Extension to the PTWV case

Γ^T

i Q+QΓ_i−C^TR_i^T −R_iC <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= (CD_y)^† y^˙ˆ−CAxˆ−CBu

(6.26)

where (CD_y)^†=^h(CD_y)^T(CD_y)ⁱ⁻¹(CD_y)^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:

e_θ=θ−θˆ=₋(CD_y)^†CAx˜ (6.27)

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 existc_1ij,c_2ij andc_3ij positive for i,j=1, ...,q, such that for allt the following inequality holds (Ioannou and Sun,1996):

c_1ijI≤

Z t0+c3ij

t0

D¯y¯_τiD¯^T_y¯

τjdt ≤c_2ijI ∀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.

4: else

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).

8: ComputeKi=Q⁻¹Ri.

9: DeduceNy,y˙ =^Γ−Ky,y˙C andLy,y˙ =Ky,y˙−Ny,y˙Hy. 10: Estimate the parametersθ.ˆ

11: End

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

∈R^p 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^˜) =n_ywheren_y< p. Also, we assume that the vectorζandζ˙are respectively defined on the hyperplanes∆ and∆¯ defined by:





∆=ζ∈R^nζ| ζimin ≤ζi≤ζimax }

∆¯ =ⁿζ^˙∈R^nζ| ζ˙_imin≤ζ˙_i≤ζ˙_imax } (6.29) where(ζ_imin,ζ_imax)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:

F

y,y, ..,˙ y⁽ⁿ⁾,θ

=D

y,y, ..,˙ y⁽ⁿ⁾

f(θ) (6.30)

where the matrixDy =D

y,y, ..,˙ y⁽ⁿ⁾

∈R^p×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^˜_ζD_yf(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 =D_yf(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 > n_y and an augmented model is reconstructed from delayed outputs as described in section6.1.

6.4. Extension to the PTWV case 117

where τ_m is them^th 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) =A_ζx_a(t) +B_ζu_a(t) +D_ζ,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) = N_ζ,ζ˙z+L_ζ,ζ˙ya+G_ζua

xˆ_a(t) = z(t)₋H_ζy_a(t) ^(6.35)

where,xˆ_aandyˆ_aare respectively the estimated state and the output vector. The matricesN_ζ,ζ˙,L_ζ,ζ˙,G_ζ and H_ζ 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˙=N_ζ,ζ˙e, must be asymptotically stable i.e.,N_ζ,ζ˙ is Hurwitz, (2) P˙_ζ,ζ˙+P_ζA_ζ−N_ζ,ζ˙P_ζ−L_ζ,ζ˙C_ζ =0.

(3) P_ζD_ζ,y=0 (4) P_ζB_ζ−G_ζ =0

From these conditions, we can state the set of LMIs to be solved in order to design the DUIO gains as follows:

N_ζ,ζ˙ =^Γ_ζ,ζ˙−K_ζ,ζ˙C_ζ (6.36)

H_ζ =₋D_ζ,y(C_ζD_ζ,y)^† (6.37)

P_ζ =I2n+H_ζC_ζ (6.38)

G_ζ =P_ζB_ζ (6.39)

whereΓ_ζ,˙

ζ =P^˙_ζ,ζ˙+P_ζA˜_ζ andK_ζ,ζ˙ =N_ζ,ζ˙H_ζ+L_ζ,ζ˙.

Since the matching condition rank(CD_ζ,y) = rank(D_ζ,y) is satisfied, the unknown parameters can be recovered by a simple algebraic inversion as following :

fˆ= (C_ζD_ζ,y)^† y^˙_a(t)₋C_ζA_ζxˆ_a−C_ζB_ζu_a(t) (6.40)

In which, the convergence offˆtowardsf can be analyzed by defining the unknown part estimation error e_f =f−fˆ=₋(C_ζD_ζ,y)^†C_ζA_ζe

Knowing thateconverges asymptotically towards zero, thene_f 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:

Γ^T_i Q+QΓ_i−C_i^TK_i^TQ−QK_iC_i<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 matrixΓ_i is saidΘ_i-stable when its spectrum λ(^Γi)belongs to regionΘ_i.

Θ_i=z= (x_z+i.yz)_∈^C_| Re(z)_{≤ −}ς ⇔z+z^¯+2ς <0 } (6.41)