Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation

Lemma 5 in [2] was incorrect. A correction is provided here. The error does not affect the rest of the results in the paper. 1 The Error and the Fix There is an error in the statement and the proof of Lemma 5 in the paper [2]. The estimate about α0(|z(t)|) towards the end of the proof of Lemma 5 is incorrect. It should be replaced by α0(|z(t)|) ≤ β(|z0| , τ(t)) + ∫ τ(t) 0 γ(|dx(s)|) ds ≤ β(|z0| , τ(t)) + ∫ t 0 γ(|d(s)|)φ(z(s)) ds. Without knowing whether φ(|z(s)|) is bounded, one cannot conclude that α0(|z(t)|) is bounded. Below we provide a correction of the lemma by posing a boundedness condition on the function φ. Lemma 1 Consider a general system ẋ = f(x, d), (1) where f is a locally Lipschitz function and d is a locally essentially bounded disturbance input. Suppose the system is iISS. Let φ : Rn → R be a bounded, positive definite and smooth function. Then the system ẋ = φ(x)f(x, d) (2) is iISS. Remark 1 The change in the lemma does not affect the consequent results in [2]. This is because in Lemma 6 we indeed require that the function φ be bounded by 1. Hence, the main result in Section 8 of [2], Theorem 1, remains valid. ∗Email: liberzon@uiuc.edu. Supported by NSF Grant ECS-0114725. †Email: sontag@control.rutgers.edu. Supported by US Air Force Grant F49620-98-1-0242. ‡Email: ywang@control.math.fau.edu. Supported by NSF Grant DMS-9457826.

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