Inverse problems for linear ill-posed equations with uncertain parameters

DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Supplement 2011
Website: www.aimSciences.org
pp. 1467–1476
INVERSE PROBLEMS FOR LINEAR ILL-POSED
DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH UNCERTAIN
PARAMETERS
Sergiy Zhuk
INRIA Paris-Rocquencourt
Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex, France
Abstract. This paper describes a minimax state estimation approach for linear differential-algebraic equations (DAEs) with uncertain parameters. The approach addresses continuous-time DAEs with non-stationary rectangular matrices and uncertain bounded deterministic input. An observation’s noise is
supposed to be random with zero mean and unknown bounded correlation
function. Main result is a Generalized Kalman Duality (GKD) principle, describing a dual control problem. Main consequence of the GKD is an optimal
minimax state estimation algorithm for DAEs with non-stationary rectangular
matrices. An algorithm is illustrated by a numerical example for 2D timevarying DAE with a singular matrix pencil.
1. Introduction. In this paper we focus on an inverse problem for a linear Differential-Algebraic Equation (DAE) in the form
d(F x)
= C(t)x(t) + f (t)
(1)
dt
where F is a rectangular matrix. We note that the case det(sF − C) 6= 0 with
C(t) ≡ C is well-understood: by a non-singular linear transformation the matrix
pencil sF − C can be converted into a Kronecker canonical form. Accordingly,
changing the basis in the state space of (1) and differentiating exactly d times (d
is an index of the pencil sF − C), one can reduce (1) to an equivalent Ordinary
Differential Equation (ODE), provided f (·) is sufficiently smooth and meets some
algebraic constraints. The latter ODE has a unique solution, provided the initial
condition belongs to some subspace. Details of the reduction process are presented
in [7]. We will refer such DAEs as causal.
One reason to study non-causal DAEs comes from the state estimation theory.
Namely, applying the optimal linear proportional feedback f = Kx to (1) with a
regular pencil (det(sF − C) 6= 0), one could arrive [19] to the system with a singular
pencil det(sF − C − BK) ≡ 0. On the other hand, due to behavioral approach[10]
dx
= C(t)x(t) + f (t) as a part of the state
one can treat an input f of the ODE
dt
so that the extended state (x, f ) verifies DAE (1) with F = [ I 0 ] and C(t) =
[ C(t) I ]. Using such a point of view one can convert parameter estimation problem
2000 Mathematics Subject Classification. Primary: 34K32,49N45x; Secondary: 49N30 .
Key words and phrases. Minimax, State estimation, Differential-algebraic equations, EulerLagrange equations, Descriptor systems.
This work was carried out during the tenure of an ERCIM “Alain Bensoussan” Fellowship
Programme.
1467
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SERGIY ZHUK
for ODE into state estimation problem for non-causal DAE (see for instance [5]
for the estimation of the unknown input). In addition, state estimation methods
for ODE, based on the Kalman filtering approach, cannot be applied directly in
the practical inverse problems due to the computational burden. For instance,
state vector in meteorological applications usually has over 106 components, which
makes it impossible to propagate a Riccati gain without appropriate reduction. A
rigorous reduced-order state estimation (allowing to control the error, introduced
by reduction) can be done by means of the state estimation framework for DAEs
(main idea and its application to an operational air quality model could be found
in [17]).
However, there are difficulties, connected with state estimation problems for
DAEs. In particular, it is not clear when the seminal Kalman duality holds, that is
the observation problem is equal to the dual control problem.
In addition, for the case of variable C(t) there is a lack of optimality criteria for an
LQ-control problems with DAE constraints. Sufficient optimality conditions in the
form of the descriptor Euler-Lagrange equation were formulated in [13]. The latter
can be solved numerically only for index-one causal DAEs. Existence of the optimal
feedback solution was studied in [12]. The authors construct the feedback control
by means of a Descriptor Riccati Equation. However, the authors point out that
“numerically qualified versions of the Riccati DAE system” require further research.
The research, presented in this paper, resolves the above problems for a class of
non-causal DAEs. In what follows, we apply a minimax estimation approach, developed in[21] for abstract linear equations in a Hilbert space, for non-causal DAE
(1) with F, C(t) ∈ Rm×n . We will be interested in the following inverse problem:
given observations y(t), t ∈ [t0 , T ] of x(·), to reconstruct1 F x(T ), provided x(·) is
a weak solution to (1), f (·) is some unknown squared-integrable function, which
belongs to a given bounded set G. We will also assume that y(t) is corrupted
by a random noise η(·) with zero mean Eη(t) = 0 and the correlation function
(t, s) 7→ Rη (t, s) := EΨ(t)Ψ0 (s) belongs to a given bounded set W .
Note that in order to reconstruct F x(T ) it is enough2 to reconstruct a linear function `(x) := h`, F x(T )i for any ` ∈ Rm . In what follows, we will be looking for
the estimate of `(x) among all linear functions of observations u(y) by means of
the minimax state estimation approach[2, 16, 15, 4, 18]. The main idea is to find
the estimate u with minimal worst-case error. Namely, for the fixed u the estimation error is E[`(x) − u(y)]2 . Since E[`(x) − u(y)]2 depends on realizations
of f and Rη , it follows that, for a fixed u, the worst-case estimation error is
the maximum of E[`(x) − u(y)]2 over all possible realizations of f , Rη and3 x:
σ(T, `, u) := supf ∈G,Rη ∈W,x {E[`(x) − u(y)]2 }. Finally, û is called a minimax meansquared estimate of `(x), iff û has a minimal worst-case error (so called minimax
error), that is σ(T, `, û) = inf u σ(T, `, u).
The minimax estimate for (1) was constructed in [18], provided F = I. The case
of deterministic η(·) and F = I was addressed in [2, 16, 15, 4], where the minimax
estimation of x(T ) is shown to be the Tchebysheff center of the ODEs reachable set,
1 Due
to the definition given at the beginning of Section 2 just F x(·) is supposed to be absolutely
continuous. Hence x(T ) is not necessary defined at the time T . Thus we are searching for the
estimate of F x(T ).
2 Having the estimate of `(x) for any ` ∈ Rm , one can set ` := e = (0, . . . , 1, . . . 0)T in order to
i
reconstruct i-th component of F x(T ).
3 As DAE is non-causal, it could have more than one solution x, corresponding to the given f
through (1).
INVERSE PROBLEMS FOR DAES
1469
consistent with observations and uncertainty description. The same result can be
proved by the method of this paper for F 6= I, provided η is deterministic and (η, f )
belongs to some ellipsoid. Main distinctive features of the problem, described above
are: 1) as F ∈ Rm×n it follows that DAE (2) cannot be converted into ODE in the
general case; the latter makes it impossible to directly apply classical minimax state
estimation methods [18, 2, 16, 15, 4] as it was done in [8, 5] for causal DAEs; 2) we
consider a quite general noise η, which can be a realization of any random process
Ψ (not necessary Gaussian as in [8]) with bounded correlation function.
The major contribution of this paper is a Generalized Kalman Duality (GKD)
principle for non-causal time-dependent DAEs (Theorem 2.2). It states that the observation problem for DAEs is equal to some control problem with DAE constraints.
The latter generalizes seminal Kalman duality for ODE and reflects the procedure
of deriving the minimax estimate: to compute the minimax error inf u σ(T, `, u) one
needs to solve a control problem for the “adjoint” DAE. GKD was previously applied in [22] in order to construct the minimax estimate for non-causal DAEs with
discrete time.
Following the GKD we derive a dual control problem for DAEs (Proposition 1)
and we describe the minimum point as a solution of the Euler-Lagrange system,
provided the matrices of DAE have ”some regularity” (Proposition 1). Finally, an
ODE, describing the minimax estimate (Corollary 1), is derived. The dimension of
the state space of this ODE is equal to the rank of F , providing a reduced order
estimate. In fact, the minimax estimate gives a reconstruction of the projection
of F x(T ) onto MOS L(T ) only. As dimL(T ) ≤ rankF , the minimax estimate
introduces an additional degree of reduction. We finish with a numerical example.
Notation: Eη denotes the mean of the random element η; int G denotes the
interior of G; f (·) or f denotes a function (as an element of some functional space);
f (t) denotes a value of function f at t; Rn denotes the n-dimensional Euclidean
space over real field; L2 (t0 , T, Rm ) denotes a space of square-integrable functions
with values in Rm ; H12 (t0 , T, Rm ) denotes a space of absolutely continuous functions
with L2 -derivative and values in Rm ; the prime 0 denotes the operation of taking
the adjoint: L0 denotes adjoint operator, F 0 denotes the transposed matrix; c(G, ·)
denotes the support function of some set G; h·, ·i denotes the inner product in
Hilbert space H; S > 0 means hSx, xi > 0 for all x; F + denotes the pseudoinverse
matrix, tr(S) denotes the trace of the matrix S.
2. Main result. We proceed with a problem statement. Consider a pair of systems
d(F x)
= C(t)x(t) + f (t),
dt
y(t) = H(t)x(t) + η(t),
F x(t0 ) = 0,
(2)
(3)
where x(t) ∈ Rn , f (t) ∈ Rm , y(t) ∈ Rp , η(t) ∈ Rp represent the state, input,
observation and observation’s noise respectively, F ∈ Rm×n , f (·) ∈ L2 (t0 , T, Rm ),
C(t) and H(t) are continuous functions of t on [t0 , T ], t0 , T ∈ R.
x(·) is said to be a weak solution to DAE (2), if F x(·) is an absolutely continuous
d
F x is a squared-integrable vector-function on (t0 , T ), x(·) verifies (2)
function,
dt
a.e. on [t0 , T ] and4 F x(t0 ) = 0.
4 We have chosen the case with zero initial condition in order to simplify the presentation. The
general case F x(t0 ) = x0 can be treated by means of the same approach.
1470
SERGIY ZHUK
We assume η(·) in (3) is a realization of a random process Ψ such that EΨ(t) = 0
on [t0 , T ] and
Z T
tr (R(t)Rη (t, t))dt ≤ 1}
(4)
Rη ∈ W = {Rη :
t0
The input f (·) in (2) is supposed to be deterministic and
Z T
hQ(t)f (t), f (t)i ≤ 1},
f (·) ∈ G := {f (·) :
(5)
t0
where Q(t) ∈ Rm×m , Q = Q0 > 0, R(t) ∈ Rp×p , R0 = R > 0; Q(t), R(t), R−1 (t)
and Q−1 (t) are continuous functions of t on [t0 , T ].
Definition 2.1. Given T < +∞, u(·) ∈ L2 (t0 , T, Rp ) and ` ∈ Rm define a meansquared worst-case estimation error
σ(T, `, u) :=
sup
x(·),f (·)∈G,Rη ∈W
{E[h`, F x(T )i − u(y)]2 }
RT
A function û(y) = t0 hû(t), y(t)idt is called an a priori minimax mean-squared
estimate in the direction ` (`-estimation) if inf u σ(T, `, u) = σ(T, `, û). The number
σ̂(T, `) = inf u σ(T, `, u) is called a minimax mean-squared a priori error in the
direction ` at time-instant T (`-error). The set L(T ) = {` ∈ Rm : σ̂(T, `) < +∞} is
called a minimax observable subspace.
2.1. Generalized Kalman Duality Principle. Definition 2.1 generalizes the notion of the a priori minimax mean-squared estimation, introduced in [18]. Next theorem generalizes the celebrated Kalman duality principle [3] to non-causal DAEs.
Theorem 2.2 (Generalized Kalman Duality). The `-error is finite iff
d(F 0 z)
= −C 0 (t)z(t) + H 0 (t)u(t), F 0 z(T ) = F 0 `
(6)
dt
for some z(·) and u(·). In this case the problem σ(T, `, u) → inf u is equal to the
following optimal control problem
Z T
Z T
−1
(7)
hR−1 u, uidt → min,
hQ (z − v), z − vidt} +
σ(T, `, u) = min{
v
t0
t0
u
with constraint (6), provided v(·) obeys (6) with u(·) = 0 and ` = 0.
Remark 1. An obvious corollary of the Theorem 2.2 is an expression for the minimax observable subspace
d(F 0 z)
+ C 0 z − H 0 u = 0 for some z(·), u(·)}
dt
In the case of stationary C(t) and H(t) the minimax observable subspace may be
calculated explicitly, using the canonical Kronecker form [7].
L(T ) = {` ∈ Rn : F 0 z(T ) = F 0 `,
Proof of Theorem 2.2. Take ` ∈ Rm , u(·) ∈ L2 (t0 , T, Rp ) and suppose `-error is
finite. The proof is based on a generalized integration-by-parts formula
hF 0 w(T ), F + F x(T )i − hF 0 w(t0 ), F + F x(t0 )i
Z T
d(F 0 w)
d(F x)
, wi + h
, xidt
h
=
dt
dt
t0
(8)
INVERSE PROBLEMS FOR DAES
1471
proved in [20] for F x(·) ∈ H12 (t0 , T, Rm ) and F 0 w(·) ∈ H12 (t0 , T, Rn ). There exists5
w(·) ∈ L2 (t0 , T, Rm ) such that F 0 w(·) ∈ H12 (t0 , T, Rn ) and F 0 w(T ) = F 0 `. Noting
that [1] F = F F + F and combining (8) with (2) one derives
h`, F x(T )i = hF 0 `, F + F x(T )i = hF 0 w(T ), F + F x(T )i
Z T
Z T
d(F x)
d(F 0 w)
d(F 0 w)
h
=
hf, wi + h
, wi + h
, xidt =
+ C 0 w, xidt
dt
dt
dt
t0
t0
Combining (9) with Eη(t) = 0 we have
Z
Z T
hH 0 u, xidt]2 + E[
E[h`(x) − u(y)]2 = [h`, F x(T )i −
t
t
(9)
T
hu(t), η(t)idt]2
0
0
(10)
Z T
d(F 0 w)
2
0
0
2
hf, wi + h
hu(t), η(t)idt]
=[
+ C w − H u, xidt] + E[
dt
t0
t0
RT
RT
RT
Using Cauchy inequality one derives E( t0 hu, ηidt)2 ≤ t0 EhRη, ηidt t0 hR−1 u, uidt.
As EhRη, ηi = tr (R(t)Rη (t, t)), it follows from (4)
Z T
Z T
Z T
Z T
hR−1 u, uidt
hR−1 u, uidt ≤
EhRη, ηidt
hu, ηidt)2 ≤
E(
Z
T
so that
sup E[
Rη ∈W
Z
t0
t0
t0
t0
T
t0
hu(t), η(t)idt]2 =
Z
T
t0
hR−1 u, uidt
This and σ̂(T, `) < +∞ imply the first term in the last line of (10) is bounded.
RT
t0 hf, widt does not depend on x(·) and is bounded due to (5). Therefore
Z T
d(F x)
d(F 0 w)
+ C 0 w − H 0 u, xidt :
= Cx + f, f (·) ∈ G} < +∞ (11)
h
sup{
dt
dt
x(·)
t0
In fact, this observation allows us to prove that there exists z(·) such that (6) holds
for the given ` and u(·). To do so we apply a general duality result6 of [21]:
sup {hF , xi, Lx ∈ G} =
x∈D(L)
inf {c(G, b), L0 b = F }
b∈D(L0 )
(12)
provided (A1) L : D(L) ⊂ H1 → H2 is a closed dense defined linear mapping, (A2)
G ⊂ H2 is a closed bounded convex set and H1,2 are abstract Hilbert spaces.
Define
d(F x)
Lx =
− Cx, x ∈ D(L) := {x(·) : F x ∈ H12 (t0 , T, Rn ), F x(t0 ) = 0}
(13)
dt
It was proved in [20], that L is closed dense defined linear mapping and
d(F 0 b)
− C 0 b, b ∈ D(L0 ) := {b(·) : F 0 b ∈ H12 (t0 , T, Rm ), F 0 b(T ) = 0} (14)
dt
d(F 0 w)
+ C 0 w − H 0 u we see from (11) that the right-hand part of (12)
Setting F :=
dt
is finite. Using (14) one derives
L0 b = −
inf{c(G, b), −
5A
d(F 0 b)
d(F 0 w)
− C 0 (t)b(t) =
+ C 0 w − H 0 u} < +∞
dt
dt
trivial example is w(t) ≡ `.
was proved in [11] for bounded L and Banach space H1,2
6 (12)
(15)
1472
SERGIY ZHUK
d(F 0 z)
+C 0 z = H 0 u, F 0 z = F 0 ` with z := (w+b) and some b(·) ∈ D(L0 ),
dt
verifying the equality in (15). This proves (6) has a solution z(·).
On the contrary, let z(·) verify (6) for the given ` and u(·). Using (10) one derives
Z T
Z T
2
σ(T, `, u) = sup
hR−1 u, uidt < ∞
(16)
hz, f idt +
(15) implies
f ∈G1
t0
t0
provided G1 = G ∩ R(L), R(L) is the range of the linear mapping L defined by (13)
(G1 is a set of all f (·) such that f (·) verifies (5) and (2) has a solution x(·)).
To prove the rest of the theorem we will apply another duality result of [21, 11]:
sup{hf, zi, f ∈ G ∩ R(L)} = inf{c(G, z − v), v ∈ N (L0 )}
(17)
provided L, G verify (A1), (A2) and int G ∩ R(L) 6= ∅. It is easy to see that the
latter inclusion holds for L and G defined by (13) and (5) respectively. Recalling
the definition of L0 (formula (14)) and noting that
Z T
hQ−1 (z − v), z − vidt
c2 (G, z − v) =
t0
we derive from (16)-(17) that σ(T, `, u) → minu is equal to (7).
3. Optimality conditions and estimation algorithm. Due to Theorem 2.2
the minimax state estimation problem is equal to the dual control problem (7) with
DAE constrain (6), provided ` ∈ L(T ). This result holds for any F ∈ Rm×n and
continuous t 7→ C(t) ∈ Rm×n . Therefore, in order to find the `-estimation û we
need to solve (7). In what follows, we formulate optimality conditions for (7) in the
form of a boundary value problem (BVP) for a descriptor Euler-Lagrange equation.
We present a condition on the matrices C(t), H(t) and F , which allow one to
convert the latter BVP for DAE into an equivalent BVP for ODE. Our approach
is a modification of a splitting method, discussed in [6]. We will split DAE into
differential and algebraic parts applying SVD decomposition [1] to F . Let D =
diag(λ1 . . . λr ) where λi are positive eigen values of F F 0 , i ∈ {1, . . . , r := rangF }.
Then there exist SL ∈ Rm×m , SR ∈ Rn×n such that
i
h
1
0
0r×n−r
(18)
F = SL ΛSR , SL SL0 = I, SR SR
= I, Λ = 0 D 2
0
m−r×r
m−r×n−r
Transforming (2) according to (18) and changing the variables one can reduce the
general case to the case F = [ I0 00 ]. Therefore, without loss of generality, we will
0
assume F = [ I0 h00 ]. We isplit `, C(t),hQ(t) and
i H (t)R(t)H(t) accordingly: C(t) =
C1 C2 Q1 Q2
S1 S2
0
C3 C4 , Q =
S 0 S4 , ` = (`1 , `2 ). Define W (t) = (S4 +
Q0 Q4 , H RH =
2
2
0 −1
0
0 −1 0
C40 Q−1
4 C4 ), A(t) = (C3 Q4 C4 + S2 ) and B(t) = (C2 − C4 Q4 Q2 ).
Proposition 1 (optimality conditions). Let R(C20 (t)) ⊆ R(C40 (t)) and
Z T
kW + (t)B(t)g(t)k2 dt < ∞, ∀g(·) ∈ L2 (t0 , T, Rr )
t0
Then the Euler-Lagrange system
d(F p)
= C(t)p(t) + Q−1 (t)z(t), F p(t0 ) = 0,
dt
d(F 0 z)
= −C 0 (t)z(t) + H 0 (t)R(t)H(t)p(t), F 0 z(T ) = F 0 `
dt
(19)
INVERSE PROBLEMS FOR DAES
1473
has a solution for any ` ∈ Rn . If p(·) and z(·) are some solution of (19) then, the
`-estimation is given by û = RHp and the `-error σ̂(T, `) = hF 0+ `, F p(T )i.
For instance, condition on W and B of the theorem is satisfied, if C(t) meets the
conditions of [9].
Proof. Taking into account the above splitting we rewrite (19) in the following form
dp1
= C1 p1 + Q1 z1 + C2 p2 + Q2 z2 , p1 (t0 ) = 0,
dt
dz1
= −C10 z1 + S1 p1 − C30 z2 + S2 p2 , z1 (T ) = `1 ,
dt
0 = C3 p1 + C4 p2 + Q02 z1 + Q4 z2 ,
(20)
0 = −C20 z1 − C40 z2 + S20 p1 + S4 p2 .
As Q−1 > 0, it follows that Q4 > 0. Hence z2 = −Q4−1 (C3 p1 + C4 p2 + Q02 z1 ) and
W (t)p2 = B(t)z1 − A0 (t)p1
R(C20 )
(21)
R(C40 )
where A, B, W were defined above. Since
⊂
it follows that (21) is
point-vise solvable (in the algebraic sense) and one solution has the form p2 =
W + (t, 0)(B(t)z1 − A0 (t)p1 ) The second assumption imply p2 ∈ L2 (t0 , T, Rn−r ).
Substituting p2 into (20) and noting that C4 (I − W + (t)W (t)) = 0 we obtain
dp1
= C+ (t)p1 + S+ (t)z1 , p1 (t0 ) = 0
dt
dz1
0
= −C+
(t)z1 + Q+ (t)p1 , z1 (T ) = `1
dt
(22)
−1
0
+
0
0
+
with C+ (t) := C1 − Q2 Q−1
4 C3 − B W (t)A , S+ (t) := Q1 − Q2 Q1 Q3 + B W (t)B,
0 −1
+
0
Q+ (t) := S1 + C3 Q4 C3 − AW (t)A . Applying simple matrix manipulations one
can prove that S+ ≥ 0, Q+ ≥ 0 so that (22) is a non-negative Euler-Lagrange
system in the Hamilton’s form. Therefore it is always solvable [14]. We continue
with the second part of the proposition.
Due to GKD (Theorem 2.2), it is sufficient to show that û, defined in the statement of Proposition, solves (7). We note that (7) is equal to
Z T
Z T
−1
(23)
hR−1 u, uidt →
min
hQ z, zidt +
N (z, u) :=
t0
t0
(z,u) verify (6)
Therefore, to conclude it is sufficient to show that N (z, u) − N (û, ẑ) ≥ 0, where
û = RHp, (p̂, ẑ) denote some solution of (19) and (z, u) verifies (6). We have
1
1
(N (z, u) − N (û, ẑ)) = (hF 0 ẑ(T ), F + F p̂(T )i − N (ẑ, û))
2
2
Z T
1
d(F 0 ẑ)
d(F p̂)
1
− (hF 0 z(T ), F +F p̂(T )i − N (z, u)) =
h
, p̂i + h
, ẑidt − N (ẑ, û)
2
dt
dt
2
t0
Z T
Z T
0
1
d(F p̂)
1
d(F z)
hH 0 û, p̂i + kẑk2 dt − N (ẑ, û)
, p̂i + h
, zidt − N (z, u)) =
h
−(
dt
dt
2
2
t0
t0
Z T
Z T
1
1
ku − ûk2 + kz − ẑk2 dt ≥ 0
hH 0 u, p̂i + hz, ẑidt − N (z, u)) =
−(
2
2
t0
t0
1474
SERGIY ZHUK
Corollary 1 (sequential `-estimation). Define C+ , Q+ and S+ as in (22) and let
dK
= C+ (t)K + KC+ (t)0 − KQ+ (t)K + S+ (t), K(t0 ) = 0,
dt
(24)
dx̂
0
+
= (C+ (t) − KQ+ (t))x̂ + K (B −KA)W (t) HRy(t), x̂(t0 ) = 0
dt
If û is defined as in Proposition 1 and ` = (`1 , `2 ) (as in (22)) then
Z T
hû, yi = h`1 , x̂(T )i, σ̂(T, `) = hK(T )`1 , `1 i
(25)
t0
Proof. The proof uses the standard reduction procedure, that is a reduction of the
Euler-Lagrange system (22) for (p1 , z1 ) to some Cauchy problem for z1 , introducing
a Riccati matrix K.
dz1
0
= −C+
(t)z1 + Q+ (t)Kz1 , z1 (T ) = `1 . Define p1 := Kz1 .
Assume z1 solves
dt
By direct calculation one finds that (p1 , z1 ) verify (22). As (22) is always uniquely
solvable, it follows that the Riccati equation in (24) has a unique solution.
Let p2 solves (21). Let us find z2 from the third equation of (20) and set p̂ =
(p1 , p2 )T , ẑ = (z1 , z2 )T . Then (ẑ, p̂) solves (19) with F = [ I0 00 ]. Proposition 1
implies that the minimax estimation is given by û = RH p̂ and the minimax error is
RT
σ̂(T, `) = hF 0+ `, F p(T )i. Now, integrating by parts t0 hû, yi and taking into account
definitions of p1 , p2 , z1 and x̂ we arrive to the first equality in (25). Again, using
integration by parts in hF 0+ `, F p(T )i one obtains the second equality in (25).
3.1. Numerical example: non-causal non-stationary DAE. Let
i
h
1
, H(t) = [ 0 1 ] .
F = [ 10 00 ] , C(t) = c−1
3 (t) 0
Then det(F − λC(t)) ≡ 0 if c3 (t) = 0. Note that the pencil sF 0 − C 0 − H 0 H is
regular. The corresponding DAE reads
dx1
= −x1 + x2 + f1 (t),
dt
(26)
0 = c3 (t)x1 (t) + f2 (t), x1 (0) = 0
Rt
Rt
Set f1 = 0 for simplicity. We have x1 (t) = 0 exp(s − t)x2 (s)ds and c3 (t) 0 exp(s −
t)x2 (s)ds = −f2 (t). From the latter formulae we see that the DAE (26) is ill-posed:
x1 is non-unique and is not continuous with respect to the input data7 .
Let us estimate x1 (t), provided y(t) = x2 (t)+η(t) is measured, (x1 , x2 ) obeys (26)
and (f1 , f2 , η) verify
√
Z T
Z T
6 2
exp( t) 2
2
f2 dt ≤ 1, E
η (t)dt ≤ 1, Eη(·) = 0
(27)
f1 +
2
0 T
0
As (19) is solvable, we can apply Corollary 1. The minimax error is given by
σ̂(t, `) := K(t)`21 and minimax estimate has the following form
2c23 (t) dx̂ √ ) x̂ + y(t), x̂(0) = 0,
= − 1 − K(t)
dt
exp( t)
T
dK
2c23 (t)
√ K 2 + (1 + ),
= −2K −
dt
6
exp( t)
K(0) = 0,
7 Since x depends on derivative of f , it follows that x is not L -continuous with respect to
2
2
2
2
f2 , implying ill-posedness. As x2 depends on an arbitrary function v from some linear subspace
we have non-uniqueness.
INVERSE PROBLEMS FOR DAES
1475
1.5
1.0
0.5
x1
1
2
3
4
5
t
x`
Σ
-0.5
Figure 1. Minimax estimate x̂(t), error σ(`1 , t) and the simulated
x1 (t) for t ∈ [0, 5].
Figure 1 reflects the result of numerical simulations with particular f2 and η, verifying (27).
4. Conclusion. The paper describes the minimax state estimation approach for
linear non-causal DAEs, that is to find an estimate u(y) of the linear function
h`, F x(T )i, minimizing the worst-case error. The case of unknown but bounded
input f and random observation error η with uncertain bounded correlation function
is considered. The background of the approach is a Generalized Kalman Duality
(GKD) principle. The GKD is used to calculate minimal worst-case (minimax)
error. In contrast to causal DAEs, the minimax error could be infinite for some
directions ` if the DAE is non-causal. In this case the observations y(t) along with
state equation (1) do not provide sufficient information for reconstructing F x(T ).
In fact, only a projection of F x(T ) onto some subspace L(T ), so called Minimax
Observable Subspace (MOS), can be reconstructed. L(T ) describes an ”observable”
(in the minimax sense) part of F x(T ): MOS consists of all `, for which minimax
error is finite. As a consequence, for any linear estimate u(y) of h`, F x(T )i the
estimation error varies8 in [0, +∞] if ` 6∈ L(T ). For the case of constant C(t) the
MOS can be efficiently calculated (Remark 1).
Restricting the matrices of DAE we present sufficient solvability conditions for the
Euler-Lagrange system, describing points of minimum of the dual control problem.
This, in turn, allows to derive a reduced-order minimax estimate in the form of
the minimax filter. The results are illustrated by a synthetic example of non-causal
ill-posed 2D DAE.
8 So that, for any natural N there is a realization of uncertain parameters f (·) and η(·) such
that the estimation error will be greater than N
1476
SERGIY ZHUK
It would be interesting to derive a sub-optimal minimax estimate for DAEs without restrictions of Proposition 1. In particular, such estimate can be useful for the
generation of the robust and mathematically justified reduced order state estimate
for systems with a high dimension of the state vector. The latter can be done by
projecting the state of the system onto some subspace (defined, for instance, by
Proper Orthogonal decomposition) and to apply minimax estimation algorithm for
a resulting DAE [17].
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Received July 2010; revised August 2010.
E-mail address: [email protected]
E-mail address: [email protected]