k))={{x^{2}_{1,1}(k)}\over {1+x^{2}_{1,1}(k)}}\\ g_{1,1}({\bar x}_{1,1}(k))=0.3 \\ f_{1,2}({\bar x}_{1,2}(k))= {{x^{2}_{1,1}(k)}\over{1+x^{2}_{1,2}(k)+x^{2}_{2,1}(k)+x^{2}_{2,2}(k)}}\\ g_{1,2}({\bar x}_{1,2}(k))=1\\ d_{1}(k)=0.1 \cos{0.05k}\cos{x_{1,1}(k)}\\ \end{cases}$

$\begin{cases} f_{2,1}({\bar x}_{2,1}(k))= {{x^{2}_{2,1}(k)}\over {1+x^{2}_{2,1}(k)}}\\ g_{2,1}({\bar x}_{2,1}(k))=0.2\\ f_{2,2}({\bar x}_{2,2}(k),u_{1}(k))={{x^{2}_{2,1}(k)}\over{1+x^{2}_{1,1}+x^{2}_{1,2}(k)+x^{2}_{2,2}(k)}}u^{2}_{1}(k)\\ g_{2,2}({\bar x}_{2,2}(k))=1\\ d_{2}(k)=0.1\cos{0.05k}\cos{x_{2,1}(k)}\\ \end{cases}$

Consider a multi-output nonlinear system in the form of

$$\begin{align*} \dot{x} &= f(x, u)\\ y &= h(x), \end{align*}$$

where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^m$ is the control input, $y \in \mathbb{R}^p$ is the output, $f$ and $h$ are smooth vector fields. The control input $u: \mathbb{R} \rightarrow \mathbb{R}^m$ is assumed to be an analytic time function. In particular, we will restrict our interest to the class of systems of the following form:

$\eqalignno{ \dot{x} &= A_ix_i+g_i(x_1,\ldots,x_i;u;y_{i+1},\ldots,y_p)\\ y &= C_ix_i & 1 \leq i \leq p, }$

where $x=[x_1^{\mathrm T}, x_2^{\mathrm T}, \ldots, x_p^{\mathrm T}]^{\mathrm T} \in \mathbb{R}^n$, $x_i=[x_{i1}, x_{i2}, \ldots, x_{in}]^T \in \mathbb{R}^{n_i}$, $y=[y_1, \ldots, y_p]^{\mathrm T} \in \mathbb{R}^p$

$A_i = \begin{bmatrix} 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots &\vdots\\ 0 & 0 & \cdots & 1\\ 0 & 0 & \cdots & 0 \end{bmatrix} \in \mathbb{R}^{n_i \times n_i}, C_i = [1, 0, \ldots, 0] \in R^{1 \times n_i}$

$g_i = \begin{bmatrix} g_{i1} (x_{[1,i-1]}x_{i1},u,y_{[i+1, p]} )\\ g_{i1} (x_{[1,i-1]}x_{i1},x_{i2},u,y_{[i+1, p]} )\\ \vdots\\ g_{i1} (x_{[1,i-1]}x_{[i1,in_i]},u,y_{[i+1, p]} ) \end{bmatrix}$

with $x_{[1,i-1]}=[x_1^{\mathrm T}, \ldots, x_{i-1}^{\mathrm T}]^{\mathrm T}$ and $y_{[i+1,p]}=[y_{i+1},\ldots,y_p]^{\mathrm T}$, and $g$ is a smooth vector field.

The system has the form

$$\begin{align*} \dot{x}_1 &= x_2 + 0.01x_1u \\ \dot{x}_2 &= -x-1 + (1-x_1^2)+x_3u \\ \dot{x}_3 &= x_4 + 0.01x_2x_3 \exp(u) \\ \dot{x}_4 &= -x_3 + (1 - x_3^2)x_4 + u \\ y_1 &= x_1 \\ y_2 &= x_3 \end{align*}$$

with $u= 2 \sin {3t}$.