On functional observers for linear time-varying ... - Semantic Scholar
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On functional observers for linear time-varying systems F. Rotella, I. Zambettakis Abstract—The paper deals with existence conditions of a functional observer for linear time-varying systems in the case where the order of the observer is equal to the number of observed variables. Constructive procedures for the design of such a linear functional observer are deduced from the existence conditions. As a specific feature, the proposed procedures do not require the solution of a differential Sylvester equation. Some examples illustrate the presented results. Index Terms—Linear time-varying system, functional observer, Luenberger observer.
I. I NTRODUCTION The interest to consider linear time-varying systems is twofold [8], [5], [17] : on the one hand as general models of linear behaviour for a plant, on the other hand as linearized models of non linear systems about a given trajectory. For state feedback control or fault diagnosis purposes the need of asymptotic observers of a given linear functional is of primary importance. Therefore, we consider the problem of observing a linear functional v(t) = L(t)x(t),
(1)
where, for every time t in R+ , L(t) is a constant full row rank (l×n) differentiable matrix, and x(t) is the n-dimensional state vector of the state space system x(t) ˙ = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t),
(2)
where u(t) is the p-dimensional control, and y(t) is the mdimensional output. For every t in R+ , A(t), B(t), and C(t) are known matrices of appropriate dimensions. To avoid tedious counts and distracting lists of differentiability requirements, we assume every time-varying matrices are such that all derivatives that appear are continuous for all t. Without loss of generality and in order to C(t) avoid useless dynamic parts in the observer, we suppose L(t) has full row rank for all t. Indeed, the rows of an arbitrary given L(t) which are linearly dependant of the rows of C(t) induce obvious estimation of the corresponding components of v(t) from the available informations. The observability matrix of (2) is defined by Ω(t) =
Ω0 (t) Ω1 (t) .. . Ωn−1 (t)
,
˙ j−1 (t) for where Ω0 (t) = C(t), and Ωj (t) = Ωj−1 (t)A(t) + Ω j = 1, 2, ...n − 1. System (2), or shortly (A(t), C(t)), is completely observable if rank (Ω(t)) = n for some t in R+ . It is uniformly observable if rank (Ω(t)) = n for every t in R+ [21], [29]. Following [8], if r states of (2) are not observable there exists a transformation which induces the following partitions for A(t) and C(t) A11 (t) A12 (t) A(t) = , 0(n−r)×r A22 (t) 0m×r C2 (t) , C(t) = F. Rotella and I. Zambettakis are with the Laboratoire de Génie de Production, Ecole Nationale d’Ingénieurs de Tarbes, France, [email protected], [email protected].
1
where (A22 (t), C2 (t)) is completely observable. Then the system is detectable when A11 (t) is a Hurwitz matrix. A matrix F (t) is said to be a Hurwitz (convergent in [25]) matrix if every solution x(t, t0 , x0 ) of the differential system x(t) ˙ = F (t)x(t), x(t0 ) = x0 , is such that limt→∞ x(t, t0 , x0 ) = 0 for every t0 and x0 . Since Luenberger’s seminal work [12] a significant amount of research is devoted to the problem of observing a linear functional in a time-invariant setting, see for instance [14], [26], [1], [24] and the references therein. Whereas, unlike the time-invariant counterpart, since [28] there are few papers dealing with the observer design for time-varying systems. The main part of the proposed developments are limited to the case of state observers design (see [18], [3], [17], [30], [23] and the references therein). For the functional observer case we can mention [18] for the design of a single-functional observer while [25] gives a necessary and sufficient existence condition of a linear functional observer. Now, it is well known that the observation of v(t) can be carried out with the design of the Luenberger observer z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = P (t)z(t) + V (t)y(t),
(3)
where z(t) is a q-dimensional state vector. The time-varying matrices F (t), G(t), H(t), P (t) and V (t) must be determined such that (3) is an asymptotic observer of (1) for the system (2). Namely, they have to ensure lim (v(t) − w(t)) = 0. t→∞
It has been shown in [25] (theorem 3.9) that the completely observable system z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = P (t)z(t),
(4)
is an asymptotic observer for system (2) and linear functional M (t)x(t) if and only if there exists a continuously differentiable matrix T (t) such that G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t), M (t) = P (t)T (t),
(5) (6)
and F (t) is a Hurwitz matrix. So, for any matrix V (t), when we choose M (t) = L(t) − V (t)C(t), the theorem stated in [25] can be read as Theorem 1. The completely observable system (3) is an asymptotic observer of linear functional (1) for system (2) if and only if there exists a continuously differentiable solution T (t) of equations G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t), L(t) = P (t)T (t) + V (t)C(t), and F (t) is a Hurwitz matrix.
(7) (8) ♦
Due to linear independence of C(t) and L(t), we know that, as in the time-invariant case ([15], [22]), l is a lower bound for the order of observer (4). In functional observer literature the words minimal and minimum are both used to denote a lower order stable observer (see for instance [18], [26]), nevertheless minimal is more frequently used. Moreover all these uses are mainly subject to the restriction that the observer eigenvalues are freely assignable. In
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[22] it has been underlined that minimality must be considered with respect to two different problems depending whether the observer poles are arbitrarily specified at the outset (the fixed poles observer problem), or are permitted to lie anywhere on the left half plane (the stable observer problem). So, in this paper, we call “minimum observer” the stable observer for which the order is equal to the size of the functional to be observed, while we imply that “minimal observer” characterizes the other cases. Therefore our main motivation is to analyze the existence conditions and to propose the design of a minimum asymptotic observer z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = z(t) + V (t)y(t),
(9)
where w(t) ∈ Rl is an asymptotic estimate for v(t). When this linear functional observer exists, its order, namely l according to our notations, is much less than the order (n − m) of a reduced-order state observer. As a consequence, observing a linear function of the state may afford a significant reduction in observer order compared to observing the entire state vector. A second motivation of our purpose is to circumvent the determination of T (t) as solution of the differential equation (7). The material in this paper is organized as follows. The first section deals with the analysis of existence conditions of a minimum observer. The second section is concerned with the proposal of design procedures. In the third section the linear time-invariant case is revisited. Indeed, the existence conditions of Darouach [6] for time-invariant systems are based on Hautus-type criteria and can’t be directly applied in the time-varying case. Theorem 3 applied to linear time-invariant systems leads to the Darouach conditions which appear then as a particular case of ours. Thus, the extension of the Darouach conditions to linear time-varying systems is an additionnal motivation of our paper. Finally, two generic examples are detailed. II. E XISTENCE CONDITIONS The application of theorem 1 leads to the following result. Corollary 2. The asymptotic observer (9) exists if and only if there exists a (l × n) matrix T (t) such that G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t),
(10)
L(t) = T (t) + V (t)C(t),
(11)
F (t) is a Hurwitz matrix, ♦
for all t.
Theorem 1 and corollary 2 require the solution T (t) of a differential Sylvester equation in which F (t) is unknown as well as the initial conditions of T (t). In order to overcome this difficulty, we propose to eliminate T (t) in conditions of corollary 2. From (11) we get T (t) = L(t) − V (t)C(t), so from (10) we obtain L(t)A(t) − V (t)C(t)A(t) − F (t)L(t) ˙ ˙ + F (t)V (t)C(t) + L(t) − V˙ (t)C(t) − V (t)C(t) = H(t)C(t).
(12)
Denoting
(12) can be written ˙ L(t) + L(t)A(t) = H(t) + V˙ (t) − F (t)V (t)
(13)
F (t)
V (t)
Σ(t).
(14)
Theorem 3. The minimum asymptotic observer (9) of linear functional (1) for system (2) exists if and only if the factorization of ˙ L(t) + L(t)A(t) ˙ L(t) + L(t)A(t) =
MC,0 (t)
ML,0 (t)
MC,1 (t)
Σ(t)
(15)
is such that MC,0 (t), ML,0 (t), and MC,1 (t) have m, l, and m columns respectively, the derivative of MC,1 (t) exists, and ML,0 (t) is a Hurwitz matrix. ♦ Proof: The if part follows from (14). The only if part is established by looking for F (t), H(t), V (t) such that in (15) : MC,0 (t)
=
H(t) + V˙ (t) − F (t)V (t),
ML,0 (t)
=
F (t),
MC,1 (t)
=
V (t).
It is obvious that, if MC,1 (t) has a derivative, the solution of this system is F (t)
=
ML,0 (t),
V (t)
=
MC,1 (t),
H(t)
=
MC,0 (t) + ML,0 (t)MC,1 (t) − M˙ C,1 (t).
Let T (t) = L(t) − MC,1 (t)C(t). This matrix fulfills the differential equation (10) and since F (t) is a Hurwitz matrix, this ends the proof. Let us underline here that the existence conditions of theorem 3 do not require the determination of T (t). Nevertheless we can get matrix T (t) as a consequence of factorization (15), without solving equation (10) in the sense of the solution of a differential equation. This standpoint has already been used in [16] to design a minimal order single functional stable observer for linear time-invariant systems. III. M INIMUM ORDER OBSERVER DESIGNS From theorem 3, two observer design methods are described in the following. However, the last step of these algorithms needs uniform observability to be assumed. A. Generalized inverse based design The first design method lays on the solution of the linear equation ˙ L(t) + L(t)A(t) = X(t)Σ(t),
(16)
for all t. As the solution X(t) exists if and only if Σ(t) rank Σ(t) = rank , ˙ L(t) + L(t)A(t) equation (16) can be solved by means of time-varying generalized inverses [11]. For example, a generalized inverse [2], [10], Σ{1} (t), for Σ(t) can be obtained from the time-varying singular value factorization of Σ(t) or from its QR factorization [7], [4]. The solution set of (16) can then be expressed a X(t)
C(t) , L(t) Σ(t) = ˙ C(t) + C(t)A(t)
2
=
=
˙ (L(t) + L(t)A(t))Σ{1} (t)
MC,0 (t)
ML,0 (t)
MC,1 (t)
+W (t)(In − Σ(t)Σ{1} (t)),
,
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where W (t) is an arbitrary (l × n) matrix. When for every t rank Σ(t) = 2m + l, the solution set is reduced to the unique ˙ element (L(t) + L(t)A(t))Σ{1} (t). Otherwise, a first design can be considered with the partitions
3
Then, z(t) ˙
=
+ML,0 (t)MC,1 (t))y(t)
SC,0 (t)
SL,0 (t)
SC,1 (t)
and
In − Σ(t)Σ{1} (t) = RC,0 (t)
RL,0 (t)
RC,1 (t)
,
where the number of columns is l for SL,0 (t) and for RL,0 (t), while SC,0 (t), RC,0 (t), SC,1 (t), and RC,1 (t) have m columns. It yields F (t)
=
ML,0 (t) = SL,0 (t) + W (t)RL,0 (t),
V (t)
=
H(t)
=
MC,1 (t) = SC,1 (t) + W (t)RC,1 (t), MC,0 (t) + ML,0 (t)MC,1 (t) − M˙ C,1 (t),
G(t)
=
(L(t) − MC,1 (t)C(t)) B(t).
1) Σ(t) has full row rank : Namely, we are in the case where rank Σ(t) = 2m + l for every t, except for some isolated instants τ . When Σ(t) < 2m + l at some instants ti , we have to split the time domain in successive time domains bounded by these isolated values ti of t, where the procedure can be applied. Then, matrices MC,0 (t), ML,0 (t), and MC,1 (t) are unique. Moreover, a derivation of (1) yields
=
= m + l + m1
In this case, in factorization (15), MC,0 (t), ML,0 (t), and MC,1 (t) are not unique. However, we can write C˙ 2 (t) + C2 (t)A(t) = NC,0 (t)C(t)+ NL,0 (t)L(t) + NC,1 (t) C˙1 (t) + C1 (t)A(t) ,
(20)
where matrices NC,0 (t), NL,0 (t), and NC,1 (t) are unique. Moreover, we have ˙ ˙ L(t) + L(t)A(t) = MC,1 (t) C(t) + C(t)A(t) + ML,0 (t)L(t) + MC,0 (t)C(t) = MC,11 (t) C˙1 (t) + C1 (t)A(t) + ML,0 (t)L(t) + MC,0 (t)C(t) + MC,12 (t) C˙2 (t) + C2 (t)A(t) = PC1,1 (t) C˙1 (t) + C1 (t)A(t) + PL,0 (t)L(t) + PC,0 (t)C(t),
(21)
where matrices PC,0 (t), PL,0 (t) and PC1,1 (t) are unique and given by PC1,1 (t) = MC,11 (t) + MC,12 (t)NC,1 (t), PL,0 (t) = ML,0 (t) + MC,12 (t)NL,0 (t),
(22)
PC,0 (t) = MC,0 (t) + MC,12 (t)NC,0 (t).
˙ (L(t) + L(t)A(t))x(t) + L(t)B(t)u(t), ˙ MC,1 (t) C(t) + C(t)A(t) x(t)
As previously, providing that the pair (PL,0 (t), NL,0 (t)) is detectable, the stability of the observer can be ensured.
+ML,0 (t)L(t)x(t) + MC,0 (t)C(t)x(t)
Nevertheless, when the pair (PL,0 (t), NL,0 (t)) is uniformly observable [20], [13], [3], we can obtain MC,12 (t) to yield uniform exponential stability at any desired rate for the observation error system η(t) ˙ = ML,0 (t)η(t) where
+L(t)B(t)u(t), ˙ MC,1 (t)C(t)x(t) + MC,1 (t)C(t)x(t) ˙ −MC,1 (t)C(t)B(t)u(t) + L(t)B(t)u(t)
ML,0 (t) = PL,0 (t) − MC,12 (t)NL,0 (t).
+ML,0 (t)v(t) + MC,0 (t)y(t), =
where C1 (t)(m1 × n) and C2 (t)(m2 × n) verify
= rank Σ(t).
The previous design procedure needs to choose a generalized inverse for Σ(t). To avoid computational burden with respect to this choice, we propose to extend the design procedure used recently to determine the minimal order single functional stable observer for linear timeinvariant systems [16]. When factorization (15) exists, we distinguish two cases.
=
2) rank Σ < 2m + l: Let us introduce, up to a possible permutation in the outputs, the partition C1 (t) , (19) C= C2 (t)
C(t) L(t) rank C˙1 (t) + C1 (t)A(t)
B. Direct design
=
Since factorization (15) is unique, when ML,0 (t) is a Hurwitz matrix the minimum stable observer problem is solved. However, the rate of convergence for the uniform exponential stability of the observation error system η(t) ˙ = ML,0 (t)η(t) cannot be arbitrarily fixed.
When the pair (SL,0 (t), RL,0 (t)) is uniformly observable, W (t) can be chosen, via eigenvalues assignement techniques ([20], [13], [3]), to yield uniform exponential stability at any desired rate[5], [17] for the observation error system η(t) ˙ = F (t)η(t) where F (t) = SL,0 (t) + W (t)RL,0 (t). When the pair (SL,0 (t), RL,0 (t)) is only detectable, we can just obtain W (t) such that F (t) is a Hurwitz matrix.
v(t) ˙
(18)
+(L(t) − MC,1 (t)C(t))B(t)u(t).
˙ (L(t) + L(t)A(t))Σ{1} (t) =
ML,0 (t)z(t) + (MC,0 (t) − M˙ C,1 (t)
Theorem 4. When in the unique factorizations (21) and (20) the pair (PL,0 (t), NL,0 (t)) is detectable, there exists a matrix MC,12 (t) such that the Luenberger observer (9) with
MC,1 (t)y(t) ˙ + ML,0 (t)v(t) + MC,0 (t)y(t) +(L(t) − MC,1 (t)C(t))B(t)u(t).
To eliminate the derivative of y(t), we define z(t) = v(t) − MC,1 (t)y(t), or v(t) = z(t) + MC,1 (t)y(t).
(17)
F (t) = PL,0 (t) − MC,12 (t)NL,0 (t), G(t) = (L(t) − MC,1 (t)C(t))B(t), H(t) = MC,0 (t) + F (t)MC,1 (t) − M˙ C,1 (t), V (t) = MC,1 (t),
(23)
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4
In the time-invariant case theorem 3 can be read as
When condition (28) is fulfilled there exist matrices MC,0 , MC,1 , and ML,0 such that (26) is satisfied. It yields C CA rank L(sIn − A) C , CA = rank sL − (MC,1 CA + ML,0 L + MC,0 C) C , CA = rank sL − ML,0 L Im Σ . sIl − ML,0 = rank Im
Corollary 5. The time-invariant system
Condition (29) can be read
where MC,1 (t) =
PC1,1 (t) − MC,12 (t)NC,1 (t)
MC,0 (t)
MC,12 (t)
,
PC,0 (t) − MC,12 (t)NC,0 (t),
=
is a minimum asymptotic observer of (1) for system (2).
♦
IV. T HE TIME - INVARIANT CASE REVISITED
z(t) ˙ = F z(t) + Gu(t) + Hy(t), w(t) = z(t) + V y(t),
is a minimum asymptotic observer of the linear functional Lx(t) for the state space system x(t) ˙ = Ax(t) + Bu(t), y(t) = Cx(t),
∀s ∈ C, <e(s) ≥ 0, Im Σ = rank Σ, sIl − ML,0 rank Im
(24)
(25)
which means that ML,0 is a Hurwitz matrix. Thus the Darouach result appears as a particular case for linear timeinvariant systems of our main result stated in the linear time-varying case.
if and only if we can write LA =
MC,0
ML,0
MC,1
V. E XAMPLES Σ,
(26)
where ML,0 is a Hurwitz matrix and
C Σ = L . CA ♦ The necessary and sufficient existence condition proposed in [6] for a minimum asymptotic observer (24) can be written ∀s ∈ C, <e(s) ≥ 0,
C C = rank L , CA rank L(sIn − A) CA C L = rank CA . LA
(27)
With 0 Σ(t) = l1 (t) 0
··· l2 (t) ···
··· ··· 0
0 ln−1 (t) 1
1 , 0 −an (t)
we get ˙ L(t)A(t) + L(t) = l2 (t) + l˙1 (t) · ·· ln−1 (t) + l˙n−2 (t) l˙n−1 (t) α(t) , P where α(t) = − n−1 i=1 ai (t)li (t). According to the previous notations we denote X(t) = mC,0 (t) mL,0 (t) mC,1 (t) .
∀s ∈ C, <e(s) ≥ 0,
C = rank Σ. CA L(sIn − A)
(30)
Our purpose is to find the conditions in terms of L that ensure the existence of a first-order observer for v(t).
and, on the other hand, an asymptotic tracking condition is given by
rank
Let us consider the observable single-output system (2) with 0 · · · · · · 0 −a1 (t) .. 1 . −a2 (t) 0 .. .. .. .. , A(t) = . . . 0 . .. .. . . . . 1 0 . 0 ··· 0 1 −an (t) T B(t) = b1 (t) b2 (t) · · · bn (t) , C = 0 ··· ··· 0 1 , where n > 2, and the linear functional (1) defined by L = l1 (t) l2 (t) · · · ln−1 (t) 0 .
Condition (27) can be split in twofold. On the one hand, the existence of the minimum observer structure is ensured with Σ rank Σ = rank , (28) LA
A. Scalar linear functional example
(29)
An examination of factorization (15) leads to express the existence conditions for a first-order observer by the following relationships
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• • •
for i = 1 to n − 2 : li+1 (t) + l˙i (t) = li (t)mL,0 (t); l˙n−1 (t) = ln−1 (t)mL,0 (t)+mC,1 (t); α(t) = mC,0 (t) − an (t)mC,1 (t).
Let us notice that when, for every time t in R+ , ∃k ∈ [1, n − 1] lk (t) = 0, we get, by induction on i > k, for every time t in R+ , ∀j ∈ [k, n − 1] lj (t) = 0. Thus when, for every time t in R+ , l1 (t) = 0 we get L = 0 and it does not exist a first-order observer for the linear functional v(t). +
Assuming l1 (t) does not vanish in R and there exists some k > 2 such that ∀t ∈ R+ , lk (t) = 0, we get mC,0 (t) = α(t) and mC,1 (t) = 0. Therefore, the existence condition for a first-order minimum functional observer is, for every t, l2 (t) + l˙1 (t) l1 (t)
lk−1 (t) + l˙k−2 (t) , lk−2 (t) l˙k−1 (t) = mL,0 (t), lk−1 (t)
··· =
= = ˆ
with
B. Multiple linear functional example Let us consider the 0 1 A(t) = 0 0 C= 0
˙ L(t)A(t) + L(t) =
mC,1 (t)
=
α(t) + an (t)mC,1 (t), l˙n−1 (t) − ln−1 (t)mL,0 (t),
and the existence condition for a first-order minimum functional observer becomes, for every t, l2 (t) + l˙1 (t) l1 (t)
l3 (t) + l˙2 (t) = ··· l2 (t) ln−1 (t) + l˙n−2 (t) = mL,0 (t), ln−2 (t)
= = ˆ
with
t
mL,0 (τ )dτ = −∞.
lim
t→∞
t0
Anyway, assuming l1 (t) does not vanish in R+ and the existence condition for mL,0 (t) is fulfilled, it yields
=
L(t) − mC,1 (t)C(t), l1 (t) · · · ln−1 (t) ln−1 (t)mL,0 (t) − l˙n−1 (t)
g(t)
=
T (t)B(t),
h(t)
=
mC,0 (t) − mL,0 (t)mC,1 (t) − m ˙ C,1 (t), ˙ α(t) + an (t)ln−1 (t) − ln−1 (t)m2L,0 (t)
T (t)
=
=
+ (m ˙ C,1 (t) − an (t)ln−1 (t)) mL,0 (t) ¨ −ln−1 (t) − ln−1 (t)m ˙ L,0 (t). Parametrized with mL,0 (t), the asymptotic observer is given by z(t) ˙
=
w(t)
=
mL,0 (t)z(t) + g(t)u(t) + h(t)y(t), z(t) + l˙n−1 (t) − ln−1 (t)mL,0 (t) y(t).
These results allow to detect the scalar functional for a single-input single-output system that can be observed with a first-order system.
l˙1 (t) l2 (t)
0 l1 (t) Σ(t) = 0 0
Otherwise, when none of the coefficients li (t) vanish, we get =
0 0
,
(31)
the nonsingular matrix 0 1 0 0 . 0 0 1 −a4 (t)
As
t0
mC,0 (t)
0 0
(2) with b1 (t) b2 (t) , b3 (t) b4 (t)
with, for every t, l1 (t)l2 (t) 6= 0. Following the given procedure we get 0 0 l1 (t) 0 Σ(t) = 0 l2 (t) 0 0
mL,0 (τ )dτ = −∞.
lim
t→∞
observable single-output system 0 0 −a1 (t) 0 0 −a2 (t) , B(t) = 1 0 −a3 (t) 0 1 −a4 (t) 0 0 1 ,
and linear functional (1) defined by l1 (t) 0 L(t) = 0 l2 (t)
and
t
5
0 l˙2 (t)
0 0 l2 (t) 0
0 0 0 1
0 0
−a1 (t)l1 (t) −a2 (t)l2 (t)
,
1 0 , 0 −a4 (t)
we obtain −1 ˙ (L(t)A(t) + L(t))Σ (t) = −a1 (t)l1 (t) −a2 (t)l2 (t)
l˙1 (t) l1 (t) l2 (t) l1 (t)
0 l˙2 (t) l2 (t)
0 . 0
Consequently, matrices MC,0 (t), ML,0 (t), and MC,1 (t) can be read as −a1 (t)l1 (t) 0 MC,0 (t) = , MC,1 (t) = , −a2 (t)l2 (t) 0 l˙1 (t) 0 l1 (t) ML,0 (t) = l2 (t) l˙2 (t) . l1 (t) l2 (t) Let us suppose that the system z(t) ˙ = ML,0 (t)z(t) is uniformly asymptotically stable (for instance, if we have l1 (t) = exp(−t) and l2 (t) = exp(−2t), the stability conditions are fulfilled), we can implement, following (18), the second-order observer for L(t)x(t) z(t) ˙ = ML,0 (t)z(t) + MC,0 (t)y(t) + L(t)B(t)u(t). v(t) = z(t). VI. C ONCLUSION In the time-varying case, we propose existence conditions for a minimum functional linear observer. When it exists, we describe a design procedure. Our algorithm uses two unique matrix factorizations based on linearly independent rows of a time-varying matrix. With respect to other procedures [18] our design method is carried out without needing to solve a differential Sylvester equation. Moreover, the proposed algorithm points out whether we can fix
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at any desired rate the convergence of the observation error. In addition, if there exists a Lyapunov transform P (t) [5] such that F (t) = P (t)ΦP (t)−1 + P˙ (t)P (t)−1 where Φ is a constant Hurwitz matrix, this can be performed by means of eigenvalues of Φ. The main result can be considered as an extension to the time-varying case of a necessary and sufficient existence condition of a minimum functional observer established previously in the time-invariant case [6]. Since such a minimum functional observer may not exist for a given linear functional, the presented results, as underlined in [9] for timeinvariant systems, can be the basis of an algorithm for the design of a minimal-order observer. This development will be the subject of a next paper. R EFERENCES [1] Aldeen, M., Trinh, H., Reduced-Order Linear Functional Observers for Linear Systems, IEE proc. control theory appl., vol. 146, n. 5, pp. 399– 405, 1999. [2] Ben-Israel, A., Greville, T.N.E., Generalized Inverses : Theory and Applications, John Wiley & Sons, 1974. [3] Chai, W., Loh, N.K., Hu, H., Observer Design for Time-Varying Systems, Int. J. Systems Sci., vol. 22, n. 7, pp. 1177–1196, 1991. [4] Ciampa, M., Volpi, A., A Note on Smooth Matrices of Constant Rank, Rend. Istit. Mat. Univ. Trieste, vol. XXXVII, pp. 155–170, 2005. [5] Chen, C.-T., Linear System Theory and Design, Holt, Rinehart and Winston, 2nd ed., 1984. [6] Darouach, M., Existence and Design of Functional Observers for Linear Systems, IEEE Trans. Aut. Control, AC–45, n. 5, pp. 940-943, 2000. [7] Dieci, L., Eirola, T., On Smooth Decompositions of Matrices, SIAM J. Matrix Anal. Appl., vol. 20, n. 3, pp. 800–819, 1999. [8] D’Angelo, H., Linear Time-Varying Systems, Allyn and Bacon, 1970. [9] Fernando, T., Trinh, H., Jennings, L., Functional Observability and the Design of Minimum Order Linear Functional Observers for Linear Systems, IEEE Trans. Aut. Control, AC-55, n. 5, pp. 1268–1273, 2010. [10] Lovass-Nagy, V., Miller, R.J., Powers, D.L., An Introduction to the Application of the Simplest Matrix-Generalized Inverse in Systems Science, IEEE Trans. Circuits and Systems, vol. CAS–25, n. 9, pp. 766– 771, 1978. [11] Lovass-Nagy, V., Miller, R.J., Mukundan, R., On the Application of Matrix Generalized Inverses to the Design of Observers for TimeVarying and Time-Invariant Linear Systems, IEEE Trans. Aut. Control, vol. AC-25, n. 6, pp. 1213–1218, 1980. [12] Luenberger, D.G., Observers for Multivariable Systems, IEEE Trans. Aut. Control, AC-11, n. 2, pp. 190–197, 1966. [13] Nguyen, C., Lee, T.N., Design of a State Estimator for a Class of TimeVarying Multivariable Systems, IEEE Trans. Aut. Control, AC–30, n. 2, pp. 179–182, 1985. [14] O’Reilly, J., Observers for Linear Systems, Academic Press, 1983. [15] Roman, J.R., Bullock, T.E., Design of Minimal Orders Stable Observers for Linear Functions of the State via Realization Theory, IEEE Trans. Aut. Control, vol. AC–20, n. 5, pp. 613–22, 1975. [16] Rotella, F., Zambettakis, I., Minimal Single Linear Functional Observers for Linear Systems, Automatica, vol. 47, n. 1, pp. 164–169, 2011. [17] Rugh, W.J., Linear System Theory, Prentice Hall, 2nd ed., 1996. [18] Shafai, B., Carroll, R.L., Minimal-Order Observer Designs for Linear Time-Varying Multivariable Systems, IEEE Trans. Aut. Control, vol. AC31, n. 8, pp. 757–761, 1986. [19] Shafai, B., Design of Single-Functional Observers for Linear TimeVarying Multivariable Systems, Int. J. Systems Sci., vol. 20, n. 11, pp. 2227–2239, 1989. [20] Silverman, L.M., Transformation of Time-variable Systems to Canonical (phase-variable) Form, IEEE Trans. Aut. Control, AC-11, pp. 303–306, 1966. [21] Silverman, L.M., Meadows, H.E., Controllability and Observability in Time-variable Linear Systems, J. SIAM. Control, vol. 5, n. 1, pp. 64–73, 1967. [22] Sirisena, H.R., Minimal Order Observers for Linear Functions of a State Vector, Int. Jour. of Control, vol. 29, n.2, pp. 235–54, 1979. [23] Tian, Y., Floquet, T., Perruquetti, W., Fast State Estimation in Linear Time-Varying Systems : an Algebraic Approach, Proc. CDC Mexico, 2008.
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[24] Trinh, H., Fernando, T., Functional Observers for Dynamical Systems, Springer, 2012. [25] Trumpf, J., Observers for Linear Time-Varying Systems, Linear Algebra and Appl., vol. 425, pp. 303–312, 2007. [26] Tsui C.C., What is the Minimum Function Observer Order, J. Franklin Inst., vol. 335B, n. 4, pp. 623–628, 1998. [27] Tsui C.C., An Overview of the Applications and Solution of a Fundamental Matrix Equation Pair, Jour. of Franklin Institute, vol. 341, pp. 465–75, 2004. [28] Wolowich, W.A., On the State Estimation of Observable Systems, Proc. 1968 Joint Automat. Contr. Conf., pp. 210–222, 1968. [29] Yüksel, Y.Ö, Bongiorno, J.J., Observers for Linear Multivariable Systems with Applications, IEEE Trans. Aut. Control, AC-16, n. 6, pp. 603–613, 1971. [30] Zhang, Q., Adaptive Observer for Multiple-Input-Multiple-Output (MIMO) Linear Time-Varying Systems, IEEE Trans. Aut. Control, vol. AC-47, n. 3, pp. 525–529, 2002.
On functional observers for linear time-varying systems F. Rotella, I. Zambettakis Abstract—The paper deals with existence conditions of a functional observer for linear time-varying systems in the case where the order of the observer is equal to the number of observed variables. Constructive procedures for the design of such a linear functional observer are deduced from the existence conditions. As a specific feature, the proposed procedures do not require the solution of a differential Sylvester equation. Some examples illustrate the presented results. Index Terms—Linear time-varying system, functional observer, Luenberger observer.
I. I NTRODUCTION The interest to consider linear time-varying systems is twofold [8], [5], [17] : on the one hand as general models of linear behaviour for a plant, on the other hand as linearized models of non linear systems about a given trajectory. For state feedback control or fault diagnosis purposes the need of asymptotic observers of a given linear functional is of primary importance. Therefore, we consider the problem of observing a linear functional v(t) = L(t)x(t),
(1)
where, for every time t in R+ , L(t) is a constant full row rank (l×n) differentiable matrix, and x(t) is the n-dimensional state vector of the state space system x(t) ˙ = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t),
(2)
where u(t) is the p-dimensional control, and y(t) is the mdimensional output. For every t in R+ , A(t), B(t), and C(t) are known matrices of appropriate dimensions. To avoid tedious counts and distracting lists of differentiability requirements, we assume every time-varying matrices are such that all derivatives that appear are continuous for all t. Without loss of generality and in order to C(t) avoid useless dynamic parts in the observer, we suppose L(t) has full row rank for all t. Indeed, the rows of an arbitrary given L(t) which are linearly dependant of the rows of C(t) induce obvious estimation of the corresponding components of v(t) from the available informations. The observability matrix of (2) is defined by Ω(t) =
Ω0 (t) Ω1 (t) .. . Ωn−1 (t)
,
˙ j−1 (t) for where Ω0 (t) = C(t), and Ωj (t) = Ωj−1 (t)A(t) + Ω j = 1, 2, ...n − 1. System (2), or shortly (A(t), C(t)), is completely observable if rank (Ω(t)) = n for some t in R+ . It is uniformly observable if rank (Ω(t)) = n for every t in R+ [21], [29]. Following [8], if r states of (2) are not observable there exists a transformation which induces the following partitions for A(t) and C(t) A11 (t) A12 (t) A(t) = , 0(n−r)×r A22 (t) 0m×r C2 (t) , C(t) = F. Rotella and I. Zambettakis are with the Laboratoire de Génie de Production, Ecole Nationale d’Ingénieurs de Tarbes, France, [email protected], [email protected].
1
where (A22 (t), C2 (t)) is completely observable. Then the system is detectable when A11 (t) is a Hurwitz matrix. A matrix F (t) is said to be a Hurwitz (convergent in [25]) matrix if every solution x(t, t0 , x0 ) of the differential system x(t) ˙ = F (t)x(t), x(t0 ) = x0 , is such that limt→∞ x(t, t0 , x0 ) = 0 for every t0 and x0 . Since Luenberger’s seminal work [12] a significant amount of research is devoted to the problem of observing a linear functional in a time-invariant setting, see for instance [14], [26], [1], [24] and the references therein. Whereas, unlike the time-invariant counterpart, since [28] there are few papers dealing with the observer design for time-varying systems. The main part of the proposed developments are limited to the case of state observers design (see [18], [3], [17], [30], [23] and the references therein). For the functional observer case we can mention [18] for the design of a single-functional observer while [25] gives a necessary and sufficient existence condition of a linear functional observer. Now, it is well known that the observation of v(t) can be carried out with the design of the Luenberger observer z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = P (t)z(t) + V (t)y(t),
(3)
where z(t) is a q-dimensional state vector. The time-varying matrices F (t), G(t), H(t), P (t) and V (t) must be determined such that (3) is an asymptotic observer of (1) for the system (2). Namely, they have to ensure lim (v(t) − w(t)) = 0. t→∞
It has been shown in [25] (theorem 3.9) that the completely observable system z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = P (t)z(t),
(4)
is an asymptotic observer for system (2) and linear functional M (t)x(t) if and only if there exists a continuously differentiable matrix T (t) such that G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t), M (t) = P (t)T (t),
(5) (6)
and F (t) is a Hurwitz matrix. So, for any matrix V (t), when we choose M (t) = L(t) − V (t)C(t), the theorem stated in [25] can be read as Theorem 1. The completely observable system (3) is an asymptotic observer of linear functional (1) for system (2) if and only if there exists a continuously differentiable solution T (t) of equations G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t), L(t) = P (t)T (t) + V (t)C(t), and F (t) is a Hurwitz matrix.
(7) (8) ♦
Due to linear independence of C(t) and L(t), we know that, as in the time-invariant case ([15], [22]), l is a lower bound for the order of observer (4). In functional observer literature the words minimal and minimum are both used to denote a lower order stable observer (see for instance [18], [26]), nevertheless minimal is more frequently used. Moreover all these uses are mainly subject to the restriction that the observer eigenvalues are freely assignable. In
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[22] it has been underlined that minimality must be considered with respect to two different problems depending whether the observer poles are arbitrarily specified at the outset (the fixed poles observer problem), or are permitted to lie anywhere on the left half plane (the stable observer problem). So, in this paper, we call “minimum observer” the stable observer for which the order is equal to the size of the functional to be observed, while we imply that “minimal observer” characterizes the other cases. Therefore our main motivation is to analyze the existence conditions and to propose the design of a minimum asymptotic observer z(t) ˙ = F (t)z(t) + G(t)u(t) + H(t)y(t), w(t) = z(t) + V (t)y(t),
(9)
where w(t) ∈ Rl is an asymptotic estimate for v(t). When this linear functional observer exists, its order, namely l according to our notations, is much less than the order (n − m) of a reduced-order state observer. As a consequence, observing a linear function of the state may afford a significant reduction in observer order compared to observing the entire state vector. A second motivation of our purpose is to circumvent the determination of T (t) as solution of the differential equation (7). The material in this paper is organized as follows. The first section deals with the analysis of existence conditions of a minimum observer. The second section is concerned with the proposal of design procedures. In the third section the linear time-invariant case is revisited. Indeed, the existence conditions of Darouach [6] for time-invariant systems are based on Hautus-type criteria and can’t be directly applied in the time-varying case. Theorem 3 applied to linear time-invariant systems leads to the Darouach conditions which appear then as a particular case of ours. Thus, the extension of the Darouach conditions to linear time-varying systems is an additionnal motivation of our paper. Finally, two generic examples are detailed. II. E XISTENCE CONDITIONS The application of theorem 1 leads to the following result. Corollary 2. The asymptotic observer (9) exists if and only if there exists a (l × n) matrix T (t) such that G(t) = T (t)B(t), T (t)A(t) − F (t)T (t) + T˙ (t) = H(t)C(t),
(10)
L(t) = T (t) + V (t)C(t),
(11)
F (t) is a Hurwitz matrix, ♦
for all t.
Theorem 1 and corollary 2 require the solution T (t) of a differential Sylvester equation in which F (t) is unknown as well as the initial conditions of T (t). In order to overcome this difficulty, we propose to eliminate T (t) in conditions of corollary 2. From (11) we get T (t) = L(t) − V (t)C(t), so from (10) we obtain L(t)A(t) − V (t)C(t)A(t) − F (t)L(t) ˙ ˙ + F (t)V (t)C(t) + L(t) − V˙ (t)C(t) − V (t)C(t) = H(t)C(t).
(12)
Denoting
(12) can be written ˙ L(t) + L(t)A(t) = H(t) + V˙ (t) − F (t)V (t)
(13)
F (t)
V (t)
Σ(t).
(14)
Theorem 3. The minimum asymptotic observer (9) of linear functional (1) for system (2) exists if and only if the factorization of ˙ L(t) + L(t)A(t) ˙ L(t) + L(t)A(t) =
MC,0 (t)
ML,0 (t)
MC,1 (t)
Σ(t)
(15)
is such that MC,0 (t), ML,0 (t), and MC,1 (t) have m, l, and m columns respectively, the derivative of MC,1 (t) exists, and ML,0 (t) is a Hurwitz matrix. ♦ Proof: The if part follows from (14). The only if part is established by looking for F (t), H(t), V (t) such that in (15) : MC,0 (t)
=
H(t) + V˙ (t) − F (t)V (t),
ML,0 (t)
=
F (t),
MC,1 (t)
=
V (t).
It is obvious that, if MC,1 (t) has a derivative, the solution of this system is F (t)
=
ML,0 (t),
V (t)
=
MC,1 (t),
H(t)
=
MC,0 (t) + ML,0 (t)MC,1 (t) − M˙ C,1 (t).
Let T (t) = L(t) − MC,1 (t)C(t). This matrix fulfills the differential equation (10) and since F (t) is a Hurwitz matrix, this ends the proof. Let us underline here that the existence conditions of theorem 3 do not require the determination of T (t). Nevertheless we can get matrix T (t) as a consequence of factorization (15), without solving equation (10) in the sense of the solution of a differential equation. This standpoint has already been used in [16] to design a minimal order single functional stable observer for linear time-invariant systems. III. M INIMUM ORDER OBSERVER DESIGNS From theorem 3, two observer design methods are described in the following. However, the last step of these algorithms needs uniform observability to be assumed. A. Generalized inverse based design The first design method lays on the solution of the linear equation ˙ L(t) + L(t)A(t) = X(t)Σ(t),
(16)
for all t. As the solution X(t) exists if and only if Σ(t) rank Σ(t) = rank , ˙ L(t) + L(t)A(t) equation (16) can be solved by means of time-varying generalized inverses [11]. For example, a generalized inverse [2], [10], Σ{1} (t), for Σ(t) can be obtained from the time-varying singular value factorization of Σ(t) or from its QR factorization [7], [4]. The solution set of (16) can then be expressed a X(t)
C(t) , L(t) Σ(t) = ˙ C(t) + C(t)A(t)
2
=
=
˙ (L(t) + L(t)A(t))Σ{1} (t)
MC,0 (t)
ML,0 (t)
MC,1 (t)
+W (t)(In − Σ(t)Σ{1} (t)),
,
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where W (t) is an arbitrary (l × n) matrix. When for every t rank Σ(t) = 2m + l, the solution set is reduced to the unique ˙ element (L(t) + L(t)A(t))Σ{1} (t). Otherwise, a first design can be considered with the partitions
3
Then, z(t) ˙
=
+ML,0 (t)MC,1 (t))y(t)
SC,0 (t)
SL,0 (t)
SC,1 (t)
and
In − Σ(t)Σ{1} (t) = RC,0 (t)
RL,0 (t)
RC,1 (t)
,
where the number of columns is l for SL,0 (t) and for RL,0 (t), while SC,0 (t), RC,0 (t), SC,1 (t), and RC,1 (t) have m columns. It yields F (t)
=
ML,0 (t) = SL,0 (t) + W (t)RL,0 (t),
V (t)
=
H(t)
=
MC,1 (t) = SC,1 (t) + W (t)RC,1 (t), MC,0 (t) + ML,0 (t)MC,1 (t) − M˙ C,1 (t),
G(t)
=
(L(t) − MC,1 (t)C(t)) B(t).
1) Σ(t) has full row rank : Namely, we are in the case where rank Σ(t) = 2m + l for every t, except for some isolated instants τ . When Σ(t) < 2m + l at some instants ti , we have to split the time domain in successive time domains bounded by these isolated values ti of t, where the procedure can be applied. Then, matrices MC,0 (t), ML,0 (t), and MC,1 (t) are unique. Moreover, a derivation of (1) yields
=
= m + l + m1
In this case, in factorization (15), MC,0 (t), ML,0 (t), and MC,1 (t) are not unique. However, we can write C˙ 2 (t) + C2 (t)A(t) = NC,0 (t)C(t)+ NL,0 (t)L(t) + NC,1 (t) C˙1 (t) + C1 (t)A(t) ,
(20)
where matrices NC,0 (t), NL,0 (t), and NC,1 (t) are unique. Moreover, we have ˙ ˙ L(t) + L(t)A(t) = MC,1 (t) C(t) + C(t)A(t) + ML,0 (t)L(t) + MC,0 (t)C(t) = MC,11 (t) C˙1 (t) + C1 (t)A(t) + ML,0 (t)L(t) + MC,0 (t)C(t) + MC,12 (t) C˙2 (t) + C2 (t)A(t) = PC1,1 (t) C˙1 (t) + C1 (t)A(t) + PL,0 (t)L(t) + PC,0 (t)C(t),
(21)
where matrices PC,0 (t), PL,0 (t) and PC1,1 (t) are unique and given by PC1,1 (t) = MC,11 (t) + MC,12 (t)NC,1 (t), PL,0 (t) = ML,0 (t) + MC,12 (t)NL,0 (t),
(22)
PC,0 (t) = MC,0 (t) + MC,12 (t)NC,0 (t).
˙ (L(t) + L(t)A(t))x(t) + L(t)B(t)u(t), ˙ MC,1 (t) C(t) + C(t)A(t) x(t)
As previously, providing that the pair (PL,0 (t), NL,0 (t)) is detectable, the stability of the observer can be ensured.
+ML,0 (t)L(t)x(t) + MC,0 (t)C(t)x(t)
Nevertheless, when the pair (PL,0 (t), NL,0 (t)) is uniformly observable [20], [13], [3], we can obtain MC,12 (t) to yield uniform exponential stability at any desired rate for the observation error system η(t) ˙ = ML,0 (t)η(t) where
+L(t)B(t)u(t), ˙ MC,1 (t)C(t)x(t) + MC,1 (t)C(t)x(t) ˙ −MC,1 (t)C(t)B(t)u(t) + L(t)B(t)u(t)
ML,0 (t) = PL,0 (t) − MC,12 (t)NL,0 (t).
+ML,0 (t)v(t) + MC,0 (t)y(t), =
where C1 (t)(m1 × n) and C2 (t)(m2 × n) verify
= rank Σ(t).
The previous design procedure needs to choose a generalized inverse for Σ(t). To avoid computational burden with respect to this choice, we propose to extend the design procedure used recently to determine the minimal order single functional stable observer for linear timeinvariant systems [16]. When factorization (15) exists, we distinguish two cases.
=
2) rank Σ < 2m + l: Let us introduce, up to a possible permutation in the outputs, the partition C1 (t) , (19) C= C2 (t)
C(t) L(t) rank C˙1 (t) + C1 (t)A(t)
B. Direct design
=
Since factorization (15) is unique, when ML,0 (t) is a Hurwitz matrix the minimum stable observer problem is solved. However, the rate of convergence for the uniform exponential stability of the observation error system η(t) ˙ = ML,0 (t)η(t) cannot be arbitrarily fixed.
When the pair (SL,0 (t), RL,0 (t)) is uniformly observable, W (t) can be chosen, via eigenvalues assignement techniques ([20], [13], [3]), to yield uniform exponential stability at any desired rate[5], [17] for the observation error system η(t) ˙ = F (t)η(t) where F (t) = SL,0 (t) + W (t)RL,0 (t). When the pair (SL,0 (t), RL,0 (t)) is only detectable, we can just obtain W (t) such that F (t) is a Hurwitz matrix.
v(t) ˙
(18)
+(L(t) − MC,1 (t)C(t))B(t)u(t).
˙ (L(t) + L(t)A(t))Σ{1} (t) =
ML,0 (t)z(t) + (MC,0 (t) − M˙ C,1 (t)
Theorem 4. When in the unique factorizations (21) and (20) the pair (PL,0 (t), NL,0 (t)) is detectable, there exists a matrix MC,12 (t) such that the Luenberger observer (9) with
MC,1 (t)y(t) ˙ + ML,0 (t)v(t) + MC,0 (t)y(t) +(L(t) − MC,1 (t)C(t))B(t)u(t).
To eliminate the derivative of y(t), we define z(t) = v(t) − MC,1 (t)y(t), or v(t) = z(t) + MC,1 (t)y(t).
(17)
F (t) = PL,0 (t) − MC,12 (t)NL,0 (t), G(t) = (L(t) − MC,1 (t)C(t))B(t), H(t) = MC,0 (t) + F (t)MC,1 (t) − M˙ C,1 (t), V (t) = MC,1 (t),
(23)
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In the time-invariant case theorem 3 can be read as
When condition (28) is fulfilled there exist matrices MC,0 , MC,1 , and ML,0 such that (26) is satisfied. It yields C CA rank L(sIn − A) C , CA = rank sL − (MC,1 CA + ML,0 L + MC,0 C) C , CA = rank sL − ML,0 L Im Σ . sIl − ML,0 = rank Im
Corollary 5. The time-invariant system
Condition (29) can be read
where MC,1 (t) =
PC1,1 (t) − MC,12 (t)NC,1 (t)
MC,0 (t)
MC,12 (t)
,
PC,0 (t) − MC,12 (t)NC,0 (t),
=
is a minimum asymptotic observer of (1) for system (2).
♦
IV. T HE TIME - INVARIANT CASE REVISITED
z(t) ˙ = F z(t) + Gu(t) + Hy(t), w(t) = z(t) + V y(t),
is a minimum asymptotic observer of the linear functional Lx(t) for the state space system x(t) ˙ = Ax(t) + Bu(t), y(t) = Cx(t),
∀s ∈ C, <e(s) ≥ 0, Im Σ = rank Σ, sIl − ML,0 rank Im
(24)
(25)
which means that ML,0 is a Hurwitz matrix. Thus the Darouach result appears as a particular case for linear timeinvariant systems of our main result stated in the linear time-varying case.
if and only if we can write LA =
MC,0
ML,0
MC,1
V. E XAMPLES Σ,
(26)
where ML,0 is a Hurwitz matrix and
C Σ = L . CA ♦ The necessary and sufficient existence condition proposed in [6] for a minimum asymptotic observer (24) can be written ∀s ∈ C, <e(s) ≥ 0,
C C = rank L , CA rank L(sIn − A) CA C L = rank CA . LA
(27)
With 0 Σ(t) = l1 (t) 0
··· l2 (t) ···
··· ··· 0
0 ln−1 (t) 1
1 , 0 −an (t)
we get ˙ L(t)A(t) + L(t) = l2 (t) + l˙1 (t) · ·· ln−1 (t) + l˙n−2 (t) l˙n−1 (t) α(t) , P where α(t) = − n−1 i=1 ai (t)li (t). According to the previous notations we denote X(t) = mC,0 (t) mL,0 (t) mC,1 (t) .
∀s ∈ C, <e(s) ≥ 0,
C = rank Σ. CA L(sIn − A)
(30)
Our purpose is to find the conditions in terms of L that ensure the existence of a first-order observer for v(t).
and, on the other hand, an asymptotic tracking condition is given by
rank
Let us consider the observable single-output system (2) with 0 · · · · · · 0 −a1 (t) .. 1 . −a2 (t) 0 .. .. .. .. , A(t) = . . . 0 . .. .. . . . . 1 0 . 0 ··· 0 1 −an (t) T B(t) = b1 (t) b2 (t) · · · bn (t) , C = 0 ··· ··· 0 1 , where n > 2, and the linear functional (1) defined by L = l1 (t) l2 (t) · · · ln−1 (t) 0 .
Condition (27) can be split in twofold. On the one hand, the existence of the minimum observer structure is ensured with Σ rank Σ = rank , (28) LA
A. Scalar linear functional example
(29)
An examination of factorization (15) leads to express the existence conditions for a first-order observer by the following relationships
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• • •
for i = 1 to n − 2 : li+1 (t) + l˙i (t) = li (t)mL,0 (t); l˙n−1 (t) = ln−1 (t)mL,0 (t)+mC,1 (t); α(t) = mC,0 (t) − an (t)mC,1 (t).
Let us notice that when, for every time t in R+ , ∃k ∈ [1, n − 1] lk (t) = 0, we get, by induction on i > k, for every time t in R+ , ∀j ∈ [k, n − 1] lj (t) = 0. Thus when, for every time t in R+ , l1 (t) = 0 we get L = 0 and it does not exist a first-order observer for the linear functional v(t). +
Assuming l1 (t) does not vanish in R and there exists some k > 2 such that ∀t ∈ R+ , lk (t) = 0, we get mC,0 (t) = α(t) and mC,1 (t) = 0. Therefore, the existence condition for a first-order minimum functional observer is, for every t, l2 (t) + l˙1 (t) l1 (t)
lk−1 (t) + l˙k−2 (t) , lk−2 (t) l˙k−1 (t) = mL,0 (t), lk−1 (t)
··· =
= = ˆ
with
B. Multiple linear functional example Let us consider the 0 1 A(t) = 0 0 C= 0
˙ L(t)A(t) + L(t) =
mC,1 (t)
=
α(t) + an (t)mC,1 (t), l˙n−1 (t) − ln−1 (t)mL,0 (t),
and the existence condition for a first-order minimum functional observer becomes, for every t, l2 (t) + l˙1 (t) l1 (t)
l3 (t) + l˙2 (t) = ··· l2 (t) ln−1 (t) + l˙n−2 (t) = mL,0 (t), ln−2 (t)
= = ˆ
with
t
mL,0 (τ )dτ = −∞.
lim
t→∞
t0
Anyway, assuming l1 (t) does not vanish in R+ and the existence condition for mL,0 (t) is fulfilled, it yields
=
L(t) − mC,1 (t)C(t), l1 (t) · · · ln−1 (t) ln−1 (t)mL,0 (t) − l˙n−1 (t)
g(t)
=
T (t)B(t),
h(t)
=
mC,0 (t) − mL,0 (t)mC,1 (t) − m ˙ C,1 (t), ˙ α(t) + an (t)ln−1 (t) − ln−1 (t)m2L,0 (t)
T (t)
=
=
+ (m ˙ C,1 (t) − an (t)ln−1 (t)) mL,0 (t) ¨ −ln−1 (t) − ln−1 (t)m ˙ L,0 (t). Parametrized with mL,0 (t), the asymptotic observer is given by z(t) ˙
=
w(t)
=
mL,0 (t)z(t) + g(t)u(t) + h(t)y(t), z(t) + l˙n−1 (t) − ln−1 (t)mL,0 (t) y(t).
These results allow to detect the scalar functional for a single-input single-output system that can be observed with a first-order system.
l˙1 (t) l2 (t)
0 l1 (t) Σ(t) = 0 0
Otherwise, when none of the coefficients li (t) vanish, we get =
0 0
,
(31)
the nonsingular matrix 0 1 0 0 . 0 0 1 −a4 (t)
As
t0
mC,0 (t)
0 0
(2) with b1 (t) b2 (t) , b3 (t) b4 (t)
with, for every t, l1 (t)l2 (t) 6= 0. Following the given procedure we get 0 0 l1 (t) 0 Σ(t) = 0 l2 (t) 0 0
mL,0 (τ )dτ = −∞.
lim
t→∞
observable single-output system 0 0 −a1 (t) 0 0 −a2 (t) , B(t) = 1 0 −a3 (t) 0 1 −a4 (t) 0 0 1 ,
and linear functional (1) defined by l1 (t) 0 L(t) = 0 l2 (t)
and
t
5
0 l˙2 (t)
0 0 l2 (t) 0
0 0 0 1
0 0
−a1 (t)l1 (t) −a2 (t)l2 (t)
,
1 0 , 0 −a4 (t)
we obtain −1 ˙ (L(t)A(t) + L(t))Σ (t) = −a1 (t)l1 (t) −a2 (t)l2 (t)
l˙1 (t) l1 (t) l2 (t) l1 (t)
0 l˙2 (t) l2 (t)
0 . 0
Consequently, matrices MC,0 (t), ML,0 (t), and MC,1 (t) can be read as −a1 (t)l1 (t) 0 MC,0 (t) = , MC,1 (t) = , −a2 (t)l2 (t) 0 l˙1 (t) 0 l1 (t) ML,0 (t) = l2 (t) l˙2 (t) . l1 (t) l2 (t) Let us suppose that the system z(t) ˙ = ML,0 (t)z(t) is uniformly asymptotically stable (for instance, if we have l1 (t) = exp(−t) and l2 (t) = exp(−2t), the stability conditions are fulfilled), we can implement, following (18), the second-order observer for L(t)x(t) z(t) ˙ = ML,0 (t)z(t) + MC,0 (t)y(t) + L(t)B(t)u(t). v(t) = z(t). VI. C ONCLUSION In the time-varying case, we propose existence conditions for a minimum functional linear observer. When it exists, we describe a design procedure. Our algorithm uses two unique matrix factorizations based on linearly independent rows of a time-varying matrix. With respect to other procedures [18] our design method is carried out without needing to solve a differential Sylvester equation. Moreover, the proposed algorithm points out whether we can fix
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at any desired rate the convergence of the observation error. In addition, if there exists a Lyapunov transform P (t) [5] such that F (t) = P (t)ΦP (t)−1 + P˙ (t)P (t)−1 where Φ is a constant Hurwitz matrix, this can be performed by means of eigenvalues of Φ. The main result can be considered as an extension to the time-varying case of a necessary and sufficient existence condition of a minimum functional observer established previously in the time-invariant case [6]. Since such a minimum functional observer may not exist for a given linear functional, the presented results, as underlined in [9] for timeinvariant systems, can be the basis of an algorithm for the design of a minimal-order observer. This development will be the subject of a next paper. R EFERENCES [1] Aldeen, M., Trinh, H., Reduced-Order Linear Functional Observers for Linear Systems, IEE proc. control theory appl., vol. 146, n. 5, pp. 399– 405, 1999. [2] Ben-Israel, A., Greville, T.N.E., Generalized Inverses : Theory and Applications, John Wiley & Sons, 1974. [3] Chai, W., Loh, N.K., Hu, H., Observer Design for Time-Varying Systems, Int. J. Systems Sci., vol. 22, n. 7, pp. 1177–1196, 1991. [4] Ciampa, M., Volpi, A., A Note on Smooth Matrices of Constant Rank, Rend. Istit. Mat. Univ. Trieste, vol. XXXVII, pp. 155–170, 2005. [5] Chen, C.-T., Linear System Theory and Design, Holt, Rinehart and Winston, 2nd ed., 1984. [6] Darouach, M., Existence and Design of Functional Observers for Linear Systems, IEEE Trans. Aut. Control, AC–45, n. 5, pp. 940-943, 2000. [7] Dieci, L., Eirola, T., On Smooth Decompositions of Matrices, SIAM J. Matrix Anal. Appl., vol. 20, n. 3, pp. 800–819, 1999. [8] D’Angelo, H., Linear Time-Varying Systems, Allyn and Bacon, 1970. [9] Fernando, T., Trinh, H., Jennings, L., Functional Observability and the Design of Minimum Order Linear Functional Observers for Linear Systems, IEEE Trans. Aut. Control, AC-55, n. 5, pp. 1268–1273, 2010. [10] Lovass-Nagy, V., Miller, R.J., Powers, D.L., An Introduction to the Application of the Simplest Matrix-Generalized Inverse in Systems Science, IEEE Trans. Circuits and Systems, vol. CAS–25, n. 9, pp. 766– 771, 1978. [11] Lovass-Nagy, V., Miller, R.J., Mukundan, R., On the Application of Matrix Generalized Inverses to the Design of Observers for TimeVarying and Time-Invariant Linear Systems, IEEE Trans. Aut. Control, vol. AC-25, n. 6, pp. 1213–1218, 1980. [12] Luenberger, D.G., Observers for Multivariable Systems, IEEE Trans. Aut. Control, AC-11, n. 2, pp. 190–197, 1966. [13] Nguyen, C., Lee, T.N., Design of a State Estimator for a Class of TimeVarying Multivariable Systems, IEEE Trans. Aut. Control, AC–30, n. 2, pp. 179–182, 1985. [14] O’Reilly, J., Observers for Linear Systems, Academic Press, 1983. [15] Roman, J.R., Bullock, T.E., Design of Minimal Orders Stable Observers for Linear Functions of the State via Realization Theory, IEEE Trans. Aut. Control, vol. AC–20, n. 5, pp. 613–22, 1975. [16] Rotella, F., Zambettakis, I., Minimal Single Linear Functional Observers for Linear Systems, Automatica, vol. 47, n. 1, pp. 164–169, 2011. [17] Rugh, W.J., Linear System Theory, Prentice Hall, 2nd ed., 1996. [18] Shafai, B., Carroll, R.L., Minimal-Order Observer Designs for Linear Time-Varying Multivariable Systems, IEEE Trans. Aut. Control, vol. AC31, n. 8, pp. 757–761, 1986. [19] Shafai, B., Design of Single-Functional Observers for Linear TimeVarying Multivariable Systems, Int. J. Systems Sci., vol. 20, n. 11, pp. 2227–2239, 1989. [20] Silverman, L.M., Transformation of Time-variable Systems to Canonical (phase-variable) Form, IEEE Trans. Aut. Control, AC-11, pp. 303–306, 1966. [21] Silverman, L.M., Meadows, H.E., Controllability and Observability in Time-variable Linear Systems, J. SIAM. Control, vol. 5, n. 1, pp. 64–73, 1967. [22] Sirisena, H.R., Minimal Order Observers for Linear Functions of a State Vector, Int. Jour. of Control, vol. 29, n.2, pp. 235–54, 1979. [23] Tian, Y., Floquet, T., Perruquetti, W., Fast State Estimation in Linear Time-Varying Systems : an Algebraic Approach, Proc. CDC Mexico, 2008.
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[24] Trinh, H., Fernando, T., Functional Observers for Dynamical Systems, Springer, 2012. [25] Trumpf, J., Observers for Linear Time-Varying Systems, Linear Algebra and Appl., vol. 425, pp. 303–312, 2007. [26] Tsui C.C., What is the Minimum Function Observer Order, J. Franklin Inst., vol. 335B, n. 4, pp. 623–628, 1998. [27] Tsui C.C., An Overview of the Applications and Solution of a Fundamental Matrix Equation Pair, Jour. of Franklin Institute, vol. 341, pp. 465–75, 2004. [28] Wolowich, W.A., On the State Estimation of Observable Systems, Proc. 1968 Joint Automat. Contr. Conf., pp. 210–222, 1968. [29] Yüksel, Y.Ö, Bongiorno, J.J., Observers for Linear Multivariable Systems with Applications, IEEE Trans. Aut. Control, AC-16, n. 6, pp. 603–613, 1971. [30] Zhang, Q., Adaptive Observer for Multiple-Input-Multiple-Output (MIMO) Linear Time-Varying Systems, IEEE Trans. Aut. Control, vol. AC-47, n. 3, pp. 525–529, 2002.
