Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements

Cross‐Scale Imperfect Data‐Based Composite H∞$$ {H}_{\infty } $$ Control of Nonlinear Two‐Time‐Scale Systems

“…Hence, we now derive the slow and fast subsystems based on time-scale decomposition principle. Referring to [8], to derive the slow subsystem, let 𝜀 = 0 in Equation (1a) and solve for 𝑥 2 (to be denoted as 𝑥 2𝑠 ) under the nonsingular property of 𝑓 22…”

“…Hence, we now derive the slow and fast subsystems based on time‐scale decomposition principle. Referring to [8], to derive the slow subsystem, let $$$ \varepsilon =0 $$$ in Equation () and solve for $$$ {x}_2 $$$ (to be denoted as $$$ {x}_{2s} $$$) under the nonsingular property of $$$ {f}_{22} $$$ $$$$ {x}_{2s}={F}_{s0}\left({x}_{1s}\right)+{G}_{s0}\left({x}_{1s}\right){u}_s+{K}_{s0}\left({x}_{1s}\right)\omega $$$$ where the subscript “s” denotes the slow component of the variable. Substituting the Equation () into () yields the slow subsystem $$$$ {\dot{x}}_{1s}={F}_{s0}\left({x}_{1s}\right)+{G}_{s0}\left({x}_{1s}\right){u}_s+{K}_{s0}\left({x}_{1s}\right)\omega $$$$ where …”

“…Therefore, the robustness of the two sub‐controllers that satisfy the attenuation indices $$$ {\gamma}_s $$$ and $$$ {\gamma}_f $$$, respectively, needs to be rigorously analyzed with respect to the original full‐order TTS system. Currently, some scholars focus on exploring the optimal perturbation attenuation level $$$ {\gamma}^{\ast } $$$ (i.e., the infimum of $$$ \gamma $$$) and designing the $$$ {H}_{\infty } $$$ suboptimal controller with the help of the zero‐sum game idea [8]. The system decomposition means given therein are of certain reference significance, but there is a noticeable difference between the control objective and that of this article.…”

Stability Analysis and Design for Nonlinear Singular Systems

“…H ∞ control problem for SPSs were studied by making use of a differential game theoretic approach in [193,40,161,162]. A class of SPSs being nonlinear only on the slow variables was examined in [163,190] and H ∞ controller design methods were proposed. In [225,155,192], H ∞ control of nonstandard SPSs was considered and some efficient design methods were proposed.…”

“…Besides the approaches based on T-S fuzzy model [137]- [233], there have been various design methods for nonlinear SPSs in particular forms. In [163]- [54], H ∞ control for nonlinear SPSs was investigated. However, these methods can be only applied to standard nonlinear SPSs since they are based on decomposing the original system into two reduced-order subsystems in different time scales.…”