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

2

## Publication details:

 Title Robust nonlinear adaptive control using neural networks Publication Type Conference Paper Year of Publication 2001 Authors Adetona, O., Sathananthan S., and Keel L.H. Conference Name Proceedings of the 2001 American Control Conference, 2001 Date Published 06/2001 Publisher IEEE Conference Location Arlington, VA ISBN Number 0-7803-6495-3 Accession Number 7092721 Keywords adaptive control, asymptotic stability, neurocontrollers, nonlinear control systems, radial basis function networks, robust control Abstract This paper provides a robust indirect adaptive control method for non-affine plants. Subject to some mild assumptions, the method can be applied to both minimum and non-minimum phase plants with operating regions of any finite size while avoiding a set of restrictions, at least one of which is imposed by all existing methods. The benefits are achieved under the following assumptions: 1) the operating region is limited to the basin of attraction of an asymptotically stable equilibrium point of the plant; 2) the desired output of the plant is sufficiently slowly varying; and 3) the output of the plant must be sufficiently sensitive to the input signal. It is shown that the adaptive control system will be stable in the presence of unknown bounded modeling errors DOI 10.1109/ACC.2001.946247

# 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*}

## Publication details:

 Title U-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems Publication Type Conference Paper Year of Publication 2006 Authors Azhar, A.S.S., Al-Sunni F.M., and Shafiq M. Conference Name 1ST IEEE Conference on Industrial Electronics and Applications, 2006 Date Published 05/2006 Publisher IEEE Conference Location Singapore ISBN Number 0-7803-9513-1 Accession Number 9097014 Keywords bilinear systems, discrete time systems, linear systems, MIMO systems, neurocontrollers, radial basis function networks Abstract Bilinear systems are attractive candidates for many dynamical processes, since they allow a significantly larger class of behaviour than linear systems, yet retain a rich theory which is closely related to the familiar theory of linear systems. A new technique for the control of unknown MIMO bilinear systems is introduced. The scheme is based on the U-model with identification based on radial basis functions neural networks which is known for mapping any nonlinear function. U-model is a control oriented model used to represent a wide range of non-linear discrete time dynamic plants. The proposed tracking scheme is presented and verified using simulation examples DOI 10.1109/ICIEA.2006.257063

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

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## Publication details:

 Title Data-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems Publication Type Journal Article Year of Publication 2011 Authors Hou, Zhongsheng, and Jin ShangTai Journal IEEE Transactions on Neural Networks Volume 22 Issue 12 Start Page 2173 Pagination 2173-2188 Date Published 11/2011 ISSN 1045-9227 ISBN Number 12409274 Keywords adaptive control, control system synthesis, convergence, discrete time systems, linearisation techniques, MIMO systems, nonlinear control systems, stability, tracking Abstract In this paper, a data-driven model-free adaptive control (MFAC) approach is proposed based on a new dynamic linearization technique (DLT) with a novel concept called pseudo-partial derivative for a class of general multiple-input and multiple-output nonlinear discrete-time systems. The DLT includes compact form dynamic linearization, partial form dynamic linearization, and full form dynamic linearization. The main feature of the approach is that the controller design depends only on the measured input/output data of the controlled plant. Analysis and extensive simulations have shown that MFAC guarantees the bounded-input bounded-output stability and the tracking error convergence. DOI 10.1109/TNN.2011.2176141

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

## Publication details:

 Title Data-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems Publication Type Journal Article Year of Publication 2011 Authors Hou, Zhongsheng, and Jin ShangTai Journal IEEE Transactions on Neural Networks Volume 22 Issue 12 Start Page 2173 Pagination 2173-2188 Date Published 11/2011 ISSN 1045-9227 ISBN Number 12409274 Keywords adaptive control, control system synthesis, convergence, discrete time systems, linearisation techniques, MIMO systems, nonlinear control systems, stability, tracking Abstract In this paper, a data-driven model-free adaptive control (MFAC) approach is proposed based on a new dynamic linearization technique (DLT) with a novel concept called pseudo-partial derivative for a class of general multiple-input and multiple-output nonlinear discrete-time systems. The DLT includes compact form dynamic linearization, partial form dynamic linearization, and full form dynamic linearization. The main feature of the approach is that the controller design depends only on the measured input/output data of the controlled plant. Analysis and extensive simulations have shown that MFAC guarantees the bounded-input bounded-output stability and the tracking error convergence. DOI 10.1109/TNN.2011.2176141

# 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 $X = [\bar{x}_{1,2}^{\mathrm T}, \bar{x}_{2,4}^{\mathrm T}]^{\mathrm T}$.

## 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