The following CSTR system developed by Liu(1967). The reaction is exothermic first-order, $A \rightarrow B$, and is given by the following mass and energy balances. One should notice that the energy balance includes cooling water jacket dynamics. The following model was identified using regression techniques on the energy balance equations:

$$\begin{align*} y(k) &= 1.3187y(k-1) - 0.2214y(k-2) - 0.1474y(k-3) \\ &- 8.6337u(k-1) + 2.9234u(k-2) + 1.2493u(k-3) \\ &- 0.0858y(k-1)u(k-1) + 0.0050y(k-2)u(k-1) \\ &+ 0.0602y(k-2)u(k-2) + 0.0035y(k-3)u(k-1) \\ &- 0.0281y(k-3)u(k-2) + 0.0107y(k-3)u(k-3). \end{align*}$$

Consider the nonlinear system

$$\begin{align*} y_{1}(k+1)&={{2.5y_{1}(k)y_{1}(k-1)}\over{1+y_{1}(k)^{2}+y_{2}(k-1)^{2}+y_{1}(k-2)^{2}}} \\ &+0.09u_{1}(k)u_{1}(k-1)+1.2u_{1}(k)+1.6u_{1}(k-2) \\ &+0.5u_{2}(k)+0.7\sin (0.5(y_{1}(k)+y_{1}(k-1))) \\ &\times\cos (0.5(y_{1}(k)+y_{1}(k-1))) \\ y_{2}(k+1)&=\displaystyle{{5y_{2}(k)y_{2}(k-1)}\over{1+y_{2}(k)^{2}+y_{1}(k-1)^{2}+y_{2}(k-2)^{2}}} \\ &+u_{2}(k)+1.1u_{2}(k-1)+1.4u_{2}(k-2) \\ &+0.5u_{1}(k). \end{align*}$$

The initial values are: $y_1(1)=y_1(3)=0$, $y_1(2)=1$, $y_2(1)=y_1(3)=0$, $y_2(2)=1$, $u(1)=u(2)=[0,0]^{\mathrm T}$