Consider the following unknown discrete nonlinear dynamic system:

$$\begin{align*} y(k+1)&=p[{\bf q}(k), u(k)]=0.2\cos[0.8(y(k)+y(k-1))] \\ & +0.4\sin[0.8(y(k)+y(k-1))+2u(k)+u(k-1)] \\ &+0.1[9+y(k)+y(k-1)]+\left[{2(u(k)+u(k-1))\over 1+\cos(y(k))}\right] \end{align*}$$

for $k=0,1,2,\ldots$ with $y(k)=0,u(k)=0$, for $k=0,-1,-2,\ldots$, $\Delta t := t(k+1)-t(k)=0.02sec$, for $k=0,1,2,\ldots$.

The suggested tracker scheme is tested with a 2-input 2-output nonlinear system given by:

$$\begin{align*} y_{1} (k) & = 0.21y_{1} (k-1)-0.12y_{2} (k-2) \\ & + 0.3y_{1} (k-1)u_{2} (k-1)-1.6u_{2} (k-1) \\ & + 1.2u_{1} (k-1), \\ y_{2} (k) & = 0.25y_{2} (k-1)-0.1y_{1} (k-2) \\ &- 0.2 y_{2} (k-1)u_{1} (k-1)-2.6u_{1} (k-1) \\ &-1.2u_{2} (k-1). \end{align*}$$

The model structure of the SDRNN have been shown in the attached image, second-order nonlinear system model is assumed as:

$$y(k+1)=\dfrac{y(k)y(k-1))[y(k)+4.5]}{1+y^2(k)+y^2(k-1)}+u(k).$$

The SDRNN(2, 7, 1) is used in simulation, that is, the input layer has 2 neurons $u(k)$ and $y(k)$, 7 neurons in hidden layer, 1 neuron $y(k +1)$ in output layer. The activation function is sigmoid function in hidden layer: this function is the commonly used bipolar function $\rho(x)=\dfrac{1-e^{-x}}{1+e^{-x}}$, initial weight is random value between -1 and 1, the learning rate $\eta=0.45$, momentum factorγ = 0.1.

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

Consider the nonlinear system

$$\begin{align*} x_{11}(k+1) &=\frac{x_{11}^2(k)}{1+x_{11}^2(k)}+0.3x_{12}(k), \\ x_{12}(k+1) &=\frac{x_{11}^2(k)}{1+x_{12}^2(k)+x_{21}^2(k)+x_{22}^2(k)}+a(k)u_{1}(k), \\ x_{21}(k+1) &=\frac{x_{21}^2(k)}{1+x_{21}^2(k)}+0.2x_{22}(k), \\ x_{22}(k+1) &=\frac{x_{21}^2(k)}{1+x_{11}^2(k)+x_{12}^2(k)+x_{22}^2(k)}+b(k)u_{2}(k), \\ y_1(k+1) &= x_{11}(k+1)+0.005 \mathrm{rand}(1), \\ y_2(k+1) &=x_{21}(k+1)+0.005 \mathrm{rand}(1), \end{align*}$$

where $a(k)=1+0.1\sin{(2\pi k/1500)}$, $b(k)=1+0.1\cos{(2\pi k/1500)}$are two time-varying parameters. This example is controlled by using neural network without time-varying parameters $a(k)$, $b(k)$, and the noise.

The initial values are: $x_{1,1}(1)=x_{1,1}(2)=x_{2,1}(1)=x_{2,1}(2)=0.5$, $x_{1,2}(1)=x_{1,2}(2)=x_{2,2}(1)=x_{2,2}(2)=0$, $u(1)=u(2)=[0,0]^{\mathrm T}.$