The generalizedH2controller synthesis problem of sampled-data systems

Robust sliding mode control for robot manipulators with analysis on trade‐off between reaching time and L∞ gain

“…Hence, these results undoubtedly demonstrate the validity of Theorem 2 with respect to the qualitative characteristic of finite-time convergence. Thus, as an immediate consequence of Corollary 1, the position and velocity errors (i.e., x(t)) converge asymptotically to zero for all the four cases, and the corresponding For a quantitative verification of the arguments provided in Theorem 2, the computation results for the reaching time t r defined as (28) and the convergence times observed from Figures 3 and 6, at which 𝜎(t) becomes zero, are provided in Tables 4 and 5. We can see from these tables that the sliding surface variable 𝜎(•) arrives at the origin no later than t = t r , and this undoubtedly verifies the overall arguments discussed in Theorem 2 for both the qualitative and quantitative aspects.…”

“…This contradicts the fact that a finite t such that V(t) > 0 can be arbitrarily taken, and thus, V(t) should converge to 0 in a finite time no larger than t r defined in (28). This completes the proof.…”

“…The rationale behind this fact might be interpreted as originating from the following two aspects. (i) Unlike the well‐known $$$ {\mathcal{L}}_{\infty }/{\mathcal{L}}_2 $$$ gain [27, 28], it is quite difficult to derive an analytical solution to the $$$ {\mathcal{L}}_{\infty } $$$ gain computation even for LTI case, and thus, various approximate approaches are developed in [29–31]. (ii) A tractable expression of the solution to () is not readily derived, due to some difficulties in its non‐uniqueness occurring from taking $$$ \operatorname{sgn}\left(\cdotp \right) $$$.…”

“…To put it another way, 𝜎(t) arrives at the origin no later than t = t r and does not deviate from the sliding surface 𝜎 = 0 after the arrival. In this sense, we call t r defined as (28) the reaching time throughout the paper.…”

The ℓ∞/p$$ {\ell}_{\infty /p} $$‐gains for discrete‐time observer‐based event‐triggered systems

“…To improve the applicability of the observer‐based ETC scheme of discrete‐time systems to various practical systems, we develop methods for computing two types of system gains from $$$ {\ell}_2 $$$ to $$$ {\ell}_{\infty } $$$ and from $$$ {\ell}_{\infty } $$$ to $$$ {\ell}_{\infty } $$$, denoted by the $$$ {\ell}_{\infty /2} $$$‐gain and the $$$ {\ell}_{\infty /\infty } $$$‐gain, respectively, in which they could be interpreted as natural extensions of the generalized $$$ {H}_2 $$$ norm 29‐33 and the $$$ {L}_{\infty } $$$‐induced norm 34‐38 for typical feedback control systems, respectively. Regarding a tractable treatment of the dynamics of observer‐based event‐triggered systems (ETSs), we first introduce a piecewise linear difference equation with which the corresponding closed‐form representation could be derived.…”