INVERSE PROBLEMS FOR LINEAR ILL-POSED DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH UNCERTAIN PARAMETERS

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.


Introduction.
In this paper we focus on an inverse problem for a linear Differential-Algebraic Equation (DAE) in the form where F is a rectangular matrix.We note that the case det(sF − C) = 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) = 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] one can treat an input f of the ODE dx dt = C(t)x(t) + f (t) as a part of the state 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 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 10 6 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) ∈ R m×n .We will be interested in the following inverse problem: given observations y(t), t ∈ [t 0 , 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) → R η (t, s) := EΨ(t)Ψ (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) := , F x(T ) for any ∈ R m .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 R η , 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 , R η and3 x: σ(T, , u) := sup f ∈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, consistent with observations and uncertainty description.The same result can be proved by the method of this paper for F = I, provided η is deterministic and (η, f ) belongs to some ellipsoid.Main distinctive features of the problem, described above are: 1) as F ∈ R m×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; R n denotes the n-dimensional Euclidean space over real field; L 2 (t 0 , T, R m ) denotes a space of square-integrable functions with values in R m ; H 1 2 (t 0 , T, R m ) denotes a space of absolutely continuous functions with L 2 -derivative and values in R m ; the prime denotes the operation of taking the adjoint: L denotes adjoint operator, F denotes the transposed matrix; c(G, •) denotes the support function of some set G; •, • denotes the inner product in Hilbert space H; S > 0 means Sx, x > 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 where a.e. on [t 0 , T ] and4 F x(t 0 ) = 0.
We assume η(•) in ( 3) is a realization of a random process Ψ such that EΨ(t) = 0 on [t 0 , T ] and The input f (•) in ( 2) is supposed to be deterministic and where 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.
Remark 1.An obvious corollary of the Theorem 2.2 is an expression for the minimax observable subspace In the case of stationary C(t) and H(t) the minimax observable subspace may be calculated explicitly, using the canonical Kronecker form [7].
Proof of Theorem 2.2.Take ∈ R m , u(•) ∈ L 2 (t 0 , T, R p ) and suppose -error is finite.The proof is based on a generalized integration-by-parts formula proved in [20] for Combining (9) with Eη(t) = 0 we have Using Cauchy inequality one derives E( This and σ(T, ) < +∞ imply the first term in the last line of ( 10) is bounded.T t0 f, w dt does not depend on x(•) and is bounded due to (5).Therefore sup 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]: provided (A1) L : D(L) ⊂ H 1 → H 2 is a closed dense defined linear mapping, (A2) G ⊂ H 2 is a closed bounded convex set and H 1,2 are abstract Hilbert spaces.Define It was proved in [20], that L is closed dense defined linear mapping and 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 provided is the range of the linear mapping L defined by ( 13) (G 1 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]: provided L, G verify (A1), (A2) and int G ∩ R(L) = ∅.It is easy to see that the latter inclusion holds for L and G defined by ( 13) and ( 5) respectively.Recalling the definition of L (formula ( 14)) and noting that we derive from ( 16)-( 17) that σ(T, , u) → min u 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 ∈ R m×n and continuous t → C(t) ∈ R m×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(λ 1 . . .λ r ) where λ i are positive eigen values of F F , i ∈ {1, . . ., r := rangF }.
Then there exist S L ∈ R m×m , S R ∈ R n×n such that 0r×n−r 0m−r×r 0m−r×n−r (18) Transforming (2) according to (18) and changing the variables one can reduce the general case to the case F = [ I 0 0 0 ].Therefore, without loss of generality, we will assume F = [ I 0 0 0 ].We split , C(t), Q(t) and H (t)R(t)H(t) accordingly: Then the Euler-Lagrange system has a solution for any ∈ R n .If p(•) and z(•) are some solution of (19) then, the -estimation is given by û = RHp and the -error σ(T, ) = F + , F p(T ) .
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 As where A, B, W were defined above.Since R(C 2 ) ⊂ R(C 4 ) it follows that ( 21) is point-vise solvable (in the algebraic sense) and one solution has the form Substituting p 2 into (20) and noting that C 4 (I − W + (t)W (t)) = 0 we obtain with 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 Therefore, to conclude it is sufficient to show that N (z, u) − N (û, ẑ) ≥ 0, where û = RHp, (p, ẑ) denote some solution of ( 19) and (z, u) verifies (6).We have Corollary 1 (sequential -estimation).Define C + , Q + and S + as in (22) and let If û is defined as in Proposition 1 and = ( 1 , 2 ) (as in ( 22)) then Proof.The proof uses the standard reduction procedure, that is a reduction of the Euler-Lagrange system (22) for (p 1 , z 1 ) to some Cauchy problem for z 1 , introducing a Riccati matrix K. Assume By direct calculation one finds that (p 1 , z 1 ) verify (22).As ( 22) is always uniquely solvable, it follows that the Riccati equation in (24) has a unique solution.
Let p 2 solves (21).Let us find z 2 from the third equation of (20) and set p = (p 1 , p 2 ) T , ẑ = (z 1 , z 2 ) T .Then (ẑ, p) solves (19) with F = [ I 0 0 0 ].Proposition 1 implies that the minimax estimation is given by û = RH p and the minimax error is σ(T, ) = F + , F p(T ) .Now, integrating by parts T t0 û, y and taking into account definitions of p 1 , p 2 , z 1 and x we arrive to the first equality in (25).Again, using integration by parts in F + , F p(T ) one obtains the second equality in (25).

Numerical example: non
Set f 1 = 0 for simplicity.We have x 1 (t) = t 0 exp(s − t)x 2 (s)ds and c 3 (t) t 0 exp(s − t)x 2 (s)ds = −f 2 (t).From the latter formulae we see that the DAE (26) is ill-posed: x 1 is non-unique and is not continuous with respect to the input data 7 .
Let us estimate x 1 (t), provided y As ( 19) is solvable, we can apply Corollary 1.The minimax error is given by σ(t, ) := K(t) 2 1 and minimax estimate has the following form dx

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 , F x(T ) , 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 , F x(T ) the estimation error varies8 in [0, +∞] if ∈ 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.
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].

Figure 1
Figure 1 reflects the result of numerical simulations with particular f 2 and η, verifying (27).