Stability of nonlinear repetitive processes with possible failures

• Main Author: J. P. Emelianova
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2014-01-01
• Series: Nauka i Obrazovanie
• Subjects: stability; 2D systems; vector Lyapunov functions; iterative learning control
• Online Access: http://technomag.edu.ru/jour/article/view/585

<p>Nonlinear discrete-time repetitive processes with Markovian jumps are considered. For such processes stability analysis is developed and this result is then applied to iterative learning control design.</p><p>Stability of nonlinear repetitive processes has not been developed previously in the current literature. This paper proposes and characterizes a stability theory for nonlinear repetitive processes that includes stability along the pass of linear examples as a special case.</p><p>For considered systems the second Lyapunov method cannot be used. Because repetitive processes belong to a class of 2D systems in which state variables are depend on two independent variables and cannot be solved using all first differences of state variables. It is not allow us to find a first difference of Lyapunov function along the trajectory of the system without finding solution of a system of equations that fully excludes a main advantage of second Lyapunov method. At the same time the use of vector Lyapunov functions and discrete-time counterpart of the divergence operator of this function along the trajectories of system instead of first difference allow us to obtain constructive results.</p><p>In this paper based on vector Lyapunov function approach sufficient conditions for pass profile exponential stability are obtained which in the linear case are obtained in terms of linear matrix inequalities and in the linear case without failures these conditions are reduced to known conditions of stability along the pass</p><p>A major application area where repetitive process stability theory can be used is Iterative Learning Control (ILC). The idea of ILC is following.</p><p>If the system repeats the same finite duration operation over and over again, it is reasonable to use the input and output variables on the current pass for improving accuracy of performance of operations on the next pass.</p><p>The new theoretical stability results are applied to ILC design under possible information failures. The ILC law convergence reduces to pass profile stability analysis. Computation and modeling of the system have been carried out using a simplified model of a vertical axis dynamics of a gantry robot.</p>