Linear

Abelian categories with applications to rings and modules by M. Popescu

By M. Popescu

Abelian different types with purposes to jewelry and Modules (London Mathematical Society Monographs)

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Example text

Large variations of these parameters are considered parametric faults. 2. 12). The equivalent fault input is defined as f (t) := ΓA (ρ)x(t). For the fault vector f (t), we can use a more structured representation using the alternative affine representation of A(ρ) as A(ρ) = A(0) + A(1) ρ1 + A(2) ρ2 , which leads to A(ρ)x(t) = A(0) x(t) + A(1) x(t) A(2) x(t) ρ . 18). 19) where, for i = 1, . . , N, x (i) (t) ∈ Rn and y(i) (t) ∈ Rp are the state vector and output vector of the i-th system, respectively.

7) can be easily obtained for LPV models whose matrices depend rationally on the components of ρ. 11) S(ρ) = LFTu (M, Δ), M11 M12 is a certain constant matrix with M11 square and Δ = M21 M22 Δ(ρ) is a diagonal matrix depending on the components of ρ such that Δ(ρ0 ) = 0. Straightforward algorithms are available to obtain the above representation. The above LFT-based representation of S(ρ) allows to immediately obtain S (0) = M22 , ΔS = M21 and ΓS (ρ) = Δ(I − ΔM11 )−1 M12 . 5(1 + ρ1 ) C= 011 , 110 Du = 00 .

An important aspect of this approach is that the resulting models with additive faults can simultaneously cover several categories of actuator and sensor faults. 1 Flight actuators with faults are often modelled as continuous-time LTI models, whose transfer-function representation is y(s) = G u (s)u(s) + G f (s)f(s) , where u(t) and y(t) are respectively, the commanded and achieved surface positions and f (t) is a fault signal. For an input (actuator) fault we can take G f (s) = G u (s), while for an output (sensor) fault G f (s) = 1.