# Time invariant stochastic bilinear system

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# Block-triangular MIMO system 1

## Model description:

\Sigma _{S_{1}}: \cases{ \begin{align*} \dot{x}_{1,1} &=f_{1,1}(\bar {x}_{1,1},\bar {x}_{2,1})+g_{1,1}(\bar {x}_{1,1},\bar{x}_{2,1})x_{1,2} \\ \dot{x}_{1,2} &=f_{1,2}(X)+g_{1,2}(\bar{x}_{1,1},\bar{x}_{2,1})u_{1} \\ \dot{x}_{2,1} &=f_{2,1}(\bar {x}_{1,1},\bar {x}_{2,1})+g_{2,1}(\bar{x}_{1,1},\bar{x}_{2,1})x_{2,2} \\ \dot{x}_{2,2} &=f_{2,2}(X,u_{1})+g_{2,2}(\bar{x}_{1,1},\bar {x} _{2,1})u_{2} \\ y_{j} &=x_{j,1}, \quad j=1,2, \end{align*}}

where $X = [\bar{x}_{1,2}^{\mathrm T}, \bar{x}_{2,2}^{\mathrm T}]^{\mathrm T}$ with $\bar{x}_{j,2}=[x_{j,1},x_{j,2}]^{\mathrm T},j=1,2$.

## Publication details:

 Title Adaptive neural control of uncertain MIMO nonlinear systems Publication Type Journal Article Year of Publication 2004 Authors Ge, Shuzhi Sam, and Wang Cong Journal IEEE Transactions on Neural Networks Volume 15 Issue 3 Start Page 674 Pagination 674-692 Date Published 05/2004 ISSN 1045-9227 Accession Number 8012935 Keywords adaptive control, closed loop systems, control system synthesis, MIMO systems, neurocontrollers, nonlinear control systems Abstract In this paper, adaptive neural control schemes are proposed for two classes of uncertain multi-input/multi-output (MIMO) nonlinear systems in block-triangular forms. The MIMO systems consist of interconnected subsystems, with couplings in the forms of unknown nonlinearities and/or parametric uncertainties in the input matrices, as well as in the system interconnections without any bounding restrictions. Using the block-triangular structure properties, the stability analyses of the closed-loop MIMO systems are shown in a nested iterative manner for all the states. By exploiting the special properties of the affine terms of the two classes of MIMO systems, the developed neural control schemes avoid the controller singularity problem completely without using projection algorithms. Semiglobal uniform ultimate boundedness (SGUUB) of all the signals in the closed-loop of MIMO nonlinear systems is achieved. The outputs of the systems are proven to converge to a small neighborhood of the desired trajectories. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. The proposed schemes offer systematic design procedures for the control of the two classes of uncertain MIMO nonlinear systems. Simulation results are presented to show the effectiveness of the approach. DOI 10.1109/TNN.2004.826130

# A nonlinear system

## Model description:

Consider a nonlinear system

\begin{align*} x_{1}(t+1) &=x_{1}(t)-x_{1}(t)x_{2}(t)+(5+x_{1}(t))u(t) \\ x_{2}(t+1) &=-x_{1}(t)-0.5x_{2}(t)+2x_{1}(t)u(t) \end{align*}

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Year of Publication 2009 Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung Conference Name International Conference on Information and Automation, 2009. ICIA '09. Date Published 06/2009 Publisher IEEE Conference Location Zhuhai, Macau ISBN Number 978-1-4244-3607-1 Accession Number 10837484 Keywords discrete time systems, fuzzy control, linear matrix inequalities, Lyapunov methods, nonlinear control systems, stability Abstract This paper deals with the stabilization problem for discrete-time nonlinear systems that are represented by the Takagi - Sugeno fuzzy model. By the multiple fuzzy Lyapunov function and the three-index algebraic combination technique, a new stabilization condition is developed. The condition is expressed in the form of linear matrix inequalities (LMIs) and proved to be less conservative than existing results in the literature. Finally, a truck-trailer system is given to illustrate the novelty of the proposed approach. DOI 10.1109/ICINFA.2009.5204890

# T-S fuzzy model

## Model description:

Consider a T-S fuzzy model

Plant Rule $i$: If $x_1(t)$ is $F_1^1(x_1(t))$

Then $x(t+1) = A_ix(t)+B_iu(t),$

where

\begin{align*} A_1 &=\left[\matrix{-a & 2\cr -0.1 & b}\right], A_2=\left[\matrix{-a & 2\cr-0.1 & b }\right], A_3=\left[\matrix{-0.9 & 0.5\cr -0.1 & -1.7}\right] \\ B_1 &=\left[\matrix{b\cr 4}\right], B_2=\left[\matrix{b\cr 4.8}\right], B_3=\left[\matrix{3\cr 0.1}\right]. \end{align*}

The parameters $a$ and $b$ are adjusted to compare the relaxation of stabilization conditions.

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Year of Publication 2009 Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung Conference Name International Conference on Information and Automation, 2009. ICIA '09. Date Published 06/2009 Publisher IEEE Conference Location Zhuhai, Macau ISBN Number 978-1-4244-3607-1 Accession Number 10837484 Keywords discrete time systems, fuzzy control, linear matrix inequalities, Lyapunov methods, nonlinear control systems, stability Abstract This paper deals with the stabilization problem for discrete-time nonlinear systems that are represented by the Takagi - Sugeno fuzzy model. By the multiple fuzzy Lyapunov function and the three-index algebraic combination technique, a new stabilization condition is developed. The condition is expressed in the form of linear matrix inequalities (LMIs) and proved to be less conservative than existing results in the literature. Finally, a truck-trailer system is given to illustrate the novelty of the proposed approach. DOI 10.1109/ICINFA.2009.5204890

# Time varying stochastic bilinear system with nonlinear feedback

## Model description:

Consider the following time varying stochastic bilinear system with nonlinear feedback.

\begin{align*} \begin{bmatrix} x_1(t+1) \\ x_2(t+1) \end{bmatrix} &= \left\{\begin{bmatrix}0.1 & 0.2 \\ 0.5 & -0.3\end{bmatrix} + \begin{bmatrix}0.36 & -0.3 \\ 0.2 & 0.42\end{bmatrix}\omega(t) \right\}\begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix} \\ &+\begin{bmatrix}0.1 & 0.9 \\ 1.5 & 1.2\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}u(t) + \begin{bmatrix}-0.3t^2\exp{(-t)} \\ 0.4t\exp{(-t)}\end{bmatrix}u(t), \\ \begin{bmatrix}y_1(t) \\ y_2(t)\end{bmatrix} &= \begin{bmatrix} 0.7\sin{t} & -0.9 \\ 0.8 & -0.6\cos{t}\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}, \end{align*}

where

$u(t)=0.2\sin{(y_1(t) + y_2(t))} + 0.3[y_1(t)+y_2(t)].$

## Publication details:

 Title Random parameter discrete bilinear system stability Publication Type Conference Paper Year of Publication 1989 Authors Yang, Xueshan, Mohler R.R., and Chen Lung-Kee Conference Name Proceedings of the 28th IEEE Conference on Decision and Control, 1989. Date Published 12/1989 Publisher IEEE Conference Location Tampa, FL Accession Number 3685072 Keywords discrete systems, feedback, linear systems, noise, nonlinear systems, stability criteria, stochastic systems Abstract Stability of discrete, time-varying, stochastic, bilinear systems is studied. Bilinear systems with output feedback are included. Mean-square stability conditions are derived for stochastic models without the assumption of stationarity for the random noise. The feedback function includes a larger class of functions than the class of linear functions or functions satisfying the Lipschitz condition. The sufficient stabilizing conditions depend only on the coefficient matrices of the bilinear system DOI 10.1109/CDC.1989.70323

# Time invariant stochastic bilinear system

## Model description:

Consider the following time invariant stochastic bilinear system:

\begin{align*} \begin{bmatrix}x_1(t+1)\\x_2(t+1)\end{bmatrix} &= \left\{ \begin{bmatrix}0.2 & 0.4 \\ 0.5 & -0.3\end{bmatrix} + \begin{bmatrix}0.3 & 0.2 \\ -0.3 & 0.4\end{bmatrix}\omega(t) \right\} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}+ \begin{bmatrix}2 & 5 \\ 3 & 9\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}u(t) + \begin{bmatrix}-0.3 \\ 0.4\end{bmatrix}u(t), \\ \begin{bmatrix}y_1(t) \\ y_2(t)\end{bmatrix}& = \begin{bmatrix} 0.7 & 0.8 \\ -0.9 & -0.6\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}, \end{align*}

where

$u(t)=0.24[y_1(t) + y_2(t)] + 0.32[y_1(t-1) + y_2(t-1)]$

and $\omega(t)$ is a white noise with zero mean and variance 0.2.

2

## Publication details:

 Title Random parameter discrete bilinear system stability Publication Type Conference Paper Year of Publication 1989 Authors Yang, Xueshan, Mohler R.R., and Chen Lung-Kee Conference Name Proceedings of the 28th IEEE Conference on Decision and Control, 1989. Date Published 12/1989 Publisher IEEE Conference Location Tampa, FL Accession Number 3685072 Keywords discrete systems, feedback, linear systems, noise, nonlinear systems, stability criteria, stochastic systems Abstract Stability of discrete, time-varying, stochastic, bilinear systems is studied. Bilinear systems with output feedback are included. Mean-square stability conditions are derived for stochastic models without the assumption of stationarity for the random noise. The feedback function includes a larger class of functions than the class of linear functions or functions satisfying the Lipschitz condition. The sufficient stabilizing conditions depend only on the coefficient matrices of the bilinear system DOI 10.1109/CDC.1989.70323