Asymptotic robust performance of the discrete tracking system in the \(\ell_1\)-metric
The author studies the robust stability and asymptotic performance of discrete linear systems with parametric and additive perturbations and driving inputs of arbitrary dimension. The goal is to determine the worst-case stable absolute value of the system output in the classes of structured perturbations with finite and damped memories. Necessary and sufficient conditions for robust stability and explicit formulas for the worst-case absolute value of the upper limit of the system output, which are obtained in the paper, are applicable, in particular, to the tracking problem for a finite-dimensional object with scalar control and scalar output (SISO system) with parametric perturbations in the output and control channels and external additive perturbations on the object and measurements: Consider a discrete linear controlled system \[ a(q^{-1})y(t)=q^{-d}b(q^{-1})u(t) + v(t),\quad t\in\mathbb N, \] where \(y(t)\), \(u(t)\), and \(v(t)\) denote, respectively, scalar output, control, and perturbation of the system, \(q^{-1}\) \((q^{-1}z(t):= z(t-1))\) is the operator of backward shift, \(d\) is the control delay, and \[ a(q^{-1}) = 1 + a_1q^{-1} +\dots + a_nq^{-n},\quad b(q^{-1}) = b_0 + b_1 +\dots + b_mq^{-m}. \] Notably, the application of any result obtained for the class of perturbations with finite and damped memories is hampered by the fact that in the chosen model an infinite sequence of observations is required to verify whether the perturbations belong to any of the above classes, which is impracticable. To overcome this difficulty, the class of bounded-memory perturbations is considered, and the formulas obtained are proved to be asymptotically precise for an unlimited growth of the perturbation memory, provided that an additional assumption about the driving input is made.
perturbations with finite and damped memories
scientific article; zbMATH DE number 2107769