A maximum likelihood estimator for switching linear systems with unknown inputs

HAL Id: hal-02184492

https://hal.archives-ouvertes.fr/hal-02184492

Submitted on 11 Jun 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A maximum likelihood estimator for switching linear systems with unknown inputs

Luc Meyer, Dalil Ichalal, Vincent Vigneron

To cite this version:

Luc Meyer, Dalil Ichalal, Vincent Vigneron. A maximum likelihood estimator for switch- ing linear systems with unknown inputs. Automatica, Elsevier, 2019, 108, pp.108490.

�10.1016/j.automatica.2019.07.003�. �hal-02184492�

A maximum likelihood estimator for switching linear systems with Unknown Inputs ?

Luc Meyer ^a , Dalil Ichalal ^b , Vincent Vigneron ^b

a

ONERA, DTIS, Universit´e Paris Saclay, 6 Chemin de la Vauve aux Granges, 91120 Palaiseau, France

b

IBISC, Univ Evry Universit´e Paris Saclay, 40, rue de Pelvoux, 91020 Courcouronnes, France

Abstract

This paper investigates the construction of an estimator for switching linear systems with unknown inputs (UI) and Gaussian noises, in the case of unknown switching sequence. First, a maximum-likelihood switching sequence estimator is constructed. Then, both state and UI estimators are proposed. An example illustrates the theoretical contributions.

Key words: State Estimation, Minimum Variance Estimation, Unknown Input, Switching Systems, Discrete-Time System

1 Introduction

A switching system is a dynamical system with both contin- uous and discrete states. Several approaches exist in order to estimate the discrete-state of such a system. If the switching sequence is known, different observers have been proposed for systems without [4] or with Unknown Inputs (UI) [6].

If the switching sequence is unknown but its dynamic fol- lows a known pattern (as a particular Markov model), then several estimators have also been proposed [7][5]. With- out any assumption made on the discrete state dynamic, [1] and [2] have proposed state estimation for discrete- time linear switching systems affected by bounded noises.

For continuous-time systems with bounded perturbations, a sliding-mode based observer is giving in [9]; in particular, this observer also deals with UI. In [3], a linear switching system affected by Gaussian noises in both state and mea- surement equations has been considered, but without the presence of UI.

The present paper focuses on discrete-time linear switching systems affected by both Gaussian noises and UI in both state and measurement equations. Its main contribution is to provide both state and UI estimators for such systems.

It is the first paper dealing with linear switching systems affected by UI in the case in which the matrix associated

?

This paper was not presented at any IFAC meeting. Correspond- ing author L. Meyer

Email addresses:

[email protected]

(Luc Meyer),

[email protected]

(Dalil Ichalal),

[email protected]

(Vincent Vigneron).

with the UI is not assumed to be constant. One of the main difficulty is to decouple the UI from the state estimation error despite of that fact. Another difficulty is to estimate the UI in such a case. The paper is organized as follows.

In section 2, the problem is stated. In section 3, a method used for a maximum likelihood estimation of the switching parameter is presented. In section 4, a practical algorithm for estimating both the continuous state and the UI and based on the switching parameter estimation is given. Finally, in section 5, an example illustrating the approach is given.

The following notations are used in the paper. For any matrix A ∈ M

n,p

( R ), A

^T

denotes its transpose, and A

^†

its pseudo- inverse. For any square matrix A ∈ M

n,n

( R ), tr(A) denotes its trace, det(A) denotes its determinant, λ (A) denotes the vector of its eigenvalues, and λ

_min

(A) (resp. λ

_max

(A)) de- notes its minimum (resp. maximum) singular value. For any vector v, ||v|| denotes its euclidean norm. For any symmet- ric positive matrix M, let set the norm ||.||

_M

such that, for any vector v of appropriate dimension, ||v||

_M

= √

v

^T

Mv. If x is a stochastic vector, its expectancy is denoted by E[x].

For any vector sequence (v

_k

)

_k=1,...

and any integers α ≥ 0 and ω ≥ 0, let set v

k−α:k+ω

= h

v

^T_k−α

v

^T_k−α+1

... v

^T_k+ω

i

T

.

2 Problem Statement

Let consider the following switching system:

x

_k+1

= A

_λk

x

_k

+D

_λ

k

d

_k

+ F

_λk

w

_k

y

_k

= C

_λ

k

x

_k

+ E

_λ

k

d

_k

+ v

_k

(1)

where x

_k

∈ R

^nx

is the continuous state, y

_k

∈ R

^ny

is the output, d

_k

∈ R

^nd

is the Unknown Input (UI), λ

k

∈ Λ := {1,2, ..., n

_Λ

} is the unknown switching parameter (also called discrete mode or discrete state). Matrices A

_λ

k

, D

_λ

k

, F

_λ

k

, C

_λ

k

and E

_λ

are switching matrices with appropriate dimensions. The

k

perturbations w

_k

and v

_k

are independent Gaussian noises with zero mean and covariance matrices denoted by W

_λk

and V

_λk

respectively. The unknown initial state follows a Gaussian distribution (independent of w

_k

and v

_k

) with known mean ˆ x

₀

and known covariance matrix P

₀

. Let assume that for any λ ∈ Λ, the matrix V

_λ

is non singular, i.e. det(V

_λ

) 6= 0 (it is a widely used assumption for Kalman filtering).

In the whole paper, let us consider two integers α ≥ 0 and ω ≥ 0, used as time indexes: for any time step k ≥ α , the measures obtained between time steps k − α and k + ω are used for the estimation of the unknown switching parameterλ

_k

. For any k ≥ α, let λ

k−α:k+ω

be the switching sequence from time step k− α to time step k +ω . Then, the observation sequence y

k−α:k+ω

gives:

y

k−α:k+ω

= A

λk−α:k+ω

x

k−α

+ D

λk−α:k+ω

d

k−α:k+ω

+ F

λk−α:k+ω

w

k−α:k+ω

+ v

k−α:k+ω

(2)

with matrices A

λk−α:k+ω

, D

λk−α:k+ω

and F

λk−α:k+ω

given in (3). The following assumption is needed for removing the UI from equation (2).

Assumption 1 For any switching sequence π ∈ Λ

^α+ω+1

, the rank r

_d

(π ) of D

π

satisfies r

_d

(π ) < (α + ω + 1)n

_y

. For all π ∈ Λ

^α+ω+1

, there exists a full row rank matrix H

π

∈ M

(α+ω+1)ny−r_d,(α+ω+1)ny

( R ) (with rank equal to r

_h

(π ) = (α + ω + 1)n

_y

− r

_d

(π )) such that H

_π

D

π

= 0. Assumption 1 ensures that r

_h

(π) > 0. Then, if π = λ

k−α:k+ω

, when left- multiplying equation (2) by H

_{λk−α:k+ω}

, it comes:

H

_{λk−α:k+ω}

y

k−α:k+ω

= H

_{λk−α:k+ω}

A

λk−α:k+ω

x

k−α

+H

_λ

k−α:k+ω

F

λ_k−α:k+ω

w

k−α:k+ω

+ H

_{λk−α:k+ω}

v

k−α:k+ω

equation in which the unknown input does not appear any- more.

3 Maximum-likelihood switching sequence estimation

This section deals with the switching sequence estimation problem. Let us start by introducing a definition of observ- ability adapted to the considered switching system.

Definition 1 π ∈ Λ

^α+ω+1

and π

⁰

∈ Λ

^α+ω+1

are saying to be jointly observable if:

rank h

H

_π⁰

A

π⁰

H

_π⁰

A

π

H

_π⁰

D

π

i

= 2n

_x

+ (α +ω + 1)n

_d

(8)

Definition 1 does not depend on the matrix H

_π⁰

chosen under Assumption 1, as this matrix is uniquely defined modulo a left-multiplication by a non singular square matrix.

Let us consider a time step k ≥ α, and let consider the following maximization problem in the presence of a UI:

λ ˆ

k−α:k+ω

∈ argmax

_π∈Λα+ω+1

h

max

x∈R^nx

l(x, π|H

_π

y

k−α:k+ω

) i (9) where l(x, π|H

_π

y

k−α:k+ω

) = p(H

_π

y

k−α:k+ω

|x,π ) is the like- lihood function to maximize.

Due to the left-multiplication of y

k−α:k+ω

by H

_π

in (9), if π = λ

k−α:k+ω

is the true switching sequence, then the likelihood l(x, λ

k−α:k+ω

|H

_{λk−α:k+ω}

y

k−α:k+ω

) is not a func- tion of the unknown input (see equation (8)). Moreover, the vector H

π

y

k−α:k+ω

is normally distributed with mean H

_π

A

λk−α:k+ω

x

k−α

and covariance matrix H

_π

Σ ˜

_{λk−α:k+ω}

H

_π^T

(where for any π

⁰

∈ Λ

^α+ω+1

, ˜ Σ

_π⁰

= F

π⁰

W

_π⁰

F

_π^T0

+V

_π0

).

Then, following the same line as in [3], the maximum like- lihood estimate can be obtained as:

λ ˆ

k−α:k+ω

∈ argmin

_π∈Λα+ω+1

h

log|Σ

_π

| + δ (H

_π

y

k−α:k+ω

, π) i (10) where Σ

π

= H

π

Σ ˜

π

H

_π^T

, and for any vector v ∈ R

^rh(π)

, δ (v,π ) = ||(I −∆

_π

)v||

²

Σ⁻¹_π

, with ∆

_π

the projection matrix on the linear span of H

π

A

π

.

Given that the true continuous state is x

k−ω

at time step k−ω and the true switching sequence is λ

k−α:k+ω

between time steps k− α and k+ ω, the probability that λ

k−α:k+ω

is more likely to be chosen by (10) than a discrete sequence π

⁰

∈ Λ

^α+ω+1

is denoted by P (λ

k−α:k+ω

π

⁰

|x

k−α

,λ

k−α:k+ω

).

The following theorem gives a lower bound of such a prob- ability.

Theorem 1 Let assumption 1 hold. Let be k ≥ α . Let π

⁰

∈ Λ

^α+ω+1

be a switching-sequence. Then, the following in- equality holds:

P (λ

_k−α:k+ω

π

⁰

|x

k−α

, λ

k−α:k+ω

) ≥ γ

_r

(

^k(λ_h(λ^{k−α:k+ω,π0)}

k−α:k+ω,π⁰)

× (||x

_k−α

||

²

+ ||d

_k−α:k+ω

||

²

) +

_h(λ ¹

k−α:k+ω,π⁰)

log

_|Σ ^|Σπ0|

λk−α:k+ω|

) (11) where γ

_r

(.) is the cumulative distribution function of a χ

²

random variable with r = (α + ω +1)n

_y

degrees of freedom.

Moreover for any switching sequences π ,π

⁰

∈ Λ

^α+ω+1

, the coefficients h(π, π

⁰

) and k(π, π

⁰

) are given by equations (4), where Σ ˜

_π

= F

π

W

_π

F

π^T

+V

_π

. Besides, if π and π

⁰

are jointly observable, k(π , π

⁰

) > 0.

PROOF. In order to simplify the notations, λ

k−ω:k+ω

is de- noted by π in the following proof, but the reader has to keep

2

A_λk−α+ω=















C

_λk−α

C

_λk−α+1

A

_λk−α

...

C

_λk+ω

A

_λk+ω−1...A_λk−α+1

A

_λk−α















,D_{λk−α:k+ω}=















E

_λk−α

0

ny,nd ...

0

ny,nd

C

_λk−α+1

D

_λk−α

E

_λk−α+1 ...

0

ny,nd

... ... ... ...

C

_λk+ω

A

_λk+ω−1...A_λk−α+1

D

_λk−α

C

_λk+ω

A

_λk+ω−1...A_λk−α+2

D

_λk−α+1 ...

E

_λk+ω













 ,

F_{λk−α:k+ω} =















0

ny,nv

0

ny,nv ...

0

ny,nv

0

ny,nv

C

_λk−α+1

F

_λk−α

0

ny,nv ...

0

ny,nv

0

ny,nv

... ... ... ... ...

C

_λk+ω

A

_λk+ω−1...A_λk−α+1

F

_λk−α

C

_λk+ω

A

_λk+ω−1...A_λk−α+2

F

_λk−α+1 ...

C

_λk+ω

F

_λk+ω−1

0

_ny,nv















(3)







k(π

,π⁰) =¹₂λmin{h Aπ Dπ

iT

H

_π^T0(I−∆_π⁰)^TΣ⁻¹

π⁰ (I−∆_π⁰)H_π⁰h Aπ Dπ

i}

h(π,π

⁰) =λmax{H_π^T(I−∆π)^TΣ⁻¹_π (I−∆π)H_π+

H

^T

π⁰(I−∆_π⁰)^TΣ⁻¹_π0 (I−∆_π⁰)H_π⁰}λmax{Σ

˜

π}

(4)

P(π_k−α:k+ωπ⁰|x_k−α,π_k−α:k+ω)≥P

z

^TΣ

˜

⁻¹

z

≤^k(π,π_h(π,π⁰0⁾)(||x_k−α||²+||d_k−α:k+ω||²) +_h(π,π¹ 0)

log

^|Σ_|Σ^π0|

π|

,

(5)

Φ_{λk−α:k−1}= h

A

_λk−1...A_λk−α

A

_λk−1...A_λk−α+1

D

_k−α ...

A

_λk−1

D

_λk−2

D

_λk−1 i

,Ψ_{λk−α:k−1}= h

A

_λk−1...A_λk−α+1

F

_k−α ...

A

_λk−1

F

_λk−2

F

_λk−1 i

(6)

[(A_λk−

A

_ˆ

λk)−

K

_k+1(C_λk+1

A

_λk−

C

_ˆ

λk+1

A

_ˆ

λk)]x_k= [(A_λk−

A

_ˆ

λk)−

K

_k+1(C_λk+1

A

_λk−

C

_ˆ

λk+1

A

_ˆ

λk)]Φ_{λk−α:k−1}





x

_k−α

d

k−α:k−1





+[(A_λk−

A

λˆk)−

K

k+1(C_λk+1

A

_λk−

C

λˆk+1

A

λˆk)]Ψ_{λk−α:k−1}

w

_k−α:k−1

(7)

in mind that it is the actual discrete sequence. The follow- ing steps are similar to [3] in the structure, however, the technical difficulties raised by the presence of the UI makes the proof very different. Using equation (10), the probabil- ity P (π π

⁰

|x

k−α

, λ

k−α:k+ω

) is equal to the probability that the following inequality holds:

log |Σ

_π

|+δ (H

_π

y

_k−α:k+ω

,π ) < log |Σ

_π⁰

|+δ (H

_π⁰

y

_k−α:k+ω

,π

⁰

) (12) Let set z = F

π

w

k−α:k+ω

+ v

k−α:k+ω

. Then z is normally distributed with zero mean and covariance matrix equal to Σ ˜

π

. On one hand, we have:

δ (H

_π0

y

k−α:k+ω

, π

⁰

) = δ (H

_π0

A

π

x

k−α

+H

_π0

D

π

d

k−α:k+ω

+H

_π0

z,π

⁰

).

(13) Besides:

δ (H

_π⁰

A

π

x

k−α

+ H

_π⁰

D

π

d

k−α:k+ω

, π

⁰

)

= δ (H

_π0

A

π

x

k−α

+ H

_π⁰

D

π

d

k−α:k+ω

+ H

_π⁰

z − H

_π⁰

z, π

⁰

)

≤ 2δ (H

_π⁰

y

k−α:k+ω

, π

⁰

) + 2δ (H

_π⁰

z, π

⁰

),

(14) the last inequality coming from one of the property of the function δ (., π): δ (u − v , π ) ≤ δ (u, π) + δ (v , π ). Thus:

δ (H

_π⁰

y

k−α:k+ω

, π

⁰

)

≥

¹₂

δ (H

_π0

A

π

x

k−α

+ H

_π⁰

D

π

d

k−α:k+ω

, π

⁰

) − δ (H

_π0

z, π

⁰

)

≥ k(π , π

⁰

)(||x

k−α

||

²

+ ||d

k−α:k+ω

||

²

) −δ (H

_π⁰

z,π

⁰

), (15)

where k(π, π

⁰

) is given by equation (4). Note, that un- der the joint observability of π and π

⁰

, it comes that (I −∆

_π⁰

)H

_π0

h A

π

D

π

i

is full column rank. Then, it follows that k(π, π

⁰

) > 0. On the other hand, we have:

δ (H

_π

y

k−α:k+ω

, π ) = δ (H

_π

A

π

x

k−α

+ H

_π

D

π

d

k−α:k+ω

+ H

_π

z, π)

= δ (H

_π

z, π) (16)

as H

_π

D

π

d

k−α:k+ω

= 0, and (I − ∆

π

)H

_π

A

π

= 0. Then it comes that:

δ (H

_π

z, π ) + δ (H

_π0

z, π

⁰

) = ||(I − ∆

_π

)H

_π

z||

²

Σ⁻¹_π

+||(I − ∆

_π⁰

)H

_π⁰

z||

²

Σ⁻¹

π0

≤ h(π, π

⁰

)||z||

²_˜

Σ⁻¹_π

(17)

where ˜ Σ

π

= F

π

W

π

F

π^T

+V

_π

and h(π, π

⁰

) is given by equa- tion (4) Finally, by combining (15), (16) and (17) it comes that

δ (H

_π

y

k−α:k+ω

, π) −δ (H

_π0

y

k−α:k+ω

, π

⁰

)

≤ h(π, π

⁰

)||z||

²_˜

Σ⁻¹_π

−k(π, π

⁰

)(||x

_k−α

||

²

+ ||d

_k−α:k+ω

||

²

) (18) and thus a sufficient condition for equation (12) to hold is:

h(π ,π

⁰

)||z||

²_˜

Σ⁻¹_π

−k(π ,π

⁰

)(||x

k−α

||

²

+||d

k−α:k+ω

||

²

) ≤ log |Σ

_π0

|

|Σ

_π

|

(19)

which leads to equation (5), and recalling that z is nor-

mally distributed with zero mean and covariance matrix

Σ ˜

_π

, it comes that z

^T

Σ ˜

⁻¹_π

z is a χ

²

random variable with dim(z) = (α + ω + 1)n

_y

degrees of freedom.

4 A maximum-likelihood estimator

The aim of this section is to construct a linear estimator based on the maximum likelihood switching sequence estimation developed in section 3. In order to do it, it is necessary (and also sufficient as it will be seen in Theorem 2) to decouple the UI from the state estimation error (otherwise the behavior of this error would depend on the UI behavior), whether the estimated switching parameter is the true one or not.

Assumption 2 The following rank condition holds:

rank "

C E DI 0

#!

= rank h C E

i

, (20)

where I = h

I

_nΛnd

... I

_n

λn_d

i

(n

_Λ

times the matrix I

_nΛnd

), D = h

D

1

... D

n_Λ

i , C = h

C

1

D ... C

n_Λ

D i

and E = h

E

1

... E

n_Λ

i . Note that this assumption differs from Assumption 1, as the former one is used in order to make the information of the output not affected by the UI, and thus to estimate at best the switching sequence, whereas assumption 2 is used in order to decouple the UI from the state estimation error for any choice of ˆ λ

k

in Λ. Under assumption 2, there exists a matrix K such that:

K h C E

i

= h DI 0

i

, (21)

with K given by:

K = G +X H , (22)

with G = h DI 0

i h C E

i

†

and H = I

ny,ny

− h C E

i h C E

i

†

, and X ∈ M

nx,ny

( R ) is any matrix (this means that for any matrix X of appropriate dimensions, equation (21) holds with K given by (22)). In fact, X is chosen differently at each time step k ≥ α. Thus, from now, it will be denoted by X

_k

and K will be denoted by K

_k

= G +X

_k

H. It is worth noting that, under Assumption 2, and according to what precedes the following equation holds for any λ , λ

⁰

∈ Λ, and any choice of X

_k

:

K

_k

D

λ,λ⁰

= h D

_λ

0

i

, (23)

where D

λ,λ⁰

= h

C

_λ⁰

D

_λ

E

_λ⁰

i

. The following assumption is needed for the UI estimation.

Assumption 3 The following rank condition holds:

rank









C E

I

_n2 Λ,n_d

0







 = rank h C E

i

(24)

where I

_n2 Λ,n_d

= h

I

_nd

... I

_nd

i

(n

²_Λ

times the matrix I

n_d

).

Following the same lines as previously, there exists a matrix L

_k

such that for any λ , λ

⁰

∈ Λ: L

_k

h

C

_λ⁰

D

_λ

E

_λ⁰

i = h

I

_nd

0

_nd

i

, where L

_k

= J + Z

_k

H with J = h

I

n_d

0

n_d

i h

C

_λ⁰

D

_λ

E

_λ⁰

i

†

, and Z

_k

is any matrix (Z

_k

will be chosen later). Saying that, the proposed state and UI estimator is the following for any k ≥ α :



 

 

 

 

 

 

 

 

 

 

 

 

 

 

λ ˆ

k−α:k+ω

= argmin

_π∈Λα+ω

log |Σ

_π

| +δ (H

_π

y

k−α:k+ω

,π ) X

_k

= [(I

_nx

−GC

_ˆ

λ_k

) P ˜

_k

C

^T_ˆ

λ_k

+GV

_ˆ

λ_k

]H

^T

[HC

_ˆ

λ_k

P ˜

_k

C

^T_ˆ

λ_k

H

^T

+HV

_ˆ

λ_k

H

^T

]

⁻¹

Z

_k

= J(C

_λ

k

P ˜

_k

C

_λ^T

k

)H

^T

[(H(C

_λ

k

P ˜

_k

C

_λ^T

k

)H

^T

]

⁻¹

K

_k

= G + X

_k

H, L

_k

= J + Z

_k

H

ˆ

x

_k

= x ˜

_k

+K

_k

(y

_k

−C

_ˆ

λ_k

x ˜

_k

), d ˆ

k−1

= L

_k

(y

_k

−C

_λk

x ˜

_k

) P

_k

= (I

_nx

− K

_k

C

_ˆ

λ_k

) P ˜

_k

(I

_nx

− K

_k

C

_ˆ

λ_k

)

^T

+ K

_k

V

_ˆ

λ_k

K

_k^T

x ˜

k+1

= A

λˆ_k

x ˆ

_k

, P ˜

k+1

= γ

²

A

λˆ_k

P

_k

A

^T_ˆ

λ_k

+ F

λˆ_k

W

λˆ_k

F

_ˆ^T

λ_k

.

(25) Remark 1 The global structure of algorithm (25) comes from the structure of the Kalman filter in the presence of a UI [8]. It is adapted using what precedes in order to take into account the case of an unknown switching sequence.

Moreover, the case of the UI estimation is also added here.

The matrix X

_k

(resp. Z

_k

) is constructed in order to minimize the trace of the covariance matrix P

_k

(resp. P

_dk

= E[e

^T_d

k

e

_dk

]), under the constraint of the unknown switching sequence.

In equations (25), γ is a scale factor chosen strictly greater than 1. The aim of such a factor is to enforce a certain convergence rate (more on this in the stability analysis given in [3]).

Assumption 4 For any switching sequences π ,π

⁰

∈ Λ

^α+ω+1

, such that π [α + 1] 6= π

⁰

[α + 1], the switching sequences π and π

⁰

are assumed to be jointly observable.

Theorem 2 Let assumptions 1, 2, 3 and 4 hold. Then, equa- tions (25) define both state and UI estimators for system (1) with estimation errors e

_k

= x

_k

− x ˆ

_k

and e

_dk

= Ld

_k

− d ˆ

_k

hav- ing the following dynamics: e

k+1

= (I −K

_k+1

C

λˆk+1

)A

_ˆ

λk

e

_k

+ ξ

k

and e

_dk

= −L

_k+1

C

_ˆ

λ_k+1

A

_ˆ

λ_k

e

_k

+ ζ

k

, where ξ

k

and ζ

k

are bounded in mean square, i.e. there exist positive constants ρ

_ξ

and ρ

_ζ

such that for all k ≥ 0, ||ξ

_k

||

²

≤ ρ

_ξ

and ||ζ

_k

||

²

≤ ρ

_ζ

.

PROOF. Due to space limitation, only the proof on the state estimation error e

_k

= x

_k

− x ˆ

_k

convergence is established, the one of the UI estimation error e

_dk

follows the same line. Using equations (25) and (23) the dynamics of the state estimation error is: e

_k+1

= (I − K

_k+1

C

_ˆ

λ_k+1

)A

_ˆ

λ

e

_k

+ ξ

k

, where ξ

_k

= [(A

_λ

k

− A

λˆk

) − K

k+1

(C

_λ

k+1

A

_λk

− C

λˆk+1

A

λˆk

)]x

_k

+ (I − K

_k+1

C

_ˆ

λ_k+1

)F

_λk

w

_k

− K

_k+1

v

_k+1

. Let prove the bound- edness in mean-square of ξ

k

. Its second and third terms

4

are clearly bounded. Let be λ

k−α:k−1

the true discrete state sequence between k − α and k − 1. It comes:

x

_k

= Φ

_{λk−α:k−1}

"

x

_k−α

d

k−α:k−1

#

+ Ψ

_{λk−α:k−1}

w

k−α:k−1

, where

Φ

_{λk−α:k−1}

and Ψ

_{λk−α:k−1}

are given by equation (6). Thus,

the first term of ξ

k

can be written as in equation (7). Here again, the second term is clearly bounded in mean square.

Following the same lines as in [3], let prove the bounded- ness in mean-square of Φ

_λk−α:k

"

x

k−α

d

k−α:k

#

. Given the state x

k−α

and the unknown input sequence d

k−α:k+ω

, the mean square error of Φ

_{λk−α:k−1}

"

x

k−α

d

k−α:k−1

#

is given by:

E

"

||Φ

_{λk−α:k−1}

"

x

k−α

d

k−α:k−1

#

||

²

|x

k−α

, d

k−α:k+ω

, π

#

= ∑

λ⁰∈Λ,λ⁰6=λ_k

P

λ ˆ

_k

= λ

⁰

|x

_k−α

, d

k−α:k+ω

, π

||Φ

_{λk−α:k−1}

×

"

x

k−α

d

k−α:k−1

#

||

²

.

(26) On one hand, it comes: ||Φ

_{λk−α:k−1}

"

x

_k−α

d

k−α:k−1

#

||

²

≤ Φ(||x

k−α

||

²

+||d

k−α:k+ω

||

²

),

where Φ = max

_πα∈Λα

λ

max

{Φ

^T_π

α

Φ

π_α

}. On the other hand, it comes:

P

λ ˆ

k

= λ

⁰

|x

k−α

, d

k−α:k+ω

, π

= ∑

π_α⁰∈Λ^α,π_ω⁰∈Λ^ω

P

λ ˆ

k−α:k+ω

= π

_α⁰

⊗λ

⁰

⊗ π

ω

|x

_k−α

, d

k−α:k+ω

, π

!

≤ card(Λ)

^α+ω

"

1 −γ

_(α+ω+1)ny

(κ (||x

k−α

||

²

+||d

k−α:k+ω

||

²

) + ν)

#

(27) the last inequality coming from theorem 1, where κ = min

_π,π0∈Λ^α+ω+1

k(π,π⁰)

h(π,π⁰)

and ν = min

_π,π0∈Λ^α+ω+1 1

h(π,π⁰)

log

_|Σ^|Σπ|

π0|

. Note that thanks to assumption 4 and theorem 1, the con- stant κ is positive. Then, using both previous items, it comes that the quantity

E

"

||Φ

_{λk−α:k−1}

"

x

k−α

d

k−α:k−1

#

||

²

|x

_k−α

, d

_k−α:k+ω

, π

# (28)

is smaller than:

card(Λ)

^α+ω

Φ(||x

k−α

||

²

+ ||d

k−α:k+ω

||

²

)

× h

1 −γ

_(α+ω+1)ny

κ(||x

k−α

||

²

+ ||d

k−α:k+ω

||

²

) + ν i

. (29)

Then, recalling that γ

_n

(s) (with any positive integer n) con- verges exponentially to 1 as s tends to infinity, we can con- clude that (29) converges to 0 as ||x

k−α

||

²

+ ||d

k−α:k

||

²

tends to infinity (κ being positive). Finally, noting that (29) de- pends continuously on ||x

_k−α

||

²

+ ||d

_k−α:k+ω

||

²

on the inter- val [0;∞), it comes that (29) can be upper bounded by some constant independent from ||x

k−α

||

²

+ ||d

k−α:k+ω

||

²

. Remark 2 Note that in order to apply equations (25), a delay equal to ω is needed. In order to avoid such a delay, it is possible to use the following equations for i = k+1...k+ω (so that x

k+ω

is estimated using y

k−α:k+ω

):



 

 

 

 



 

 

 

 



X

_i

= [(I − GC

_ˆ

λi

) P ˜

_i

+ GV

_k

]C

^T_ˆ

λi

H

^T

[HC

_ˆ

λi

P ˜

_i

C

^T_ˆ

λi

H

^T

+ HV

_i

H

^T

]

⁻¹

Z

_i

= J(C

_λi

P ˜

_i

C

^T

λ_i

)H

^T

[(H(C

_λi

P ˜

_i

C

^T

λ_i

)H

^T

]

⁻¹

K

_i

= G +X

_i

H, L

_i

= J + Z

_i

H

ˆ

x

_i

= x ˜

_i

+K

_i

(y

_i

−C

_ˆ

λ_i

x ˜

_i

), d ˆ

i−1

= L

_i

(y

_i

−C

_λi

x ˜

_i

) P

i

= (I − K

i

C

λˆi

) P ˜

i

(I −K

_i

C

λˆi

)

^T

+ K

i

V

i

K

_i^T

˜ x

_i+1

= A

_ˆ

λ

x ˆ

_i

, P ˜

_i+1

= γ

²

A

_ˆ

λ_i

P

_i

A

^T_ˆ

λ_i

+ F

_ˆ

λ_i

W

_i

F

^T_ˆ

λ_i

.

(30) Remark 3 The stability results stated in [3] remain true by considering X

_k

instead of K

_k

(as in [3]) for the gain matrix, and (I −GC

_ˆ

λ_k+1

−X

_k

HC

_ˆ

λ_k+1

)A

_ˆ

λ_k

instead of (I −K

_k

C

_ˆ

λ_k+1

)A

_ˆ

λ_k

(as in [3]) for the transition matrix of the state error vector, as well as some minor adaptations. They are not detailed here for a lack of space.

5 Illustrative Example

Let consider the following academic example (adapted from [3]) of the form (1) where λ ∈ Λ = {1; 2}, ∆ = 0.1, A

1

=

"

cos(ω

₁

∆) −ω

₁

sin(ω

₁

∆)

1

ω1

sin(ω

₁

∆) cos(ω

₁

∆)

#

, with ω

1

= 0.5,

A

₂

=

"

cos(ω

₂

∆) −ω

₂

sin(ω

₂

∆)

1

ω2

sin(ω

₂

∆) cos(ω

₂

∆)

#

, with ω

2

= 1, D

₁

=

"

0 1

# ,

D

2

=

"

0 2

#

, F

1

= F

2

= I

2

, C =







 0 1 2 1 1 0







 , E =







 0 1 0









, w

_k

and v

_k

are Gaussian noises with covariance matrices W = 0.01I

_nx

and V = 0.01I

_ny

respectively, the unknown input is set to

d

_k

= 3 sin(π /100k) for k ≥ 0, the simulation is launched for

400 time steps, with γ = 1.1. By setting α = 3 and ω = 2, we

can check that assumptions 1, 2 and 4 are satisfied. The De-

layed Maximum Likelihood Estimator with UI (DMLEUI)

of equations (25) is compared with its Undelayed version

presented in remark 2 (’UMLEUI’), and with the Switching

Kalman Filter with Perfect Information (’SKFPI’), i.e. the

same filter applied with the true switching parameter at each

time step. Table 1 gives the performance of each estimator in

terms of the Root Mean Square Error (RMSE), defined for

the ith component by RMSE(x[i]) = s

1 400

399

∑

k=0

(x

_k

[i] − x ˆ

_k

[i]).

It is clear that the SKFPI is only slightly better to the DM- LEUI. This is due to the possible error made by estimating the switching sequence in the MLEUI estimator. Besides, this error increases when there are less measurements avail- able as in the case of the UMLEUI, which explains the slightly worse performance of this former.

Table 1

RMSE (average on 100 simulations)

RMSE on... SKFPI DMLEUI UMLEUI

x[1] 0.1000 0.1000 0.7011

x[2] 5.1225 5.1689 5.2434

6 Conclusions

The main contribution of this paper is the development of an observer for switching systems affected by Gaussian noises and Unknown Inputs (UI) with unknown switching sequence. The main contribution holds in the way of dealing with the UI in order to decouple it from the state estimation error. First, a method that estimates the switching parameter with maximum likelihood is proposed despite the presence of a UI. Then, this method is used in both state and UI es- timators.

References

[1] Angelo Alessandri, Marco Baglietto, and Giorgio Battistelli. Receding- horizon estimation for switching discrete-time linear systems. IEEE Transactions on Automatic Control, 50(11):1736–1748, 2005.

[2] Angelo Alessandri, Marco Baglietto, and Giorgio Battistelli.

Luenberger observers for switching discrete-time linear systems.

International Journal of Control, 80(12):1931–1943, 2007.

[3] Angelo Alessandri, Marco Baglietto, and Giorgio Battistelli. A maximum-likelihood kalman filter for switching discrete-time linear systems. Automatica, 46(11):1870–1876, 2010.

[4] Angelo Alessandri and Paolo Coletta. Switching observers for continuous-time and discrete-time linear systems. In Proceedings of the 2001 American Control Conference.(Cat. No. 01CH37148), volume 3, pages 2516–2521. IEEE, 2001.

[5] Yaakov Bar-Shalom, Subhash Challa, and Henk AP Blom. Imm estimator versus optimal estimator for hybrid systems. IEEE Transactions on Aerospace and Electronic Systems, 41(3):986–991, 2005.

[6] Francisco Javier Bejarano, Alessandro Pisano, and Elio Usai. Finite- time converging jump observer for linear switched systems with unknown inputs. IFAC Proceedings Volumes, 42(17):352–357, 2009.

[7] Oswaldo Luiz V Costa and S Guerra. Stationary filter for linear minimum mean square error estimator of discrete-time markovian jump systems.IEEE Transactions on Automatic Control, 47(8):1351–

1356, 2002.

[8] Mohamed Darouach, Michel Zasadzinski, and Mohamed Boutayeb.

Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39(5):867–876, 2003.

[9] H´ector R´ıos, Jorge Davila, and Leonid Fridman. High-order sliding mode observers for nonlinear autonomous switched systems with unknown inputs.Journal of the Franklin Institute, 349(10):2975–3002, 2012.

6