On an extension of homogeneity notion for differential inclusions

Bernuau, Emmanuel; Efimov, Denis; Perruquetti, Wilfrid; Polyakov, Andrey

doi:10.23919/ecc.2013.6669525

Cited by 51 publications

(91 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Theorem Let F be an r ‐homogeneous set‐valued map with degree m , satisfying the standard assumptions. Then, the next claims are equivalent System is strongly globally asymptotically stable. For all

k > \max false(- m, 0 false)

, there exists a pair ( V , W ) of continuous functions, such that

V \in C^{\infty} false(R^{n} false)

is positive definite and homogeneous with degree k ,

W \in C^{\infty} false(R^{n} ∖ false{0 false} false)

is strictly positive outside the origin and homogeneous with degree k + m , and

\max_{ν \in F false(x false)} D V false(x false) ν \leq - W false(x false)

for all x ≠ 0. …”

Section: Preliminariesmentioning

confidence: 95%

See 1 more Smart Citation

On finite‐time robust stabilization via nonlinear state feedback

Zimenko

Polyakov

Efimov

2018

Intl J Robust & Nonlinear

View full text Add to dashboard Cite

Summary A nonlinear control law is designed for finite‐time stabilization of a chain of integrators. The method is based on the implicit Lyapunov function technique and homogeneity properties. The scheme of control parameter selection is presented by a linear matrix inequality. The method is simple in implementation and does not assume an online procedure for the computation of the implicit Lyapunov function value at the current state that is an improvement with respect to the works of Polyakov et al. The control law is presented in an explicit form and allows finding the values of all control parameters that make the solution one of the most constructive. The theoretical results are supported by numerical examples.

show abstract

k > \max false(- m, 0 false)

, there exists a pair ( V , W ) of continuous functions, such that

V \in C^{\infty} false(R^{n} false)

is positive definite and homogeneous with degree k ,

W \in C^{\infty} false(R^{n} ∖ false{0 false} false)

is strictly positive outside the origin and homogeneous with degree k + m , and

\max_{ν \in F false(x false)} D V false(x false) ν \leq - W false(x false)

for all x ≠ 0. …”

Section: Preliminariesmentioning

confidence: 95%

“…Assume also that F is strongly globally asymptotically stable. Then, F is strongly globally finite‐time stable, and the settling‐time function is continuous at zero and locally bounded …”

Section: Preliminariesmentioning

confidence: 99%

On finite‐time robust stabilization via nonlinear state feedback

Zimenko

Polyakov

Efimov

2018

Intl J Robust & Nonlinear

View full text Add to dashboard Cite

show abstract

“…Under Hypothesis 1, given the observer (7) and the control (13), provided that the initial value of the estimation statex is close enough to the initial state x, the reference trajectory is reached and followed in finite-time.…”

Section: Robust Trajectory Trackingmentioning

confidence: 99%

“…In this paper, we will use continuous as well as discontinuous vector fields. For a more detailed introduction on the properties of continuous homogeneous objects, we refer to [9]; about discontinuous homogeneous objects, to [13].…”

Section: Homogeneitymentioning

confidence: 99%

Finite-time observation and trajectory tracking for a direct wave energy converter

Bernuau

Glumineau

Plestan

et al. 2015

2015 IEEE Conference on Control Applications (CCA)

Self Cite

View full text Add to dashboard Cite

In this paper, we address the problem of finite-time observation and trajectory tracking for the control of a direct wave energy converter device. We first consider a simplified model of the system on which we develop an homogeneitybased finite-time observer. Then we design a control able to make the system follow a given reference trajectory in finitetime. Simulation results finally demonstrate the applicability of the method.

show abstract

“…X(t, x 0 ) ∈ S \ ∂S for all t > 0 and all x 0 ∈ ∂S). The requirement on continuity of the function f has been relaxed in [25] (the function V can still be selected smooth).…”

Section: A Weighted Homogeneitymentioning

confidence: 99%