# A smooth nonlinear system (1)

## Model description:

The system

\begin{align*} \dot{x}_1 & = x_2 \\ \dot{x}_2 & = x_3^2 + x_4 + u_1 + au_2 \\ \dot{x}_3 & = x_4 + bu_1 + u_2 \\ \dot{x}_4 & = -x_4 \\ y_1 &= x_1 \\ y_2 &=x_3 \end{align*}

with $ab \neq 1$, has a well-defined vector relative degree (2, 1) and a nonsingular decoupling matrix $\begin{bmatrix} 1 & a \\ b & 1 \end{bmatrix}$.

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## Publication details:

 Title Input-output models for a class of nonlinear systems Publication Type Conference Paper Authors Atassi, A.N., and Khalil H.K.

# MIMO nonlinear system

## Model description:

\begin{align*} y_{1}(k+1) &=0.9y_{1}(k)-0.3y_{1}(k-1)/[1+y_{2}^{2}(k-1)]+0.7u_{1}(k) \\ &+0.1y_{1}^{2}(k-1)y_{2}^{2}(k)+0.3\sin(u_{1}(k-1))-0.7u_{2}(k)+0.6u_{2}(k-1) \\ y_{2}(k+1) &=-0.1y_{2}(k-1)+0.3y_{1}(k-1)y_{2}(k)+0.8\sin(u_{1}(k)) \\ &+0.1u_{1}(k-1)+0.9u_{2}(k)+0.2u_{2}(k-1)+0.1u_{2}^{2}(k-1) \end{align*}

## Publication details:

 Title Stable adaptive neural network control of MIMO nonaffine nonlinear discrete-time systems Publication Type Conference Paper Authors Zhai, Lianfei, Chai Tianyou, Yang Chenguang, Ge S.S, and Lee Tong Heng

# Recurrent Trainable Neural Network

## Model description:

The RTNN model is described bythe following equations:

\begin{align*} X(k+1) &= JX(k) + BU(k)\\ Z(k) &= S[X(k)]\\ Y(k) &= S[CZ(k)]\\ J &\doteq \mathrm{blockdiag}(J_i); |J_i| <1, \end{align*}

here $X(\cdot)$ is a $n$-state vector of the RTTN; $U(\cdot)$ is a $m$-input vector; $Y(\cdot)$ is a $l$-output vector; $Z(\cdot)$ is an auxiliary vector variable with $l$ dimension; $S(\cdot)$ is a vector-valued smooth activation function (sigmoid, $tanh$, saturation) with appropriate dimensions; $J$ is a weigh-state block-diagonal matrix with $(1 \times 1)$ and $(2 \times 2)$ blocks; $J_i$ is an $i-th$ block of $J$ and $|J_i|<1$ is a stability condition.

## Publication details:

 Title Adaptive Neural Control of Nonlinear Systems Publication Type Conference Paper Authors Garrido, Ruben Editor Baruch, Ieroham, Flores Jose Martin, and Thomas Federico

# MAGLEV

## Model description:

The model of the MAGLEV system is unstable and nonlinear

$$m\ddot{x}=mg-\dfrac{K_{c}V^{2}}{x^{2}},$$

where $x$ is the metal ball position being the system output, $V$ is the system input as the voltage. Other parameters are $m$ as the mass of the metal ball, $K_c$ as constant for magnet circuit, and $g$ is the gravitational acceleration of 9.8 m/s$^2$. A free-body diagram is shown also in the attached image.

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## Publication details:

 Title Identification of a class of unstable processes Publication Type Conference Paper Authors Shahab, M., and Doraiswami R.

# A Truck-Trailer System

## Model description:

Consider a truck-trailer system depicted in the attached image. Its dynamics is described by

\begin{align*} x_{1}(t+1) &=\left(1-\frac{vT}{L}\right)x_{1}(t)+\frac{vT}{l}u(t) \\ x_{2}(t+1) &=\frac{vT}{L}x_1(t)+x_{2}(t) \\ x_{3}(t+1) &=x_{3}(t)+vT\sin\left(\frac{vT}{2L}x_{1}(t)+x_{2}(t)\right)x_{1}(t), \end{align*}

where $x_1(t)$ : angle difference between truck and trailer. $x_2(t)$ : angle of trailer. $x_3(t)$ : vertical position of rear of trailer, $u(t)$ : steering angle, $T$ : sampling time. In this example, the parameters are $T=2.0s$, $l=2.8m$, $L=5.5m$, $v=-1.0m/s$.

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## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung