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On feedback transformation and integral input-to-state stability in designing robust interval observers

Abstract : Interval observers offer useful guarantees in monitoring the state of dynam-ical systems in the presence of large disturbances [1]. This talk addresses the problem of designing such observers for the controlled plant of the forṁ x(t) = A(y(t))x(t) + B(y(t))u(y(t),x + (t)) + δ(t), y(t) = Cx(t) (1) with the state x(t) ∈ R n , the output feedback control input u(y(t),x + (t)) ∈ R q , the measurement output y(t) ∈ R p and the disturbance δ(t) ∈ R n. The signalx + (t) ∈ R n denotes an estimate of x(t). Assume that x − 0 , x + 0 ∈ R n and δ − , δ + : R ≥0 → R n satisfying x − 0 ≤ x 0 ≤ x + 0 and δ − (t) ≤ δ(t) ≤ δ + (t) for all t ∈ R ≥0 are known, while x(0) = x 0 and δ(t) are unknown. Note that all the inequalities must be understood component-wise. To generate x − (t), x + (t) ∈ R n such that x − (t) ≤ x(t) ≤ x + (t), ∀t ∈ R ≥0 (2) holds, the following interval observer has been proposed in [2]: x + =A(y)x + + B(y)u + H(y)[Cx + − y] + S[R + δ + − R − δ − ] (3a) x − =A(y)x − + B(y)u + H(y)[Cx − − y] + S[R + δ − − R − δ + ], (3b) where S = R −1 , R + = (max{R i,j , 0}) n,n i,j=1,1 , R − = R + − R, x + = S + Rx + − S − Rx − and x − = S + Rx − − S − Rx +. Indeed, if there exists a matrix R such that for each fixed y ∈ R p ,
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Thach Ngoc Dinh, Hiroshi Ito. On feedback transformation and integral input-to-state stability in designing robust interval observers. The 5th International Symposium on Positive Systems, Sep 2016, Rome, Italy. 2 p. ⟨hal-02435251⟩

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