Attached image shows the principal structure of the three-tank system. The plant consists of three cylinders $T_1$, $T_2$, and $T_3$ with the cross section $S_A$. These tanks are connected serially with each other by pipes with the cross section $S_n$. A single outflow valve with the cross section $S_n$ is located at tank 2. The outflowing liquid (usually distilled water) is collected in a reservoir, which supplies the pumps 1 and 2. $H_{max}$ denotes the highest possible liquid level. The control input signals are the pump liquid flow rates $Q_1$ and $Q_2$ , the output signals are the liquid levels $h_1$ and $h_2$ .

Define the following variables and the parameters: $az_i$ : outflow coefficients of tank $i$ ; $h_1$ , $h_2$ , $h_3$ : liquid levels (m); $Q_{13}$ : flow rate from tank 1 to tank 3 $(m^3/sec)$ ; $Q_{32}$ : flow rate from tank 3 to tank 2 $(m^3/sec)$ ; $Q_{20}$ : flow rate from tank 2 to reservoir $(m^3/sec)$ ; $Q_1$ , $Q_2$ : supplying flow rates $(m^3/sec)$ ; $S_A$ : section of cylinder $(m^2)$ ; $S_1$ : section of leak opening $(m^2)$ ; $S_n$ : section of connection pipe $(m^2)$. Then, the dynamics of the three-tank system is expressed by a set of differential equations

$$\eqalignno{S_{A}\displaystyle{{dh_{1}}\over{dt}}=&\,Q_{1}(t)-Q_{13}(t)\cr S_{A}\displaystyle{{dh_{3}}\over{dt}}=&\, Q_{13}(t)-Q_{32}(t)\cr S_{A}\displaystyle{{dh_{2}}\over{dt}}=&\, Q_{2}(t)+Q_{32}(t)-Q_{20}(t)\cr Q_{13}(t)=&\,\alpha z_{1}S_{n}sgn(h_{1}(t)-h_{3}(t))\sqrt{(2g\left\vert h_{1}(t)-h_{3}(t)\right\vert}\cr Q_{32}(t)=&\,\alpha z_{3}S_{n}sgn(h_{3}(t)-h_{2}(t))\sqrt{2g\left\vert h_{3}(t)-h_{2}(t)\right\vert}\cr Q_{20}(t)=&\,\alpha z_{2}S_{n}sgn(h_{2}(t))\sqrt{2g\left\vert h_{2}(t)\right\vert},}$$

where $\alpha_1$, $\alpha_2$, $\alpha_3$: outflow coefficients (dimensionless, real values ranging from 0 to 1), $g$: earth acceleration $(m/s^2)$, $sgn(z)$: sign of the argument $z$.

In the simulation, the discretized model of three-tank system is obtained by first-order Euler's method, the sampling period is 1 s and the simulation time is 1500 s. The parameters of three-tank system are given in the table below. The initial conditions are $h_1(1)=0$, $h_2(1)=0$, $h_3(1)=0$, $Q_1(1)=0$, and $Q_2(1)=0$.

In the operation of a power plant superheater, exacting demands are made on the steam temperature maintenance at the outlet. For temperature control at the outlet of a superheater, the relevant system state is the temperature pattern along the superheater tube. This is described by a distributed-parameter system, which involves an infinite number of state variables. To derive a simplified model for control purposes, the superheater is divided into segments, and a lumped model is derived, which represents a finite number of intermediate temperatures.

Assuming that the pressure inside the tube is constant, the enthalpy of the steam satisfies the relation $dH = C_pdT(kcal/kg)$ , where $C_p(kcal/kg^{\circ}C)$ is the constant-pressure specific heat. Hence, we conclude that the heat supplied to the following fluid(steam) only increases its enthalpy, $dH = dQ$ , where $Q$ denotes the heat. In the above equations, it is assumed that convection is the exclusive heat transfer mode for the superheater. Hence the heat transfer from to metal $Q_{ms}(kcal/s)$ and from gas to metal $Q_{gm}(kcal/s)$ are expressed in terms of the heat transfer rates from gas to metal $\alpha_{gm}(kcal/m^2s^{\circ}C)$ and from metal to steam $\alpha_{ms}(kcal/m^2s^{\circ}C)$ and heating surface $S(m^2)$ :

$$\begin{align*} \alpha_{ms}S_1(T(l,t)-T(l,t)) &=Q_{ms} \\ \alpha_{gm}S_2(T_m(l,t)-T(l,t)) &=Q_{gm}. \end{align*}$$

It is also assumed that the heat transfer rates $\alpha_{gm}$ and $\alpha_{ms}$ are constants.

Now, to simulate the profile of superheated steam precisely, it is necessary to divide the superheater into $n$ segments as shown in the attached image.

In the first segment, the desuperheater is included and system is modified as follows:

$$\begin{align*} V_s\rho C_p\frac{{\mathrm d} x_1}{{\mathrm d}t} &={C_{p}}{T_{i}}{w_{i}}-{C_{p}}({w_{i}}+{w_{d}}){x_{1}} +{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}})+{C_{pd}}{T_{d}}{w_{d}}\\ M_mC_m \frac{{\mathrm d}z_1}{{\mathrm d}t} &={\alpha_{gm}}{S_{2}}(T{g_{1}}-{z_{1}})-{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}}), \end{align*}$$

where $x=[x_1,x_2,\ldots,x_n]^{\mathrm T}=[T_1,T_2,\ldots,T_n]^{\mathrm T}$, $z=[z_1,z_2,\ldots,z_n]^{\mathrm T}=[T_{m1},T_{m2},\ldots,t_{mn}]^{\mathrm T}$, and $T_{mi}(^{\circ}C)$ are metal temperature, $T_i(^{\circ}C)$ are steam temperature, $i=1,\ldots,n$.

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

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

The process dynamic model consists of six nonlinear ordinary differential equations:

$$\begin{align*} \dot x_{11} &= b_{11} x_{12} \\ \dot x_{12} &= b_{12} u_1 \\ \dot x_{21} &= b_{21} x_{22} + \phi _{21} \left({x_{11},x_{21} } \right) + \Phi x_{31} \\ \dot x_{22} &= b_{22} u_2 + \phi _{22} \left({x_{21},x_{22} } \right) \\ \dot x_{31} &= b_{31} x_{32} + \phi _{31} \left({x_{11},x_{12},x_{21},x_{31} } \right) + \Psi w \\ \dot x_{32} &= b_{32} u_3 + \phi _{32} \left({x_{31},x_{32} } \right) \\ y &= \left[{y_1,y_2,y_3 } \right] = \left[{x_{11},x_{21},x_{31} } \right], \end{align*}$$

where

$\eqalignno{b_{11} &= 1,b_{12} = 1,b_{21} = {{UA} \over {\rho c_p V}},b_{22} = {{F_{j2} } \over {V_j}},b_{31} = {{UA} \over {\rho c_p V}}\cr b_{32} &= {{F_{j1} } \over {V_j}},\Psi = {{F_0 } \over V},\Phi = {{F + F_R } \over V} \cr \phi _{21} &= {{F + F_R } \over V}T_1^d - {{F + F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}\left({x_{11} + C_{A2}^d } \right)e^{- \left({{E \over {R\left({x_{21} + T_2^d } \right)}}} \right)}\cr &\quad - {{UA} \over {\rho c_p V}}\left({x_{21} + T_2^d - T_{j2}^d } \right) \cr \phi _{22} &= {{F_{j2} } \over {V_j}}\left({T_{j20}^d - x_{22} - T_{j2}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{21} + T_2^d - x_{22} - T_{j2}^d } \right) \cr \phi _{31} &= {{F_0 } \over V}T_0^d - {{F + F_R } \over V}\left({x_{31} + T_1^d } \right) + {{F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}C_A e^{- \left({{E \over {R\left({x_{31} + T_1^d } \right)}}} \right)} - {{UA} \over {\rho c_p V}}\left({x_{31} + T_1^d - T_{j1}^d } \right) \cr \phi _{32} &= {{F_{j1} } \over {V_j}}\left({T_{j10}^d - x_{32} - T_{j1}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{31} + T_1^d - x_{32} - T_{j1}^d } \right) \cr C_A &= {V \over {F + F_R}}\Bigg(x_{12} + {{F + F_R } \over V}({x_{11} + C_{A2}^d })\cr &\quad + \alpha ({x_{11} + C_{A2}^d })e^{- \Big({{E \over {R({x_{21} + T_2^d })}}} \Big)} \Bigg). }$

The values of the process parameters are

$\eqalignno{& \alpha = {\rm 7}{\rm .08} \times {\rm 10}^{{\rm 10}} {\rm h}^{- 1},\quad \rho = 800.9189\,{\rm kg/m}^{\rm 3}\cr & \rho _j = 997.9450\,{\rm kg/m}^3,\quad \lambda = - 3.1644 \times {\rm 10}^{\rm 7} {\rm J/mol}\cr & R = 1679.2\,{\rm J/(mol} {\cdot} {}^{\circ} {\rm C)},\quad E = 3.1644 \times 10^7 {\rm J/mol}\cr & c_\rho = 1395.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)},\quad c_j = 1860.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)}\cr & U = 1.3625 \times 10^6{\kern1pt} {\rm J/(h} {\cdot} {\rm m}^{\rm 2} {\cdot} {}^{\circ} {\rm C)},\quad F_0 = F_2 = F = 2.8317\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_R = 1.4158\,{\rm m}^{\rm 3}\!{\rm /h},\quad F_{j1} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_{j2} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h},\quad T_0^d = 703.7\,{}^{\circ} {\rm C},\quad T_1^d = 750\,{}^{\circ} {\rm C}\cr & T_2^d = 737.5\,{}^{\circ} {\rm C},\quad T_{j1}^d = 740.8\,{}^{\circ} {\rm C},\quad T_{j2}^d = 727.6\,{}^{\circ} {\rm C}\cr & T_{j10}^d \! = \! 629.2\,{}^{\circ} {\rm C},\quad T_{j20}^d \!=\! 608.2\,{}^{\circ} {\rm C},\quad C_{A0}^d \!=\! 18.3728\,{\rm mol/m}^{\rm 3}\cr & C_{A1}^d = 12.3061\,{\rm mol/m}^{\rm 3},\quad C_{A2}^d = 10.4178\,{\rm mol/m}^{\rm 3}\cr & V_1 = V_2 = V = 1.3592\,{\rm m}^{\rm 3},\quad V_{j1} = V_{j2} = V_j = 0.1090\,{\rm m}^{\rm 3}\cr & A = 23.2\,{\rm m}^{\rm 3} . }$

The system, that could describe a biochemical reaction network, is represented by twenty differential equations, twenty two parameters, and all the states are assumed to be measured:

$$\begin{align*} \dot{x}_1&=-\dfrac{v_{max}x_1}{(k_m+x_1)}-p_1x_1 + u(t),\\ \dot{x}_{2}&=p_{1}x_{1}-p_{2}x_{2},\\ \dot{x}_{3}&=p_{2}x_{2}-p_{3}x_{3},\\ \dot{x}_{4}&=p_{3}x_{3}-p_{4}x_{4},\\ \dot{x}_{5}&=p_{4}x_{4}-p_{5}x_{5},\\ \dot{x}_{6}&=p_{5}x_{5}-p_{6}x_{6},\\ \dot{x}_{7}&=p_{6}x_{6}-p_{7}x_{7},\\ \dot{x}_{8}&=p_{7}x_{7}-p_{8}x_{8},\\ \dot{x}_{9}&=p_{8}x_{8}-p_{9}x_{9},\\ \dot{x}_{10}&=p_{9}x_{9}-p_{10}x_{10},\\ \dot{x}_{11}&=p_{10}x_{10}-p_{11}x_{11},\\ \dot{x}_{12}&=p_{11}x_{11}-p_{12}x_{12},\\ \dot{x}_{13}&=p_{12}x_{12}-p_{13}x_{13},\\ \dot{x}_{14}&=p_{13}x_{13}-p_{14}x_{14},\\ \dot{x}_{15}&=p_{14}x_{14}-p_{15}x_{15},\\ \dot{x}_{16}&=p_{15}x_{15}-p_{16}x_{16},\\ \dot{x}_{17}&=p_{16}x_{16}-p_{17}x_{17},\\ \dot{x}_{18}&=p_{17}x_{17}-p_{18}x_{18},\\ \dot{x}_{19}&=p_{18}x_{18}-p_{19}x_{19},\\ \dot{x}_{20}&=p_{19}x_{19}-p_{20}x_{20},\\ \end{align*}$$

where $x_i, i = 1, …, 20$ are the states. The unknown parameters are $p_i, i = 1, …, 20$, plus the two Michaelis-Menten parameters $ν_{max}$ and $k_m$. All the states are assumed to be measured.