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

Induction motor is represented by fifth order nonlinear differential equation as

$$\begin{align*} \dot{i}_{sa} &={MR_{r} \over \sigma L_{s}L_{r}^{2}}\phi_{ra}+{n_{p} M\over \sigma L_{s}L_{r}} \omega\phi_{rb}- \gamma i_{sa}+{1\over \sigma L_{s}}u_{sa} \\ \dot{i}_{sb} &={MR_{r} \over \sigma L_{s}L_{r}^{2}}\phi_{rb}- {n_{p} M \over \sigma L_{s}L_{r}}\omega\phi_{ra}-\gamma i_{sb}+{1\over \sigma L_{s}}u_{sb} \\ \dot{\phi}_{ra} &=-{R_{r}\over L_{r}}\phi_{ra}-n_{p}\omega \phi_{rb}+{MR_{r} \over L_{r}}i_{sa} \\ \dot{\phi}_{rb} &=n_{p}\omega\phi_{ra}-{R_{r}\over L_{r}}\phi_{rb}+{MR_{r}\over L_{r}} i_{sb}\cr \dot{\omega} &={n_{p} M \over JL_{r}}(\phi_{ra}i_{sb}-\phi_{rb}i_{sa})-{fv\over J}\omega-{1\over J}T_{l}, \end{align*}$$

where $i_{sa}, i_{sb}, \phi_{ra},\phi_{rb}$ and $\omega$ denote stator currents, rotor fluxes, and angular velocity, respectively, and $u_{sa}$ and $u_{sb}$ denote stator voltage inputs. The parameters $\sigma$ and $\gamma$ are defined as $\sigma = 1-M^2/L_sL_r, \gamma = (L_r^2r_s+M^2R_r)/\sigma L_s L_r^2 \cdot M, L_s, L_r, R_s$ and $R_r$ denote the mutual inductance, the self-inductances, the resistances, respectively. The subscript $a$ and $b$ denote the components of a vector with respect to a fixed stator reference frame and $s, r$ stand for stator and rotor of motor. $n_p, f_v, J, T_l$ are the number of pole-pair, the co-efficient of viscous damping, the inertia of rotor, and the load torque. We assume that the state variables $i_{sa}, i_{sb}, \omega$ are available for measurement and $T_l$ has a unknown constant value, that is, $\dot{T}_l=0$ . As a result, the model of induction motor can be rewritten into the form

$\eqalignno{ \dot{x}_{i} & =A_{i} (u, y_{i+1}, \cdots,y_{p})x_{i}\cr & +g_{i}(x_{1}, \cdots, x_{i},; u; y_{i+1}, \cdots, y_{p}) \cr & y_{i}=C_{i}x_{i}, 1\leq i \leq p}$

as follows:

$\eqalignno{ & \dot{x}_{1}=\left(\matrix{ 0 & A_{11}(y_{2})\cr 0 & 0}\right)x_{1}+g_{1}(x_{1}, u, y_{2})\cr & \dot{x}_{2}=\left(\matrix{ 0 & A_{21}\cr 0 & 0}\right) x_{2}+g_{2}(x_{1}, x_{2}, u)\cr & y_{1}=C_{1}x_{1}\cr & y_{2}=C_{2}x_{2},}$

where $x_1=[i_{sa},i_{sb},\phi_{ra},\phi_{rb}]^T$, $x_2=[\omega,T_l]^T$, $y_1=[i_{sa},i_{sb}]^T$, $y_2=\omega$, $u=[u_{sa},u_{sb}]^T$ and

$\eqalignno{ & A_{11}= \left(\matrix{ MR_{r}/ \sigma L_{s}L_{r}^{2} & (n_{p}M/\sigma L_{s}L_{r})y_{2}\cr -(n_{p}M/ \sigma L_{s}L_{r})y_{2} & M R_{r}/\sigma L_{s}L_{r}^{2}}\right)\cr & A_{21}=\left(\matrix{ -{1\over J}}\right)\cr & g_{1}=\left(\matrix{ -\gamma i_{sa} +(1/ \sigma L_{s})u_{sa}\cr -\gamma i_{sb} + (1/\sigma L_{s})u_{sb}\cr -(R_{r}/L_{r})\phi_{ra}-n_{p}y_{2}\phi_{rb}+(MR_{r}/L_{r})i_{sa}\cr n_{p}y_{2}\phi_{ra} -(R_{r}/L_{r})\phi_{rb}+(MR_{r}/L_{r})i_{sb}}\right)\cr & g_{2}=\left(\matrix{(n_{p}M/JL_{r})(\phi_{ra}i_{sb})-(\phi_{rb} i_{sa}) - (f_{v}/J)\omega)\cr 0}\right)}$

Consider the system

$$\begin{align*} \dot{x}_1 &= p_{13}x_3 + p_{12}x_2 - p_{21}x_1+u \\ \dot{x}_2 &= -p_{12}x_2 + p_{21}x_1 \\ \dot{x}_3 &= -p_{13}x_3 \\ y &= x_2, \end{align*}$$

where $x=[x_1, x_2, x_3]$ is the state vector, e.g. $x_1,x_2,x_3$ are drug masses in compartment 1, 2 and 3, respectively; initial conditions are $x_1(0)=0$, $x_2(0)=0$, $x_3(0)=0$; $u$ is the drug input; $y$ is the measured drug output; $p=[p_{12},p_{21},p_{13}]$ is the rate parameter vector (assumed constant).

The hydrostatic transmission dynamics is represented by a nonlinear fourth order state-space model

$$\begin{align*} \dot{q}_{1}(t) &= -a_{11}q_{1}(t)+b_{11}u_{1}(t) \\ \dot{q}_{2}(t) &= -a_{22}q_{2}(t)+b_{22}u_{2}(t) \\ \dot{q}_{3}(t) &= a_{31}q_{1}(t)p(t)-a_{33}q_{3}(t)-a_{34}q_{2}(t)q_{4}(t) \\ \dot{q}_{4}(t) &=a_{43}q_{2}(t)q_{3}(t)-a_{44}q_{4}(t), \end{align*}$$

where $q_1(t)$ is the normalized hydraulic pump angle, $q_2(t)$ is the normalized hydraulic motor angle, $q_3(t)$ is the pressure difference [bar], $q_4(t)$ is the hydraulic motor speed [rad/s], $p(t)$ is the speed of hydraulic pump [rad/s], $u_1(t)$ is the normalized control signal of the hydraulic pump, and $u_2(t)$ is the normalized control signal of the hydraulic motor. It is supposed that the external variable $p(t)$ , as well as the second state variable $q_2(t)$ are measurable. In given working point the model parameters are

$\eqalignno{& a_{11}=7.6923 \qquad a_{22}=4.5455 \quad a_{33}=7.6054.10^{-4} \cr &a_{31}=0.7877 \qquad a_{34}=0.9235\quad\ b_{11}=1.8590.10^{3} \cr &a_{43}=12.1967 \quad\ \ a_{44}=0.4143\quad b_{22}= 1.2879.10{}^{{3}}}$