Validaci´on de la arquitectura de QoS de la pasarela residencial

4.2.1 Fundamental Problem of Residual Generation

Consider the LPV plant G(ρ) given in state space form as ˙

x = A(ρ)x + Bu(ρ)u + Bd(ρ)d + Bf(ρ)f (4.1)

y = C(ρ)x + Du(ρ)u + Dd(ρ)d + Df(ρ)f (4.2)

where x∈ Rnx _{is the plant state, y} ∈ Rny _{is the measured output, u}∈ Rnu _{is the input,}

4.2 Brief Review of Model-Based Residual Generation

It can be equivalently expressed in transfer function form as

y = Gyu(ρ)u + Gyd(ρ)d + Gyf(ρ)f (4.3)

where Gyu(ρ), Gyd(ρ) and Gyf(ρ) are the parameter-dependent transfer function matrices

from u, d and f respectively to y.

The fundamental problem of residual generation (FPRG) concerns the search for a stable filter F (ρ) whose output is sensitive to the occurrence of specified faults, while remaining insensitive to the disturbances. F (ρ) takes ¯y = [yT uT]T as input and computes the residual r, that is

r = F (ρ) [ y u ] . (4.4)

The residual signal r has the following properties: i) r(t) = 0 if f (t) = 0 for all u(t) and d(t) ii) r(t)̸= 0 if fj(t)̸= 0 for any j = 1, . . . , nf

where fj denotes the j-th element of the fault vector, also referred to as the j-th fault.

From (4.3) and (4.4), r can be written as

r = Gru(ρ)u + Grd(ρ)d + Grf(ρ)f (4.5) where Gru(ρ) = F (ρ) [ Gyu(ρ) I ] , Grd(ρ) = F (ρ) [ Gyd(ρ) 0 ] , Grf(ρ) = F (ρ) [ Gyf(ρ) 0 ] .

The j-th column of Grf, i.e. the transfer function column vector from fjto r is denoted

Grf [j]. Condition ii) requires that this is non-zero for all j = 1, . . . , nf while condition i)

requires Gru = 0 and Grd = 0. [Var07] defines a stronger condition which requires that

persistent (constant) faults produces asymptotically persistent (constant) residuals. This is referred to as strong detectability and requires ∥Grf [j](0)∥ > 0. A detectable fault that

does not fulfil this condition is known as weakly detectable.

4.2.2 Perfect and Approximate Decoupling

The problem formulation described in the previous section refers to perfect decoupling, in which the residual is completely insensitive to disturbances. The existence of such a solution is conditional on the subspace of the fault direction(s) lying outside of the minimal unobservability subspace containing the disturbance directions. This is shown by [MVW89] for LTI systems, [BB04] for LPV systems, and [PI01] for nonlinear systems. Geometric methods for residual generator synthesis based on these subspaces are proposed

in the respective works. Further approaches are available for solving the perfect decoup- ling problem, including unknown input observers and parametric eigenvalue assignments [CP99]. By approximating the effects of model uncertainties by appropriate disturbance inputs, robust residual generation may also be tackled using these methods [CP99].

When the condition for perfect decoupling is not satisfied, or it is desirable to handle model uncertainties directly, then approximate decoupling approaches for residual gen- eration can be applied. In this formulation, the decoupling constraint is relaxed to an attenuation of the disturbance effects. The fault sensitivity constraint is also modified to ensure that the effect of the fault on the residual is not merely non-zero, but sufficiently large to be distinguishable from the effect caused by disturbances and uncertainties. To facilitate the discussion, let the function R(·) be a suitable measure of system input-to- output gain, withS(·) a compatible sensitivity measure. Considering only the case without uncertainty for now, conditions i) and ii) from the previous section are modified into

i)R(Grd) < γd, R(Gru) < γu (4.6a)

ii)S(Grf) > γf, with γf > max(γd, γu) (4.6b)

For uncertain systems, the above conditions need to hold for all permissible uncertain- ties. These conditions guarantee that for d and u with upper bounded sizes, one can find a lower-bound for f such that faults with sizes exceeding this lower bound have effects on

r that are distinguishable from those due to the bounded disturbance and inputs.

The design objectives i) and ii) are generally conflicting, leading to a multi-objective optimisation for residual generation synthesis, i.e., minimising the influence of the dis- turbances and inputs on the residual while simultaneously maximising the effects of the possible faults. Some approaches fuseR(·) and S(·) into a single metric by means of sub- tractions or a quotient to reduce the multi-objective optimisation problem into a single objective one. [Din13] provides examples of these metrics:

• JS−R = αfS(Grf)− αdR(Grd)− αuR(Gru)

• JS/R=

αfS(Grf)

αdR(Grd) + αuR(Gru)

where αd, αu, αf ∈ R+. These two metrics, which should be maximised, are shown to

be equivalent to the multi-objective one in a certain sense in [Din13]. Instead of using either of these in this work, the trade-off between attenuation and sensitivity measures is kept explicit at this stage as motivated by the mixed H_∞ / H₋ approaches in literature [JLM05; WYL07]. The measures will be combined later in the worst-case induced-L2

norm framework.

The abovementioned norm is used in this work directly as the R(·) measure. It was defined for uncertain LPV systems in the previous chapter in (3.17), and for nominal LTI systems it is equivalent to the H_∞ norm. In the next sections, it will be shown that recasting the sensitivity measure S(·) in terms of a reference model matching constraint