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Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases
Automatica (
)
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Technical Communique
Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases✩ Mohamed Darouach a , Francesco Amato b , Marouane Alma a a
CRAN-CNRS (UMR 7039), Université de Lorraine, IUT de Longwy, 186, Rue de Lorraine, 54400 Cosnes et Romain, France School of Computer and Biomedical Engineering, Experimental Clinical Medicine Department, Università degli Studi Magna Grcia di Catanzaro, Campus di Germaneto ‘‘Salvatore Venuta’’, 88100 Catanzaro, Italy b
article
info
Article history: Received 19 August 2016 Received in revised form 17 May 2017 Accepted 22 June 2017 Available online xxxx
a b s t r a c t This paper investigates the design of functional observers for linear time-invariant descriptor systems. A new method for designing these observers is given by using an LMI (Linear matrix inequality) formulation. The obtained result unifies the design, it considers the continuous-time and discrete-time cases and concerns the observers of various orders. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Functional observer Descriptor systems Stability Existence conditions LMI
1. Introduction In the present paper, based on the results of Darouach (2012), a new functional observers design method for descriptor systems is presented. The novelty of the presented results lies in the formulation of the dynamics of the error which is represented in a descriptor system form. It permits the simplification of the design for the uncertain systems and gives more degrees of freedom for this design Yong-Yan & Zongli (2004) (see also Gao Huijun & Li, 2014 and references therein). It also unifies the design for different observer orders and concerns the continuous-time as well as the discrete-time systems. Necessary and sufficient conditions for the existence of these observers are given in LMIs form. The observers design in the LMI framework allows to integrate our approach with existing ones (optimal control, pole placement, etc.) that nowadays are all expressed in this way. The proposed approach also presents the advantage that LMIs can be easily tested by using standard convex optimization algorithms. The paper is organized as follows. Section 2 presents some preliminary results on descriptor systems which will be used in the sequel of the paper. The functional observers design of different orders for descriptor systems, including particular cases and the ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tong Zhou under the direction of Editor André L. Tits. E-mail addresses: [email protected] (M. Darouach), [email protected] (F. Amato), [email protected] (M. Alma).
procedure of this design, is given in Section 3. Section 4 concludes the paper. 2. Preliminary results In this section we recall some results on linear descriptor systems which are used in the sequel of the paper (see Dai, 1989; Darouach, 2012). Consider the linear time-invariant multivariable system described by E σ x(t) = Ax(t) + Bu(t)
(1a)
y(t) = Cx(t)
(1b)
z = Lx(t)
(1c)
where σ denotes the derivative operator σ x(t) = dx(t)/dt for continuous-time systems and the forward-shift operator σ x(k) = x(k + 1), where k ∈ Z, for discrete-time systems, x ∈ Rn and y ∈ Rp are the semi state vector and the output vector of the system, u ∈ Rm is the known input and z ∈ Rr is the vector to be estimated, where r ⩽ n. Matrix E ∈ Rn1 ×n , when n1 = n matrix E is singular, matrices A, B, C , and L are known constant and of appropriate dimensions. It is assumed that u(t) and x(0) = x0 are admissible, i.e they are such that there exists at least one trajectory satisfying (1a). By using the Laplace transform (z transform for the discrete-time case) we have the following solvability conditions Ishihara & Terra (2001):
http://dx.doi.org/10.1016/j.automatica.2017.08.016 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
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M. Darouach et al. / Automatica (
1. The pair (u(t), x0 ) is admissible if and only if normal − [ ] rank sE − A BU(s) Ex0 = normal − rank(sE − A) for the continuous-time case, 2. The pair (u(t), x0 ) is admissible if and only if normal − ] [ rank zE − A BU(z) Ex0 = normal − rank(zE − A) for the discrete-time case where the normal-rank of the matrix pencil (λE − A) is defined as the rank(λE − A) for almost all λ ∈ C. In the sequel of this paper we shall assume that the pair (u(t), x0 ) is admissible. Also we shall use the following notations: The symbol Σ + denotes any generalized inverse of the matrix Σ , i.e. verifies ΣΣ + Σ = Σ . This generalized inverse matrix is also denoted by Σ − or Σ (1) in the literature. The symbol E ⊥ denotes a maximal row rank matrix such that E ⊥ E = 0. When E is of full row rank, E ⊥ = 0 by definition. Before presenting the observer design for system (1), we shall recall the following results, which extend the notion of impulse (causal for the discrete-time case) observability to the partial impulse (causal) observability (see reference Darouach, 2012). Definition 1. The descriptor system (1) with u(t) = 0, or the triplet (C , E , A), is said to be partially impulse observable with respect to L, if y(t) is impulse free for t ⩾ 0, only if Lx(t) is impulse free for t ⩾0. Definition 2. The descriptor system (1) with u(t) = 0, or the triplet (C , E , A), is said to be partially causal observable with respect to L, if Lx(k) at any time point k is uniquely determined by the initial condition and the former measurement y(i), i = 0, 1, . . . .., k. The following lemma gives the conditions for the partial impulse observability (partial causal observability for the discretetime case) Darouach (2012). Lemma 1. The following statements are equivalent 1. The triplet (C , E , A) is partially impulse (causal) observable with respect ; ⎡ to L ⎤ 2.
E 0 rank ⎣0 0
⎡
A C⎦ E L
L E
[E
= rank
⎤
3. rank ⎣E ⊥ A⎦ = rank C
[
A C E
0 0 E E⊥A C
]
;
] .
The next assumption is used in the sequel of the paper. Assumption I : We assume that system (1) is partial impulse (causal) observable with respect to L. Now we can give the following definitions for descriptor systems (see Dai, 1989; Xu & Lam, 2006). Definition 3. Let us consider the following descriptor system : E σ x(t) = Ax(t) where E ∈ Rn×n and A ∈ Rn×n , then we have the following definitions 1. The pair (E , A) is regular if det(λE − A) is not identically zero. 2. The pair (E , A) is impulse free (causal for the discrete-time case) if deg(det(λE − A)) = rankE. 3. The pair (E , A) is said to be stable if the roots of det(λE − A) = 0 have negative real parts (are inside the unit circle for the discrete-time case). 4. The pair (E , A) is said to be admissible if it is regular, impulse free (causal for the discrete-time case) and stable. The following lemma gives the necessary and sufficient conditions for the admissibility of the pair (E , A), in the strict LMI formulation (see Xu & Lam, 2006).
)
–
Lemma 2. The pair (E , A) is admissible if and only if there exist matrices X > 0 and Q such that (XE + E ⊥T Q )T A + AT (XE + E ⊥T Q ) < 0
(2)
for the continuous-time case, and AT XA − E T XE + QE ⊥ A + AT E ⊥T Q T < 0
(3)
for the discrete-time case. Let us consider the following reduced order observer.
[ ⊥ ] −E Bu(t)
σ ζ (t) = N ζ (t) + F
y(t)
[
ˆ z(t) = P ζ (t) + Q
−E ⊥ Bu(t)
+ Hu(t)
(4a)
] (4b)
y(t)
where ζ (t) ∈ Rq is the state of the observer, ˆ z(t) ∈ Rr is the estimate of z(t). Matrices N, F , H, P and Q are constant and of appropriate dimensions to be determined such that limt →∞ (ˆ z(t) − z(t)) = 0. The following theorem gives the conditions for system (4) to be a qth order observer for the functional z(t) in system (1). Theorem 1. The qth-order observer (4) will estimate (asymptotically) z(t) if there exists a matrix parameter T such that the following conditions hold (1) NTE − TA + F (2)
[
P
]
⎡
TE
]
Q
|
[
E⊥A C
= 0, ⎤
⎣ E⊥A ⎦ = L , C
(3) H = TB. (4) The pair (E, A) is admissible, where E =
[
N P
0 −I
[
I 0
]
0 0
and A =
]
.
Proof. Let ϵ (t) be the error between ζ (t) and TEx(t), i.e ϵ (t) = ζ (t) − TEx(t), then its dynamic is given by ( [ ⊥ ]) E A σ ϵ (t) = N ϵ (t) + NTE − TA + F x(t) C
+ (H − TB)u(t).
(5)
On the other hand from (4b) the estimate of z(t) can be written as
⎡ ˆ z(t) = P ϵ (t) +
[
P
|
Q
]
⎤
TE ⎣ E ⊥ A ⎦ x(t). C
(6)
If conditions (1), (2) and (3) are satisfied, then (5) and (6) reduce to σ ϵ (t) = N ϵ (t), and e(t) = ˆ z(t) − z(t) = ˆ z(t) − Lx(t) = P ϵ (t), which can be written as Eσ ξ (t) = Aξ (t) where ξ (t) =
ϵ (t)
[
e(t)
(7)
]
. In addition if condition (4) is satisfied we have
limt →∞ ϵ (t) = 0 and limt →∞ e(t) = 0 for any x(0), ˆ z(0) , and u(t). Hence ˆ z(t) in (4) is an estimate of z(t). This completes the proof. From Theorem 1, the design of the observer (4) is reduced to find the matrices T , N, P, Q , F and H such that conditions (1)–(4) are satisfied. In the sequel of this paper we shall use the same notations as⎤in ⎡ E Darouach (2012). Define the following matrices Γ = ⎣ E ⊥ A ⎦ C
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
M. Darouach et al. / Automatica (
[ ] R
and let R ∈ Rq×n be a full row rank matrix such that rank Γ rankΓ . Also define the following matrices α = RΓ + RΓ +
[] 0 I
, β = (I − Γ Γ + )
[] I 0
, β1 = (I − Γ Γ + )
[] 0 I
[]
=
I 0
, α1 =
⎡
⎤
R
)
–
3
and only if there exist matrices X1 = X1T > 0, X2 , Q1 , Q2 , ΩZ and ΩZ1 such that the following LMI
Σ1 Σ2T
] Σ2 <0 −Q2 − Q2T
[
(10)
, Π = ⎣ E⊥A ⎦ C
where Σ1 = AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 and Σ2 = AT4 Q2 − BT4 ΩZT −
⎢ E⊥A ⎥ ,Σ = ⎣ ⎦ and Θ = α A. Also from Darouach (2012) we C βA
Proof. From Lemma 2, the pair (E, A) is admissible if and only if there exist matrices X > 0 and Q such that
R
⎡
⎤
have the following results:
(X E + E⊥T Q )T A + AT (X E + E⊥T Q ) < 0.
Lemma 3. The necessary and sufficient conditions for the existence of the observers matrices, which satisfy conditions (1)–(3) of Theorem 1 are given by
⎡
R
⎤
R
⎡
X2 − Q1T , is satisfied; in this case Z = X1−T ΩZ and Z1 = Q2−T ΩZ1 .
Let X =
[
[
X1 X2T
]
0
X2 X3
]
⎢E ⊥ A⎥ ⎢ ⎥ ⎢E ⊥ A⎥ rank ⎢ C ⎥ = rank ⎣ ⎦ C ⎣ βA ⎦ βA αA
(8)
[
R
⎤
⎡
R
Φ3
[
⎤
⎢E A⎥ rank ⎣ ⎦ = rank ⎣E ⊥ A⎦ C
(9)
C
L
]
P T Q2 − (X2 + Q1T ) <0 −Q2 − Q2T
I 0 , [00] ΣΣ + ) I , 0
B1 = (I − ΣΣ + ) A3 = ΘΣ + (I − Π Π + )
[0] 0 I
[] I 0
[I ]
[0]
0 0
I 0
, A2 = ΘΣ +
, B3 = (I − ΣΣ + )
, A5 = LΠ +
[] 0 I
, B2 = (I −
[0] 0 I
, A4 = LΠ +
and B5 = (I − ΠΠ + )
[] I 0
, B4 =
[] 0 I
(12)
multiplying and post-multiplying this inequality by I P
[
F = K1 + NK , Q = K2 + PK , H = TB, where A1 = ΘΣ +
]
where Φ3 = N T X1 + X1 N + P T (X2T + Q1 ) + (X2 + Q[1T )P. By] pre-
[
Now if these conditions are satisfied, then we obtain the following matrices N = A1 − Z B1 , K1 = A2 − Z B2 , Y = A3 − Z B3 , P = A4 − Z1 B4 , K2 = A5 − Z1 B5 , T = α − Y β , K = α1 − Y[β]1 ,
P T Q2 −Q2
N T X1 + P T (X2T + Q1 ) −(X2T + Q1 )
−(X2T + Q1 ) + Q2T P ⊥
Q2 . On the other hand E⊥ =
and Q = Q1
which inserted into (11) leads to
and
⎡
(11)
]
[
I , then we obtain
AT (X E + E⊥T Q ) =
⎤
1
0 I
]
I 0
PT I
and
respectively, we obtain the following inequality
] −P T Q2T − X2 − Q1T < 0. −Q2 − Q2T
N T X1 + X1 N −Q2 P − X2T − Q1
(13)
The LMI (10) can be obtained from (13) by inserting the values of matrices N and P, and by putting ΩZ = X1T Z and ΩZ1 = Q2T Z1 . □
3.2. Discrete-time descriptor systems In this section we shall present an LMI condition for system (7) to be admissible in the discrete-time case.
. Matrices
Z and Z1 are arbitrary of appropriate dimensions. The problem of the design of the functional observer (4) for descriptor system (1) is reduced to the determination of the parameter matrices Z and Z1 such that the condition (4) of Theorem 1 is satisfied. This can be obtained from the study of the admissibility of system (7). In the following section we shall present a strict LMI formulation for the determination of these matrices.
Theorem 3. Under assumption I and conditions (8) and (9) there exist two parameter matrices Z and Z1 such that system (7) is admissible if and only if there exist matrices X1 = X1T > 0, X¯ > 0, X2 , Q1 , Q2 , ΩZ and ΩZ1 such that the following LMI
3. Observer design
where Φ1 = −Q1 − AT4 Q2 + BT4 ΩZT and Φ2 = AT1 X1 − BT1 ΩZT , is
Φ1
⎡ − X1 ⎣ Φ1T Φ2T
X¯ − Q2T − Q2 − X2
⎤ Φ2 −X2T ⎦ < 0 −X1
(14)
1
In the previous section, we presented the parameterization of all the observer matrices. The determination of these matrices is based on the knowledge of two parameter matrices Z and Z1 . So, in this section we shall present an LMI formulation for their determination. 3.1. Continuous-time descriptor systems
satisfied; in this case Z = X1−T ΩZ and Z1 = Q2−T ΩZ1 .
Proof. From Lemma 2, the pair (E, A) is admissible in the discretetime case if and only if there exist matrices X > 0 and Q such that AT X A − ET X E + Q E⊥ A + AT E⊥T Q T < 0. Let X =
[
]
0
The following theorem gives the necessary and sufficient conditions for (7) to be admissible in the continuous-time case.
[
Theorem 2. Under assumption I and conditions (8) and (9) there exist two parameter matrices Z and Z1 such that system (7) is admissible, if
−
[
X1 X2T
X2 X3
]
> 0 and Q =
[ ] Q1 Q2
(15)
. On the other hand E⊥ =
I , then we obtain
NT 0
[
X1 0
PT −I
][ ]
X1 X2T
X2 X3
[ ]
][
0 Q [ + 1 P 0 Q2
N P
0 −I
]
[ T] ] P [ T −I + Q1 −I
Q2T < 0.
]
(16)
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
4
M. Darouach et al. / Automatica (
By pre-multiplying inequality (16) by it by
[
I P
0 I
]
[
I 0
T
P I
]
and post-multiplying
we obtain the following LMI
[
N T X1 N − X1 −X2T N − Q2T P − Q1T
] −Q1 − P T Q2 − N T X2 < 0. X3 − Q2 − Q2T
(17)
The Schur complement applied to this inequality leads to the following LMI
⎡
−X1 ⎣−X T N − Q T P − Q T 1 2 2 X1 N
Now by pre-multiplying this LMI by
[
⎡
−X1 ⎣−Q T P − Q T 2 1 X1 N
N T X1 0 ⎦ < 0. (18) −X1
0
[
multiplying it by
⎤
−Q1 − P T Q2 − N T X2 X3 − Q2 − Q2T 0
0
0 0
I 0
X2T X1−1 I
and by post-
0 I
0 0
0
X1−T X2
I
3.3. Design procedure
• Step 1: Let Γ =
−Q1 − P T Q2 X3 − X2T X1−1 X2 − Q2T − Q2 −X2
⎤
N T X1 −X2T ⎦ < 0. −X1
(19)
AT4 Q2 − Q1T −Q2 − Q2T
]
<0
(20)
and
⎤
AT1 X1 − BT1 ΩZT ⎦<0 0 −X1
−Q1 − AT4 Q2 −Q2T − Q2 0
(21)
respectively. Now, if in addition we choose the parameter matrix Q1 = −Q2T A4 for the continuous-time case and Q1 = −AT4 Q2 for the discrete-time case, we obtain the following simple LMIs: AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 < 0
(22)
and AT1 X1 − BT1 ΩZT −X1
E E⊥A C
[
] and let R be a full row rank matrix
with respect to R and R(L) ⊂ R(Π ), with Π =
AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 Q2T A4 − Q1
−X1 X1 A1 − ΩZ B1
Let the symbol R(L) denotes the row space of a matrix L, then the procedure of the presented functional observer design can be summarized as follows:
such that system (2) is partially impulse (causal)[observable ]
we obtain
[
[
Theorems 2 and 3 general, necessary and sufficient, and permit to have the parameterization of all the observers (4) for system (1). It can also be used in large scale and uncertain systems (see the references Yong-Yan and Zongli, 2004; Gao Huijun and Li, 2014 ). The presentation of the results in LMIs formulation presents the advantage that they can be easily solved by standard convex optimization algorithms.
]
I 0
Remark 1. Eqs. (10) give the parameterization of all the functional observers in the form (4). Then the LMIs (10) and (14) contain many slack variables; to reduce the number of these variables we can consider the following particular case: If we take the matrix parameter Z1 = 0 and the matrix X diagonal or equivalently X2 = 0, which leads to sufficient conservative conditions, the LMIs (10) and (14) become:
−X1 ⎣−Q T A4 − Q T 2 1 X1 A1 − ΩZ B1
–
]
I
Let X¯ = X3 − X2T X1−1 X2 , then since X1 > 0 and X > 0, we have X¯ > 0. The LMI (14) can be obtained from (19) by inserting the values of matrices N and P, and by putting ΩZ = X1T Z and ΩZ1 = Q2T Z1 . □
⎡
)
]
< 0.
(23)
From these results one can see that the LMIs (10) and (14) contain the slack parameters X2 , Q1 , Q2 and X¯ which make the conditions of
R E⊥A C
, com-
pute the matrix parameters α , α1 , β and β1 , then compute matrices Σ and Θ . • Step 2: Verify that conditions (8) and (9) are satisfied, then compute matrices A1 , B1 , A2 , B2 , A3 , B3 , A4 , B4 , A5 and B5 . • Step 3: Solve the LMI (10) for the continuous-time case and the LMI (14) for the discrete-time case to obtain the parameter matrices Z and Z1 . Once these parameters are determined, all the parameter matrices of the functional observers can be determined from Eqs. (10). 4. Conclusion In this paper, we have presented a novel method to design functional observers for continuous-time and discrete-time descriptor systems. We formulated the estimation error in a descriptor form which permitted the generalization of the existing results. The order of the considered observers may be different of the dimension of the functional to be estimated. The existence and stability conditions are given in LMIs formulation and generalize those generally presented for the standard and descriptor systems. References Dai, L. (1989). Impulse modes and causality in singular systems. International Journal of Control, 40, 1267–1281. Darouach, M. (2012). On the functional observers for linear descriptor systems. Systems & Control Letters, 3, 427–434. Gao Huijun, H., & Li, X. (2014). Robust filtering for uncertain systems. Springer Verlag. Ishihara, J. Y., & Terra, M. H. (2001). Impulse controllability and observability of rectangular descriptor systems. IEEE Transactions on Automatic Control, 6, 991–994. Xu, S., & Lam, J. (2006). Robust control and filtering of singular systems. Springer Verlag. Yong-Yan, C., & Zongli, L. (2004). A descriptor system approach to robust stability analysis and controller design. IEEE Transactions on Automatic Control, 11, 2081–2084.
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
)
–
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical Communique
Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases✩ Mohamed Darouach a , Francesco Amato b , Marouane Alma a a
CRAN-CNRS (UMR 7039), Université de Lorraine, IUT de Longwy, 186, Rue de Lorraine, 54400 Cosnes et Romain, France School of Computer and Biomedical Engineering, Experimental Clinical Medicine Department, Università degli Studi Magna Grcia di Catanzaro, Campus di Germaneto ‘‘Salvatore Venuta’’, 88100 Catanzaro, Italy b
article
info
Article history: Received 19 August 2016 Received in revised form 17 May 2017 Accepted 22 June 2017 Available online xxxx
a b s t r a c t This paper investigates the design of functional observers for linear time-invariant descriptor systems. A new method for designing these observers is given by using an LMI (Linear matrix inequality) formulation. The obtained result unifies the design, it considers the continuous-time and discrete-time cases and concerns the observers of various orders. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Functional observer Descriptor systems Stability Existence conditions LMI
1. Introduction In the present paper, based on the results of Darouach (2012), a new functional observers design method for descriptor systems is presented. The novelty of the presented results lies in the formulation of the dynamics of the error which is represented in a descriptor system form. It permits the simplification of the design for the uncertain systems and gives more degrees of freedom for this design Yong-Yan & Zongli (2004) (see also Gao Huijun & Li, 2014 and references therein). It also unifies the design for different observer orders and concerns the continuous-time as well as the discrete-time systems. Necessary and sufficient conditions for the existence of these observers are given in LMIs form. The observers design in the LMI framework allows to integrate our approach with existing ones (optimal control, pole placement, etc.) that nowadays are all expressed in this way. The proposed approach also presents the advantage that LMIs can be easily tested by using standard convex optimization algorithms. The paper is organized as follows. Section 2 presents some preliminary results on descriptor systems which will be used in the sequel of the paper. The functional observers design of different orders for descriptor systems, including particular cases and the ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tong Zhou under the direction of Editor André L. Tits. E-mail addresses: [email protected] (M. Darouach), [email protected] (F. Amato), [email protected] (M. Alma).
procedure of this design, is given in Section 3. Section 4 concludes the paper. 2. Preliminary results In this section we recall some results on linear descriptor systems which are used in the sequel of the paper (see Dai, 1989; Darouach, 2012). Consider the linear time-invariant multivariable system described by E σ x(t) = Ax(t) + Bu(t)
(1a)
y(t) = Cx(t)
(1b)
z = Lx(t)
(1c)
where σ denotes the derivative operator σ x(t) = dx(t)/dt for continuous-time systems and the forward-shift operator σ x(k) = x(k + 1), where k ∈ Z, for discrete-time systems, x ∈ Rn and y ∈ Rp are the semi state vector and the output vector of the system, u ∈ Rm is the known input and z ∈ Rr is the vector to be estimated, where r ⩽ n. Matrix E ∈ Rn1 ×n , when n1 = n matrix E is singular, matrices A, B, C , and L are known constant and of appropriate dimensions. It is assumed that u(t) and x(0) = x0 are admissible, i.e they are such that there exists at least one trajectory satisfying (1a). By using the Laplace transform (z transform for the discrete-time case) we have the following solvability conditions Ishihara & Terra (2001):
http://dx.doi.org/10.1016/j.automatica.2017.08.016 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
2
M. Darouach et al. / Automatica (
1. The pair (u(t), x0 ) is admissible if and only if normal − [ ] rank sE − A BU(s) Ex0 = normal − rank(sE − A) for the continuous-time case, 2. The pair (u(t), x0 ) is admissible if and only if normal − ] [ rank zE − A BU(z) Ex0 = normal − rank(zE − A) for the discrete-time case where the normal-rank of the matrix pencil (λE − A) is defined as the rank(λE − A) for almost all λ ∈ C. In the sequel of this paper we shall assume that the pair (u(t), x0 ) is admissible. Also we shall use the following notations: The symbol Σ + denotes any generalized inverse of the matrix Σ , i.e. verifies ΣΣ + Σ = Σ . This generalized inverse matrix is also denoted by Σ − or Σ (1) in the literature. The symbol E ⊥ denotes a maximal row rank matrix such that E ⊥ E = 0. When E is of full row rank, E ⊥ = 0 by definition. Before presenting the observer design for system (1), we shall recall the following results, which extend the notion of impulse (causal for the discrete-time case) observability to the partial impulse (causal) observability (see reference Darouach, 2012). Definition 1. The descriptor system (1) with u(t) = 0, or the triplet (C , E , A), is said to be partially impulse observable with respect to L, if y(t) is impulse free for t ⩾ 0, only if Lx(t) is impulse free for t ⩾0. Definition 2. The descriptor system (1) with u(t) = 0, or the triplet (C , E , A), is said to be partially causal observable with respect to L, if Lx(k) at any time point k is uniquely determined by the initial condition and the former measurement y(i), i = 0, 1, . . . .., k. The following lemma gives the conditions for the partial impulse observability (partial causal observability for the discretetime case) Darouach (2012). Lemma 1. The following statements are equivalent 1. The triplet (C , E , A) is partially impulse (causal) observable with respect ; ⎡ to L ⎤ 2.
E 0 rank ⎣0 0
⎡
A C⎦ E L
L E
[E
= rank
⎤
3. rank ⎣E ⊥ A⎦ = rank C
[
A C E
0 0 E E⊥A C
]
;
] .
The next assumption is used in the sequel of the paper. Assumption I : We assume that system (1) is partial impulse (causal) observable with respect to L. Now we can give the following definitions for descriptor systems (see Dai, 1989; Xu & Lam, 2006). Definition 3. Let us consider the following descriptor system : E σ x(t) = Ax(t) where E ∈ Rn×n and A ∈ Rn×n , then we have the following definitions 1. The pair (E , A) is regular if det(λE − A) is not identically zero. 2. The pair (E , A) is impulse free (causal for the discrete-time case) if deg(det(λE − A)) = rankE. 3. The pair (E , A) is said to be stable if the roots of det(λE − A) = 0 have negative real parts (are inside the unit circle for the discrete-time case). 4. The pair (E , A) is said to be admissible if it is regular, impulse free (causal for the discrete-time case) and stable. The following lemma gives the necessary and sufficient conditions for the admissibility of the pair (E , A), in the strict LMI formulation (see Xu & Lam, 2006).
)
–
Lemma 2. The pair (E , A) is admissible if and only if there exist matrices X > 0 and Q such that (XE + E ⊥T Q )T A + AT (XE + E ⊥T Q ) < 0
(2)
for the continuous-time case, and AT XA − E T XE + QE ⊥ A + AT E ⊥T Q T < 0
(3)
for the discrete-time case. Let us consider the following reduced order observer.
[ ⊥ ] −E Bu(t)
σ ζ (t) = N ζ (t) + F
y(t)
[
ˆ z(t) = P ζ (t) + Q
−E ⊥ Bu(t)
+ Hu(t)
(4a)
] (4b)
y(t)
where ζ (t) ∈ Rq is the state of the observer, ˆ z(t) ∈ Rr is the estimate of z(t). Matrices N, F , H, P and Q are constant and of appropriate dimensions to be determined such that limt →∞ (ˆ z(t) − z(t)) = 0. The following theorem gives the conditions for system (4) to be a qth order observer for the functional z(t) in system (1). Theorem 1. The qth-order observer (4) will estimate (asymptotically) z(t) if there exists a matrix parameter T such that the following conditions hold (1) NTE − TA + F (2)
[
P
]
⎡
TE
]
Q
|
[
E⊥A C
= 0, ⎤
⎣ E⊥A ⎦ = L , C
(3) H = TB. (4) The pair (E, A) is admissible, where E =
[
N P
0 −I
[
I 0
]
0 0
and A =
]
.
Proof. Let ϵ (t) be the error between ζ (t) and TEx(t), i.e ϵ (t) = ζ (t) − TEx(t), then its dynamic is given by ( [ ⊥ ]) E A σ ϵ (t) = N ϵ (t) + NTE − TA + F x(t) C
+ (H − TB)u(t).
(5)
On the other hand from (4b) the estimate of z(t) can be written as
⎡ ˆ z(t) = P ϵ (t) +
[
P
|
Q
]
⎤
TE ⎣ E ⊥ A ⎦ x(t). C
(6)
If conditions (1), (2) and (3) are satisfied, then (5) and (6) reduce to σ ϵ (t) = N ϵ (t), and e(t) = ˆ z(t) − z(t) = ˆ z(t) − Lx(t) = P ϵ (t), which can be written as Eσ ξ (t) = Aξ (t) where ξ (t) =
ϵ (t)
[
e(t)
(7)
]
. In addition if condition (4) is satisfied we have
limt →∞ ϵ (t) = 0 and limt →∞ e(t) = 0 for any x(0), ˆ z(0) , and u(t). Hence ˆ z(t) in (4) is an estimate of z(t). This completes the proof. From Theorem 1, the design of the observer (4) is reduced to find the matrices T , N, P, Q , F and H such that conditions (1)–(4) are satisfied. In the sequel of this paper we shall use the same notations as⎤in ⎡ E Darouach (2012). Define the following matrices Γ = ⎣ E ⊥ A ⎦ C
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
M. Darouach et al. / Automatica (
[ ] R
and let R ∈ Rq×n be a full row rank matrix such that rank Γ rankΓ . Also define the following matrices α = RΓ + RΓ +
[] 0 I
, β = (I − Γ Γ + )
[] I 0
, β1 = (I − Γ Γ + )
[] 0 I
[]
=
I 0
, α1 =
⎡
⎤
R
)
–
3
and only if there exist matrices X1 = X1T > 0, X2 , Q1 , Q2 , ΩZ and ΩZ1 such that the following LMI
Σ1 Σ2T
] Σ2 <0 −Q2 − Q2T
[
(10)
, Π = ⎣ E⊥A ⎦ C
where Σ1 = AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 and Σ2 = AT4 Q2 − BT4 ΩZT −
⎢ E⊥A ⎥ ,Σ = ⎣ ⎦ and Θ = α A. Also from Darouach (2012) we C βA
Proof. From Lemma 2, the pair (E, A) is admissible if and only if there exist matrices X > 0 and Q such that
R
⎡
⎤
have the following results:
(X E + E⊥T Q )T A + AT (X E + E⊥T Q ) < 0.
Lemma 3. The necessary and sufficient conditions for the existence of the observers matrices, which satisfy conditions (1)–(3) of Theorem 1 are given by
⎡
R
⎤
R
⎡
X2 − Q1T , is satisfied; in this case Z = X1−T ΩZ and Z1 = Q2−T ΩZ1 .
Let X =
[
[
X1 X2T
]
0
X2 X3
]
⎢E ⊥ A⎥ ⎢ ⎥ ⎢E ⊥ A⎥ rank ⎢ C ⎥ = rank ⎣ ⎦ C ⎣ βA ⎦ βA αA
(8)
[
R
⎤
⎡
R
Φ3
[
⎤
⎢E A⎥ rank ⎣ ⎦ = rank ⎣E ⊥ A⎦ C
(9)
C
L
]
P T Q2 − (X2 + Q1T ) <0 −Q2 − Q2T
I 0 , [00] ΣΣ + ) I , 0
B1 = (I − ΣΣ + ) A3 = ΘΣ + (I − Π Π + )
[0] 0 I
[] I 0
[I ]
[0]
0 0
I 0
, A2 = ΘΣ +
, B3 = (I − ΣΣ + )
, A5 = LΠ +
[] 0 I
, B2 = (I −
[0] 0 I
, A4 = LΠ +
and B5 = (I − ΠΠ + )
[] I 0
, B4 =
[] 0 I
(12)
multiplying and post-multiplying this inequality by I P
[
F = K1 + NK , Q = K2 + PK , H = TB, where A1 = ΘΣ +
]
where Φ3 = N T X1 + X1 N + P T (X2T + Q1 ) + (X2 + Q[1T )P. By] pre-
[
Now if these conditions are satisfied, then we obtain the following matrices N = A1 − Z B1 , K1 = A2 − Z B2 , Y = A3 − Z B3 , P = A4 − Z1 B4 , K2 = A5 − Z1 B5 , T = α − Y β , K = α1 − Y[β]1 ,
P T Q2 −Q2
N T X1 + P T (X2T + Q1 ) −(X2T + Q1 )
−(X2T + Q1 ) + Q2T P ⊥
Q2 . On the other hand E⊥ =
and Q = Q1
which inserted into (11) leads to
and
⎡
(11)
]
[
I , then we obtain
AT (X E + E⊥T Q ) =
⎤
1
0 I
]
I 0
PT I
and
respectively, we obtain the following inequality
] −P T Q2T − X2 − Q1T < 0. −Q2 − Q2T
N T X1 + X1 N −Q2 P − X2T − Q1
(13)
The LMI (10) can be obtained from (13) by inserting the values of matrices N and P, and by putting ΩZ = X1T Z and ΩZ1 = Q2T Z1 . □
3.2. Discrete-time descriptor systems In this section we shall present an LMI condition for system (7) to be admissible in the discrete-time case.
. Matrices
Z and Z1 are arbitrary of appropriate dimensions. The problem of the design of the functional observer (4) for descriptor system (1) is reduced to the determination of the parameter matrices Z and Z1 such that the condition (4) of Theorem 1 is satisfied. This can be obtained from the study of the admissibility of system (7). In the following section we shall present a strict LMI formulation for the determination of these matrices.
Theorem 3. Under assumption I and conditions (8) and (9) there exist two parameter matrices Z and Z1 such that system (7) is admissible if and only if there exist matrices X1 = X1T > 0, X¯ > 0, X2 , Q1 , Q2 , ΩZ and ΩZ1 such that the following LMI
3. Observer design
where Φ1 = −Q1 − AT4 Q2 + BT4 ΩZT and Φ2 = AT1 X1 − BT1 ΩZT , is
Φ1
⎡ − X1 ⎣ Φ1T Φ2T
X¯ − Q2T − Q2 − X2
⎤ Φ2 −X2T ⎦ < 0 −X1
(14)
1
In the previous section, we presented the parameterization of all the observer matrices. The determination of these matrices is based on the knowledge of two parameter matrices Z and Z1 . So, in this section we shall present an LMI formulation for their determination. 3.1. Continuous-time descriptor systems
satisfied; in this case Z = X1−T ΩZ and Z1 = Q2−T ΩZ1 .
Proof. From Lemma 2, the pair (E, A) is admissible in the discretetime case if and only if there exist matrices X > 0 and Q such that AT X A − ET X E + Q E⊥ A + AT E⊥T Q T < 0. Let X =
[
]
0
The following theorem gives the necessary and sufficient conditions for (7) to be admissible in the continuous-time case.
[
Theorem 2. Under assumption I and conditions (8) and (9) there exist two parameter matrices Z and Z1 such that system (7) is admissible, if
−
[
X1 X2T
X2 X3
]
> 0 and Q =
[ ] Q1 Q2
(15)
. On the other hand E⊥ =
I , then we obtain
NT 0
[
X1 0
PT −I
][ ]
X1 X2T
X2 X3
[ ]
][
0 Q [ + 1 P 0 Q2
N P
0 −I
]
[ T] ] P [ T −I + Q1 −I
Q2T < 0.
]
(16)
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.
4
M. Darouach et al. / Automatica (
By pre-multiplying inequality (16) by it by
[
I P
0 I
]
[
I 0
T
P I
]
and post-multiplying
we obtain the following LMI
[
N T X1 N − X1 −X2T N − Q2T P − Q1T
] −Q1 − P T Q2 − N T X2 < 0. X3 − Q2 − Q2T
(17)
The Schur complement applied to this inequality leads to the following LMI
⎡
−X1 ⎣−X T N − Q T P − Q T 1 2 2 X1 N
Now by pre-multiplying this LMI by
[
⎡
−X1 ⎣−Q T P − Q T 2 1 X1 N
N T X1 0 ⎦ < 0. (18) −X1
0
[
multiplying it by
⎤
−Q1 − P T Q2 − N T X2 X3 − Q2 − Q2T 0
0
0 0
I 0
X2T X1−1 I
and by post-
0 I
0 0
0
X1−T X2
I
3.3. Design procedure
• Step 1: Let Γ =
−Q1 − P T Q2 X3 − X2T X1−1 X2 − Q2T − Q2 −X2
⎤
N T X1 −X2T ⎦ < 0. −X1
(19)
AT4 Q2 − Q1T −Q2 − Q2T
]
<0
(20)
and
⎤
AT1 X1 − BT1 ΩZT ⎦<0 0 −X1
−Q1 − AT4 Q2 −Q2T − Q2 0
(21)
respectively. Now, if in addition we choose the parameter matrix Q1 = −Q2T A4 for the continuous-time case and Q1 = −AT4 Q2 for the discrete-time case, we obtain the following simple LMIs: AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 < 0
(22)
and AT1 X1 − BT1 ΩZT −X1
E E⊥A C
[
] and let R be a full row rank matrix
with respect to R and R(L) ⊂ R(Π ), with Π =
AT1 X1 − BT1 ΩZT + X1 A1 − ΩZ B1 Q2T A4 − Q1
−X1 X1 A1 − ΩZ B1
Let the symbol R(L) denotes the row space of a matrix L, then the procedure of the presented functional observer design can be summarized as follows:
such that system (2) is partially impulse (causal)[observable ]
we obtain
[
[
Theorems 2 and 3 general, necessary and sufficient, and permit to have the parameterization of all the observers (4) for system (1). It can also be used in large scale and uncertain systems (see the references Yong-Yan and Zongli, 2004; Gao Huijun and Li, 2014 ). The presentation of the results in LMIs formulation presents the advantage that they can be easily solved by standard convex optimization algorithms.
]
I 0
Remark 1. Eqs. (10) give the parameterization of all the functional observers in the form (4). Then the LMIs (10) and (14) contain many slack variables; to reduce the number of these variables we can consider the following particular case: If we take the matrix parameter Z1 = 0 and the matrix X diagonal or equivalently X2 = 0, which leads to sufficient conservative conditions, the LMIs (10) and (14) become:
−X1 ⎣−Q T A4 − Q T 2 1 X1 A1 − ΩZ B1
–
]
I
Let X¯ = X3 − X2T X1−1 X2 , then since X1 > 0 and X > 0, we have X¯ > 0. The LMI (14) can be obtained from (19) by inserting the values of matrices N and P, and by putting ΩZ = X1T Z and ΩZ1 = Q2T Z1 . □
⎡
)
]
< 0.
(23)
From these results one can see that the LMIs (10) and (14) contain the slack parameters X2 , Q1 , Q2 and X¯ which make the conditions of
R E⊥A C
, com-
pute the matrix parameters α , α1 , β and β1 , then compute matrices Σ and Θ . • Step 2: Verify that conditions (8) and (9) are satisfied, then compute matrices A1 , B1 , A2 , B2 , A3 , B3 , A4 , B4 , A5 and B5 . • Step 3: Solve the LMI (10) for the continuous-time case and the LMI (14) for the discrete-time case to obtain the parameter matrices Z and Z1 . Once these parameters are determined, all the parameter matrices of the functional observers can be determined from Eqs. (10). 4. Conclusion In this paper, we have presented a novel method to design functional observers for continuous-time and discrete-time descriptor systems. We formulated the estimation error in a descriptor form which permitted the generalization of the existing results. The order of the considered observers may be different of the dimension of the functional to be estimated. The existence and stability conditions are given in LMIs formulation and generalize those generally presented for the standard and descriptor systems. References Dai, L. (1989). Impulse modes and causality in singular systems. International Journal of Control, 40, 1267–1281. Darouach, M. (2012). On the functional observers for linear descriptor systems. Systems & Control Letters, 3, 427–434. Gao Huijun, H., & Li, X. (2014). Robust filtering for uncertain systems. Springer Verlag. Ishihara, J. Y., & Terra, M. H. (2001). Impulse controllability and observability of rectangular descriptor systems. IEEE Transactions on Automatic Control, 6, 991–994. Xu, S., & Lam, J. (2006). Robust control and filtering of singular systems. Springer Verlag. Yong-Yan, C., & Zongli, L. (2004). A descriptor system approach to robust stability analysis and controller design. IEEE Transactions on Automatic Control, 11, 2081–2084.
Please cite this article in press as: Darouach, M., et al., Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.08.016.