Robust ǫ-contaminated filter for discrete-time systems E. Blanco
Ph. Neveux
A. Boukrouche
Laboratoire AMPERE UMR CNRS 5005 Ecole Centrale de Lyon 36 avenue Guy de Collongue 69134 Ecully Cedex - France Email:
[email protected]
EMMAH UMR UAPV-INRA Universit´e d’Avignon UFR-ip Sciences et Technologies 84000 Avignon - France Email:
[email protected]
Universit´e de Guelma Laboratoire PI:MIS BP 401 24000 Guelma - Algeria Email:
[email protected]
Abstract—The robust filtering problem for uncertain discretetime systems is treated in this paper. An ǫ-contaminated framework is adopted to design the robust filter in presence of model uncertainties. The result is presented in term of Toeplitz matrix equations. The proposed approach does not require a formal modelling of uncertainties. Meanwhile, the ǫ-contaminated robust filter is equivalent to the minimum variance filter with a stochastic representation of the uncertainties. An example shows the effectiveness of the approach.
I. I NTRODUCTION Discrete-time signal processing problems have been widely treated by Wiener and Kalman filters. The design of these filters requires a precise description of the noises and the model of the system. Noise uncertainty can lead to too smooth estimates when the noise level has been over-estimated. On the other hand, system uncertainty may lead to dramatically poor performances due to large estimation errors over a timeinterval [3]. Ensuring a good and almost constant performance level, in term of estimation error, over a specified uncertainty set, defines the robustness of a given filter. Filters having this property are termed as robust filters. Various approaches have been adopted over the decades in order to design robust filters. The most widespread approaches consist in modelling the uncertainties by means of so-called norm bounded uncertainties, polytopes or IQC. Riccati equation are commonly used to solve problems with norm bounded uncertainties [12] and some IQC problems [10]. LMIs permit to define robust filters in presence of polytopic uncertainties [9][11] and IQC [4][1]. However, if no consistent idea to model the uncertainties is available, then an alternative solution has to be found. The introduction of the sensitivity of the estimation error with respect to the uncertain parameters by means of an ǫcontaminated approach has been shown to be an interesting alternative (in the frequency domain [6] for discrete-time systems, [7] for continuous systems; and the time domain, [13] for discrete-time systems). In this context, a formal modelling of the uncertainties is not required. c 978-1-4799-6773-5/14/$31.00 2014 IEEE
In this article, an ǫ-contaminated robust filter is developed for discrete-time linear systems described by their impulse response. Toeplitz matrix representation of the convolution product is used. Hence, the filter has a finite impulse response. Finally, this work is an extension of the result in [5] to uncertain systems. The following notation are adopted in the sequel: T • X denotes the transpose of the matrix X, • E{·} is the expectation operator, • ∂X f is the partial derivative of the function f w.r.t. variable X. II. P ROBLEM STATEMENT Consider a linear causal and stable discrete-time system depicted in Fig. 1 and described by the convolution product: sk = (h ⋆ n)k yk = s k + v k (1) nk = (u + w)k
with: • the finite impulse response is known explicitly, i.e. hk = hk (Θ) where Θ = {θi } ∈ Rp is the vector of uncertain parameters. The nominal value of Θ is denoted Θ, and the nominal impulse response of the system is hk . • uk is a measured input signal, sk is the signal to be estimated and yk is the measured output, • wk and vk are independent zero mean white noises with 2 variance σw and σv2 respectively. wk uk
hk
vk
❄ n ❄ s ❄ yk ✲ +✐ k✲ ⋆✐ k ✲ +✐ ✲
Fig. 1.
Schematic view of the system.
The objective of the present paper is to find the robust estimator for the signal sk given the measured input uk and output yk , and the statistical properties of the noises when
the impulse response hk is uncertain. In order to fulfil this objective, the previous expressions will be written in matrix form. Hence, after N + 1 recorded samples, we have the relation: s = Hn (2) y =s+v T and H = T (h) with T (·) where y = y0 y1 · · · yN an operator that generates the Toeplitz matrix from a given vector, i.e. h0 0 ··· ··· ··· ··· 0 .. .. .. . . . h0 .. .. . . .. .. . . .. .. .. H = hN . . . . . . . .. .. .. .. 0 . .. .. .. .. . . . 0 0
···
0
hN
···
···
h0
In the sequel, uppercase variables are Toeplitz matrices of their corresponding lowercase variable; bold face variables are vectors of N + 1 samples of their corresponding signal.
filter. The robustness is obtained by realising the balance between the two parts of the criterion, in others words, confidence in (2) is represented by the choosing a value of ǫ In order to minimise criterion (3), we express the estimation error ek in terms of the available information for the filtering operation. From (1), (4) and (5), it follows that: ek = ((f ⋆ h − h) ⋆ n)k + (g ⋆ u)k + (f ⋆ v)k
(6)
and the matricial form is : e = (F − I)Hn + Gu + F v
(7)
∂θj ek = ∂θj (h ⋆ (f − δ) ⋆ n)k
(8)
Considering
with δ the Dirac delta function The sensitivity of the estimation error w.r.t. the uncertain parameters is then obtained: •
∂θj ek = ((f ⋆ ∂θj h − ∂θj h) ⋆ n)k By denoting dH j = T ∂θj h , the matricial form is: ∂θj e = (F − I)dH j n
(9)
(10)
The optimal robust ǫ-contaminated filter is given as follows: III. ROBUST ǫ- CONTAMINATED FILTER The robustness of the filter can be obtained by modelling the uncertainties. But if the designer does not have enough information to model the uncertainties, another approach has to be adopted. The ǫ-contaminated approach permits to overcome the lack of information on the uncertainties. The optimality criterion to be minimised in this context is:
Theorem 1: The filter of the form (5) for the uncertain linear system (1) minimising the ǫ-contaminated criterion (3) is given by:
with
T
J = (1 − ǫ)E e e + •
•
p X j=1
ǫj E
n
∂ θj e
T
∂ θj e
o
(3)
the sample ek of estimation error e is
G = (I − F )H −1 F = XX T XX T + σv2 I
(11)
T
2 2 XX T = σw HH + (σu2 + σw )HHT p X ǫj T dH j dH j HHT = 1 − ǫ j=0
(12)
σu2 = E{u}
ek = sˆk − sk
(4)
sˆk = (f ⋆ y)k + (g ⋆ u)k
(5)
the estimate sˆk is:
with f and g to be determined such as a minimum for (3) exists • the weights ǫj ∈ R verify: 0 ≤ ǫj < 1 and 0 ≤ ǫ < 1 P with ǫ = ǫj . This criterion can be divided in two parts, namely: 1) the minimum variance of the estimation error weighted by (1 − ǫ), 2) the weighted sum of the variance of the estimation error sensitivity w.r.t. each uncertain parameter. In an ǫ-contaminated formulation, the second term plays the role of the contaminating agent of the minimum variance
▽ Proof: In a first step, we will optimise criterion (3) w.r.t. G and then F . As the nominal expression of the impulse response is used to compute the optimal filter, we will use its Toeplitz matrix in this proof. Thus, the criterion (3) is rewritten : J = (1 − ǫ){((F − I)H + G)T σu2 ((F − I)H + G) 2 +σw ((F − I)H)T (F − I)H + σv2 F T F } p X 2 ǫj ((F − I)dH j )T ((F − I)dH j ) +(σu2 + σw )
(13)
j=0
The optimal filter G is obtained by means of differentiation of (13) with respect to G. We should verify that: ∂G J = 0 and
2 ∂G J ≻0
Computing these quantities leads to: ∂G J = 2 (F − I)H + G σu2 = 0 ⇒ (11) 2 ∂G J = 2 σu2 I ≻ 0
with
and the matricial form is :
Consequently, the expression of G minimises (3).
e = (F − I)Hn + Gu + F v
Introducing this expression in criterion (13) leads to: 2 ((F − I)H)T (F − I)H + σv2 F T F J = (1 − ǫ) σw p X 2 +(σu2 + σw ) ǫj ((F − I)dH j )T ((F − I)dH j ) (15) j=0
The optimal filter F is obtained by means of differentiation of (13) with respect to F . We should verify that: ∂F J = 0 and
∂F2 J
≻0
p X
the first variation of the estimation error is dek = (dh ⋆ (f − δ) ⋆ n)k
j=1
ηkj = ∂θj hk (Θ) θ
T
ǫj dH j dH j = 0 (16)
j
αk = (dh ⋆ n)k =
p X
dhµ nk−µ
(26)
µ=1
Introducing (24) in (26), the following equation is obtained
The second derivative of (3) w.r.t. F is given by: T 2 HH + σv2 I 2(1 − ǫ) σw
: αk =
p X
dθj (η j ⋆ n)k
(27)
j=1
T
ǫj dH j dH j = 0 (17)
j=0
This real-valued matrix is symmetric positive definite. Consequently, the solution of the optimisation problem is a minimum. This completes the proof.
In the matrix formulation, noting that η j ⋆ n = dH j n with dH j = T (η j ), we get that: α=
p X
dθj dH j n
IV. R ELATION WITH STOCHASTIC DESCRIPTION OF
So, we finally obtain:
with:
UNCERTAINTIES
In this section, the estimate of sk is obtained by minimising: (18)
considering the uncertain parameters Θ to be modelled as follows: (19) Θ = Θ + dΘ
(28)
j=1
de = (F − I)dHM V n
JM V = E{eT e}
(25)
In order to express the variation dek in the matrix form, we introduce the variable α and compute the following convolution product:
After manipulations, the result in Theorem 1 is obtained.
p X
(23)
The term dhk is the first variation of hk (Θ) w.r.t. to dΘ expressed as: p X ηkj dθj dhk = (24)
j=0
2 + 2(σu2 + σw )
(22)
with
The condition on the first derivative leads to: T 2 (F − I)HH + σv2 F 2(1 − ǫ) σw 2 + 2(σu2 + σw )(F − I)
ek = ((f ⋆ h + g − h) ⋆ u)k + ((f ⋆ h − h) ⋆ w)k + (f ⋆ v)k (21)
(14)
dHM V =
p X
dθj dH j
(29)
(30)
j=1
The robust minimum variance filter for sk is given by: Theorem 2: The robust minimum variance filter for the uncertain system (1) with stochastic uncertainty (19) is given in Theorem 1 with ǫj = σj2 (31) 1−ǫ Proof: From the expression of the first variation of the estimation error, we obtain:
where dΘ is a vector of p independent white processes with variance Σ. By construction, the matrix Σ is diagonal and its diagonal elements are denoted σj2 . This kind of representation of uncertainties has been adopted in [2] to design Wiener and Kalman filters. Due to the model of the uncertainties, we will consider the first variation of the estimation error e w.r.t. Θ in order to design the robust filter. It comes that:
Noting that eT de depends linearly w.r.t. to dΘ, the expectation of this term is null. Hence, we get:
ek = ek + dek
JM V = E{eT e} + E{deT de}
(20)
JM V = E{eT e} + 2E{eT de} + E{deT de}
(32)
(33)
Developing this expression and comparing it with (13), it is clear that: J = (1 − ǫ)JM V . After differentiation, the result in Theorem 2 is straightforward. This result has a practical interest for the design of the ǫ-contaminated robust filter. As a matter of fact, the designer can evaluate the sensitivity of each parameter on the estimation error and decide to set for each parameter an amount of probable uncertainty. Doing so, values of σj can be given and then, values of ǫj are known. V. R ELATION WITH BIAS AWARE FILTERING In [8], the authors presented an approach to overcome uncertain estimation problem with bias. In this section, it will been showed that the result in [8] is a special case of the result proposed here. Introduce the signal i = s − Hu. Now consider the system with finite impulse response x with the measured output ξ defined by:
An hydrocarbon concentration sensor has been characterised by its impulse response: h(t) = 1 − e−θ1 kTs 1 − e−kTs e−0.396kTs
with
θ1 = 0.110 sampling period Ts = 0.5s 2 2 • σw = 2 and σv = 2. The input u and the signal s are given in figure 2, the measurement noise characteristics results in SNR = 18dB. •
•
This sensor is used to evaluate the cracking performance of a boiler by analysing the concentration of unburned hydrocarbon in its smokes. In presence of uncertainty in the impulse response, the evaluation is not reliable. Hence, robust filtering should be performed in order to improve the quality of the data. We consider a range of variation δθ1 = ±10% of its 30
(34)
with ψ is a white noise with unit variance ˆ of the input signal ψ is given by the Then, the estimate ψ inverse Wiener filter, i.e.: ˆ = X T XX T + σv2 I ξ ψ (35)
25 20 15 magnitude
5
with X = T (x). Furthermore, the estimate ˆi of the signal i is:
0 −5
(36)
−10 0
Combining this expression with the estimate sˆ of the signal s, we obtain: (37)
Furthermore, if we assume that the actual system impulse response is such that H = H + ∆H then the expression of the signal i is given by,
80
100
120
Input u (dashed) vs Signal s (solid) for δθ1 = −9%.
nominal value. The performance will be evaluated in term of the Relative Mean Square Error (RMSE) over 100 realisations of noises.
(38) 0.23717 0.21813 0.19908 0.18004
0.16099 195
0.11 0.108
0.23717 0.21813 0.19908 0.18004 0.16099
0.14195
0.10386
0.112
0.1229
0.114
0.106 0.10386
0.104
0.1229
θ1
0.102
5
which means that we indirectly construct an estimator of the bias signal generated by the presence of the uncertainty.
0.116
419
(40)
0.118
0.1
where s = H(u + w). Hence, the signal b can be viewed as an unknown bias signal and the estimated signal ˆi is :
0.14
0.12
This term is compounded of the two terms, namely • a nominal stochastic signal ζ = Hw • a signal generated by the mismatch in the impulse response b = ∆H(u + w). If we replace the latter in the expression of the signal s, we get that: (39) s=s+b
ˆ ˆi = ζˆ + b
60 time
4816
i = ∆H(u + w) + Hw
Fig. 2.
40
0.1
ˆ = Hu + F ξ = Hu + ˆi s
20
0.08
ˆ ˆi = X ψ
10
0.16 099 0.1 419 5 22 9
ik = (x ⋆ ψ)k ξ k = ik + v k
VI. I LLUSTRATIVE EXAMPLE
0.1 −4
−3
10
−2
10
10
−1
10
ε1
Fig. 3.
Mapping of the RMSE along ǫ1 and θ1 .
From figures 3-4, we can see that the proposed robust filter permits to reduce the RMSE compared to the nominal
filter. From figure 4, for values of ǫ1 close to 3 · 10−2 , the RMSE varies in a narrow range for all possible values of θ1 . The optimal choice of ǫ1 seems to be in the vicinity of ǫ∗1 = 3 · 10−2 . On the other hand, considering that the
1.5 1 0.5
magnitude
0
RMSE
0.8
−0.5
0.7
−1
0.6
−1.5
0.5
−2
0.4
−2.5 0
20
40
60
80
100
120
time 0.3
Fig. 6. Signal i (dashed) and its optimal estimate ˆi (solid) for δθ1 = −9%.
0.2 0.1 0 −4 10
−3
−2
10
−1
10
0
10
10
ε1
Fig. 4.
Mapping of the RMSE along ǫ1 and θ1 .
distribution of θ1 is uniform in δθ1 , it turns out that the uncertainty is 2.5 · 10−2 . As ǫ∗1 /(1 − ǫ∗1 ) ≡ ǫ∗1 , this result consolidates the relation between the ǫ-contaminated design and the stochastic representation of the uncertainty. Finally, in figure 5, we have plotted the estimation errors of the nominal filter and the robust filter (with ǫ1 = 0.0346) when δθ1 = −9%. It appears that the estimation error of the robust filter almost centred on zero while the nominal filter is severely biased. 1
magnitude
0.5
0
−0.5
−1
−1.5 0
20
40
60 time
80
100
120
Fig. 5. Estimation error of the nominal filter (dashed) vs optimal robust filter with ǫ1 = 0.0346 (solid) for δθ1 = −9%.
In addition, we have plotted in figure 6, the estimate of the signal i obtained for the optimal filter and δθ1 = −9%. It appears that the estimate ˆi is relevant with the original signal i which clearly enlightens the interpretation given previously. VII. C ONCLUDING REMARKS In this article, the robust filtering problem for uncertain discrete-time systems has been treated. The approach adopted is an ǫ-contaminated one where the contaminating agent is
the sensitivity of the estimation error w.r.t. the uncertain parameters. Based on the impulse response of the system, the result is presented in term of Toeplitz matrix equations. It has been shown that this approach is equivalent to the stochastic modelling of the uncertainty. An example has shown the efficiency of the approach and the effectiveness of the equivalence of the approaches. R EFERENCES [1] B. Bayon, G. Scorletti, E. Blanco, ”Robust L2-Gain Observation for structured uncertainties: an LMI approach”, 50th IEEE Conf. on Decision and Control and European Control Conference CDC-ECC, Orlando, USA, pp. 4949-4954, December, 2011 [2] M.J. Grimble, ”Wiener and Kalman Filters for Systems with Random Parameters”, IEEE Trans. on Autom. Control, vol. 29, pp. 552–554, 1984. [3] F.L. Levis, L. Xie and D. Popa, ”Optimal and Robust Estimation”, CRC Press , second edition, 2008 [4] H. Li and M. Fu, ”A Linear Matrix Inequality Approach to H∞ Robust Filtering”, IEEE Trans. Signal Processing, vol. 45, pp. 2338–2350, 1997. [5] Ph. Neveux, E. Sekko and G. Thomas, ”A Constrained Iterative Deconvolution Technique with an Optimal Filtering: Application to a Hydrocarbon Concentration Sensor”, IEEE Trans. on Instrumentation ans Measurement, vol. 49, pp. 852-856, 2000. [6] Ph. Neveux and G. Thomas, ”Robust filtering for uncertain systems”, Signal Processing, vol. 81, pp. 809–817, 2001. [7] Ph. Neveux, E. Blanco and G. Thomas, ”Robust filtering for linear time invariant continuous systems”, IEEE Trans. Signal Processing, vol. 55, pp. 4752-4757, 2007. [8] Ph. Neveux, ”A Polynomial Approach to Bias Aware Fixed-Lag Smoothing Problem”, IEEE Transactions on Automatic Control, vol. 54, pp. 10681072, 2009. [9] R.M. Palhares and P.L.D. Peres, ”Robust H∞ filter design with pole constraints for discrete-time systems”, Journal of the Franklin Institue, vol. 337, pp. 713-723, 2000. [10] I.R. Petersen and A.V. Savkin, Robust Kalman Filtering For Signals and Systems with Large Uncertainties, Birkha¨user, 1999. [11] C.E. de Souza, K.A. Barbosa and A. Trofino Neto, ”Robust H∞ filtering for discrete-time linear systems with uncertain time-varying parameters”, IEEE Trans. on Signal Processing, vol. 54, pp.2110-2118, 2006. [12] Y. Theodor, U. Shaked and C.E. de Souza, ”A game theory approach to robust discrete-time H∞ -estimation”, IEEE Trans. on Signal Processing, vol. 42, pp. 1486-1495, 1994. [13] T. Zhou, ”Sensitivity penalization based robust state estimation for uncertain linear systems”, IEEE Trans. Automat. Contr., vol. 55, pp. 1018– 1024, 2010.
