Modeling and Deterioration Diagnosis of Catalyst for Automobile Exhaust Gas by On-Line Identification Method With Variable Forgetting Factor

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Hiroyuki Okuda Non-member (Utsunomiya University) Shuichi Adachi Member (Keio University, adachi@appi.keio.ac.jp) Shinji Niwa Non-member (Daihatsu Motor Co., Ltd., Shinji Niwa@mail.daihatsu.co.

jp) Keywords: catalyst, deterioration diagnosis, system identification, variable forgetting factor, recursive least squares method The environmental preservation is a critical task for the automotive

1 Extended Summary pp Modeling and Deterioration Diagnosis of Catalyst for Automobile Exhaust Gas by On-Line Identification Method With Variable Forgetting Factor Yasushi Okada Non-member (Utsunomiya University) Hiroyuki Okuda Non-member (Utsunomiya University) Shuichi Adachi Member (Keio University, Shinji Niwa Non-member (Daihatsu Motor Co., Ltd., Shinji Mitsunobu Kajitani Non-member (Daihatsu Motor Co., Ltd., Mitsunobu Seiji Hashimoto Member (Gunma University, Keywords: catalyst, deterioration diagnosis, system identification, variable forgetting factor, recursive least squares method The environmental preservation is a critical task for the automotive industry. In relation to this matter, regulations for automobile exhaust gas are enforced. The present exhaust gas purification system has been composed of an electric fuel-injection system and a three-way catalytic converter, that simultaneously oxidizes excess levels of exhaust carbon monoxide (CO) and unburned HCs, and reduces excess levels of the oxides of nitrogen (NO x ). In such system, the purification ability of the catalyst drastically lowers when the catalyst deteriorated. On the other hand, since the operability of the automobile is not affected by the deterioration, there is a danger which keeps using the car with deteriorated catalyst. Therefore, on-board diagnosis for the catalysts is strongly required. The object of this study is to construct an on-board catalyst diagnosis system to detect the deterioration under the actual driving conditions. In this paper, the on-line identification and deterioration diagnosis of the catalyst based on the recursive least squares (RLS) method with variable forgetting factor (VFF) is proposed. Fig. 1 shows the automobile exhaust gas purification system. It is well known that the air-fuel ratio (A/F) is controlled in order to effectively utilize the purification performance of the catalyst. The control is carried out by constructing double feedback loop which uses outputs of two exhaust gas sensors installed on upstream and downstream of the catalyst. In this study, from the viewpoint of the cost reduction and the practical utilization on the market, the oxygen sensor which possesses the non-linearity to the A/Fisemployed instead of the more costly linear A/F sensor. For the non-linearity of the sensor, the hysteresis property is dominant. The catalyst also has a dynamic non-linearity which is caused by the chemical reaction. Input and output signals in the modeling of the catalyst are the voltage signal from the oxygen sensors. Therefore, the identified object includes both non-linearities. Since the hysteresis property is dominant in the non-linearities of the identified object, we adopt the on-line identification based on RLS method with VFF which can adaptively identify the change of the characteristic. The advantage to introduce the forgetting factor into the RLS method is that the data in a past can be exponentially forgotten in the parameter estimation. In addition, by making the forgetting factor variable, it is expected that the hysteresis characteristic of the object, which divisionally changes in proportion to the piecewise regions, can be accurately identified. Moreover, the main contribution to introduce the RLS method with VFF is that the deterioration diagnosis of the catalyst become possible through the time ratio of VFF defined as R VFF. The identification and diagnosis experiments were carried out with three kinds of catalysts that differ in deterioration and two driving patterns. The introduced R VFF with respect to the deterioration degree is shown in Fig. 2. The horizontal axis represents the deterioration degree normalized by the deterioration of the 60,000 km driving. It can be confirmed that R VFF linearly increases with the deterioration degree. The validity of utilizing this value for the catalyst deterioration diagnosis has been experimentally verified. Fig. 1. Automobile exhaust gas purification system Fig. 2. Rates of change of forgetting factor 6

2 Modeling and Deterioration Diagnosis of Catalyst for Automobile Exhaust Gas by On-Line Identification Method With Variable Forgetting Factor Yasushi Okada, Non-member, Hiroyuki Okuda, Non-member, Shuichi Adachi,Member, Shinji Niwa, Non-member, Mitsunobu Kajitani, Non-member, Seiji Hashimoto,Member The majority of the conventional system purifying exhaust gas is composed of a three-way catalyst and an electronic fuel injection. However, harmful pollutants are increasingly emitted when the catalyst becomes aged.therefore, it is necessary to detect the deterioration of the catalyst by means of on board diagnosis. In this paper, we propose a diagnosis method of the aged catalyst using recursive system identification method with variable forgetting factor. We focus on the parameter of identified model which represents a characteristic of the catalyst, and possibility to describe the diagnosis of the aged catalyst. 2 Keywords: catalyst, deterioration diagnosis, system identification variable forgetting factor recursive least squares method HC, CO NO x Utsunomiya University 7-1-2, Yoto, Utsunomiya Keio University , Hiyoshi, Kohoku-ku, Yokohama Daihatsu Motor Co., Ltd Yamanoue, Ryuo-cho, Gamo-gun, Shiga Gunma University 1-5-1, Tenjin-cho, Kiryu OBD On Board Diagnosis (1) VFF Variable Forgetting Factor RLS Recursive Least Squares (2) D

3 VFF RLS JC08 VFF RLS (3) 1 1 A/F 2 A/F 2 2 A/F V OXAD OX2AD Front A/F Rear A/F ũ ỹ OXAD OX2AD u y P S 1 S 2 ỹ = Pũ (1) u = S 1 ũ (2) y = S 2 ỹ (3) u y 2 Fig. 2. Characteristics of output voltage of oxygen sensor. 1 Fig. 1. Composition of system purifying exhaust gas. 3 Fig. 3. System identification problem of catalyst IEEJ Trans. IA, Vol.126, No.12, 2006

4 (a) Dynamic nonlinear system (b) Wiener model {u(k),y(k) :k = 1, 2,...} ARX Auto-Regressive with exogenous input y(k) = θ T φ(k) + w(k) (5) (c) Piecewise dynamic linear model 4 Fig. 4. Conversion from dynamic nonlinear to piecewise and dynamic linear. y = S 2 PS 1 1 u (4) (4) S 2 PS 1 1 (4) 4(a) RLS 4(a) RLS 4(b) Wiener (5) 4 6 4(c) (6) VFF RLS (2) RLS θ = [a 1,...,a n, b 1,...,b n ] T (6) φ(k) = [ y(k 1),..., y(k n), u(k 1),...,u(k n)] T (7) w(k) n J k (θ) = k λ k i (i)ε 2 (i) (8) i=1 ε(i) i VFF RLS Step 1 ε(k) ε(k) = y(k) φ T (k)ˆθ(k 1) (9) Step 2 K(k) P(k 1)φ(k) K(k) = λ(k 1) + φ T (k)p(k 1)φ(k) (10) Step 3 ˆθ(k) ˆθ(k) = ˆθ(k 1) + K(k)ε(k) (11) Step 4 λ(k) λ(k) = 1 1 φt (k)k(k) Σ 0 ε 2 (k) (12) λ(k) <λ min λ(k) = λ min Step 5 P(k) P(k) = 1 λ(k) [I K(k)φT (k)]p(k 1) (13) Step 4 Σ 0 Σ 0 λ min D

5 3 2 2 NO x N 2 O 2 HC CO H 2 O CO 2 5 OX2AD OX2AD OX2AD λ(k i) λ b, i = 0,, l (14) λ k l λ b R VFF R VFF = T λ T exp 100 [%] (15) T λ (14) T exp R VFF VFF RLS F. B. Feedback TTAUS Front A/F Rear A/F A/F OXAD OX2AD F. B. FAF TTAUS % FAF TTAUS FACF F. B. F. B. OXAD* OX2AD* OXAD OX2AD 0.6 V OXAD OX2AD 7 TTAUS 5 Fig. 5. Conversion efficiency versus A/F ratio. Fig Configuration of tested equipment IEEJ Trans. IA, Vol.126, No.12, 2006

6 (a) mode 7 Fig. 7. Block diagram of identification experiment for catalyst. (7) km /10 60 km 2 OBD JC s JC08 JC JC ,400 OXAD OX2AD 9 9 JC s OBD (b) JC08 mode JC08 Fig. 8. Changes of vehicle speed of mode and JC08 mode. OX2AD OX2AD VFF RLS 0.1 V 0.8 V 2 10 OBD 0.1 s RLS 4 3 VFF RLS 9 ū(k) ȳ(k) (5) ū(k) = ū(k 1) + 1 k [um (k) ū(k 1)] (16) D

7 ȳ(k) = ȳ(k 1) + 1 k [ym (k) ȳ(k 1)] (17) u m (k) y m (k) 9 (5) u(k) = u m (k) ū(k) (18) y(k) = y m (k) ȳ(k) (19) 10 Fig. 10. Coherence functions between input and output data. 12 Fig. 12. Rates of change of forgetting factor. 9 Fig. 9. Input and output data. (a) Old Catalyst (b) OBD Catalyst (c) New Catalyst 11 Fig. 11. RLS Identification results by RLS method with VFF IEEJ Trans. IA, Vol.126, No.12, 2006

2 2 ARX 1 Σ 0 VFF RLS Σ 0 = 1.0 λ min = 0.9 (8) 10 τ(k) = 1 1 λ(k) (20) VFF RLS 11 VFF RLS ARX [â 1, â 2, ˆb 1, ˆb 2 ] LS 12 λ b = 0.9999 l = 10 12 1 2 VFF RLS 6 km 1 1 O 2 FFT, 8,83/85 (2002) 2 T.R. Fortescue, L.

Shelef, and R.G. Delosh: Laboratory Evaluation of Three-way Catalysts, SAE Paper, No.760201 (1976) 5, (1993) 6 G.F. Trecate, M. Muselli, D. Liberati, and M.

8 2 2 ARX 1 Σ 0 VFF RLS Σ 0 = 1.0 λ min = 0.9 (8) 10 τ(k) = 1 1 λ(k) (20) VFF RLS 11 VFF RLS ARX [â 1, â 2, ˆb 1, ˆb 2 ] LS 12 λ b = l = VFF RLS 6 km 1 1 O 2 FFT, 8,83/85 (2002) 2 T.R. Fortescue, L.S. Kershenbaum, and B.E. Ydstie: Implementation of Self-tuning Regulators with Variable Forgetting Factors, Automatica, Vol.17, No.6, pp (1981) 3, (1997) 4 H.S. Gandhi, A.G. Piken, M. Shelef, and R.G. Delosh: Laboratory Evaluation of Three-way Catalysts, SAE Paper, No (1976) 5, (1993) 6 G.F. Trecate, M. Muselli, D. Liberati, and M. Morari: A Clustering Technique for the Identification of Piecewise Affine Systems, Automatica, Vol.39, No.2, pp (2003) 7 MATLAB, (2004) 8 MATLAB, (1996) 9 J.M. Maciejowski, (2005) 10 Y. Okada, S. Adachi, and J.M. Maciejowski: A System Identification in the Presence of Nonlinear Sensors, T. of SICE, Vol.41, No.2, pp (2005-2) (in Japanese) J.M. Maciejowski,, 41, 2, pp (2005-2) 11 L. Ljung: System Identification Theory for the user (2nd edition), Englewood Cliffs, Nj: Prentice Hall PTR (1999) 12 H. Maki, S. Akazaki, Y. Hasegawa, I. Komoriya, Y. Nishimura, and T. Hirota: Real Time Engine Control Using STR in Feedback System, SAE Paper, No (1995) 13 H. Yakabe: A/F Control of a Gas Engine Using Response Characteristics of Three-Way Catalysts under Dynamic Operation, T. of JSME, Vol.57, No.539, pp (1991-7) (in Japanese), B, 57, 539, pp (1991-7) D

9 SICE IEEE IEEE IES Best Presentation Award IEEE 1644 IEEJ Trans. IA, Vol.126, No.12, 2006

Frequency Tracking: LMS and RLS Applied to Speech Formant Estimation

Aldebaro Klautau - http://speech.ucsd.edu/aldebaro - 2/3/. Page. Frequency Tracking: LMS and RLS Applied to Speech Formant Estimation ) Introduction Several speech processing algorithms assume the signal