Block-triangular MIMO system 2

SISO NLTI plant

Model description: 

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 $k=0,1,2,\ldots$.

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2

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Publication details: 

TitleRobust nonlinear adaptive control using neural networks
Publication TypeConference Paper
AuthorsAdetona, O., Sathananthan S., and Keel L.H.

2-input 2-output nonlinear system

Model description: 

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

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

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Publication details: 

TitleU-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems
Publication TypeConference Paper
AuthorsAzhar, A.S.S., Al-Sunni F.M., and Shafiq M.

Nonlinear System (2)

Model description: 

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

Type: 

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Model order: 

3

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Autonomity: 

Publication details: 

TitleData-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems
Publication TypeJournal Article
AuthorsHou, Zhongsheng, and Jin ShangTai

Nonlinear System (1)

Model description: 

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

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Publication details: 

TitleData-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems
Publication TypeJournal Article
AuthorsHou, Zhongsheng, and Jin ShangTai

Block-triangular MIMO system 2

Model description: 

$$\Sigma _{S_{2}}: \cases{\begin{align*} \dot{x}_{1,1} &=f_{1,1}(\bar {x}_ {1,1},\bar {x}_{2,3})+g_{1,1}(\bar {x}_{1,1},\bar{x}_{2,3})x_{1,2} \\ \dot{x}_{1,2} &=f_{1,2}(X)+g_{1,2}(\bar {x}_{1,1},\bar{x}_ {2,3})u_{1} \\ \dot{x}_{2,1} &=f_{2,1}(\bar {x}_{2,1})+g_{2,1} (\bar {x}_{2,1})x_{2,2} \\ \dot{x}_{2,2} &=f_{2,2}(\bar {x}_{2,2})+g_{2,2}(\bar {x}_{2,2})x_{2,3} \\ \dot{x}_{2,3} &=f_{2,3} (\bar {x}_{1,1},\bar {x}_{2,3})+g_{2,2}(\bar {x}_{1,1},\bar{x}_{2,3})x_{2,4} \\ \dot{x}_{2,4} &=f_{2,4}(X, u_{1})+g_{2,4}(\bar {x}_ {1,1},\bar{x}_{2,3})u_{2} \\ y_{j} &=x_{j,1}, \quad j=1,2, \end{align*}}$$

where $\bar{x}_{j,i_j}=[x_{j,1},\dots,x_{j,i_j}]^{\mathrm T},j=1,2, i_1=1,2, i_2=1,\dots,4$, and $X = [\bar{x}_{1,2}^{\mathrm T}, \bar{x}_{2,4}^{\mathrm T}]^{\mathrm T}$.

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Publication details: 

TitleAdaptive neural control of uncertain MIMO nonlinear systems
Publication TypeJournal Article
AuthorsGe, Shuzhi Sam, and Wang Cong

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