Christopher Schneider
email: christopher.schneider@uni-jena.de

Walter Alt
email: walter.alt@uni-jena.de

Regularization of Linear-Quadratic Control Problems with L 1 -Control Cost

Keywords: Optimal Control, Bang-Bang Control, L 1 -Minimization, Nonsmooth Analysis, Regularization, Discretization

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The regularization of optimal control problems by a L 2 -term α 2 u 2 L 2 is often used in order to get a smoother optimal control. In this cases α can be viewed as a regularization parameter and one is interested in the question how the solutions depend on this parameter. For the special case that the control variable appears linearly in the control problem and the optimal control without regularization (α = 0) has bang-bang structure this question has been investigated in Deckelnick/Hinze [START_REF] Deckelnick | A note on the approximation of elliptic control problems with bang-bang controls[END_REF] for a class of elliptic control problems and in Alt/Seydenschwanz [START_REF] Alt | Regularization and discretization of linear-quadratic control problems[END_REF] for a general class of linear-quadratic control problems governed by ordinary differential equations.

Maurer/Vossen [START_REF] Vossen | On L 1 -minimization in optimal control and applications to robotics[END_REF] investigate first order necessary and second order sufficient optimality conditions for a class of nonlinear control problems involving a L 1 -term in the cost functional, where the parameter β is kept fixed. They also propose some numerical algorithms for the solution of such problems. Sakawa [START_REF] Sakawa | Trajectory planning of a free-flying robot using the optimal control[END_REF] also considers a special numerical algorithm for a fixed parameter β > 0. Stadler [START_REF] Stadler | Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices[END_REF] and Casas et al. [START_REF] Casas | Approximation of sparse controls in semilinear equations by piecewise linear functions[END_REF][START_REF] Casas | Optimality conditions and error analysis of semilinear elliptic control problems with L 1 -cost functional[END_REF] investigate classes of elliptic control problems with a L 1 -term in the cost functional, which is interpreted as a regularization term. They derive results on the dependence of the solutions on the parameter β and error estimates for discretizations, but an additional L 2 -regularization term with fixed parameter α is used in order to get smoother solutions. In Wachsmuth/Wachsmuth [START_REF] Wachsmuth | Convergence and regularization results for optimal control problems with sparsity functional[END_REF] the dependence of solutions of a class of elliptic control problems on the regularization parameter α is studied while the parameter β is kept fix.

Results for the dependence of the solutions on the parameter β and error estimates for discretizations for a general class of linear-quadratic control problems governed by ordinary differential equations have been recently derived in [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF]. In the present paper, we investigate the regularization of such control problems and the dependence of solutions on the parameter β and the regularization parameter α assuming that for a fixed parameter β * the corresponding optimal control is of bang-zero-bang type.

Problem formulation

With X = X 1 × X 2 , X 1 = W 1 ∞ (0, t f ; R n ), X 2 = L ∞ (0, t f ; R m ),
we consider the following family of L 2 -regularized linear-quadratic control problems with L 1 -control cost depending on the parameters α ≥ 0 and β ≥ 0: min

(x,u)∈X f α,β (x, u) s. t. ẋ(t) = A(t)x(t) + B(t)u(t) a.e. on [0, t f ] , x(0) = a , u(t) ∈ U a.e. on [0, t f ] , (PQ α,β )
where f α,β is a linear-quadratic cost functional with an additional nonsmooth L 1 -term defined by

f α,β (x, u) = 1 2 x(t f ) T Qx(t f ) + q T x(t f ) + t f 0 1 2 x(t) T W (t)x(t) + w(t) T x(t) + r(t) T u(t) dt + β u L 1 + α 2 u 2 L 2 .
Here, u(t) ∈ R m is the control, and x(t) ∈ R n is the state of the system at time t, where t ∈ [0, t f ]. Further Q ∈ R n×n is a symmetric and positive semidefinite matrix, q ∈ R n , and the functions W : [0,

t f ] → R n×n , w : [0, t f ] → R n , r : [0, t f ] → R m , A : [0, t f ] → R n×n , and B : [0, t f ] → R n×m are Lipschitz contin- uous.
The matrices W (t) are assumed to be symmetric and positive semidefinite, and the set U ∈ R m is defined by lower and upper bounds, i.e.

U = {u ∈ R m | b ≤ u ≤ b u } with b , b u ∈ R m , b < b u
, where all inequalities are to be understood componentwise.

While the regularization term α 2 u 2 L 2 leads to a smooth optimal control for α > 0 the term β u L 1 may be interpreted as both a regularization or some (nonsmooth) L 1 -control cost. We are interested in the behavior of a solution u α,β of Problem (PQ α,β ) depending on both parameters α and β.

Optimality conditions

We denote by

U = {u ∈ X 2 | u(t) ∈ U a.e. on [0, t f ]}
the set of admissible controls, and by

F = {(x, u) ∈ X | u ∈ U, ẋ(t) = A(t)x(t) + B(t)u(t) a.e. on [0, t f ], x(0) = a}
the feasible set of (PQ α,β ). Since U is nonempty, the feasible set F is nonempty, too. And since U is bounded, it follows that ẋ is bounded for any feasible pair (x, u) ∈ F, and therefore F ⊂ X. Moreover, there is some constant c such that x 1,∞ ≤ c u L ∞ for any solution x of the system equation, which implies that

F is bounded. A feasible pair (x α,β , u α,β ) ∈ F is called a minimizer for Problem (PQ α,β ) if f α,β (x α,β , u α,β ) ≤ f α,β (x, u) for all (x, u) ∈ F.
Since the feasible set F is nonempty, closed, convex and bounded, and the cost functional is convex and continuous, a minimizer (

x α,β , u α,β ) ∈ W 1 2 (0, t f ; R n ) × L 2 (0, t f ; R m ) of (PQ α,β ) exists (see [10, Chap. II, Prop. 1.2]), and since U is bounded we have (x α,β , u α,β ) ∈ X = W 1 ∞ (0, t f ; R n ) × L ∞ (0, t f ; R m ). Let (x α,β , u α,β ) ∈ F be a minimizer of (PQ α,β ). Then there exist an ele- ment γ α,β ∈ ∂ u α,β L 1 of the subdifferential of u α,β L 1 and a function λ α,β ∈ W 1 ∞ (0, t f ; R n ) such that the adjoint equation -λα,β (t) = A(t) T λ α,β (t) + W (t)x α,β (t) + w(t) a.e. on [0, t f ] , λ α,β (t f ) = Qx α,β (t f ) + q , (1)
and the minimum principle

B(t) T λ α,β (t) + r(t) + α u α,β (t) + β γ α,β (t) T u -u α,β (t) ≥ 0 ∀u ∈ U (2)
hold a.e. on [0, t f ] (compare e.g. [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF]Theorem 10.47] or [3, Sect. 2]).

Remark 1. Since (PQ α,β ) is a convex optimization problem for all α ≥ 0 and β ≥ 0, a pair (x α,β , u α,β ) ∈ F satisfying the minimum principle (2) and solving the adjoint equation ( 1) with some functions γ α,β and λ α,β is a solution of (PQ α,β ) (compare [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF]Propositon 4.12]).

Provided α = 0 we are able to evaluate the minimum principle (2) in more detail (compare [START_REF] Vossen | On L 1 -minimization in optimal control and applications to robotics[END_REF] and [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF]) and obtain

u 0,β i (t) =                b u,i , if ξ β i (t) < -β , undetermined ∈ ]0, b u,i ] , if ξ β i (t) = -β , 0 , if ξ β i (t) ∈ ]-β, β[ , undetermined ∈ [b ,i , 0[ , if ξ β i (t) = β , b ,i , if ξ β i (t) > β , (3)
where ξ β (t) := B(t) T λ 0,β (t) + r(t). If we assume that the set of switching times

M β i = t ∈ [0, t f ] ξ β i (t) = β or ξ β i (t) = -β .
is finite, then by (3) the i-th component of the optimal control has a bang-zerobang structure.

Problem transformation

In common with [START_REF] Vossen | On L 1 -minimization in optimal control and applications to robotics[END_REF] and [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF] we formulate a transformed problem (TQ α,β ) in order to study the dependence of the optimal control on the parameters α and β. This is a well known augmentation approach from linear programming wherewith we obtain a linear-quadratic control problem with smooth cost functional (see e.g. [START_REF] Murty | Operations Research: Deterministic Optimization Models[END_REF]).

Introducing new controls v ∈ X2 := L ∞ (0, t f ; R 2m ) and using the matrix

M :=      1 -1 1 -1 . . . . . . 1 -1      ∈ R m×2m (4)
we have min

(x,v)∈X1× X2 fα,β (x, v) s. t. ẋ(t) = A(t)x(t) + B(t)v(t) a.e. on [0, t f ] , x(0) = a , v(t) ∈ V a.e. on [0, t f ] , (TQ α,β )
where B(t) := B(t)M . There are new box constraints for the controls,

V := v ∈ R 2m | v ≥ 0 , v 2i-1 ≤ b u,i , v 2i ≤ -b ,i , i = 1, . . . , m ,
and f α,β is a linear-quadratic cost functional:

fα,β (x, v) = 1 2 x(t f ) T Qx(t f ) + q T x(t f ) + t f 0 1 2 x(t) T W (t)x(t) + w(t) T x(t) + r(t) T M v(t) dt + β M v L 1 + α 2 M v 2 L 2 .
With the same argumentation as above for Problem (PQ α,β ) we are able to show that a minimizer of Problem (TQ α,β ) exists. We denote the set of admissible controls by

V = v ∈ X2 | v(t) ∈ V a.e. on [0, t f ]
and the feasible set of Problem (TQ α,β ) by T ⊂ X 1 × X2 , where

T = {(x, v) | v ∈ V , ẋ(t) = A(t)x(t) + B(t)v(t) a.e. on [0, t f ] , x(0) = a} .
Although Problem (TQ α,β ) admits controls with components v 2i-1 , v 2i being positive simultaneously, such controls cannot be optimal (see [START_REF] Vossen | On L 1 -minimization in optimal control and applications to robotics[END_REF]Sect. 4], [9, Sect. 3], [12, p. 42 et seq.]). Therefore, all optimal controls satisfy

v α,β 2i-1 (t) = max 0, u α,β i (t) , v α,β 2i (t) = max 0, -u α,β i (t) . (5)
The optimality conditions also prove this result. By [START_REF] Stadler | Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices[END_REF] and v(t) ≥ 0 we now are able to simplify

M v L 1 = v L 1 = t f 0 2m i=1 v i (t) dt and M v 2 L 2 = v 2 L 2 ,
which nicely shows, that a L 1 -or L 2 -regularization of the original problem implies the same regularization of the transformed problem. We finally introduce the minimum principle of Problem (TQ α,β )

σ α,β T v -v α,β (t) ≥ 0 ∀v ∈ V , (6)
where

σ α,β := M T B(t) T λ α,β (t) + r(t) + α v α,β (t) + β e , (7)
with e := (1, . . . , 1) T ∈ R 2m . The adjoint equation ( 1) as well as the adjoint variables λ α,β do not change im comparison to Problem (PQ α,β ). A detailed discussion of the optimality conditions can be found in [START_REF] Vossen | On L 1 -minimization in optimal control and applications to robotics[END_REF] and [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF].

Uniqueness of solutions

It is well known that the solution of Problem (TQ α,β ) is uniquely determined for each β ≥ 0, if α > 0 (compare e.g. [13, Satz 3.2.5]). This extends with [START_REF] Stadler | Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices[END_REF] to Problem (PQ α,β ).

In the case of α = 0 we consider a fixed parameter β * ≥ 0 and assume that the optimal control v 0,β * of Problem (TQ 0,β * ) is of bang-bang type which implies an optimal control u 0,β * of bang-zero-bang type for Problem (PQ 0,β * ) by [START_REF] Stadler | Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices[END_REF]. To ensure this we assume that (B1) There exists a solution (x 0,β * , v 0,β * ) ∈ T of (TQ 0,β * ) such that the set Σ of zeros of the components of the switching function σ 0,β * defined by ( 7) is finite and 0, t f / ∈ Σ, i.e. Σ = {s 1 , . . . , s l } with 0 < s 1 < . . . < s l < t f . Let I(s j ) := {1 ≤ i ≤ 2m | σ 0,β * i (s j ) = 0} be the set of active indices for the components of the switching function. In order to get stability of the bang-bang structure under perturbations we need an additional assumption (compare [START_REF] Felgenhauer | On Stability of Bang-Bang Type Controls[END_REF]):

(B2) The functions B and r are differentiable, Ḃ and ṙ are Lipschitz continuous, and there exists σ > 0 such that min

1≤j≤l min i∈I(sj ) σ0,β * i (s j ) ≥ 2σ .
Remark 2. Assumption (B2) can be slightly relaxed (see e.g. [START_REF] Alt | Error Bounds for Euler Approximation of Linear-Quadratic Control Problems with Bang-Bang Solutions[END_REF][START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF]).

The following result is extracted from [14, Proof of Lemma 3.3]. Proofs can also be found in [START_REF] Alt | Regularization and discretization of linear-quadratic control problems[END_REF], [START_REF] Alt | Error Bounds for Euler Approximation of Linear-Quadratic Control Problems with Bang-Bang Solutions[END_REF] and [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF]. Lemma 1. Let (x 0,β * , v 0,β * ) be a minimizer for Problem (TQ 0,β * ) and let the switching function σ 0,β * (t) be defined by [START_REF] Casas | Optimality conditions and error analysis of semilinear elliptic control problems with L 1 -cost functional[END_REF]. If Assumptions (B1) and (B2) are satisfied, then there are constants ω, γ, δ > 0 independent of β such that for any feasible pair (x, v)

t f 0 σ 0,β * (t) T v(t) -v 0,β * (t) dt ≥ ω v -v 0,β * 2 L 1 (8) if v -v 0,β * L 1 ≤ 2γ δ, and
t f 0 σ 0,β * (t) T v(t) -v 0,β * (t) dt ≥ ω v -v 0,β * L 1 (9) if v -v 0,β * L 1 ≥ 2γ δ.
By the help of standard arguments this result implies uniqueness of the solution of (TQ 0,β * ) (compare [START_REF] Felgenhauer | On Stability of Bang-Bang Type Controls[END_REF]Theorem 2.2]). It follows with (5) that Problem (PQ 0,β * ) has a unique solution, too.

Calmness of solutions

In this section for α ≥ 0 and β ≥ 0 we denote by (x α,β , u α,β ) and (x α,β , v α,β ) the solutions of (PQ α,β ) and (TQ α,β ), respectively. We want to study the dependence of solutions on α and β. We derive estimates which show that the solutions as functions of the regularization parameters α and β are calm at α = 0 and β = β * (compare Dontchev/Rockafellar [16, Sect. 1C]). For this purpose we combine the results achieved in [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF] and [START_REF] Alt | Regularization and discretization of linear-quadratic control problems[END_REF].

Theorem 1. Let (B1) and (B2) be satisfied for some β * ≥ 0. Then for any α ≥ 0 and β ≥ 0 the estimate

v α,β -v 0,β * L 1 ≤ c 1 (α + |β -β * |) (10)
holds, where the constant c 1 is independent of α and β.

Proof. We only consider the case v α,β -v 0,β * L 1 ≤ 2γ δ and refer to [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF] and [START_REF] Alt | Regularization and discretization of linear-quadratic control problems[END_REF] for the case v α,β -v 0,β * L 1 ≥ 2γ δ which can be handled analogously. Since Assumptions (B1) and (B2) are satisfied, for α, β ≥ 0 by [START_REF] Wachsmuth | Convergence and regularization results for optimal control problems with sparsity functional[END_REF] we have

t f 0 σ 0,β * (t) T v α,β (t) -v 0,β * (t) dt ≥ ω v α,β -v 0,β * 2 L 1 (11)
with ω > 0. By the minimum principle ( 6) we obtain

t f 0 σ α,β (t) T v 0,β * (t) -v α,β (t) dt ≥ 0 . (12)
Adding ( 12) and ( 11) it follows that

t f 0 σ 0,β * (t) -σ α,β (t) T v α,β (t) -v 0,β * (t) dt ≥ ω v α,β -v 0,β * 2 L 1 . (13)
Since

σ 0,β * (t) -σ α,β (t) = T λ 0,β * (t) -λ α,β (t) + (β * -β) e -α v α,β (t) ,
and due to the fact that x α,β , x 0,β * satisfy the system equation, and λ α,β , λ 0,β * satisfy the adjoint equation we obtain

t f 0 B(t) T λ 0,β * (t) -λ α,β (t) T (v α,β (t) -v 0,β * (t)) dt = x 0,β * (t f ) -x α,β (t f ) T Q x α,β (t f ) -x 0,β * (t f ) + t f 0 x 0,β * (t) -x α,β (t) T W (t) x α,β (t) -x 0,β * (t) dt .
Together with [START_REF] Alt | EAGLE-STARTHILFE Optimale Steuerung. Theorie und Verfahren[END_REF] this implies

ω v α,β -v 0,β * 2 L 1 + x α,β (t f ) -x 0,β * (t f ) T Q x α,β (t f ) -x 0,β * (t f ) + t f 0 x α,β (t) -x 0,β * (t) T W (t) x α,β (t) -x 0,β * (t) dt ≤ t f 0 (β * -β) e -α v α,β (t) T v α,β (t) -v 0,β * (t) dt ≤ |β -β * | v α,β -v 0,β * L 1 + α v α,β L ∞ v α,β -v 0,β * L 1 .
Since the matrices Q and W (t), t ∈ [0, t f ], are assumed to be positive semidefinite and ω > 0, we obtain

ω v α,β -v 0,β * 2 L 1 ≤ |β -β * | + α v α,β L ∞ v α,β -v 0,β * L 1 .
We now get [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF] with some constant c 1 independent of α and β.

Remark 3. By Theorem 1 we also obtain estimates for the optimal states

x α,β -x 0,β * 1,1 ≤ c1 (α + |β -β * |)
and for the optimal controls u α,β of the original problem (PQ α,β ), by using the matrix (4) and the relation ( 5) between u α,β and v α,β

u α,β -u 0,β * L 1 = M v α,β -M v 0,β * L 1 ≤ M 1 v α,β -v 0,β * L 1 ≤ c 1 (α + |β -β * |) .
If we choose some β in a sufficiently small neighborhood of β * this result can even be improved.

Theorem 2. Let (B1) and (B2) be satisfied for some β * ≥ 0. Then there exist ρ > 0 and a constant c 2 independent of α ≥ 0 and ρ, such that for any

β i ∈ R, i = 1, 2, with β i ≥ 0 and |β i -β * | < ρ the estimate v α,β1 -v 0,β2 L 1 ≤ c 2 (α + |β 1 -β 2 |) (14) holds.
Proof. We use [9, Theorem 6.3, Remark 10], which proved the local Lipschitzcontinuity of the optimal control depending on β, where the constant c is independent of β:

u 0,β1 -u 0,β2 L 1 ≤ c |β 1 -β 2 | . (15)
In addition to this we are able to extend the result of [2, Theorem 4.1] using the problem transformation introduced in Sect. 4 and obtain

u α,β1 -u 0,β1 L 1 ≤ c α (16)
with some constant c independent of α. Together ( 15) and ( 16) lead to

u α,β1 -u 0,β2 L 1 ≤ u α,β1 -u 0,β1 L 1 + u 0,β1 -u 0,β2 L 1 ≤ c α + c |β 1 -β 2 | ,
which implies [START_REF] Felgenhauer | On Stability of Bang-Bang Type Controls[END_REF].

Discretization

For the numerical solution of Problem (PQ α,β ) we use the Euler discretization scheme described in [START_REF] Alt | Error Bounds for Euler Approximation of Linear-Quadratic Control Problems with Bang-Bang Solutions[END_REF] and [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF]. Given a natural number N and let h N = t f /N be the meshsize, we approximate the cost functional f α,β by

f α,β,N (x, u) = 1 2 x T N Qx N + q T x N + h N N -1 i=0 1 2 x T i W (t i )x i + w(t i ) T x i + r(t i ) T u i + h N   β N -1 i=0 m j=1 |u j,i | + α 2 N -1 i=0 m j=1 u 2 j,i   ,
and Problem (PQ α,β ) by min f α,β,N (x, u)

s. t. x i+1 = x i + h N (A(t i )x i + B(t i )u i ) , i = 0, . . . , N -1 , x 0 = a , u i ∈ U , i = 0, . . . , N -1 . (PQ N α,β )
Remark 4. Note that analogously to [START_REF] Alt | Linear-Quadratic Control Problems with L 1 -Control Cost[END_REF] we solve a transformed discretized problem (compare also Sect. 4) to compute the solution of Problem (PQ α,β ) numerically.

u α,β h -u 0,β * L 1 ≤ u α,β h -u 0,β L 1 + u 0,β -u 0,β * L 1 ≤ c α h + cβ |β -β * |
with some constant cβ independent of β, which implies the assertion.

Example 1 (The Rocket Car). We consider the popular example of the rocket car, driving from some starting point to it's destination (0, 0). a. e. on [0, 5] .

Table 1 shows numerical results for different meshsizes which confirm the theoretical findings of Theorem 3. To solve the discretized problems we used Ipopt [START_REF] Wächter | On the implementation of an interior-point filter linesearch algorithm for large-scale nonlinear programming[END_REF].

Theorem 3 .

 3 Let (x 0,β * , u 0,β * ) be the solution of Problem (PQ 0,β * ) for which Assumptions (B1) and (B2) are satisfied. Then, for sufficiently large N , choosing α = c α h N and β = β * + c β h N with constants c α and c β , any optimal control u α,β h of Problem (PQ N α,β ) can be estimated byu α,β h -u 0,β * L 1 ≤ c u h N ,where the constant c u is independent of N . Proof. Using [17, Theorem 5.2] and [9, Theorem 5.1, Remark 8] we have

min 1 2 x 1 ( 5 ) 2 + x 2 ( 5 ) 2 + β u L 1 + α 2 u 2 L 2 s

 15225222 . t. ẋ1 (t) = x 2 (t) , ẋ2 (t) = u(t) a. e. on [0, 5] , x 1 (0) = 6 , x 2 (0) = 1 , u(t) ∈ [-1, 1]

Table 1 .

 1 Discretization for different N , β * = 1, β = β * + hN and α = 10 hN .

	N	125	250	500	1000	2000	4000
	u α,β h u α,β -u 0,β * h -u 0,β * h N	L 1 0.2644 0.1344 0.0644 0.0331 0.0177 0.0083 L 1 6.6098 6.7177 6.4409 6.6123 7.0826 6.6752


Ewa Skubalska-Rafaj Lowicz

Ewaryst Rafaj
email: ewaryst.rafajlowicz@pwr.wroc.pl

Deployment of sensors according to quasi-random and well distributed sequences for nonparametric estimation of spatial means of random fields

Keywords: sensor networks, quasi-random sequences, space-filling curves, nonparametric estimation

) and their sub-class recently proposed by the authors [17] that are well-distributed (WD) as sensors' positions in estimating the spatial mean. UD and WD sequences have many interesting properties that are useful both for wireless sensors networks (coverage an and connectivity) and for large area networks such as radiological or environment pollution monitoring stations. In opposite to most popular parameter estimation approaches, we consider a nonparametric estimator of the spatial mean. We shall prove the estimator convergence in the integrated mean square-error sense.

Introduction

Spatial sampling is a crucial issue for proper estimation of parameters in spatio-temporal dynamical models [START_REF] Patan | Optimal sensor network scheduling in identification of distributed parameter systems[END_REF], [START_REF] Uciǹski | Optimal Measurement Methods for Distributed Parameter System Identification[END_REF] and for estimation of spatial fields (see [START_REF] Rasch | Optimal Experimental Design with R[END_REF]). For wireless sensor network (WSN) at least three requirements are crucial, namely, efficient energy usage, coverage and connectivity. Here, we concentrate on the last two of them from the view-point of sensors' deployment. Coverage (or information coverage) is the ability of WSN to cover the whole area, assuming that a single sensor has the ability of collecting information from its (usually circular) neighborhood. The connectivity requires that wireless senors can transmit information from one to another and finally to a sink. Both connectivity and coverage require sensors to be evenly placed in the area. Our aim is to discuss advantages of equidistributed (EQD) (also known as uniformly distributed (UD) or quasi-random points [START_REF] Niederreiter | Random Number Generation and QuasiMonte Carlo Methods[END_REF]) and their sub-class recently proposed by the authors [START_REF] Skubalska-Rafaj Lowicz | Sampling multidimensional signals by a new class of quasi-random sequences[END_REF] that are well-distributed (WD). Furthermore, we shall prove that it is possible to construct a nonparametric estimator of the mean of (possibly correlated) random field that is based on observations from such points. We refer the reader to [START_REF] Akyildiz | Wireless sensor networks: a survey[END_REF] and [START_REF] Chong | Sensor networks: Evolution, opportunities, and challenges[END_REF] for surveys on WSN and to [START_REF] Griffith | Statistical efficiency of model-informed geographic sampling designs[END_REF], [START_REF] Pilz | Spatial sampling design for prediction taking account of uncertain covariance structure. 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences[END_REF], [START_REF] Rasch | Optimal Experimental Design with R[END_REF], [START_REF] Rafaj | Sensors allocation for estimating scalar fields by wireless sensor networks[END_REF], [START_REF] Uciǹski | An algorithm to configure a large-scale monitoring network for parameter estimation of distributed systems[END_REF] for an excerpt of approaches recently proposed for sensors placement.

Equidistributed sequences -good candidates for sensors' sites

Equidistributed sequences, also called uniformly distributed, quasi-random or quasi Monte-Carlo sequences, are well known in the theory of numerical integration (see, e.g., [START_REF] Niederreiter | Random Number Generation and QuasiMonte Carlo Methods[END_REF], [START_REF] Ch | Monte Carlo and Quasi-Monte Carlo Sampling[END_REF]). Here, we summarize some of their basic properties, putting emphasis on those, which indicate that they are good candidates for sensors' positions in WSN.

Definition

Define I d = [0, 1] d as d-dimensional unit cube, which is our space for sensors' deployment. Clearly, most WSN are considered for d = 2, but it can also be of interest in some applications to consider WSN in 3D space, e.g., for air pollution. A deterministic sequence (xi

) n i=1 is called EQD sequence in I d iff lim n→∞ n -1 n i=1 g(xi) = I d g(x) dx (1)
holds for every g continuous on I d . Thus, formally, EQD sequence behave as uniformly distributed random sequences, since (1) mimics a law of large numbers. As we shall see later, EQD sequences are in some sense "more uniform" than uniformly distributed random sequences. The well known EQD sequences include Corput, Halton, Hammersley, Korobov, Zaremba and Sobol (see, e.g., [START_REF] Kuipers | Uniform Distribution of Sequences[END_REF], [7] [8]). For our purposes, the following example is important. Important example -the Weyl sequence is defined as follows:

ti = fractional part(i θ), i = 1, 2, . . . ,
where θ is selected irrational number. E.g., quadratic irrational -θ = ( √ 5-1)/2 behaves quite well in practice. In general, as it is more difficult to approximate θ by rational numbers, then it is better candidate for generating the Weyl sequence.

Discrepancy as indicator of WSN coverage and connectivity

A discrepancy Dn of EQD sequence is well known measure of its uniformity (see, e.g., [START_REF] Kuipers | Uniform Distribution of Sequences[END_REF], [START_REF] Ch | Monte Carlo and Quasi-Monte Carlo Sampling[END_REF]). Dn discrepancy of (xi) n i=1 is defined as follows:

Dn = sup A⊂I d µ d (A) - Nn(A) n , (2)
where A is any parallelepiped and supremum is taken with respect to all such A ⊂ I d , µ d (A) is d-dimensional Lebesgue measure (the area or volume) of A, while Nn(A) is number of xi's in A.

As in the classical Monte-Carlo method, one can expect that for fairly spaced sensors µ d (A) is close to the fraction of sensors in A. For interpreting [START_REF] Niederreiter | Random Number Generation and QuasiMonte Carlo Methods[END_REF] assume that the supremum is attained for a certain set Ân ⊂ I d . Then, Ân is the set that is most unevenly covered by sensors, i.e., it contains too small or too large number of sensors in comparison to other areas. For this reason, in our opinion, Dn is a a good measure for connectivity and coverage of a sensors' net. It is, however, not easy to find Ân numerically and for this reason in the theory of numerical integration a simplified, but still useful, version of discrepancy, called D * n discrepancy, is more frequently used. Let Π(x) be the parallelepiped in I d with vertices in (0, . . . 0) and x. D * n discrepancy of (xi) n i=1 is defined as follows:

D * n = sup x∈I d µ d (Π(x)) - Nn(x) n , (3)
where Nn(x) is the number of xi's in Π(x).

Notice that D * n is an analog to Kolmogorov-Smirnoff statistics for testing the uniformity of a distribution. D * n can be efficiently calculated for a given set of points in 1D, 2D. Furthermore, it can be proved (see [START_REF] Ch | Monte Carlo and Quasi-Monte Carlo Sampling[END_REF]) that

D * n ≤ Dn ≤ 2 d D * n . (4)
For "good" known EQD sequences:

D * n = log d (n) n .
This is much better than for "usual" uniformly distributed random variables, for which

D * n ∼ 1 √ n .
The same order 1/ √ n is obtained when equidistant grid is considered in R 2 .

Basic properties of space-filling curves

Our idea is to transform a sequence of one dimensional EQD sequence by a space-filling curve (SFC) in order to obtain multidimensional EQD sequence with good properties. Note that a similar construction has been already proposed and used in the theory of numerical integration, but the transformed sequence was equidistant, i.e., i/n, i = 1, 2, . . . , n. Below, we summarize known properties of space filling curves that are either directly used in the rest of the paper or indicate why we can expect that our construction leads to sensors' site with good coverage and connectivity. Space-filling curve is a mapping

Φ : [0, 1] onto → I d such that • Φ(t) a continuous function in I1 = [0, 1] • maps I1 = [0, 1] onto I d -d-dim. cube.
The Hilbert, Peano and Sierpinski are well known examples of SFCs [START_REF] Sagan | Space-filling Curves[END_REF]. All these curves: -preserve areas and neighbors (see SFC2, SFC 3 below), -fill the space uniformly and as densely as desired.

-can be generated recursively and efficiently. For the Peano, Hilbert and Sierpiński curves the following properties hold: SFC 1) ∀g :

I d → R, g -continuous I d g(x) dx = 1 0 g(Φ(t)) dt, (5)
where x = [x (1) , x (2) , . . . , x (d) ] SFC 2) Hölder continuity:

∃C Φ >0 ||Φ(t) -Φ(t )|| ≤ CΦ|t -t | 1/d , (6)
where

||.|| is the Euclidean norm in R d .
It is well known that Φ does not have the inverse. However, for any Borel set A ⊂ I d we can define Φ -1 (A) as preimage of A.

SFC 3) Φ preserves the Lebesgue measure in the sense that

∀A⊂I d , µ d (A) = µ1(Φ -1 (A)), (7)
where µ1, µ d are the Lebesgue measures in R1, R d , respectively. In other words, volume of Borel set A in I d is equal to the length of Φ -1 (A) in I1.

We stress that the Peano, Hilbert and Sierpiński curves can be approximated efficiently, using O d ε arithmetic operations, where ε > 0 is the approximation accuracy (see [START_REF] Butz | Alternative algorithm for Hilbert's space-filling curve[END_REF], [START_REF] Skubalska-Rafaj Lowicz | Pattern recognition algorithm based on space-filling curves and orthogonal expansion[END_REF]). Notice that for our purpose we shall calculate Φ(t) once for t = t1, t2, . . . , tn. Approximation of the Hilbert SFC is shown in Fig. 1.

Proposed EQD sequences

The following algorithm generates EQD sequences that can be used as sensors' positions. Algorithm 1

Step 1) Generate EQD sequence in [0, 1] by the Weyl method: ti = frac(i θ), i = 1, 2, . . . , n, where θ is irrational number, e.g., θ = ( √ 5 -1)/2.

Step 2*) Sort ti's and get t (1) < t (2) < . . . < t (n) . This step is optional, it serves mainly for theoretical purposes, but it can also be used to determine ordering of senors along SFC, which -in turn -can be used to indicate the ordering of information transmission between sensors.

Step 3) Select SFC and generate xi's as follows: xi = Φ(t (i) ), i = 1, 2, . . . , n. Below we state properties of the above generated sequences that justify their usefulness as sensors' positions. Algorithm 1 generates sequences that are extendable, i.e., one can add points without recalculating positions of earlier This property is

n -1 n i=1 g(xi) = n -1 n i=1 g(Φ(ti)) → → 1 0 g(Φ(t))dt = I d g(x) dx, (8)
since {ti} n i=1 are EQD, g(Φ(.)) is also continuous, while last equality follows from SFC 1). The proof of WD property uses the Weyl criterion, generalized to WD that is too complicated to be presented here (see [START_REF] Skubalska-Rafaj Lowicz | Sampling multidimensional signals by a new class of quasi-random sequences[END_REF] for details). Corollary 1. Under the same assumptions as in Thm. 1, for points generated by Algorithm 1 we have:

Dn → 0 and D * n → 0 as n → ∞ (9)
The first statement follows from ( 8) by selecting g as indicator functions of parallelepipeds. The second one is a direct consequence of (4).

Points generated by Algorithm 1, using the Hilbert SFC and θ = ( √ 5 -1)/2 are shown in Fig. 2. We do not have a closed form formula for (11) They follow a general pattern of "good" EQD sequences

D * n (d) = O log d (n) n . (12)
5 Nonparametric estimation of spatial means of random fields

Our aim in this section is to show how to estimate the spatial mean of a random field using WD points generated by Algorithm 1. The proposed estimator is similar to those proposed earlier in [START_REF] Rafaj | Halton and Hammersley sequences in multivariate nonparametric regression[END_REF] and [START_REF] Rafaj | Equidistributed designes in nonparametric regression[END_REF], but it differs in the following two respects.

-In [START_REF] Rafaj | Halton and Hammersley sequences in multivariate nonparametric regression[END_REF] and [START_REF] Rafaj | Equidistributed designes in nonparametric regression[END_REF] points were generated using the Halton-Hammersley sequences, which are EQD, but it is not known whether they are WD or not, while our sensors' positions are WD. -Here, we allow correlated observations. Let us assume that observations yi of a scalar random field with unknown mean f (x) are collected at spatial points xi's , generated by Algorithm 1. More precisely,

yi = f (xi) + εi, i = 1, 2, . . . , n, (13)
where εi are random variables (r.v.'s) for which the following conditions hold: ERR1) εi's have zero mean, finite variance σ2 , ERR2) for each i = 1, 2, . . . , n the covariance E(εi εj) is not equal to zero only for a finite number, 0 ≤ G(n) < n of indices j = 1, 2, . . . , n.

This assumption means that we allow for correlations between observations from i-th sensor and observations from only a finite number G(n) of its nearest neighborhood sensors with indices in a set denoted by O(i).

As we shall see later, we may allow G(n) to grow with n, but rather slowly. It is a reasonable assumption since the statistical dependence between sensors fades out rapidly as a function of a distance between them. Clearly, if errors are uncorrelated, we have O(i) = {i}.

Our aim is to estimate f from (xi, yi), i = 1, 2, . . . , n in a nonparametric way, i.e, without imposing any finite parametrization of f . Instead, we impose some smoothness assumptions on f and our aim is to construct estimator fn(x) such that the integrated mean-square error (IMSE) I(f, fn)

def = I d E(f (x) -fn(x)) 2 dx → ∞ as n → ∞.
We select the following estimator fn of f . Algorithm 2

fn(x) = N k=1 âkn v k (x), (14) âkn
= 1 n n i=1 yiv k (xi), k = 1, 2, . . . , N, (15)
where v1, v2, . . . is a complete sequence of orthonormal functions1 in the space L2(I d ) of square integrable functions on I d . Denote by a k =

I d f (x)v k (x) dx, k = 1, 2, .
. ., the coefficients of the Fourier series of f in the basis (v k ), i.e.,

f (x) ∼ ∞ k=1 a k v k (x), ( 16
)
where convergence is understand in L2 norm. In [START_REF] Rasch | Optimal Experimental Design with R[END_REF] one can recognize the truncated version of ( 16). In practice, the truncation point N is also estimated. In asymptotical considerations below, we admit a slowly growing dependence of N on the number of sensors n and we shall write N (n).

Remark 1.

Notice that âkn is the estimator of a k , which is asymptotically unbiased. Indeed, equidistribution of xi's implies

E(â kn ) = n -1 n i=1 f (xi) v k (xi) → I d f v k = a k . (17)
But, due tu WD property of xi's, we can say more, namely

E(â kn ) = n -1 n+p i=p f (xi) v k (xi) → a k , (18)
uniformly w.r.t p. This property seems to be important for WSN, because it frequently happens that a group of sensors fails (due to, e.g., a battery or communication faults) and can be replaced by differently located sensors. From ( 18) it follows that WSN based on sensors placed at WD points still can provide estimators with a small bias, provided that n, i.e., the number of active sensors, is sufficiently large. Remark 2. Notice that (15) has the form that is well suited for collecting data along a network, because coefficients can be calculated recursively. Indeed, v k (xi)' s, (k = 1, 2, . . . , N ) can be pre-computed and stored in i-h sensor. Sensors can be ordered along SFC (see Fig. 2) and values of partial sums, after adding yi v k (xi) to previous ones can be passed from one sensor to another. The role of the sink is to calculate [START_REF] Rasch | Optimal Experimental Design with R[END_REF] . Now, our aim is to sketch the proof of IMSE consistency of fn. For simplicity of formulas, we assume that d = 2 and ONS) v k (x (1) , x (2) ) are ordered and normalized trigonometric functions of the form: 1, sin(x (1) ) sin(x (2) ), sin(x (1) ) cos(x (2) ), cos(x (1) ) sin(x (2) ), cos(x (1) ) cos(x (2) ), . . .. We have N = N (n) such products in common and they are commonly bounded by H > 0, say, while their derivatives are commonly bounded by N (n) H. Notice that N (n) implicitly depends on d, so for d = 2, N (n) = M1(n) M2(n), where M1(n) and M2(n) are the numbers of trigonometric functions in x (1) and x (2) variables, respectively. Below, when we write N (n) → ∞, we require that M1(n) → ∞ and M2(n) → ∞ One can weaken this assumption by allowing H be dependent on k, as it is, e. g., for the Legendre polynomials.

The orthonormality of (v k ) implies

IM SE( fn, f ) = Wn + B 2 n + R(N, f ) (19)
where R(N, f

) def = ∞ k=N +1 a 2 k , Wn def = N k=1
Var(â kn ), (20)

B 2 n def = N k=1 (E(â kn ) -a k ) 2 , (21)

Fig. 1 .

 1 Fig. 1. Approximation of the Hilbert SFC using 1024 points.

Fig. 2 .

 2 Fig.2. n = 256 points generated by Algorithm 1, using the Hilbert SFC, and linked by thin lines, indicating their ordering on SFC that can be used as ordering for transmitting information between sensors.

Orthonormality means that v k (x)

dx = 1 and v k (x)v (x) dx = 0, for all k =

To estimate the variance, let us note that

Due to ONS) and ERR2) we have

. Denote by V (I d ) the space of functions having bounded variation and let V(g) denote the total variation of g ∈ V (I d ). Then, by the Koksma-Hlavka inequality (see, e.g., [START_REF] Kuipers | Uniform Distribution of Sequences[END_REF]) we obtain for f ∈ V (X) ∩ C(X)

where c1 > 0 is a constant independent of n, while the second inequality can be obtained in a way similar to the one that was used in [START_REF] Rafaj | Halton and Hammersley sequences in multivariate nonparametric regression[END_REF]. Thus,

and finally, we obtain

then IM SE( fn, f ) → 0.

In the case of bivariate random fields, a particular choice of

, where > 0 is arbitrarily small. Thus, selecting M1(n) = M2(n) = c n β/2 , c > 0, 0 < β < 2 (1 -)/3, we can assure that for N (n) = M1(n) M2(n) conditions (26) hold.


Christoph Weise

Kai Wulff

Marc-Hinrik Höper

Romain Hurtado

Continuous-Time Local Model Network for the Boost-Pressure Dynamics of a Turbocharger

Keywords: Local model network, modelling, turbocharger

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Mathematical modelling is an essential tool for system analysis and control. In this regard we motivate our aim to derive a mathematical model that describes the considered process in an adequate fashion. Precise physical models, however, are often hard to obtain and are frequently challenged by tedious parameter estimation. Moreover, physical models can be very sensitive with respect to parameter variations such that an appropriate set of parameters is very hard to obtain. In particular nonlinear dynamical processes suffer from this problem. Resorting to grey-box models that are driven by experimental data can often resolve this problem. For processes with nonlinear dynamics local-model networks can be used to represent the global dynamics by identifying simple linear models locally and combine them to a network that matches the nonlinear global dynamics sufficiently. This is particular useful in industrial practice where simple controller structures with few tunable parameters are often preferred. Therefore we aim for an continuous-time LMN such that most common manual controldesign methods are readily applicable.

The basic concepts for local model networks have been developed in the nineteens, see e.g. [START_REF]Multiple model approaches to modelling and control[END_REF][START_REF] Nelles | Nonlinear system identification with local linear neuro-fuzzy models[END_REF], for early work. Ever since local model networks have been studied for various system classes and applications range from mechatronical systems to process engineering. Typical issues that arise in most studies pertain to the implementation of the local model network. In [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF][START_REF] Hentabli | State-space local model networks based continuous-time gpc. application to induction motor[END_REF] the local models are realised each having their own local state-space. The overall output of the model network is then obtained by interpolation of the local model outputs. The complement approach is taken in [START_REF] Verdult | Non linear system identification : a state-space approach[END_REF], where the state is shared by all local models. Here the parameters are interpolated in appropriate fashion to emulate the nonlinear dynamics of the process. The vast majority of the publications, however, consider discrete-time systems.

An important roll for the complexity of the local model network plays the choice of the operating regimes of the individual models. In this regard several methods have been proposed to partition the scheduling space. A summary on existing partitioning strategies is given in [START_REF] Nelles | Axes-oblique partitioning strategies for local model networks[END_REF] and [START_REF] Hartmann | Recent partitioning strategies for local model networks[END_REF].

More details on local models trees based on recursive orthogonal splitting (LoLiMoT) give [8-10, 2, 11]. This partitioning strategy uses hyper-rectangles and is time-consuming regarding large scheduling problems. Axis oblique partitioning strategies using the HiLoMoT algorithm can be found in [START_REF] Nelles | Axes-oblique partitioning strategies for local model networks[END_REF] and [START_REF] Hametner | Neuro-fuzzy modelling using a logistic discriminant tree[END_REF]. Other publications focus on the optimisation of the partitions for a fixed number of models [START_REF] Verdult | Non linear system identification : a state-space approach[END_REF] itself. Based on the chosen partitioning the used weighting functions have an additional effect on the performance of the model [START_REF] Hartmann | On the smoothness in local model networks[END_REF].

In this contribution we investigate the suitability of local model network for continuous-time domain. In regard to the controller design the dynamics and the gain of the process are equally important. In particular whenever the dominating time-constant of the dynamics vary within the operating range, good knowledge of the latter is vital in order to design high-performing controllers. Therefore we choose an approach that is able to map individual dynamics to different points in the scheduling space, as well as individual gains.

For the resulting local model network we address in particular issues regarding the offsets of the local models and suitable partitioning techniques. For processes that exhibit strong nonlinear dynamics, the local model network may need a very large number of models to match the behaviour appropriately, when the local operating points are evenly distributed. In order to keep the number of local models moderately large, we apply the LoLiMoT algorithm and propose a novel technique that is able to introduce multiple partitions in each iteration and may lead to partitions that fit the nonlinearity better.

The described techniques are applied to model the boots-pressure dynamics of a turbo charged combustion engine. The experimental data for modelling and verification are obtained from a full-scale test rig.

Local model networks

The local model network (LMN) approach [START_REF]Multiple model approaches to modelling and control[END_REF] uses the strategy of divide and conquer to describe a complex and non-linear behaviour of a dynamical system, given by:

y = h(x(t), u(t), t) . (1)
First of all the operating range of the system is divided in M regions, wherein the dynamics can be approximated by a more simple (e.g. linear or affine) model hi (x(t), u(t), t).

In order to cover the complete operating range the interaction of the local models is controlled by the scheduler Φ, a vector of exogenous or system variables prior chosen. Each model is weighted with a functions ρ i (Φ) ∈ [0, 1], representing the range of validity of the i-th model. The system dynamics (1) are then approximated by the LMN by the interpolation of the local models

ỹ = M i=1 ρ i (Φ) hi x(t), u(t), t with M i=1 ρ i (Φ) = 1 .
In this paper we examine the application of the LMN approach to nonlinear dynamical systems of the form

ẋ(t) = f (x(t), u(t)), x(0) = x 0 with: x ∈ IR n , y ∈ IR q , u ∈ IR p y(t) = h(x(t), u(t)) .
(

) 2
The nonlinear dynamics shall be approximated by linearisations at M stationary solutions (

x * i , u * i ) with f (x * i , u * i ) = 0: ẋ(t) = f (x * i , u * i ) + A i (x(t) -x * i ) + B i (u(t) -u * i ) y(t) = h(x * i , u * i ) + C i (x(t) -x * i ) + D i (u(t) -u * i ) . (3)
Following [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF] we can define M models with local states xi (t

) = x(t) -x * i , inputs ũi (t) = u(t) -u * i and outputs ỹi (t) = y(t) -h(x * i , u * i ).
This leads to a localstate representation of the LMN where the output is a weighted sum of the single models' outputs. This description proves sufficient for controller design based on local observes as discussed in [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF].

However, simulation purposes or even stability analysis may pose further requirements onto the LMN [START_REF] Leith | Issues in the design of switched linear control systems: A benchmark study[END_REF][START_REF] Wulff | A control design method for a class of SISO switched linear systems[END_REF]. In a local-state architecture, the initial state of the local model may induce strong transient responses at the switching instance which are not desirable and may affect stability [START_REF] Wulff | On the stabilisation of a class of SISO switched linear systems[END_REF]. Furthermore, whenever the scheduling variable switches different output signals between the subsequent local models will cause discontinuities of the output, which will not be observed in the original nonlinear process behaviour.

Therefore we shall choose a global-state representation of the LMN, where the state-vector x is shared by all local models. Then the linearisation (3) becomes an affine system with local system-offset K i and the local output-offset L i :

K i = -(A i x * i + B i u * i ) L i = h(x * i , u * i ) -(C i x * i + D i u * i ) . (4)
For the global-state LMN we obtain the parameter-varying affine system

ẋ = A(Φ)x + B(Φ)u + K(Φ) y = C(Φ)x + D(Φ)u + L(Φ) , (5)
where

A(Φ) = M i=1 ρ i (Φ)A i and B(Φ), K(Φ), C(Φ), D(Φ)
and L(Φ) with according interpolations.

Note that the state-space of the global-state LMN has M -time smaller dimension than that of the local-state LMN. Furthermore, the state is continuous for all variations in Φ. With some mild assumptions the same holds for the output.

Experimental identification of the pressure-dynamics

Modern combustion engines are frequently equipped with an exhaust turbocharger that boosts the pressure of the intake air in order to increase the amount of oxygen in the cylinder. Often the compressor power can be manipulated by varying the geometry of the turbine, so-called variable-geometry turbine (VGT). This allows for influencing the charging pressure within certain constraints. While the physical relations of the quantities are well known, e.g. [START_REF] Moran | Fundamentals of engineering thermodynamics[END_REF], deriving a dynamical model of the pressure dynamics and identifying its parameters is very time-consuming as such models exhibit strong non-linearities and are very sensitive with respect to certain parameter variations [START_REF] Jankovic | Constructive lyapunov control design for turbocharged diesel engines[END_REF][START_REF] Jung | Calibratable linear parameter-varying control of a turbocharged diesel engine[END_REF][20]. A data-driven black-box approach may therefore prove valuable in this context and, thus, the pressure dynamics are a very suitable process to investigate properties and challenges of the LMN approach. The data-base for identification consists of 16 time-series of experimental input-output data at various engine operating points (n i , q i ). Within one testcycle the engine speed n i and injection rate q i was maintained constant whereas the compressor power P C was increased stepwise, see Fig. 1 for a sample timeseries.

LMN identification

The engine speed and injection rate are natural scheduling variables from an engineering point of view. We can further observe that the dynamics vary significantly for different levels of the compressor power P C , cf. Fig. 1. Therefore we choose Φ = [n, q, P C ] T as scheduling vector; the system's input is the compressor power u := P C , the output is boost-pressure y := p 2 .

Based on the data structure the most simple partitioning approach of a regular grid is used to divide the total working range into 256 local regions. For each of these local regions we choose a grey-box approach to identify the local dynamics. In our case physical considerations lead to the following parametrisation of the local models:

G(s) = V τ 1 s + 1 (τ 2 s + 1)(τ 3 s + 1)
.

The parameters (V i , τ 1i , τ 2i , τ 3i ) of the local models i = 1, . . . , M are obtained by least-square optimisation for each individual step in the time-series with removed offsets. In order to cast the local models in into state-space representation (5) we choose the observer canonical form which has constant output matrix that renders the output continuous for all variations of Φ. The regions of validity of each model are defined by triangular weighting functions ρ i .

The offset parameters K i and L i in (4) are chosen to account for the removed offset in the local identification. Without loss of generality we choose L i = 0 for all i. For calculating K i we note, that the second state-component is the output and thus can be taken from the data. For the first local model of each time-series then we have:

K i = -B i u(0) -A i x 1 (0) y(0) ,
where only x 1 (0) can be chosen arbitrarily. For the second step of a given timeseries we obtain the initial condition from the final state using the already identified parameters of the first model, etc. Determining the offsets in this fashion yields a continuous output for each time-series as shown in Fig. 1.

Verification of the LMN

In order to verify the LMN's ability to match the non-linear process dynamics, we simulate the LMN using a highly dynamic verification-cycle covering the full operation range. Fig. 2 shows the evolution of the scheduling variable's components during that test-cycle featuring steep sloped jumps as well as various smooth transitions. The overall performance of the LMN is quite satisfactory with a mean square error of M SE = 2.72 × 10 3 hPa 2 and a maximum error |e max | = 223.6 hPa. Fig. 3 shows two details of the full cycle. The model matches the experimental data very well for dynamical (right) and stationary (left) areas. However, we note that for high compressor powers the stationary error may rise as high as 100 hPa. This may be due to high noise-levels in the data for these operating points that prohibit a precise estimate of the DC-gain in this regions.

Local model trees

The grid-based approach of the previous section typically leads to a large number of local models that are equally spaced in the operating range, irrespective of the character of the nonlinearity. Local model tree algorithms address these issues by starting with a low number of models and improving the model network iteratively by introducing additional models, if the error is large in a certain region. Thereby, only few models will be placed in regions where the process behaves in an almost linear fashion and local over-fitting is avoided. While there are a number of partitioning strategies available, an orthogonal partitioning appears to be a natural choice in our case, as the data for identification is distributed in a grid-like fashion within the scheduling space. The classical LoLiMoT-algorithm presented in [START_REF] Nelles | Local basis function networks for identification of a turbocharger[END_REF], [START_REF] Nelles | Basis function networks for interpolation of local linear models[END_REF] uses such orthogonal partitioning. In each iteration the region of the worst performing local model is split in 2 hyper-rectangles wherein the parameters for new local models are identified. Every possible division (one for each dimension of the scheduling space) is analysed and the best division is chosen.

Local error based partitioning (LEB)

The classical LoLiMoT considers the accumulated quadratic error to evaluate the performance of the model and thus does not use the full available information at each iteration. In this section we propose a novel partitioning strategy that uses the signed error at each point in the scheduling space in relation to certain thresholds τ -and τ + dividing the scheduling space into 3 regions in one iteration. As illustrated in Fig. 4 this may lead to a better approximation of the nonlinearity by using a smaller number of models compared to the classical bisection. Our optimisation data is given at discrete points of the scheduling space. Therefore we can define the local error

e l (Φ k ) := t end 0 [ŷ(t) -y(t)] ρ k (Φ(t)) dt
for each local model k = 1, 2, . . . , M max . In order to find the best cut in the scheduling space the local error is projected onto one component by the weighted sum of the local errors of the remaining dimensions.

For the boost-pressure dynamics we obtained either a quadratic or linear distribution of this projected local error (PLE), cf. Finally we embedded this approach into the LoLiMoT algorithm [START_REF] Nelles | Local basis function networks for identification of a turbocharger[END_REF] (see Fig. 5):

1. The weighting functions of the initial model are calculated. If only one global model is used, the global weight is set to ρ 1 (Φ) = 1. 2. The partitioning criterion is evaluated choosing the worst model. the chosen model covers the smallest possible region given by the data structure its partitioning criteria is set to zero and the next model is chosen. The inner loop can be evaluated using parallel computing with one threat covering each dimension of the scheduling vector. If linear triangular weighting functions are used, only a part of the weighting functions needs to be updated. Because every iteration leads to 2 new local models (best case) the approach reduces the number of optimisation tasks from 4N (M -1)/2 + 1 optimisation task for the classical LoLiMoT to 3N (M -1)/2 + 1 where M denotes the total odd number of models and N the dimension of the scheduling vector.

LM2 LM1 LM3 LM1' LM2' LM3' LM1 LM2 LM3 LM4 LM5 LM1' LM2' LM3' LM5' LM4' Φ1 Φ1 Φ1 ep,2 Φ1 Φ1 Φ1 Φ1 Φ2 Φ2 Φ2 Φ2 Φ2 Φ2 Φ2 initial model ep,2 ep,1 ep,1                        2nd iteration: error evaluation                        1st iteration: error evaluation                        1st iteration: select best partition                        2nd

Results and comparison of the partitioning techniques

We used the LoLiMoT with LEB partitioning technique as well as the classical bisection method to obtain a reduced LMN. To have a fair comparison we ran the latter for more iterations to obtain the same number of local models. Both networks are then compared to the full-grid LMN obtain in Section 3 using the verification cycle, see Fig. 6. Both networks (with only 10 local models each) are able to match the nonlinear dynamics reasonably well. However, the stationary error is significantly larger compared to the full-grid LMN with 256 models. Increasing the number of models reduces both algorithms can reduce the error measure as to be expected. Often the LEB-network shows better stationary behaviour than the bisectionnetwork, but the data does not allow for a strong statement here. However, the overall error of the LEB-network is significantly better than the errors produced by the bisection-model, see Table 1. This holds in particular for the maximum error which is up to 25% lower for the LEB-network.

Conclusions

In this work we apply a continuous-time global-state local model network to identify the boost-pressure dynamics of an exhaust turbocharger using experimental input-output data. The approach is feasible to match the strongly nonlinear dynamics and may therefore be suitable as a basis for control design. We discuss several implementation issues such as the choice of state-space representation, global-state implementation and the use of discontinuous time-series at various operation points. We propose a novel partition strategy that is computationally more efficient compared to the classical bisection method and also yields better results in the global performance using an independent verification-cycle.

Fig. 1 .

 1 Fig. 1. Sample of data time-series and identification result

Fig. 2 .

 2 Fig. 2. Evolution of the scheduling variables during the validation cycle.

Fig. 3 .

 3 Fig. 3. Simulation result using the validation cycle with 256 local models.

Fig. 4 .

 4 Fig. 4. Illustration of local error based partitioning (left) and classical LoLiMoT partitioning (right) for a simple static nonlinearity.

Fig. 5 .

 5 Based on this classification of the PLE distribution an adaptive threshold τ is calculated: 1. If the PLE shows a parabolic distribution (with 2 zero-crossings) the threshold is set to zero τ = 0 defining three regions, based on the assumption that the area of validity of a local model should be compact. 2. If the PLE shows a linear distribution (with a single zero-crossing) the symmetric thresholds |τ + | = |τ -| are calculated based on the PLE: τ ± = ±κ • min{e l,min ; e l,max } with a constant factor κ < 1 three regions are set. 3. If the PLE distribution cannot be categorised clearly, the classical bisection is applied defining two new regions.

Fig. 5 .

 5 Fig. 5. LoLiMoT algorithm with local error based (LEB) partitioning algorithm.

3 .

 3 The local error for the complete operation range is calculated. 4. For every scheduling variable Φ 1 , . . . , Φ N do:(a) The projected local error and the partitioning thresholds are calculated, if the original region is minimal regarding the dimension i, the temporary partitioning criteria is set to infinity and the following steps are skipped. (b) The model parameters are optimized locally. (c) The weighting function and the partitioning criteria of the new models are determined. 5. The best partitioning is chosen and the number of local models increased. 6. If any abort criterion is reached (e.g. maximum number of models), the LMN is completely defined. Otherwise the iteration starts again at point 1.

Fig. 6 .

 6 Fig. 6. Comparison of the LMN performances using the verification cycle.

Table 1 .

 1 LMN error development regarding the validation cycle.

		LoLiMoT	Local error based
	M	M SE/hPa 2	|emax|/hPa M SE/hPa 2	|emax|/hPa
	10	4.1e3	253.9	2.4e3	203.8
	30	2.7e3	200.5	2.8e3	165.5
	45	2.4e3	208.9	2.3e3	157.5
	60	2.4e3	208.9	2.2e3	159.6


Dimitri Blueschke

Viktoria Blueschke-Nikolaeva

Reinhard Neck
email: reinhard.neck@uni-klu.ac.at

Stochastic control of econometric models for Slovenia

Keywords: Optimal control, stochastic control, algorithms, policy applications, nonlinear models

This paper considers the optimal control of two small stochastic models of the Slovenian economy applying the OPTCON algorithm. OPTCON determines approximate numerical solutions to optimum control problems for nonlinear stochastic systems and is particularly applicable to econometric models. We compare the results of applying the OPTCON2 version of the algorithm to the nonlinear model SLOVNL and the linear model SLOVL. The results for both models are similar, with open-loop feedback controls giving better results on average but with more 'outliers' than open-loop controls.

Introduction

Optimum control theory has been applied in many areas of science, from engineering to economics. An algorithm that provides (approximate) solutions to optimum control problems for nonlinear dynamic systems with different kinds of stochastics is OPTCON, which was first introduced in [START_REF] Matulka | OPTCON: An Algorithm for the Optimal Control of Nonlinear Stochastic Models[END_REF]. An extension has been developed in [START_REF] Blueschke-Nikolaeva | Optimal Control of Nonlinear Dynamic Econometric Models: An Algorithm and an Application[END_REF], which includes passive learning or open-loop feedback control policies.

OPTCON was implemented in MATLAB and can deliver numerical solutions to problems with real economic data. Two such applications are described and analyzed in this paper. We develop two macroeconomic models of the Slovenian economy, a nonlinear model called SLOVNL and a (comparable) linear model called SLOVL. The algorithm with both open-loop and open-loop feedback strategies is applied to these models and the influence of the control scheme and the nonlinearity of the model on the optimum solution is investigated in some optimization experiments.

The problem

The OPTCON algorithm is designed to provide approximate solutions to optimum control problems with a quadratic objective function (a loss function to be minimized) and a nonlinear multivariate discrete-time dynamic system under additive and parameter uncertainties. The intertemporal objective function is formulated in quadratic tracking form, which is quite often used in applications of optimum control theory to econometric models. It can be written as

J = E T t=1 L t (x t , u t ) , (1)
with

L t (x t , u t ) = 1 2 x t -xt u t -ũt W t x t -xt u t -ũt . ( 2
)
x t is an n-dimensional vector of state variables that describes the state of the economic system at any point in time t. u t is an m-dimensional vector of control variables, xt ∈ R n and ũt ∈ R m are given 'ideal' (desired, target) levels of the state and control variables respectively. T denotes the terminal time period of the finite planning horizon. W t is an ((n + m) × (n + m)) matrix, specifying the relative weights of the state and control variables in the objective function. W t (or W ) is symmetric.

The dynamic system of nonlinear difference equations has the form

x t = f (x t-1 , x t , u t , θ, z t ) + ε t , t = 1, ..., T, (3)
where θ is a p-dimensional vector of parameters the values of which are assumed to be constant but unknown to the decision maker (parameter uncertainty), z t denotes an l-dimensional vector of non-controlled exogenous variables, and ε t is an n-dimensional vector of additive disturbances (system error). θ and ε t are assumed to be independent random vectors with expectations θ and O n respectively and covariance matrices Σ θθ and Σ εε respectively. f is a vectorvalued function fulfilling some differentiability assumptions, f i (.....), is the i-th component of f (.....), i = 1, ..., n.

The optimum control algorithm

The OPTCON1 algorithm [START_REF] Matulka | OPTCON: An Algorithm for the Optimal Control of Nonlinear Stochastic Models[END_REF] determines policies belonging to the class of openloop controls. It either ignores the stochastics of the system altogether or assumes the stochastics to be given once and for all at the beginning of the planning horizon. The nonlinearity problem is tackled iteratively, starting with a tentative path of state and control variables. The tentative path of the control variables is given for the first iteration. In order to find the corresponding tentative path for the state variables, the nonlinear system is solved numerically. After the tentative path is found, the iterative approximation of the optimal solution starts. The solution is sought from one time path to another until the algorithm converges or the maximal number of iterations is reached. During this search the system is linearized around the previous iteration's result as a tentative path and the problem is solved for the resulting time-varying linearized system. The criterion for convergence demands that the difference between the values of current and previous iterations be smaller than a pre-specified number. The approximately optimal solution of the problem for the linearized system is then used as the tentative path for the next iteration, starting off the procedure all over again. The more recent version OPTCON2 [START_REF] Blueschke-Nikolaeva | Optimal Control of Nonlinear Dynamic Econometric Models: An Algorithm and an Application[END_REF] incorporates both open-loop and open-loop feedback (passive-learning) controls. The idea of passive learning corresponds to actual practice in applied econometrics: at the end of each time period, the model builder (the control agent) observes what has happened, that is, the current values of state variables, and uses this information to re-estimate the model and hence improve his/her knowledge of the system.

The passive-learning strategy implies observing current information and using it in order to adjust the optimization procedure.

The SLOVNL model

We estimated two simple macroeconometric models for Slovenia, one nonlinear (SLOVNL) and one linear (SLOVL). The SLOVNL model (SLOVenian model, Non-Linear version) is a small nonlinear econometric model of the Slovenian economy consisting of 8 equations, 4 behavioral equations and 4 identities. SLOVNL includes 8 state variables, 3 control variables, 4 exogenous noncontrolled variables and 16 unknown (estimated) parameters. We used quarterly data for the time periods 1995:1 to 2006:4; this data base with 48 observations admits a full-information maximum likelihood (FIML) estimation of the expected values and the covariance matrices for the parameters and the system errors. The starting period for the optimization is 2004:1; the terminal period is 2006:4 (12 periods).

Model variables used in SLOVNL:

Endogenous (state) variables:

x[1] CR real private consumption x[2] IN V R real investment x[3] IM P R
GDP Rt = CRt + IN V Rt + GRt + EXRt -IM P Rt V Rt = GDP Rt + IM P Rt P Vt = GDP R t V R t • GDP DEFt + IM P R t V R t • IM P DEFt P i4t = P V t -P V t-4 P V t-4 • 100 Table 1. 'Ideal' values of objective variables, SLOVNL CR IN V R IM P R ST IRLN GDP R V R P V P i4 T axRate GR M 3N 1% 1% 2% -0.25 1% 1.5% 0.75% 3 25.2 1% 1.75%
The objective function penalizes deviations of objective variables from their 'ideal' (desired, target) values. The 'ideal' values of the state and control variables (x t and ũt respectively) are chosen as shown in Table 1. The 'ideal' values for most variables are defined in terms of growth rates (denoted by % in Table 1) starting from the last given observation (2003:4). For P i4 and T axRate, constant 'ideal' values are used; for ST IRLN , a linear decrease of 0.25 per quarter is assumed to be the goal.

The weights for the variables, i.e. the constant matrix W in the objective function, are first chosen as shown in Table 2a ('raw' weights) to reflect the relative importance of the respective variable in the (hypothetical) policy maker's objective function. These 'raw' weights have to be scaled or normalized according to the levels of the respective variables to make the weights comparable. The normalized ('correct') weights are shown in Table 2b.

---------- ---------- CR 1 CR 3.457677 IN V R 1 IN V R 12.16323 IM P R 1 IM P R 1.869532 ST IRLN 1 ST IRLN 216403.9 GDP R 2 GDP R 2 V R 1 V R 0.333598 P V 1 P V 423

The SLOVL model

To analyse the impact of the nonlinearity of the system we developed a linear pendant to the SLOVNL model. This 'sister model' is called SLOVL (SLOVenian model, Linear version) and consists of 6 equations, 4 of them behavioral and 2 identities. The model includes 6 state variables, 3 exogenous non-controlled variables, 3 control variables, and 15 unknown (estimated) parameters. We used the same data base as for SLOVNL and a specification as close as possible to that of SLOVNL in order to make comparisons between the results of the algorithm for a linear and a nonlinear model. Again, we used full-information maximum likelihood (FIML) to estimate the expected values and the covariance matrices for the parameters and the system errors. The starting and the terminal period for the optimization are again 2004:1 and 2006:4.

Model variables used in SLOVL:

Endogenous (state) variables:

x

GDP Rt = CRt + IN V Rt + GRt + EXRt -IM P Rt V Rt = GDP Rt + IM P Rt Table 3. 'Ideal' values of objective variables, SLOVL CR IN V R IM P R ST IRLN GDP R V R T axes GR M 3R 1% 1% 2% -0.25 1% 1.5% 25.2 1% 1.75%
The objective function is analogous as for SLOVNL, where the 'ideal' values of the state and control variables (x t and ũt respectively) are chosen as shown in Table 3. For the weights for the variables, Table 4a shows the 'raw' weights and Table 4b gives the normalized weights. -------------------

- CR 1 CR 3.457677 IN V R 1 IN V R 12.16323 IM P R 1 IM P R 1.869532 ST IRLN 1 ST IRLN 216403.9 GDP R 2 GDP R 2 V R 1 V R 0.

Optimization experiments

The OPTCON2 algorithm is applied to the two econometric models SLOVNL and SLOVL. Two different experiments are run for both models: in experiment 1, two open-loop solutions are compared, a deterministic one where the variances and covariances of the parameters are ignored, and a stochastic one where the estimated parameter covariance matrix is taken into account. In experiment 2, the properties of the open-loop and the open-loop feedback solutions are compared. Furthermore, by comparing the results for the SLOVNL and the SLOVL models we want to analyze the impact of nonlinearity on the properties of the optimal solution.

Experiment 1: open-loop optimal policies

For experiment 1, two different open-loop solutions are calculated: a deterministic and a stochastic one. The deterministic solution assumes that all parameters of the model are known with certainty and are equal to the estimated values.

In the stochastic case, the covariance matrix of the parameters as estimated by FIML is used but no updating of information occurs during the optimization process.

The results (for details, see [START_REF] Blueschke | Stochastic Control of Linear and Nonlinear Econometric Models: Some Computational Aspects[END_REF][START_REF] Blueschke-Nikolaeva | Optimal Control of Nonlinear Dynamic Econometric Models: An Algorithm and an Application[END_REF]) show that both the deterministic and the stochastic solutions follow the 'ideal' values fairly well but fiscal policies are less expansionary and hence real GDP is mostly below its 'ideal' values. The values of the objective function show a considerable improvement in system performance obtained by optimization and only moderate costs of uncertainty.

An interesting result is that the deterministic and the stochastic open-loop solutions are very similar. Furthermore, one can see that the SLOVL model is a good 'linear approximation' of the nonlinear SLOVNL model because the results for both models are nearly identical. This fact can be used for isolating the impact of nonlinearity on finding the optimum control solution, especially for the case of open-loop feedback policies. The aim of experiment 2 consists in comparing open-loop (OL) and open-loop feedback (OLF) optimal stochastic controls. Figures 1 and2 show the results of a representative Monte Carlo simulation, displaying the value of the objective function arising from applying OPTCON2 to the SLOVNL and the SLOVL models respectively, under 1000 independent random Monte Carlo runs. The graphs plot the values of the objective function for OL policies (x-axis) and OLF policies (y-axis) against each other. In the 'zoom in' panels of the figures, we cut the axes so as to show the mass of the points and omitting 'outliers', i.e. results where the value of the objective function becomes extremely large.

One can see that in most cases the values of the objective function for the open-loop feedback solution are smaller than the values of the open-loop solution, indicated by a greater mass of dots below the 45 degree line. This means that open-loop feedback controls give better results (lower values of the cost function) in the majority of the cases investigated. For the SLOVNL model, the OLF policy gives better results than the OL policy in 66.4% of the cases, for the SLOVL model in 65.4% of the cases considered here.

However, one can also see from these figures (especially in the left-hand panels with a 'normal' view) that there are many cases where either control scheme results in very high losses, indicated by dots which are significantly distant from the main mass of the dots. These cases are called 'outliers' and can be seen even more clearly in Figure 3 The results mean that (passive) learning does not necessarily improve the quality of the final results; it may even worsen them. One reason for this is the presence of the two types of stochastic disturbances: additive (random system error) and multiplicative error ('structural' error in the parameters). The decision maker cannot distinguish between realizations of errors in the parameters and in the equations as he just observes the resulting state vector. Based on this information, he learns about the values of the parameter vector, but he may be driven away from the 'true' parameter vector due to the presence of the random system error.

On the impact of nonlinearity

In the previous section we saw that there is a severe problem of what we called 'outliers' -cases with very high losses or values of the objective function. In a similar framework of optimum stochastic control, [START_REF] Tucci | The Parameter Set in an Adaptive Control Monte Carlo Experiment: Some Considerations[END_REF] investigated numerical reasons for outliers. It was not possible to confirm that the sources of the problem found by these authors were decisive for the outlier problem in our framework. We suspect that there are other reasons for the outliers. One possible reason is the stochastics of the dynamic system itself. In our SLOVNL and SLOVL models all the parameters (including all the intercepts) are considered to be stochastic, which may make this reason more likely to work.

The second possible reason is based on the nonlinear nature of the models for which the OPTCON algorithm was created. The SLOVL model was created mainly in order to test this possibility. The graphical results in the previous section show that the outliers occur in the linear as well as in the nonlinear model version. Moreover, in some of the experiments with 1000 Monte Carlo runs for the SLOVNL model, it turned out that the algorithm did not converge in some runs. In these cases, the algorithm starts to diverge and results in some nonreasonable or even complex numbers for some variables. In the 1000 Monte Carlo runs experiment considered above this happened six times. On the contrary, under the SLOVL model, not one single case of non-convergence out of the 1000 Monte Carlo runs occurred. Thus we arrive at the conclusion that nonlinearity is not the reason for the 'outliers', but it can worsen the problem.

Conclusion

A comparison of open-loop control (without learning) and open-loop feedback control (with passive learning) shows that open-loop feedback control outperforms open-loop control in the majority of the cases investigated for the two small econometric models of Slovenia. But it suffers from a problem of 'outliers' which is present for both policy schemes. When comparing the results for the nonlinear SLOVNL model and the linear SLOVL model, we found that the nonlinearity of the system is not responsible for the 'outliers' but may worsen their influence in some cases.

  For the purpose of comparing open-loop and open-loop feedback results, it is not possible to observe current and true values, so one has to resort to Monte Carlo simulations. Large numbers of random time paths for the additive and multiplicative errors are generated, representing what new information could look like in reality. In this way 'quasi-real' observations are created and both types of controls, open-loop and passive-learning (open-loop feedback), are compared.

6. 2 Fig. 1 .Fig. 2 .

 212 Fig. 1. OL and OLF control, value of objective function; SLOVNL; 1000 Monte Carlo runs; left: 'normal', right: 'zoom in'

Fig. 3 .

 3 Fig. 3. Open-loop vs. open-loop feedback control, value of objective function (1000 Monte Carlo runs) (left: SLOVNL, right: SLOVL)

  Standard deviations are given in brackets.CRt = 240.9398 + 0.740333 CRt-1 + 0.111727 GDP Rt (1 -T axRate t

					Control variables:
					u[1] T axRate net tax rate
					u[2] GR	real public consumption
		real imports of goods and services	u[3] M 3N	money stock, nominal
	x[4] ST IRLN short term interest rate
	x[5] GDP R	real gross domestic product
	x[6] V R x[7] P V	real total aggregate demand general price level		Exogenous non-controlled variables:
	x[8] P i4	rate of inflation		z[1] EXR	real exports of goods and services
					z[2] IM P DEF import price level
					z[3] GDP DEF domestic price level
					z[4] SIT EU R nominal exchange rate SIT/EUR
	Model equations:
					100	)
	(189.7449)	(0.1115)	(0.0330)
	-1.007353 (ST IRLNt -P i4t) -4.773533 P i4t
		(2.5848)			(2.4966)
		(176.8549)	(0.1423)		(0.0924)
		-0.455511 (ST IRLNt -P i4t) -2.981241 P i4t
		(6.9044)		(3.1277)
			(0.0724)		(18.9336)
	ST IRLNt = 0.811606 ST IRLNt-1 -0.000877 (M 3N ) t P V t	• 100
		(0.1375)	(0.0008)
		+ 0.002746 GDP Rt
		(0.0026)

IN V Rt = 75.41731 + 0.932211 IN V Rt-1 + 0.264523 (V Rt -V Rt-1) IM P Rt = IM P Rt-1 + 0.826449 (V Rt -V Rt-1) -38.14954 SIT EU Rt

Table 2 .

 2 Weights of objective variables, SLOVNL

	2a: 'raw' weights	2b: 'correct' weights
	variable	weight	variable	weight

Table 4 .

 4 Weights of objective variables, SLOVL

	4a: 'raw' weights	4b: 'correct' weights
	variable	weight	variable	weight