A New Approach to Robust Linear Filtering Problems

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

A new approach to robust linear filtering problems ⋆ Marco H. Terra ∗

∗∗

Jo˜ ao Y. Ishihara ∗ Roberto S. Inoue ∗∗

Department of Electrical Engineering, University of S˜ ao Paulo at S˜ ao Carlos, S˜ ao Paulo, Brazil (e-mail: terra,[email protected]) ∗∗ Department of Electrical Engineering, Faculty of Technology, University of Bras´ılia, DF, Brazil (e-mail: [email protected])

Abstract: This paper deals with optimal filtering problem for uncertain discrete-time systems. The parametric uncertainties are assumed to be norm bounded. It is proposed a robust penalty game approach to solve this problem. The optimum robust filter with the respective recursive Riccati equation proposed are written in a unified framework defined in terms of blocks of matrices. This framework does not depend on any auxiliary parameter to be adjusted. Simulation results show the effectiveness of the robust filter proposed. Keywords: Robust filtering, least squares, penalty functions, game theory. 1. INTRODUCTION The Kalman filter has been widely used to solve estimation problems in attitude and position determination, robotics, communications, control, economics, signal processing, computer vision and other fields, see for instance Kaylath et al. (2000), Farrel (2008), Brown and Hwang (1997), Hasan and Azim-Sadjani (1995), Mills and Goldenberg (1989) and Stevens and Lewis (1991). It was developed in the 1960’s years based on the assumption that all parameter matrices of the state-space model are not subject to uncertainties. This assumption guarantees an optimum estimation of the state. However, when the model considered in the filtering process is uncertain this central premise of the Kalman filter is violated. In this case its performance can deteriorate appreciably, see Sayed (2001) and references therein. This kind of problem motivates the use of robust estimation methods to limit the performance degradation of optimal filters. In the last forty years dozens of researchers have dedicated efforts to solve this problem. Four representative approaches that deal with this problem were developed based on H∞ filtering, set-valued estimation, guaranteed-cost, and robust regularized least-squares. All these approaches were compared in Sayed (2001). Therefore important issues of these filters can be seen there and in the references therein. We are interested in this paper in deal with optimum robust recursive filtering of uncertain systems. The idea is provide solutions as close as possible to the Kalman filter developed for systems with accurate models. In this sense the filters proposed in Sayed (2001) are designed to minimize the worst-possible regularized residual norm over an admissible class of uncertainties at each iteration. ⋆ This work was supported by FAPESP under grants 07/03484-8, 04/03826-8, 03/12574-0, and Cnpq grant 310852/2006-4.

Copyright by the International Federation of Automatic Control (IFAC)

Similar to the standard Kalman filter, they are useful to be used in online applications. In this case the stability can always be guaranteed. However, to obtain a robust optimum performance it is necessary to perform offline computations to adjust these filters. It is necessary to adjust a λ parameter which is related with a minimization of a function that depends on the state to be estimated. Despite the author provides an alternative to adjust these filters to work in an optimum point, the procedure to find this optimum continues been an offline procedure. We propose in this paper a robust Kalman filter for linear discrete-time state-space systems subject to normbounded parameter uncertainties based on a penalty game approach. It is based on the combination of robust regularized least-squares problem and penalty functions, see for instance Sayed and Nascimento (1999) and Luenberger (2003). We have applied this approach in robust control problems for standard state-space systems and for Markovian jumps linear systems, see for instance Cerri et al. (2010) and Cerri et al. (2009). The robust filter proposed in this paper does not depend on any auxiliary parameter to be adjusted, only on the parameters and weighting matrices which are known a-priori. We present the stability and convergence analysis of the steady-state robust filter and the associated Riccati recursion. Comparative results, based on simulations, among the proposed robust filter, the bounded data uncertainties (BDU) filter of Sayed (2001), and the standard Kalman filter are shown to demonstrate the effectiveness of the approach proposed. We show in these simulations that the ensemble-average error variance of the proposed robust filter is equivalent to the optimum BDU filter. Experimental results of this robust filter to estimate the attitude of a vehicle are shown in a companion paper also submitted to the 18th IFAC World Congress, Inoue and Terra (2011). This paper is organized as follows: the problem we are dealing with is defined in Section 2; auxiliary results are shown in Section 3; the optimal robust filter is developed

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

in Section 4; stability and convergence of the robust filter are presented in Section 5; a numerical example is given in Section 6 and some concluding remarks are provided in Section 7. The notation we use in this paper is standard: ℜ is the set of real numbers, ℜn is the set of n-dimensional vectors whose elements are in ℜ, ℜm×n is the set of m × n real matrices, AT is the transpose of the matrix A, P > 0 (P ≥ 0) denotes a positive definite (semi definite) matrix, kxk is the Euclidean norm of x, kxkP is the weighted norm 1 of x defined by (xT P x) 2 , A† is a pseudo-inverse of A and the notation Y T XY = Y T X[·]. 2. PROBLEM STATEMENT

that constrained optimization problem can be solved by an equivalent non-constrained quadratic optimization problem. In this case the original constrain is interpreted in this new functional as a penalized quadratic term. The use of the penalizion approach turns out to be more convenient than the more commonly used Lagrange multiplier based approach. It contributes in the unconstrained functional with a linear term. As a result, the obtained robust filter parameterized by the penalty variable, has a strikingly resemblance with the non-robust Kalman filter and so, the convergence and stability analysis can be extended directly. In the next section we present more precise statements of these facts which motivated the proposed functional (2). 3. AUXILIARY RESULTS

Consider the following uncertain discrete-time dynamic system: xk+1 = (Fk + δFk ) xk + (Gk + δGk ) wk , zk+1 = (Hk+1 +δHk+1 ) xk+1+(Kk+1 +δKk+1 ) vk+1 ,(1) whose parametric uncertainties are modeled by      ∆1 0 M1,k 0 δFk δGk 0 0 := 0 ∆2 0 M2,k 0 0 δKk δHk+1   0 N F k N Gk 0 × , 0 0 NKk NHk+1

where k∆j k ≤ 1, j = 1, 2, xk ∈ ℜn is the state variable; zk+1 ∈ ℜp the measured output variable, wk and vk+1 are errors with known positive-definite weighting matrices Qk and Rk , respectively. We are given in this state-space model a deterministic interpretation for the variables wk and vk+1 . They are considered in general, in the Kalman filter literature, as stochastic variables where Qk and Rk are variances of the respective noises. Further we consider that do not exist any weighting matrices among {x0 , w0 , v1 }. In the stochastic interpretation it means that they are uncorrelated variables. The recursive optimal robust filter we develop in this paper aims to solve the following unconstrained optimization problem to find an estimate x bk+1|k+1 to the system (1): min max Jk ,

xk ,xk+1

k+1

h

0

0

0

vk+1 xk+1

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{δA, δb}

2

2

f (x) := {kxkQ + k(A + δA) x − (b + δb)kW }, (4) where A is the data matrix, b is the measurement vector which are assumed to be known, x is the unknown vector, Q = QT ≥ 0 and W = W T > 0 are given weighting matrices, δA, δb are perturbations modeled by [ δA δb ] = M ∆ [ NA Nb ] , k∆k ≤ 1.

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The next result is proved in Sayed (2001) and in the references therein. It is a fundamental lemma to the theory of robust filtering, in which is used a purely deterministic approach. Lemma 1. The optimization problem (3)-(5) has a unique solution x ˆ given by:  −1   ˆ T Nb , ˆ + AT W ˆA ˆ b + λN x ˆ= Q AT W (6) A

ˆ and W ˆ are where the modified weighting matrices Q defined as ˆ T NA ˆ := Q + λN Q A †  ˆ − M T W M M T W, ˆ := W + W M λI W

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ˆ is a nonnegative scalar parameter obtained by and λ following optimization problem ˆ := arg λ

xk+1

  T h i i xk − xˆk|k  −I  wk − −Fδ,k xˆk|k µI • ,

Fδ,k Gδ,k 0 0 0 Kδ,k Hδ,k

x

k+1

k+1

0

min max f (x)

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δk

where δk := {δFk , δGk , δKk , δHk } and the cost function Jk associated is defined as   x − xˆ T P −1 0 0 0  xk − x ˆk|k k k|k k|k −1 wk wk 0 Qk 0 0    + Jk :=   0  v v 0 R−1 0 xk+1

In this section the problem of regularized least squares for models with bounded data uncertainties will be solved in the form of array of matrices. This form is suitable for the robust filter developed in this paper. Consider the following optimization problem:

Γ (λ) ,

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where

zk+1

and Fδ,k = (Fk + δFk ), Gδ,k = Gk + δGk , Hδ,k = Hk+1 + δHk+1 , and Kδ,k = Kk + δKk .

min

λ≥kM T W M k

2

2

2

Γ(λ) := kx(λ)kQ+λ kNA x(λ)−Nb k +kAx(λ)−bkW (λ) (10) and the auxiliary functions are defined by

We formulate this optimization problem in this way based on two important issues: first, we use the fact that the stochastic robust estimation problem can be solved through a deterministic approach and second, we consider 1175

 −1  T  x(λ) := Q(λ) + AT W (λ)A A W (λ)b + λNAT Nb ,

Q(λ) := Q + λNAT NA ,

W (λ) := W + W M λI − M T W M


†

M T W.

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

⋄ We can relate the optimization problem (2) with (3) through the following identifications: 



 0 0 , 0 0

 −1 Pk|k 0 0 xk − x ˆk|k −1   wk  0 Q 0 k x← , Q← −1  0 vk+1  0 Rk+1 xk+1 0 0 0     F G 0 −I δFk δGk 0 0 A← k k , δA ← , 0 0 Kk Hk+1 0 0 δKk δHk+1     −δFk x ˆk|k −Fk x ˆk|k . b← , δb ← zk+1 0 It is known that there exists a global minimum to the Γ (λ) function in (9), see Sayed et al. (2002), which should be minimized in order to obtain the optimum solution to the optimization problem (3)-(5). However, Γ (λ) depends on x which is an unknown variable. It is not an easy task to perform this minimization online. Before we present a solution for this problem, we provide in the next lemma equivalent solutions for this robust optimization problem. Lemma 2. The following statements are equivalent: (i) x ˆ ∈ arg min max f (x), (11) x

Lemma 3. Let V ∈ ℜn×n positive definite and G ∈ ℜk×n . Consider the constrained minimization problem x ˆ = arg min xT V x subject to Gx = u, (16) where u ∈ ℜk×1 and x ∈ ℜn×1 . This problem can be transformed in an unconstrained minimization problem considering an auxiliary variable µ: T

x ˆ (µ) = arg min ( Gx − B) V (µ) ( Gx − B) , x

      I V 0 0 , V (µ) = and B = , G= G 0 µI u µ > 0. Then, limµ→∞ x ˆ (µ) always exists and it is equal to lim x ˆ (µ) = x ˆ0 ,

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µ→∞

where x ˆ0 is solution of (15)-(16) and x ˆ (µ) is given by:  T  −1   −1 0 B V (µ) G x ˆ (µ) = . I 0 GT 0

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Therefore, T

lim ( G x ˆ (µ) − B) V (µ) ( G x ˆ (µ) − B) = (ˆ x0 ) T V x ˆ0 .

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ˆ γˆ ) is a solution of the system (iii) (x, β, γ) = (ˆ x, β, 

 0 0 I 0 0 0 β1   0  ˆ 0 0 I 0 0  β2   0  W   ˆ 0 0 I 0   β3  0 λI  0    γ 0 (13) 0 0 0 0 0 I   1 =  .  γ    I 0 0 0 0 A   2  b   Nb 0 I 0 0 0 NA  γ 3 x 0 0 0 I AT NAT 0 ˆ T NA ) is invertible, it follows that x ˆ A + λN If (Q + AT W ˆ A is the unique solution of (ii). Moreover, Q 0  0  I  0 0 0

ˆ T NA ) = − [0 0 0 0 0 0 I] ˆ A + λN (Q + AT W A  −1   Q 0 0 I 0 0 0 0 ˆ 0 0 I 0 0  0 0 W  ˆ 0 0 I 0   0 0 λI   0      ×  I 0 0 0 0 0 I  0 .   0 0 I 0 0 0 0 A    0 0 I 0 0 0 N  0 A I 0 0 0 I AT NAT 0

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where

{δA,δb}

where δA and δb are given by (5). " # " I # 0 T A x− b (ii) x ˆ ∈ arg min x NA Nb   Q 0 0   ˆ 0  × 0 W • . ˆ 0 0 λI

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x

µ→∞

⋄ Remark 4. A useful property of this lemma, which will be ˆ in (9), is that the quadratic term used to adjust λ (Gˆ x(µ) − u)T µI(Gˆ x(µ) − u) (20) goes to zero when µ → ∞. Remark 5. Originally the filtering problem we defined in (2) was considered as a constrained optimization problem: 

T xk − x ˆk|k  wk  min max  vk+1  xk ,xk+1 δk xk+1



−1 0 0 Pk|k  0 Q−1 0  k −1  0 0 Rk+1 0 0 0

  0 xk − x ˆk|k wk  0   v ,  k+1 0 xk+1 0

subject to (1). Thanks to the Lemma 3 we can reformulate it as a unconstrained optimization problem. 4. OPTIMAL ROBUST FILTER (14)

⋄

The next lemma combines Lemma 1 with the results presented in Albert (1972) and Luenberger (2003). It provides expressions for optimal solutions of optimization problems under constraints. Its proof can be obtained following the results provided in Albert (1972).

To deduce the robust filter it is necessary to redefine some variables and perform complementary identifications in order to use the lemmas proposed in the previous section,         −I Fk Gk 0 0 Ek := , Fk:= , Gk:= , Zk+1= , Hk+1 0 0 Kk zk+1       0 NFk N Gk 0 NE,k := , NF ,k := , NG,k = , NHk+1 0 0 NK k     M1,k 0 Qk 0 Mk := , Rk = , (21) 0 M2,k 0 Rk+1

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

and A ← [Fk Gk Ek ] , NA ← [NFk NGk NEk ] .

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Now we are in position to present the main result of this paper. Theorem 6. The optimal robust filtered estimates for the System (1) and the corresponding recursive Riccati equaˆk tion are given by (24). The optimum scalar parameter λ is obtained by the minimization of the function (10) over the interval ˆk ≥ λ for a given value of µ.

kMkT µIMk k,

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⋄ Remark 7. In order to find the robust optimal solution to the filter proposed in Theorem 6 we consider the whole game penalty approach we are proposing, which is the combination of Lemmas 2 and 3. For the following limits ˆ −1 → 0, µ → ∞, µ−1 → 0, λ (26)

matrix at the i-th block position and zero matrices at the other block positions. The steady-state value of the matrix Pk|k is denoted as P . ¯ F, ¯ G, ¯ and R are time-invariant counterThe matrices E, parts of the corresponding variables defined in (21). With these new notations we can show that the steady-state filter is given by: x bk+1|k+1 = Y61 x bk|k + Y63 Zk+1|k+1 ,

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x bk+1|k+1 = −Y63 F x bk|k + Y63 Zk+1|k+1 ,

(30)

P = −Y66 .

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The steady-state filter (28)-(29) can be rewritten as:

T = Y61 P Y61

T Y62 RY62 .

P + (31) Definition 1. P is a stabilizing solution of the algebraic Riccati equation (ARE) (29) if P satisfies (29) and Y61 is stable (the matrix Y61 has all eigenvalues inside the unit circle |λ| < 1).

k

and after some algebra, we can show that (24) tends to (25). In this case, the existence condition of this filter is guaranteed if   Fk Gk Ek NF ,k NG,k NE,k

is full row rank. Note that this new framework does not depend on any auxiliary parameter to be adjusted, only on the parameters and weighting matrices which are known a-priori. This result is useful for online applications. Note also that we provide at the same time, through a unified structure, the variables x ˆk+1|k+1 and Pk+1|k+1 . Remark 8. The robust filter presented in (25) collapses to the standard Kalman filter if the uncertainties go to zero. We omit in this paper the algebra that show this reduction due to lack of space. The importance of this optimum robust Kalman filter is that we recover the original features of the Kalman filter proposed in the 1960’s years, related with optimality and recursiveness, for systems subject to uncertainties.

The next theorem shows that (29) has a stabilizing semidefinite solution P whose proof can be seen in Ishihara et al. (2010) and in the references therein.   Theorem 9. Suppose that λE¯ + F¯ G¯ has full row rank for |λ| ≥ 1, R > 0 and E has full column rank. Let P be a solution of the ARE (29). If P ≥ 0 then P is the unique stabilizing solution for (29). ⋄ Based on deterministic arguments we can show the convergence of the generalized Riccati recursion. We can verify that it is a monotone nondecreasing sequence bounded by P + ≥ 0. Lemma 10. Consider a sequence {Pk|k } generated by the recursion (25). If Pk|k ≥ 0, then Pk+1|k+1 is positive semidefinite. ⋄ Lemma 11. Consider the following matrices, for i = 1, 2

5. STABILITY AND CONVERGENCE OF THE STEADY-STATE ROBUST FILTER

i i i Pk|k := −Y66,k and Ffi k := Y61,k .

The stability of the robust filter (25) can be proved based on the arguments developed in Ishihara et al. (2010). Considering the parameters of (25) time-invariant we introduce the following notation:   P 0 0 I 0 0  0 R 0 0 I 0  0 0 0 F¯ G¯ E¯   Ω (P ) :=  I 0 F¯ T 0 0 0  ,    0 I G¯T 0 0 0  (27) 0 0 E¯T 0 0 0 T

ei := [0 ... I ... 0] , Y (P ) := Ω−1 (P ) ,       G E F ¯ ¯ ¯ , , G= , E= F= NG NE NF

⋄

where ei and Y (P ) are partitioned in blocks according to the block partition of Ω (P ). Y (.) is partitioned as Yij , i, j = 1, ..., 6 and the vector of blocks ei has the identity

(32)

Then, we obtain   1 2 1 2 (Ff2k )T ,(33) −Pk|k (i) Pk+1|k+1 −Pk+1|k+1 = Ff1k Pk|k   1 1 2 Y11,k . (34) − Pk|k (ii) Ff1k − Ff2k = Ff2k Pk|k ⋄ 1 2 Lemma 12. Consider (32) for i = 1, 2 . If Pk|k ≥ Pk|k ≥ 0, 1 2 then Pk+1+|k+1 ≥ Pk+1|k+1 ≥ 0. ⋄ Lemma 13. Suppose that R > 0 and define {Pk|k }∞ by k=0
T , Pk+1|k+1 := Y63,k FPk|k F T + GRG T Y63,k

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with P0 = 0. Then {Pk|k }∞ k=0 ıs a nondecreasing monotone sequence.

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011



x ˆk+1|k+1

 T 0 0   0 0      0  Pk+1|k+1 =  0 0   0   0 0 I





Pk|k  0   0   0   0   0   0   0   I   0 0

0 0 0 Rk 0 0 ˆ −1 I 0 λ 0 k ˆ −1 I 0 0 −λ k 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 I 0 0 0 0 0

0T Pk|k 0 0  0  0        Pk+1|k+1 = 0  0 I 0     0  0 I 0

x ˆk+1|k+1

Pk+1|k+1

Pk 0 I = − [0 0 I] 0 R 0 I 0 0

#−1 " # 0 0 , I

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where we define Lk := [I −Lp,k ] and  FT LTk , GT
−1 Lp,k := F Pk|k H T HPk|k H T + R ,   T  Q 0 G R := [G K] . 0 R KT Pk := Lk [F G]



Pk|k 0 0 R

0 0 0 0 0 0 I 0

µ−1 I 0 0 MkT FkT NFT ,k T GkT NG,k T T Ek NE,k



¯T

(25)

The proofs of Lemmas 15 and 16 are extensions of Lancaster and Rodman (1995), which in turn is due to Caines and Mayne (1970). Theorem 17. Assume that λE¯ + F¯ has full column rank for |λ| ≥ 1 and R > 0. Let a sequence {Pk+1|k+1 }∞ k=0 generated by (25), with P0|0 = 0. Then, it is a nondecreasing sequence and converges to P + ≥ 0 which satisfies the Riccati equation (38) ⋄

The detailed proofs of the convergence and stability results listed in this section, can be seen in Ishihara et al. (2010). 6. NUMERICAL EXAMPLE Consider the system (1) reduced to the model presented in Sayed (2001): xk+1 = (Fk + δFk ) xk + Gk wk , zk+1 = Hk+1 xk+1 + vk+1 ,



  Sk|k 0 F Sk := Ls,k F¯ G¯ LTs,k , 0 R G¯T    Q 0 GT . R := [G K] 0 R KT

where Fk =

Let P0|0 be a matrix for which 0 ≤ P0|0 ≤ S0|0 and define a sequence {Pk|k }∞ k=0 (36). Then 0 ≤ Pk+1|k+1 ≤ Sk+1|k+1 for k = 0, 1, 2, .... ⋄ Lemma 16. Let the sequence be defined as in Lemma 14. If λE¯ + F¯ has full column rank for |λ| ≥ 1, then there exists a matrix P + such that 0 ≤ Pk ≤ P + for k = 0, 1, 2, ..., ∞. {Pk }∞ k=0

 0 0  0  0 0  0 −I

 0 0  0 0  0  0  , (24) 0  0  0 0 −I

⋄

where we define Ls,k := [I −Ls,k ] and 

−1  0 I 0 0 x ˆk|k 0 0 I 0    0  0 0 0 0    0  I 0 0 0    0  Mk Fk Gk Ek   Zk+1  0 NF ,k NG,k NE,k    0 0 0 0 0   0   0 0 0 0   0   0 0 0 0   0   0 0 0 0 0  0 0 0 0 0


T ¯ F¯ T + GR ¯ G¯T Y63 P = Y63 FP .

⋄ Lemma 15. Let R > 0. Consider a given arbitrary matrix S0|0 ≥ 0, and let a sequence {Sk|k }∞ k=0 be defined by: " #−1 " # Sk 0 I 0 0 , Sk+1|k+1 = − [0 0 I] 0 R 0 (37) I 0 0 I 

0 0 I 0 0 I 0 0 0 0 0

−1  0 0 I 0 0 x ˆk|k 0 0 0 I 0   0  0 0 Fk Gk Ek   Zk+1  0 0 NF ,k NG,k NE,k    0 T  Fk NFT ,k 0 0 0    0 T T  0  Gk NG,k 0 0 0 T T 0 Ek NE,k 0 0 0

0 Rk 0 0 0 I 0

⋄ Lemma 14. The Riccati recursion given by (25) can be rewritten in the following form "

0 0 0 0



   0.9802 0.0196 1 0 , Gk = , Hk+1 = [1 −1] , 0 0.9802 0 1

and the covariance of wk and vk (or weighting matrices) are given respectively by   1.9608 0.0195 Qk = , Rk+1 = 1. 0.0195 1.9605 The parametric uncertainties are modeled by:   0.198 M1 = , Nf = [0 5] . 0

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

We compare in Figure 1 the robust optimum (OR) of the Equation (25) with the BDU filter proposed in Sayed (2001). It is also a robust optimum filter whose parameter α = 0.5 was chosen after some simulations. Note that the OR filter does not depend on any extra parameter to be adjusted. The results of Figure 1 show an equivalence of both filters and a significative deterioration of the standard Kalman filter performance. These curves represent the error variances computed via the ensemble-average Ekxk − x ˆk k ≈

T 1 X (j) (j) kx − x ˆk k. T j=1 k

40 Kalman filter

35

Error Variance (dB)

30 25 20 15

OR filter (µ → ∞)

10 Optimal 5 0 0 10

BDU filter (α = 0.5) 1

2

10

10

3

10

k

Fig. 1. Error variance with ∆ selected uniformly within the interval [−1, 1]. Each point at instant k in each variance curve is the ensemble-average calculated over T experiments (T = 5000 trajectories with N = 1000 points were generated). For each experiment j, the matrices ∆1 and ∆2 with norm less or equal than one are selected randomly and fixed for all k. 7. CONCLUSION This paper has developed a Kalman-type recursive formulation for robust filtering problem for general discrete-time systems subject to uncertainties on all system matrices. It is useful to online applications mainly due to the fact that it does not need any auxiliary parameter to be adjusted.

Cerri, J.P., Terra, M.H., and Ishihara, J.Y. (2009). Recursive robust regulator for discrete-time state-space systems. Proceedings of the American Control Conference, St. Louis, Missouri, USA. Cerri, J.P., Terra, M.H., and Ishihara, J.Y. (2010). Robust recursive regulator for discrete-time markovian jump linear systems via penalty game approach. Proceedings of the IEEE Conference on Decision and Control, Atlanta, Georgia, USA. Farrel, J.A. (2008). Aided Navigation GPS with High Rate Sensors. The McGRaw-Hill Companies. doi: 10.1036/0071493298. Hasan, M.A. and Azim-Sadjani, M.R. (1995). Noncausal image modeling using descriptor approach. IEEE Transactions on Circuits and Systems II, 42(8), 536–540. Inoue, R.S. and Terra, M.H. (2011). Robust recursive kalman filtering for attitude estimation. In 18th World Congress of the International Federation of Automatic Control (IFAC). Milano, Italy. Ishihara, J.Y., Terra, M.H., and Bianco, A.F. (2010). Recursive linear estimation for general discrete-time descriptor systems. Automatica, 46, 761–766. Kaylath, T., Sayed, A.H., and Hassibi, B. (2000). Linear Estimation. Prentice-Hall. Lancaster, P. and Rodman, L. (1995). Algebraic Riccati Equations, volume 1. Oxford Science Publications, Oxford and New York. Luenberger, D.G. (2003). Linear and Nonlinear Programming. Kluwer Academic Publishers. Mills, J.K. and Goldenberg, A.A. (1989). Force and position control of manipulators during constrained motion tasks. IEEE Transactions Robotics and Automation, 68, 30–46. Sayed, A., Nascimento, V., and Cipparrone, F. (2002). A regularized robust design criterion for uncertain data. SIAM Journal on Matrix Analysis and Its Applications, 23(4), 1120–1142. Sayed, A.H. (2001). A framework for state-space estimation with uncertain models. IEEE Transaction on Automatic Control, 46, 998–1013. Sayed, A.H. and Nascimento, V.H. (1999). Design criteria for uncertain models with structured and unstructured uncertainties. In A. Garulli, A. Tesi, and A. Vicino (eds.), Robustness in Identification and Control, volume 245, 159–173. Springer-Verlag, London. Stevens, B.L. and Lewis, F.L. (1991). Aircraft Modeling, Dynamics and Control. Wiley, New York.

REFERENCES Albert, A. (1972). Regression and the Moore-Penrose Pseudoinverse. Academic Press, New York and London. Brown, R.G. and Hwang, P.Y.C. (1997). Introduction to random signal and applied Kalman filtering: with Matlab exercises and solutions. John Wiley & Sons, New York, USA, 3rd edition. Caines, P. and Mayne, D. (1970). On the discrete time matrix Riccati equation of optimal control. International Journal of Control, 12(5), 785–794. 1179