Dissipativity and stabilization of nonlinear repetitive processes

Abstract

Repetitive processes are characterized by repeated executions of a task defined over a finite duration with resetting after each execution is complete. Also the output from any execution directly influences the output produced on the next execution. The repetitive process model structure arises in the modeling of physical processes and can also be used to effect in the control of other systems, such as iterative learning control where experimental verification of designs has been reported. The existing systems theory for them is, in the main, linear model based. This paper considers nonlinear repetitive processes using a dissipative setting and develops a stabilizing control law with the required conditions expressed in terms of vector storage functions. This design is then extended to stabilization plus disturbance attenuation.

Passivity Based Iterative Learning Control Design in the Discrete Repetitive Process Setting

“…where M f > 0 is a finite scalar, κ d > 0 is independent of T and 0 < ζ d < 1 determines the rate of convergence of the pass state initial vector sequence. Throughout this paper, it is assumed that the boundary conditions considered satisfy (7) and no further explicit mention of these conditions is made. The unique control problem for repetitive processes is the possible presence of oscillations that increase in amplitude from pass-to-pass.…”

“…This section summarizes the required results from the existing passivity based stability theory, which follows in the main [7] for nonlinear dynamics…”

“…For example, the stability of nonlinear 2D systems written in the Roesser or Fornasini-Marchesini state-space model form have been considered in, e.g., [6]. In [7] a stability theory was developed for discrete nonlinear repetitive processes where vector Lyapunov functions were used to characterize the practically motivated stability properties.…”

“…In the case of standard, termed 1D in the multidimensional systems literature, nonlinear systems, dissipativity theory [8] is one of the most powerful methods for control design, where a particular form, known as passivity (and its generalizations) see, e.g., [9], [10], can be used to solve the global feedback stabilization problem for a wide class of systems. In [7] new results on a dissipativity approach to the stabilization of discrete and differential nonlinear repetitive processes were obtained by using a vector storage function approach that is different from that in, e.g., [11] and the resulting design based on divergence properties of this function.…”

Finite-region boundedness and stabilization for 2D continuous-discrete systems in Roesser model

“…As such, the research on two-dimensional continuous-discrete systems has been a hot area in control field. Especially, the stability analysis for two-dimensional continuous-discrete systems has attracted much attention of some researchers over the last few decades, and several interesting findings in linear and non-linear frameworks have been obtained (see Xiao, 2001;Benton et al, 2002;Knorn & Middleton, 2013a;Chesi & Middleton, 2014;Knorn & Middleton, 2016;Galkowski et al, 2016;Pakshin et al, 2016;Wang et al, 2017 and references therein). For example, in the linear setting, Xiao (2001) considered three models of two-dimensional continuous-discrete systems and gave sufficient and necessary conditions for their Lyapunov asymptotic stability (LAS)-based two-dimensional characteristic polynomial.…”