Recurrent-neural-network-based identification of a ... - Springer Link

Journal of Mechanical Science and Technology 26 (5) (2012) 1599~1606 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0304-z

Recurrent-neural-network-based identification of a cascade hydraulic actuator for closed-loop automotive power transmission control† Seung-Han You1 and Jin-Oh Hahn2,* 1

Hyundai Motor Company, Seoul, Korea Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada

2

(Manuscript Received June 9, 2011; Revised October 2, 2011; Accepted January 20, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract By virtue of its ease of operation compared with its conventional manual counterpart, automatic transmissions are commonly used as automotive power transmission control system in today’s passenger cars. In accordance with this trend, research efforts on closed-loop automatic transmission controls have been extensively carried out to improve ride quality and fuel economy. State-of-the-art power transmission control algorithms may have limitations in performance because they rely on the steady-state characteristics of the hydraulic actuator rather than fully exploit its dynamic characteristics. Since the ultimate viability of closed-loop power transmission control is dominated by precise pressure control at the level of hydraulic actuator, closed-loop control can potentially attain superior efficacy in case the hydraulic actuator can be easily incorporated into model-based observer/controller design. In this paper, we propose to use a recurrent neural network (RNN) to establish a nonlinear empirical model of a cascade hydraulic actuator in a passenger car automatic transmission, which has potential to be easily incorporated in designing observers and controllers. Experimental analysis is performed to grasp key system characteristics, based on which a nonlinear system identification procedure is carried out. Extensive experimental validation of the established model suggests that it has superb one-step-ahead prediction capability over appropriate frequency range, making it an attractive approach for model-based observer/controller design applications in automotive systems. Keywords: Closed-loop power transmission control; Automatic transmission; Hydraulic actuator; Pressure control; Recurrent neural network; Nonlinear system identification ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction In today’s passenger car industry, the use of an automatic transmission as power transmission control system keeps increasing by virtue of its ease of operation compared with a conventional manual transmission. The automatic transmission automatically performs two main power transmission functions, i.e. gear shift clutch control and torque converter clutch control. Extensive research on the development of hardware and software aimed at high-quality power transmission control has been carried out following this trend. Jung et al. (1999) proposed a novel gear shift control algorithm for an automatic transmission based on a proportional solenoid valve. Zheng et al. (1999) investigated the design of a gear shift control system during inertia phase based on a low-order linear model. Shin et al. (2000) developed and experimentally validated a supervisory gear shift control system adaptable to a wide range of vehicle operating conditions. Hahn et al. *

Corresponding author. Tel.: +1 780 492 6534, Fax.: +1 780 492 2200 E-mail address: [email protected] † Recommended by Associate Editor Yang Shi. © KSME & Springer 2012

(2002a) developed a robust nonlinear control algorithm for a torque converter clutch slip system using open-loop and closed-loop torque estimators. Hahn et al. (2002b) also developed a self-learning supervisory system for inertia-phase gear shift control. More recently, Hibino et al. (2009) presented a simplified design method for a robust torque converter clutch slip control system. Due to the extreme complexity of its first-principle modeling, the dynamics of the hydraulic actuator has been neglected in the majority of the existing research efforts on closed-loop and supervisory power transmission controls for automotive systems. Instead, only its steady-state input-output relationship has been used as a map to convert the commanded target pressure to the actual input fed to the hydraulic actuator (e.g. duty cycle signal), which essentially leads to an open-loop control scheme. Since the ultimate viability of closed-loop power transmission control is dominated by precise pressure control at the level of hydraulic actuator, closed-loop control can potentially attain superior efficacy in case the hydraulic actuator can be easily incorporated into model-based observer/ controller design. Recently, several researchers suggested the use of linear

1600

S.-H. You and J.-O. Hahn / Journal of Mechanical Science and Technology 26 (5) (2012) 1599~1606

empirical models to overcome the aforementioned issue. Cho et al. (1998) proposed a low-order linear model of a proportional solenoid valve for gear shift control of an automatic transmission using the linear system identification method. Hahn et al. (2002c) adopted a system identification technique to obtain a Hammerstein-type linear empirical model, which was incorporated in the design of a robust pressure observer. Besides, Gao et al. (2010) used a simple dynamic model of a hydraulic actuator to design an automatic transmission pressure observer. Despite its local validity and ease of use in observer/controller design, the applicability of linear modeling techniques applied to automotive hydraulic actuators can end up with limited value, due to large amount of uncertainty and nonlinearity inherent in these hydraulic actuators. Nonlinear system identification methods based on neural networks can be an attractive alternative to yield a desirable compromise between complexity (first-principle modeling) versus limited fidelity (linear system identification). The resulting nonlinear empirical models will be relatively complex in comparison with their linear counterparts. However, neural network models can leverage many existing observer/controller design methods available in the literature (e.g. Henriques et al. (1999), Zhai et al. (2010)). Regardless, existing investigations on the use of neural networks for hydraulic actuator modeling is very rare. In this paper, we present an empirical model of a cascade hydraulic actuator in a passenger car automatic transmission based on the recurrent neural network (RNN). The steadystate and the dynamic characteristics of the cascade hydraulic actuator are experimentally analyzed, which guides the appropriate selection of the model structure. Then, a nonlinear empirical model is obtained by a system identification procedure. Extensive experimental validation of the established model suggests that it has superb one-step-ahead prediction capability over appropriate frequency range, making it an attractive approach for model-based observer-controller design applications in automotive systems.

Fig. 1. Main pressure control system.

2. Cascade hydraulic actuator system The cascade hydraulic actuator system considered in this paper consists of a main pressure control system (MPC) and an actuating pressure control system (APC) (see Figs. 1 and Fig. 2). It is noted that many hydraulic actuator systems used in automotive power transmission control applications consist of an MPC and an APC. MPC sets up the maximum pressure supplied to the port #2 of APC by controlling the pressure control solenoid valve (PCSV). APC controls the pressure supplied to its port #2 via on-off control of the PWM (Pulse-Width-Modulation) solenoid valve (SOL), thereby managing the engagement and the disengagement of the clutch. The effective pressure applied to the clutch is the difference between the engaging pressure and the disengaging pressure (see Fig. 2). Figs. 1 and 2 suggest that the cascade hydraulic actuator system, consisting of MPC

Fig. 2. Actuating pressure control system.

and APC, can be regarded as a two-input/one-output system, where the inputs to the system are the control signals for PCSV and SOL, and the output is the effective pressure applied to the clutch.

3. Experimental analysis Experimental analysis on the cascade hydraulic actuator system was conducted in order to grasp its steady-state and dynamic response characteristics with respect to the two system inputs, and some significant experimental results are described in the following.


Fig. 3. Steady-state effective pressure response to PCSV.

1601

Fig. 5. Steady-state effective pressure response to SOL under constant PCSV duty ratios.

(a) 0.1 Hz Sweeping frequency Fig. 6. A modular model structure for cascade hydraulic actuator system.

appear if the sweeping frequency of PCSV duty ratio exceeds 1Hz (see Fig. 4). (b) 0.2 Hz Sweeping frequency

(c) 1.0 Hz Sweeping frequency Fig. 4. Dynamic effective pressure response to PCSV.

3.1 Effective pressure response to PCSV control The experimental results on the effective pressure response to PCSV control are shown in Figs. 3 and 4. In the experiments, the range of duty ratio signal fed to PCSV was varied within 10%-30%, which corresponds to nominal supply pressure of 5bar-8bar. The duty ratio signal fed to APC was fixed at 100% so that it is forced to generate maximum possible effective pressure for a given MPC supply pressure. The steady-state characteristics of the effective pressure with respect to PCSV duty ratio shows that it is in inverse proportion to PCSV duty ratio (see Fig. 3). The effective pressure assumes values between 5.5 bar and 7.4 bar. The dynamic characteristics of the effective pressure with respect to PCSV duty ratio suggests that significant dynamic delay will

3.2 Effective pressure response to SOL control The steady-state characteristics of the effective pressure with respect to SOL duty ratio when MPC duty ratio is fixed at several different values is shown in Fig. 5. It can be seen that within the available SOL duty ratio envelope (where the effective pressure is not saturated), the effective pressure response is independent of PCSV duty ratio applied to MPC. On the other hand, the variation of PCSV duty ratio affects the available SOL duty ratio envelope, i.e. PCSV duty ratio is related only to the saturating pressure level but it does not affect the relationship between SOL duty ratio and the effective pressure in the absence of saturation. Based on the experimental analysis, the effect of MPC on APC is just to limit the maximum available effective pressure, which makes it possible to model MPC and APC independently. This finding yields a modular model structure shown in Fig. 6 for the cascade hydraulic actuator system considered in this paper. It is also noted in Fig. 5 that although the steady-state effective pressure responses to SOL duty ratio are consistent with respect to different PCSV duty ratio levels in the available SOL duty ratio envelope, there is some degree of variability in responses. In particular, a relatively large variability can be observed in the neighborhood of 65% SOL duty ratio (which corresponds to t = 20 sec). Considering that the fidelity of the hydraulic actuator system model is especially crucial for the range of SOL duty ratio associated with clutch engagement (i.e. positive effective pressure), the potential modeling errors due to the variability in this region are not practically important, because this region corresponds to negative effective

1602


(a) 0.2 Hz Sweeping frequency

(b) 0.5 Hz Sweeping frequency

Fig. 8. Dynamic input-output recurrent neural network.

Fig. 7. Dynamic effective pressure response to SOL.

pressure in which the target clutch is fully disengaged. The dynamic characteristics of the effective pressure with respect to SOL duty ratio is shown in Fig. 7. In this experiment, PCSV duty ratio was fixed at 10% to yield maximum supply pressure. There exists apparent dynamic delay in the effective pressure response to SOL duty ratio that is larger than that associated with PCSV duty ratio. Comparing the effective pressure responses shown in Figs. 5 and 7 clearly demonstrates that the transient delay in effective pressure response increases as the SOL duty ratio sweeping frequency increases.

4. Nonlinear system identification based on recurrent neural network One of the most popular approaches to obtain linear empirical models is to exploit linear system identification theory (Ljung, 1999). However, it is not the most attractive approach to the problem at hand, because the linear empirical model of the cascade hydraulic actuator may be valid only in the neighborhood of a specific operating point due to the nonlinear characteristics discussed in Section 3. In essence, it is reasonable to claim that linear empirical models suffer from limited performance in “globally” representing the nonlinear characteristics of the cascade hydraulic actuator. To resolve this drawback, this paper proposes to use a RNN as a dynamic model structure for nonlinear system identification of the cascade hydraulic actuator system. Among a variety of RNN structures that have been suggested up to now, an input-output RNN (Haykin, 1999) shown in Fig. 8 was used in this paper. Input-output RNN is a transformed version of a multi-layer neural network. It is made up of a non-linear multi-layer neural network and time delay operators. Mathematically, the RNN model in Fig. 8 can be represented as follows:

.

(1)

The model structure shown in Fig. 6 suggests that MPC and APC can be modeled separately from each other. In accordance with this reasoning, two nonlinear RNN models were developed for MPC and APC, respectively, instead of identifying the overall dynamics of the multi-input cascade hydraulic actuator system using a single RNN. The reason for this separate model identification strategy was to reduce the possibility of RNN falling into local minima: it is expected that large number of neurons are required in RNN if it has to model the entire cascade system, compared with the case where MPC and APC are first modeled separately and later combined together to represent the overall system dynamics. Sine wave was chosen as the excitation signal for the nonlinear system identification task in this paper. The frequency envelope of the system identification was selected to be within 10Hz, which was chosen based on the speed of system response obtained by the experimental analysis. In fact, the effective pressure response of the cascade hydraulic actuator system to the sine wave inputs above 10 Hz showed significant time delays, which suggests that the frequency range beyond 10 Hz is not supposed to be its nominal operating frequency envelope. Within this frequency band, the excitation signal was designed as a linear combination of 50 sine waves with random magnitudes and phases, which was then digitized to carry out the system identification procedure in the discrete-time domain. On the other hand, the number of the delay operators used in RNN was determined in accordance with the order of the state-space representation of the cascade hydraulic actuator system derived using the Newton’s 2nd law of motion, which dictates that (1) MPC is expressed as a set of nonlinear differential equations with 10 state variables, whereas (2) APC is


1603

represented as a set of nonlinear differential equations with 9 state variables. Were a linearization procedure to be applied to a set of nonlinear differential equations with q states, the resulting input-output relationship assumes the following form in the discrete-time domain: (2) which suggests that the numbers of the time delay operators required in the input and the output channels are q − 1 and q , respectively. Following this result for the linear systems, 9 and 10 time delay operators were used in the input and the output channels of RNN model for MPC, respectively, and 8 and 9 time delay operators were used in the input and the output channels of RNN model for APC. Based on the universal approximation theorem (Haykin, 1999), only RNN with a single hidden layer was used for system identification. The RNN was equipped with sigmoidal activation functions in the hidden layer neurons and linear activation functions in the output layer neuron. The experimental data were filtered with a low-pass filter having 10 Hz cut-off frequency before used for RNN-based nonlinear system identification. Using the training data, the RNN models were optimized with the Levenberg-Marquardt algorithm.

(a) Identification result

(b) Error of identification system Fig. 9. Training result for MPC RNN model.

(a) Observation result

5. Cross-validation of identified MPC-APC system dynamics To investigate the generalization capability of the “modular” nonlinear RNN model of the cascade hydraulic actuator system, the following experimental validations were carried out: (1) The prediction performance of the MPC RNN model for the effective pressure response to slow PCSV duty ratio change (steady-state prediction). (2) The prediction performance of the MPC RNN model for the effective pressure response to step PCSV duty ratio change (transient prediction). (3) The prediction performance of the APC RNN model for the effective pressure response to slow SOL duty ratio change (steady-state prediction). (4) The prediction performance of the APC RNN model for the effective pressure response to step SOL duty ratio change (transient prediction). (5) The prediction performance of the combined MPC-APC RNN models for the effective pressure response to randomly combined sinusoids inputs to PCSV and SOL. The experimental data were filtered with a low-pass filter having 10Hz cut-off frequency before used for model validation. The RNN training result for MPC shows that the prediction error is very small, i.e. the training error is mostly bounded within 0.2 bar (Fig. 9). Figs. 10 and 11 illustrate the prediction performance of the MPC RNN model for the steady-state and

(b) Observation error Fig. 10. Steady-state prediction performance of MPC RNN model.


(b) Observation error Fig. 11. Step response prediction performance of MPC RNN model.

1604


(a) Identification result

(b) Error of identification system Fig. 12. Training result for APC RNN model.


(b) Observation error Fig. 13. Steady-state prediction performance of APC RNN model.

the dynamic characteristics of real MPC, where both the steady-state and the dynamic response of MPC can be predicted with errors less than 0.2 bar-0.3 bar. Fig. 12 shows the RNN training result for APC, where the prediction error is bounded by about 0.3 bar-0.4 bar. Figs. 13 and 14 illustrate the prediction performance of the APC RNN model for the steady-state and the dynamic characteristics of real APC. The prediction error in this case is within 0.2bar0.3bar, which is again reasonably small. There are isolated instants (especially in Figs. 10 and 13) that show relatively large prediction errors. However, most of these errors are due to the measurement noise in the relatively long (e.g. the responses in Fig. 10 and Fig. 13 are approximately 20 sec long) experiment data. Essentially, both the MPC and the APC RNN models could provide sufficient de-


(b) Observation error Fig. 14. Step response prediction performance of APC RNN model.


(b) Observation error Fig. 15. Prediction performance of combined MPC-APC RNN models.

gree of prediction accuracy over all of the training and test data. Finally, the prediction performance of the combined MPCAPC RNN models (which consists of modularly constructed RNN models for MPC and APC (see Fig. 6)) in response to simultaneous PCSV and SOL duty ratio challenges was examined. One such result based on randomly combined sinusoids is illustrated in Fig. 15, where the RNN model is shown to predict the effective pressure response of the cascade hydraulic actuator system accurately with errors mostly within 0.3bar, with occasional isolated exceptions. Fig. 15 also supports the validity of our key assumption on the cascade hydraulic actuator system that the entire system model can be established as a combination of two independent subsystem models, i.e. MPC and APC. In our previous study (Hahn et al. (2003)), we examined the


relative accuracy of the empirical models of the APC derived as a pure linear model, a Hammerstein model, and an RNN model, in which we demonstrated that the RNN model could yield approximately 64% and 18% reductions in the rootmean-squared prediction errors compared with the pure linear and Hammerstein models, respectively. Though MPC was not taken into account in our previous work, the RNN model we developed in this paper for the combined MPC-APC system is expected to achieve comparable superiority to its linear counterparts, because the major role of MPC is to set an upper bound on the effective pressure. Summarizing, this paper demonstrated that RNN can be a viable approach to nonlinear empirical modeling of complex automotive hydraulic actuator systems. It is expected that the RNN-based models established in this paper can provide an attractive alternative to the linear empirical models in designing observers and controllers to improve the closed-loop performance of automotive power transmission systems. There are several limitations in this study. First, this paper focused on the one-step-ahead prediction as the performance metric. This metric is appropriate if the objective is to use the RNN models as observer/controller design models. However, additional metrics such as pure prediction (i.e. simulation) must also be examined if the model is to be used as an openloop virtual sensor. Second, the frequency range of the excitation signals used in nonlinear system identification was chosen somewhat empirically. Although our choice is based on the findings from experimental analysis, future study on the systematic excitation signal design may be rewarding. Third, the measurement noise was not rigorously excluded in the system identification process conducted in this paper (except for the 10Hz low-pass filtering at the pressure transducer). As such, it is possible that the fidelity of the RNN models may have been negatively affected by the presence of residual noise. From this perspective, strict quantification of the signalto-noise ratio and pre-processing of measurements may further improve the quality of the resulting RNN models.

6. Conclusions High-fidelity and mathematically tractable models of hydraulic actuator systems has potential to improve the closedloop control performance of automotive power transmission systems, by facilitating the design of model-based observers and controllers. This paper presented the use of RNN for nonlinear system identification of complex automotive hydraulic actuator systems. The feasibility of the approach was demonstrated via rigorous experimental analysis and crossvalidations. Future work will investigate strategies and algorithms to design RNN-model-based observers and controllers for automotive power transmission control applications.

Acknowledgment This work was supported in part by Research Program sup-

1605

ported by the Ministry of Education, Science and Technology the Natural Science and Engineering Research Council of Canada.

References [1] G. H. Jung, B. H. Cho and K. I. Lee, Shifting control method for ef-automatic transmission with proportional control solenoid valve, Transaction of KSAE, 4 (1999) 251-259. [2] Q. Zheng, K. Srinivasan and G. Rizzoni, Transmission shift controller design based on a dynamic model of transmission response, Control Engineering Practice, 7 (8) (1999) 1007-1014. [3] B. K. Shin, J. O. Hahn, K. Yi and K. I. Lee, A supervisorbased neural-adaptive shift controller for automatic transmissions considering throttle opening and driving load, Journal of Mechanical Science and Technology, 14 (4) (2000) 418-425. [4] J. O. Hahn and K. I. Lee, Nonlinear robust control of torque converter clutch slip system for passenger vehicles using advanced torque estimation algorithms, Vehicle System Dynamics, 37 (3) (2002) 175-192. [5] J. O. Hahn, J. W. Hur, G. W. Choi, Y. M. Cho and K. I. Lee, Self-learning approach to automatic transmission shift control in a commercial construction vehicle during the inertia phase, Journal of Automobile Engineering, 216 (D11) (2002) 909-919. [6] R. Hibino, M. Osawa, K. Kono and K. Yoshizawa, Robust and simplified design of slip control system for torque converter lock-up clutch, ASME Dynamic Systems, Measurement and Control, 131 (1) (2009) 011008. [7] B. H. Cho, G. H. Jung, J. W. Hur and K. I. Lee, Modeling of proportional control solenoid valve for automatic transmission using system identification theory, Proceedings of the SAE Transmission and Driveline Systems Symposium, USA (1999) 329-336. [8] J. O. Hahn, J. W. Hur, Y. M. Cho and K. I. Lee, Robust observer-based monitoring of a hydraulic actuator in a vehicle power transmission control system, Control Engineering Practice, 10 (3) (2002) 327-335. [9] B. Gao, H. Chen, H. Zhao and K. Sandra, A reduced-order nonlinear clutch pressure observer for automatic transmission, IEEE Transactions on Control Systems Technology, 18 (2) (2010) 446-453. [10] J. Henriques and A. Dourado, Multivariable adaptive control using an observer based on a recurrent neural network, International Journal of Adaptive Control and Signal Processing, 13 (4) (1999) 241-259. [11] Y. Zhai, D. Yu, H. Guo and D. L. Yu, Robust air/fuel ratio control with adaptive DRNN model and AD tuning, Engineering Applications of Artificial Intelligence, 23 (2) (2010) 283-289. [12] L. Ljung, System identification: Theory for the user, Second Ed. Prentice-Hall, Upper Saddle River, USA (1999). [13] S. Haykin, Neural networks: A comprehensive foundation, Second Ed. Prentice-Hall, Upper Saddle River (1999).

1606


[14] J. O. Hahn, J. W. Hur, Y. M. Cho and K. I. Lee, Empirical modeling of a hydraulic actuator in a vehicle power transmission control system, Mathematical and Computer Modeling of Dynamical Systems, 9 (2) (2003) 193-208.

Seung-Han You received B.S, M.S and Ph.D degrees in mechanical and aerospace engineering from Seoul National University, Korea, in 1999, 2001, and 2006, respectively. Since 2006, he has been with Hyundai Motor Company, where he is Senior Research Engineer. His current research interests are controller/observer design and system optimization with applications to automotive chassis control systems.

Jin-Oh Hahn received B.S and M.S degrees in mechanical engineering from Seoul National University, Korea, in 1997 and 1999, and Ph.D degree in mechanical engineering from Massachusetts Institute of Technology in 2008. Since 2010, he has been with the University of Alberta, where he is Assistant Professor in the Department of Mechanical Engineering, Assistant Adjunct Professor in the Department of Biomedical Engineering, and Research Affiliate at the Glenrose Rehabilitation Hospital. His current research interests include control systems approach to health monitoring, diagnostics and maintenance of dynamic systems, with applications to biomedicine and bionic systems, automotive systems, and energy systems.