MIMO discrete-time system with triangular form inputs

Model description: 

Simulation studies are carried out for the following MIMO discrete-time system with triangular form inputs

$$\begin{align*} x_{1,1}(k+1) &= f_{1,1}({\bar x}_{1,1}(k))+g_{1,1}({\bar x}_{1,1}(k))x_{1,2}(k) \\ x_{1,2}(k+1) &= f_{1,2}({\bar x}_{1,2}(k))+g_{1,2}({\bar x}_{1,2}(k))u_{1} (k)+d_{1}(k) \\ x_{2,1}(k+1) &= f_{2,1}({\bar x}_{2,1}(k)) +g_{2,1}({\bar x}_{2,1}(k))x_{2,2}(k) \\ x_{2,2}(k+1) &= f_{2,2}({\bar x}_{2,2}(k),u_{1}(k)) +g_{2,2}({\bar x}_{2,2}(k))u_{2}(k)+d_{2}(k) \\ y_{1}(k) &= x_{1,1}(k) \\ y_{2}(k) &= x_{2,1}(k), \end{align*}$$


$\begin{cases} f_{1,1}({\bar x}_{1,1}(k))={{x^{2}_{1,1}(k)}\over {1+x^{2}_{1,1}(k)}}\\ g_{1,1}({\bar x}_{1,1}(k))=0.3 \\ f_{1,2}({\bar x}_{1,2}(k))= {{x^{2}_{1,1}(k)}\over{1+x^{2}_{1,2}(k)+x^{2}_{2,1}(k)+x^{2}_{2,2}(k)}}\\ g_{1,2}({\bar x}_{1,2}(k))=1\\ d_{1}(k)=0.1 \cos{0.05k}\cos{x_{1,1}(k)}\\ \end{cases}$

$\begin{cases} f_{2,1}({\bar x}_{2,1}(k))= {{x^{2}_{2,1}(k)}\over {1+x^{2}_{2,1}(k)}}\\ g_{2,1}({\bar x}_{2,1}(k))=0.2\\ f_{2,2}({\bar x}_{2,2}(k),u_{1}(k))={{x^{2}_{2,1}(k)}\over{1+x^{2}_{1,1}+x^{2}_{1,2}(k)+x^{2}_{2,2}(k)}}u^{2}_{1}(k)\\ g_{2,2}({\bar x}_{2,2}(k))=1\\ d_{2}(k)=0.1\cos{0.05k}\cos{x_{2,1}(k)}\\ \end{cases}$



Time domain: 


Publication details: 

TitleAdaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time
Publication TypeJournal Article
Year of Publication2004
AuthorsGe, S.S., Zhang Jin, and Lee Tong Heng
JournalIEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
Start Page1630
Date Published08/2004
Accession Number8111571
Keywordscascade systems, closed loop systems, discrete time systems, Lyapunov methods, MIMO systems, neural nets, nonlinear systems, stability
AbstractIn this paper, adaptive neural network (NN) control is investigated for a class of multiinput and multioutput (MIMO) nonlinear systems with unknown bounded disturbances in discrete-time domain. The MIMO system under study consists of several subsystems with each subsystem in strict feedback form. The inputs of the MIMO system are in triangular form. First, through a coordinate transformation, the MIMO system is transformed into a sequential decrease cascade form (SDCF). Then, by using high-order neural networks (HONN) as emulators of the desired controls, an effective neural network control scheme with adaptation laws is developed. Through embedded backstepping, stability of the closed-loop system is proved based on Lyapunov synthesis. The output tracking errors are guaranteed to converge to a residue whose size is adjustable. Simulation results show the effectiveness of the proposed control scheme.