# 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