Delayed Unknown Input Observer
6.4 Extension to the PTWV case
i Q+QΓi−CTRiT −RiC <0 (6.25) Thus, from the Lypunov stability theory, if the LMI condition (6.21) is satisfied, the system (6.13) is expo-nentially asymptotically stable. This completes the proof of Theorem6.1.
Once, the state vector is estimated, ﬁnding the unknown parameters is straightforward by using an algebraic inversion as following:
θˆ= (CDy)† y˙ˆ−CAxˆ−CBu
where (CDy)†=h(CDy)T(CDy)i−1(CDy)T. However, the feasibility of this inversion is conditioned by a convenient selection of the time delay to fulﬁll the rank condition. By substituting the output derivative in the inversion model (6.26), the parameter estimation error is deﬁned as:
It is easy, from this equation, to show that the convergence ofθˆtowardsθ is ensured from the asymptotic stability of the state estimation errors x˜(t) under suitable persistence excitation described in the following deﬁnition.
Definition 6.2. The Persistent Excitation Condition is fullfiled if there existc1ij,c2ij andc3ij positive for i,j=1, ...,q, such that for allt the following inequality holds (Ioannou and Sun,1996):
τjdt ≤c2ijI ∀i,j =1, ...,q
Finally, the design procedure of the full order DUIO observer is summarized in the following Algorithm3.
An example is proposed in the appendixBto illustrate the estimation performance of DUIO observer.
Algorithm 3: Observer design procedure DUIO Require:
1: Check if system (A,Dy,C) is observable or detectable. If so, go to Step 2; Otherwise, stop.
2: Check the decoupling conditions :
3: if rank(C¯D¯y) =rank(D¯y) ∀y∈∆holds then m=0, go to step 10.
Find the minimum integer(m)←according to (6.8).
Reconstruct the augmented matricesA,B,C,Din (6.9).
5: Compute matricesHy,Py and deduce the matrixGy. 6: Compute the matrixP˙y,y˙ and deduceΓ=PyAy+P˙y,y˙. 7: Solve the LMIs (6.21).
9: DeduceNy,y˙ =Γ−Ky,y˙C andLy,y˙ =Ky,y˙−Ny,y˙Hy. 10: Estimate the parametersθ.ˆ
6.4 Extension to the PTWV case
6.4.1 Problem Statement
In the state space represenation described in equation (6.1), the matricesA,¯ C¯ andB¯ are supposed to be time invariant. However, we have seen in section 5.1 that matrixA in the PTWV dynamics model of equation
116 Chapter 6. Delayed Unknown Input Observer (5.1) is parameter varying with resepect to ζ. So, in this section we aim to extend the full order DUIO design procedure to the case of the following LPV class :
x˙(t) = A˜(ζ)x(t) +Bu˜ (t) +D˜(ζ)F(y,y, ..,˙ y(n),θ)
y = Cx˜ (t) (6.28)
where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector and y(t) ∈ Rny denotes the output vector. The vector F
y,y, ..,˙ y(n),θ
∈Rp incloses all non-linear terms which depend on the known time varying parameter ζ and the unknown constant parameterθ. The matricesA˜(ζ) and D˜(ζ) are parameter varying matrices of appropriate dimensions while we suppose that the matricesC˜ andB˜ are constant.
Without loss of generality, here we assume thatrank(D˜(ζ)) =pandrank(C˜) =nywhereny< p. Also, we assume that the vectorζandζ˙are respectively deﬁned on the hyperplanes∆ and∆¯ deﬁned by:
∆=ζ∈Rnζ| ζimin ≤ζi≤ζimax }
∆¯ =nζ˙∈Rnζ| ζ˙imin≤ζ˙i≤ζ˙imax } (6.29) where(ζimin,ζimax)and(ζ˙imin,ζ˙imax)are the lower and upper bounds of the forward speed and its derivative respectively.
Assumption 6.1. In the following, assume that F can be written in an affine linear form with respect to unknown parameters such that:
y,y, ..,˙ y(n),θ
y,y, ..,˙ y(n)
where the matrixDy =D
y,y, ..,˙ y(n)
∈Rp×p is full column rank, i.e. rank(Dy) =p.
From the aforementioned assumption, and for simple readability, the system in equation (6.28) is rewritten as:
x˙(t) = A˜ζx(t) +Bu˜ (t) +D˜ζDyf(θ)
y = Cx˜ (t) (6.31)
Theorem 6.2. The well-known rank condition for the existence of the UIO is given by the following state-ments (Darouach and Boutat-Baddas,2008and Trentelman, Stoorvogel, and Hautus,2012):
1. The statexis bounded (stable or stabilized), 2. The system(A˜ζ,D˜ζ,C˜)is observable or detectable,
3. The relative degreerof the system with respect to the unknown part F =Dyf(θ)exists.
Usually, this theorem can be summarized into two well-known conditions for the existence of the UIO, the observability condition of the pair (A˜ζ,C˜) and the matching condition rank(C˜D˜ζ) = rank(D˜ζ) on the system matrices. Based on system deﬁned by (6.31), one considers the case whenrank(C˜D˜ζ)6=rank(D˜ζ),
∀ ζ ∈ ∆ and ζ˙ ∈∆¯. In this case, the matching condition is not satisﬁed since p > ny and an augmented model is reconstructed from delayed outputs as described in section6.1.
6.4. Extension to the PTWV case 117
where τm is themth time delay, and m represents the number of delayed outputs to recover the matching condition. The parametermcan be easily computed fromp=ϑ×n+β, where,
|β|< n andm=
ϑ if β ≤0
ϑ+1 Otherwise (6.32)
The augmented state-space representation of the system is as follow:
In a simple form, the augmented system takes the following structure : x˙a(t) =Aζxa(t) +Bζua(t) +Dζ,yf(θ)
For the augmented system (6.33) we propose the LPV unknown input observer of the form:
118 Chapter 6. Delayed Unknown Input Observer ( z˙(t) = Nζ,ζ˙z+Lζ,ζ˙ya+Gζua
xˆa(t) = z(t)−Hζya(t) (6.35)
where,xˆaandyˆaare respectively the estimated state and the output vector. The matricesNζ,ζ˙,Lζ,ζ˙,Gζ and Hζ are parameter varying. The gains design is addressed in the same manner as in section6.3. Therefore, if e=xa−xˆa is the estimation error and∀ζ∈∆and∀ζ˙∈∆¯, if the following conditions hold, the estimation error tends asymptotically towards zero.
(1) e˙=Nζ,ζ˙e, must be asymptotically stable i.e.,Nζ,ζ˙ is Hurwitz, (2) P˙ζ,ζ˙+PζAζ−Nζ,ζ˙Pζ−Lζ,ζ˙Cζ =0.
(3) PζDζ,y=0 (4) PζBζ−Gζ =0
From these conditions, we can state the set of LMIs to be solved in order to design the DUIO gains as follows:
Nζ,ζ˙ =Γζ,ζ˙−Kζ,ζ˙Cζ (6.36)
Hζ =−Dζ,y(CζDζ,y)† (6.37)
Pζ =I2n+HζCζ (6.38)
Gζ =PζBζ (6.39)
ζ =P˙ζ,ζ˙+PζA˜ζ andKζ,ζ˙ =Nζ,ζ˙Hζ+Lζ,ζ˙.
Since the matching condition rank(CDζ,y) = rank(Dζ,y) is satisﬁed, the unknown parameters can be recovered by a simple algebraic inversion as following :
fˆ= (CζDζ,y)† y˙a(t)−CζAζxˆa−CζBζua(t) (6.40)
In which, the convergence offˆtowardsf can be analyzed by deﬁning the unknown part estimation error ef =f−fˆ=−(CζDζ,y)†CζAζe
Knowing thateconverges asymptotically towards zero, thenef also converges asymptotically towards zero.
The stability analysis of the system is studied in the same way as in section (6.3) and yields to the following BMI conditions ensuring asymptotic stability:
ΓTi Q+QΓi−CiTKiTQ−QKiCi<0, i=1, ...,r.
Finally, the performances of the DUIO can be improved by pole assignment in an LMI region to ensure an acceptable transient response. The poles of the estimator are considered in the complex plane region, and can be represented as an LMI region given by the stability marginς >0 in a subsetΘof the complex plane.
In this case, the matrixΓi is saidΘi-stable when its spectrum λ(Γi)belongs to regionΘi.
Θi=z= (xz+i.yz)∈C| Re(z)≤ −ς ⇔z+z¯+2ς <0 } (6.41)