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

Type: 

Form: 

Model order: 

2

Time domain: 

Linearity: 

Publication details: 

TitleRobust nonlinear adaptive control using neural networks
Publication TypeConference Paper
Year of Publication2001
AuthorsAdetona, O., Sathananthan S., and Keel L.H.
Conference NameProceedings of the 2001 American Control Conference, 2001
Date Published06/2001
PublisherIEEE
Conference LocationArlington, VA
ISBN Number0-7803-6495-3
Accession Number7092721
Keywordsadaptive control, asymptotic stability, neurocontrollers, nonlinear control systems, radial basis function networks, robust control
AbstractThis 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
DOI10.1109/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:

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

Type: 

Form: 

Time domain: 

Linearity: 

Publication details: 

TitleU-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems
Publication TypeConference Paper
Year of Publication2006
AuthorsAzhar, A.S.S., Al-Sunni F.M., and Shafiq M.
Conference Name1ST IEEE Conference on Industrial Electronics and Applications, 2006
Date Published05/2006
PublisherIEEE
Conference LocationSingapore
ISBN Number0-7803-9513-1
Accession Number9097014
Keywordsbilinear systems, discrete time systems, linear systems, MIMO systems, neurocontrollers, radial basis function networks
AbstractBilinear 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
DOI10.1109/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: $y_1(1)=y_1(3)=0$, $y_1(2)=1$, $y_2(1)=y_1(3)=0$, $y_2(2)=1$, $u(1)=u(2)=[0,0]^{\mathrm T}$

Type: 

Form: 

Model order: 

3

Time domain: 

Linearity: 

Autonomity: 

Publication details: 

TitleData-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems
Publication TypeJournal Article
Year of Publication2011
AuthorsHou, Zhongsheng, and Jin ShangTai
JournalIEEE Transactions on Neural Networks
Volume22
Issue12
Start Page2173
Pagination2173-2188
Date Published11/2011
ISSN1045-9227
ISBN Number12409274
Keywordsadaptive control, control system synthesis, convergence, discrete time systems, linearisation techniques, MIMO systems, nonlinear control systems, stability, tracking
AbstractIn 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.
DOI10.1109/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: $x_{1,1}(1)=x_{1,1}(2)=x_{2,1}(1)=x_{2,1}(2)=0.5$, $x_{1,2}(1)=x_{1,2}(2)=x_{2,2}(1)=x_{2,2}(2)=0$, $u(1)=u(2)=[0,0]^{\mathrm T}.$

Type: 

Form: 

Time domain: 

Linearity: 

Publication details: 

TitleData-Driven Model-Free Adaptive Control for a Class of MIMO Nonlinear Discrete-Time Systems
Publication TypeJournal Article
Year of Publication2011
AuthorsHou, Zhongsheng, and Jin ShangTai
JournalIEEE Transactions on Neural Networks
Volume22
Issue12
Start Page2173
Pagination2173-2188
Date Published11/2011
ISSN1045-9227
ISBN Number12409274
Keywordsadaptive control, control system synthesis, convergence, discrete time systems, linearisation techniques, MIMO systems, nonlinear control systems, stability, tracking
AbstractIn 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.
DOI10.1109/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 $X = [\bar{x}_{1,2}^{\mathrm T}, \bar{x}_{2,4}^{\mathrm T}]^{\mathrm T}$.

Type: 

Form: 

Time domain: 

Linearity: 

Publication details: 

TitleAdaptive neural control of uncertain MIMO nonlinear systems
Publication TypeJournal Article
Year of Publication2004
AuthorsGe, Shuzhi Sam, and Wang Cong
JournalIEEE Transactions on Neural Networks
Volume15
Issue3
Start Page674
Pagination674-692
Date Published05/2004
ISSN1045-9227
Accession Number8012935
Keywordsadaptive control, closed loop systems, control system synthesis, MIMO systems, neurocontrollers, nonlinear control systems
AbstractIn 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.
DOI10.1109/TNN.2004.826130

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