The following time series is modeled using RBF networks

$$y(t)=\left(0.8-0.5e^{-y^{2}(t-1)}\right)y(t-1)-\left(0.3+0.9e^{-y^{2}(t-1)}\right)y(t-2)+0.1\sin(\pi y(t-1))+\xi(t),$$

where $\xi(t)$ is a zero-mean Gaussian white noise sequence with variance 0.01.

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

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

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

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

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

Consider the model described in Two van der Pol oscillators coupled via a bath (1).

The current model is using a slightly different notation:

$$\begin{align*} \dot{\xi}_{1}^{1} &= \dot{x}_1 \\ \dot{\xi}_{2}^{1} &= \dot{x}_2 \\ \dot{\xi}_{1}^{2} &= \dot{x}_3 \\ \dot{\xi}_{2}^{2} &= \dot{x}_4 \\ \dot{\eta}_{1} &= \dot{x}_5 \end{align*}$$

Note that this system is decouplable by static state feedback because the decoupling matrix of this system is

$D_{1}(\xi,\eta)=\left[\matrix{1 &0 \cr 0 &1}\right]$

The authors have proposed the following Yuz and Goodwin type approximate model which is more accurate than the Euler model.

$$\begin{align*} x_{1,k+1}&=x_{1,k}+T_{x_{2,k}}+\frac{T^2}{2}\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{2,k+1}&=x_{2,k}+T\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{3,k+1}&=x_{3,k}+T_{x_{4,k}}+\frac{T^2}{2}\{u_{2,k}-x_{1,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{1,k})\} \\ x_{4,k+1}&=x_{4,k}+T_{x_{4,k}}+T\{u_{2,k}-x_{2,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{3,k})\} \\ x_{5,k+1}&=x_{5,k}+T\{k(x_{1,k}-x_{5,k})+k(x_{3,k}-x_{5,k})\} \\ y_{1,k}&=x_{1,k} \\ y_{2,k}&=x_{3,k} \end{align*}$$

The system consists of a DC water pump feeding a conical flask which in turn feeds a square tank, giving the system second-order dynamics. The controllable input is the voltage to the pump motor and the system output is the height of the water in the conical flask. The aim, under simulation conditions, is for the water height to follow some demand signal. The plant model was identified as

$$\begin{align*}z(t) &=0.9722z(t-1)+0.3578u(t-1)-0.1295u(t-2)-\\ &-0.3103z(t-1)u(t-1)-0.04228z^6(t-2)+0.1663z(t-2)u(t-2)+\\ &+0.2573z(t-2)e(t-1)-0.03259z^2(t-1)z(t-2) - 0.3513z^2(t-1)u(t-2)+\\ &+0.3084z(t-1)z(t-2)u(t-2)+0.2939z^2(t-2)e(t-1)+\\ &+0.1087z(t-2)u(t-1)u(t-2)+0.4770z(t-2)u(t-1)e(t-1)+\\ &+0.6389u^2(t-2)e(t-1)+e(t), \end{align*}$$

where $e(t)$ is a noise.