Multirate Closed-Loop System Identification of a Variable Valve Timing Actuator for an Internal Combustion Engine

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1 h American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, WeA9.3 Multirate Closed-Loop System Identification of a Variable Valve Timing Actuator for an Internal Combustion Engine Zhen Ren and uoming. Zhu Abstract: This paper applies the multi-rate closed-loop system identification (ID) technique to a continuously variable valve timing (VVT) actuator system. Limited by the sample rate of the crank-based cam position sensor, function of engine speed, the actuator control sample rate is different from that of cam position sensor. Multi-rate system identification is a necessity for this application. On the other hand, it is also difficult to maintain the desired actuator operational condition with an open-loop control. Therefore, system identification in a closed-loop is required. In this study, PRBS q-markov Cover identification is used to obtain the closed-loop model, and the open-loop system model is calculated based upon information of closed loop controller and identified closed-loop system model. Both open and closed-loop identifications are performed in an HIL simulation environment with a given reference model as a validation process. An intake VVT actuator system test bench is used to conduct the proposed multi-rate system identification using PRBS as excitation signals. Key words: Closed-loop system identification, Multirate system identification, PRBS, Variable valve timing system, Powertrain control C I. INTRODUCTION ontinuously variable valve timing (VVT) system was developed in early nineties []. The benefits of using VVT for internal combustion engines include improved fuel economy and reduced emissions at low engine speed, as well as increased power and torque at high engine speed. Vane-type VVT phaser system [] is a hydraulic mechanic actuator controlled by a solenoid. Electric cam phase actuators become available recently due to its fast responses [3]. There are two approaches to obtain a VVT model for control development and validation: physics based system modeling [4] and system identification. In this paper, the closed-loop system identification approached is employed to obtain the VVT system model. System identification using closed-loop experimental data was developed in seventies [5] and it has been widely used in engineering practice ([6], [7], [8]). Closed-loop system identification can be used to obtain the open-loop system models when the open-loop plant cannot be excited at the conditions ideal for system identification. For instance, the open-loop plant could be unstable. In this Both Ren and Zhu are with Mechanical Engineering of Michigan State University, 555 Engineering Building, East Lansing, MI 4884, USA ( renzhen@msu.edu for Ren and zhug@msu.edu for Zhu). paper, closed-loop identification was selected due to many factors. One main reason is that the system gain of a VVT actuator is a function of engine speed, load, oil pressure and temperature, which made it impossible to maintain the cam phase at a desired value with constant input. Therefore, open loop system identification at a desired cam phase is not practical. In order to maintain at a desired operational condition for identifying the VVT actuator system, closed-loop identification was selected. The purpose of this paper is to obtain linear system models for VVT actuator system at certain operating conditions using the indirect closed-loop system identification method that is discussed in [7]. In this paper, the q-markov COVariance Equivalent Realization (q- Markov Cover) system identification method ([9], [], []) using PRBS (Pseudo-Random Binary Signals) was used to obtain the closed-loop system models. The q- Markov cover theory was originally developed for model reduction. It guarantees that the reduced order system model preserves the first q-markov parameters of the original system. The realization of all q-markov Covers from input and output data of a discrete time system is used for system identification. Q-Markov Cover for system identification uses pulse, white noise, or PRBS as input excitations. It can be used to identify linear model representing same input/output sequence for a nonlinear system []. It was also been extended to identify multirate discrete-time systems when input and output sampling rates are different []. For the proposed study, the multirate system identification is required because the actuator control signal is updated at a different sample rate from that of cam position sensor which is a function of engine speed. For our test bench setup, the cam position sample rate is limited to eight samples per engine cycle. That is, when the engine is operated at 5 RPM, the sample period is ms, while the control output is updated at a fixed period of every 5ms. In this paper, the multirate PRBS closedloop identification was used to conduct closed-loop system identification on a VVT actuator HIL (Hardware-In-the- Loop) system for debugging and validation, and then, the HIL simulator was replaced by the VVT test bench and closed-loop system identification was repeated. The test bench consists of an AC motor driven crank shaft that is connected the cam shaft on cylinder head through a VVT //$6. AACC 664

2 actuator. System identification results show that the PRBS q-markov Cover system identification can be applied to our multirate VVT system when input and output sampling rates are different. The paper is organized as the following. Section II provides framework and formulation of closed-loop identification for the VVT actuator system. Section III shows system identification results using an HIL simulator and Section IV presents the results obtained from the test bench, along with the discussions of the experiment results. Conclusions are provided in Section V. II. SYSTEM IDENTIFICATION FRAMEWORK Consider a general form of linear time-invariant closedloop system in Fig., where r is the reference signal, n is the measurement noise, u and y are input and output. As discussed in the Introduction section, there are many approaches for the closed-loop identification, which are categorized as direct, indirect, and joint input-output approaches. In this paper, we utilize the knowledge of the controller to calculate open-loop plant from identified closed-loop plant, which is called indirect approach. To ensure the quality of identified plant, controller in this paper is set to be proportional ([3] and [4]). r n K(s) (s) u y Fig. : Closed-loop identification framework The input and output relationship of the generalized closed-loop system, shown in Fig., can be expressed below: y=h r=ki+k ( ) r () Let Ĥ be identified closed-loop transfer functions from r to y. The open-loop system model ID can be calculated using identified Ĥ, assuming that ˆ - (I - H) is invertible. The closed-loop controller transfer function is used to solve for the open-loop system models. We have ˆ ˆ - - ID = H(I - H) K () PRBS signal is used in the identification of the system. It is a digital signal and easy to generate for engineering practice. The most commonly used PRBSs are based on maximum length sequences (called m-sequences) [5] for which the length of the PRBS signals is m= n -, where n is an integer (order of PRBS). Let z - represent a delay operator, and define pz ˆ( ) and pz ( ) to be polynomials n pz ( ) az ˆ n az a pz ( ) z, (3) where a i is either or, and obeys binary addition, i.e., = = & = =, (4) and the non-zero coefficients a i of the polynomial are defined in the following table and also in [5]. Table Nonzero coefficients of PRBS polynomial Polynomial order (n) Period of sequence (m) Non-zero Coefficient 63 a 5, a a 4, a a, a 3, a 4, a a 5, a a 7, a 7 47 a 9, a Then the PRBS can be generated by the following formula uk ˆ( ) pz ˆ( ) uk ˆ( ), k,,,, (5) where uˆ() and uˆ ( ) uˆ( ) uˆ( n). Defined the following sequence a; If k is even; sk ( ) (6) a; If k is odd. Then, the signal uk ( ) sk ( ) [- a+ auˆ ( k)] (7) is called the inverse PRBS, where obeys a a = -a = -a -a & a -a = a = -a a. (8) It is clear after some analysis that u has a period m and u (k) = -u (k+m). The mean of the inverse PRBS is m mu Emuk ( ) uk ( ) m (9) k and the autocorrelation ( Ruu ( ) Emuk ( ) uk ( )) of u is a, ; m a, m; Ruu ( ) u( k ) u( k) () m k a / m, even; a / m, odd. The inverse PRBS is used in the q-markov Cover identification algorithm. For convenience, in the rest of this paper, the term PRBS is used to represent the inverse PRBS. In this paper, system models are identified in discrete time domain using PRBS-UI [] developed for multirate PRBS q-markov Cover. The advantage of using the UI in [] is that the number of Markov parameter and the order of the identified model can be adjusted based upon identification results. III. SYSTEM ID USIN AN HIL SIMULATOR Closed-loop system identification was conducted using an HIL simulator. A plant model, described in [6], is loaded into a dspace based HIL simulator..348( s ) () s 6.96 The pulse signal of the cam position sensor is generated in the HIL simulator. In every engine cycle, the HIL simulator generates 4 evenly positioned pulses, which are sampled by an Opal-RT real-time controller. The real-time controller processes the cam position signal and calculates the cam phase position. Figure shows the architecture of the closed-loop system identification. 665

3 that the responses between the original and identified models are fairly close. 5 Fig. : Closed-loop system ID using an HIL simulator After the error between the calculated cam phase and the reference PRBS signal is calculated, the cam actuator command is generated and sampled by the dspace HIL simulator. Reference PRBS signal r and the measured cam phase signal y are recorded for system identification. In this study, a proportional controller K.5 is used. Note that at different engine speeds, the ratio between input and output sample rates is different. Recorded system response data was processed using MATLAB PRBS-UI. Number of Markov parameter was selected based on the quality of the identified model. Model order is selected based on the dominant dynamics of the recorded data, see [9] for details. The open-loop plant models shown in Equation () are obtained from identified closed loop models using (). s rpm () s 5rpm s rpm () s Phase (deg). 6.3s s 3.98s 6.7.5s 6.3s 339 () s.998s 5.74s s 9.6 Magnitude (db) s 78.86s 4. 5 _ Bode Diagram - Frequency (Hz) () Fig. 3: Identified model frequency responses of the HIL simulator Table Closed-loop PRBS q-markov cover system ID results Engine Speed (rpm) 5 DC ain Error (db) ¼ Sample Rate limit (Hz) Plant Resp. Error at Freq. above ain (db) Phase (deg) From Bode diagrams, all identified models are accurate at low frequency, and the accuracy improves as the engine speed increases (See Fig.3 and Table ). Figure 4 compares the responses of original plant and close-loop identified plants at rpm using PRBS excitation. The figure shows one fourth of th order PRBS length (about 5 seconds of the responses). From the plot, one can observe -5 Recorded response Identified model response Fig. 4: Identified model and physical system responses with PRBS input IV. BENCH TEST RESULTS This section discuss the closed system identification results using a test bench, see Fig. 5. Since in this case, the identified system models cannot be compared with a given plant, the system time responses of identified system models are compared with these of the test bench system. A. VVT Phaser Identification Bench Test Setting Fig. 5: VVT phase actuator test bench For identifying the VVT actuator system models using the test bench, the control system setup is the same as the HIL simulation case. The HIL system, shown in Fig., was replaced by the physical test bench (Fig. 5), and the Opal-RT real-time prototype controller adjusts the cam phase by adjusting the actuator solenoid duty cycle (Fig. 6). A Ford 5.4L V8 engine head was modified and mounted on the test bench. The cylinder head has a single cam shaft with a VVT actuator for two intake valves and one exhaust valve. The cam shaft is driven by an electrical motor through a timing belt, and an encoder is install on the motor shaft (simulating the crank shaft), which generates crank angle signal with one degree resolution, along with a so-called gate signal (36 degrees per pulse). Two cam position sensors are installed at other side of the extended cam shaft. One is used to calculate engine position, along with the encoder signals, and the other calculates the cam phase. An electrical oil pump was used to supply pressurized oil for both lubrication and as hydraulic actuating fluid of the cam phase actuator. Intake and exhaust valves are installed to simulate cyclic torque load to the cam shaft. The cam phase actuator system consists of a solenoid driver circuit converting DC voltage command to PWM signal, a solenoid actuator, and hydraulic cam actuator, see Fig. 6. The cam phaser command voltage signal is generated by the Opal-RT prototype controller and sent to the solenoid driver. The PWM duty cycle is proportional to 666

4 input voltage with maximum duty cycle (99%) corresponding to 5V. The solenoid actuator controls the hydraulic fluids (engine oil) flow and changes the cam phase. The cam position sensor signals are sampled by the Open-RT prototype controller and the corresponding cam phase is calculated. The identified cam phase system model is transfer function between the control input u (voltage) and calculated cam phase signal y (degree), see Fig. 6. process, is almost impossible. Therefore, closed loop system identification is a necessity for this application. In order to ensure the closed loop system identification, a proportional controller is selected for the closed loop system identification due to [3] and [4]. Cam phase (deg) 3 - Rising Falling - Fig. 6: Closed-loop system identification architecture B. VVT Phaser Open-loop properties CamPhase (deg) Time (s) Fig. 7: Cam phase actuator open-loop step response The cam phase actuator has an output range of 3 degrees. Fig. 7 shows an open-loop step response of the VVT phaser. It can be found that the cam phase system has a settling time about.5 seconds for advancing (rising) and. second for retard (falling), demonstrating its nonlinear characteristics. This is mainly due to the fact that the VVT actuator has quite different dynamics during advancing and retarding. When the actuator is advancing, the engine oil pressure is the main actuating torque to move cam phase forward. Therefore, the advance rate is dependent on the oil pressure; while during the retard operation, the oil bleeds back to the oil reserve with the help of the cam shaft load. This difference leads to the response rate difference for advance and retard operations, which makes the system nonlinear. This phenomenon will affect the identified model dynamics and will be discussed in next section. Fig. 8 shows the VVT system steady-state responses via open loop constant inputs. It can be observed that for openloop control, the cam phase actuator behaves almost like a binary state and is very difficult for the VVT phaser to maintain at a desired non-saturated valve timing position due to the actuator physical system structure, cam load and engine oil pressure variation. This indicates that open loop system identification, which requires maintaining the actuator operation location during the system identification Solenoid Driver Input (volt) Fig. 8: Cam phase actuator open-loop steady-state responses C. Bench Test Results Bench test setup is quite similar to that of the HIL simulation. However, there are two main differences between the bench and HIL setup. Firstly, the control input is saturated by the physical solenoid drive circuit. The solenoid drive circuit has an operational range of and 5 volts that corresponding to and 99 percent of the solenoid PWM duty cycle. Therefore, in order to avoid saturation, we have to select the phase actuator operation condition carefully; otherwise, the control input could be saturated, leading to high system identification error. Secondly, the control transfer function used for the HIL simulation was unit less, while the control signal for bench tests has a unit of (volt/degree). Therefore, the PRBS signal magnitude is selected to be, nominal operational condition is centered at -4 cam phase, and the control signal is. (volt/degree) Table 3 PRBS q-markov cover system ID parameters Engine Speed (rpm) 5 Input Sample Rate (ms) 5 5 Output Sample Rate (ms) 3 Output/Input Sample Ratio.67.5 PRBS order 3 3 Signal length (s) Number of Markov parameters 9 6 ID open-loop model order 4 and 4 Since the engine speed was limited to 75rpm speed due to the speed limitation of the electrical motor, the system identification bench tests were conducted at only engine speeds ( and 5rpm). Recorded reference signal and system response data are processed using MATLAB PRBS-UI. Number of Markov parameters was selected to optimize the system identification quality and corresponding responses (Fig. 9), and identified model order is determined by the dominant dynamics of PRBS response data (see Fig. ). Fig. shows the order selection chart, generated in PRBS system ID UI, at 5rpm, the chart shows a dominant st order dynamic because the order index chart has the largest gap between the st and nd dots. ap between 4 th and 5 th dots are larger 667

5 than the gap between nd and 3 rd order gaps. Therefore, the order of the identified model is selected to be 4 in order to keep the model order low without losing too much model quality. The rest of system identification parameters are shown in Table 3. Measured Phase (Degree) Recorded Response Identified Model Response Time (Second) Fig. 9: Comparison of identified model and physical responses Order Index ID model order Fig. : Identified model order selection Using the identified closed-loop model and equation (), the open-loop plant model (Table 4 in Appendix section) at 5rpm is obtained. The corresponding open loop Bode diagram (see Fig. ) shows that there exist dynamic modes at around.5 Hz, which is at the engine cycle frequency of.5 Hz (8ms/cycle). We believe this dynamic mode is introduced by the cyclic cam load variation introduced by both the cam drive belt and cyclic torque load. For the purpose of phase controller design, the dynamics introduced by the cam cyclic load can be eliminated. Therefore, a second order model is obtained with all the other parameters to be the same. Plant model calculated from identified nd order model has almost identical behavior to the 4 th order model but without the dynamic mode close to.5 Hz (see Fig. and Table 4 in Appendix section). Only 4th order closed-loop model is identified at rpm engine speed. Similar to the case at 5rpm, the identified model has a dynamic mode at about 8 Hz, which is corresponding to engine cam cyclic load frequency at 8.3 Hz (ms/cycle). However, in this case, a nd order open loop model was not obtained directly from system identification. In the identified model, there exists a pair of nonminimal phase pole-zero at the frequency close to engine cycle frequency. To eliminate the dynamics at this frequency, a nd order model is obtained by removing the pole-zero pairs from the 4th order plant model (Table 4 in Appendix section). The nd order plant model has very similar frequency response as the 4th order plant model except without the dynamics introduced by the cyclic engine cam load (see Fig. ). Magnitude (db) Phase (deg) nd order Model 4th order Model Bode Diagram Frequency (Hz) Fig. : Bode diagram of identified open-loop plant at 5rpm Magnitude (db) Phase (deg) th order Model nd order Model Bode Diagram - - Frequency (Hz) Figure : Bode diagram for identified open-loop plant at 5rpm D. Validation of Identified Model Identified 4th order Model. Identified nd order Model Recorded Data Fig. 3: Closed-loop step response comparison at rpm To evaluate the quality of these identified models, their step responses are compared with responses of the bench physical system. Since the open-loop step response cannot be obtained for the VVT actuator, their closed-loop responses are compared in this section. The controller used for the step responses are the same as those used for the system identification. A step input of crank degrees is used. For bench tests, a reference step is applied when the phaser is operating at - crank degrees. For the identified models, the same operation was simulated in Simulink using identified with the same controller. The normalized step responses are compared in Fig. 3 and

6 response of the physical nonlinear system. We are going to study the effect of this approximation when we evaluate these controller designed based upon the identified models Identified 4th Order Model. Identified nd Order Model Recorded Data Fig. 4: Closed-loop step response comparison at 5rpm From both plots, it can be observed that the system DC gains of both actual system and identified model are fairly close; and for the transient response, the step down responses are very close for both model and actual system in both engine speeds, but the step up responses of the identified model at 5rpm is fast than the actual system. This is mainly due to the nonlinear characteristics of the VVT actuator. When camshaft is moving toward the advance side, VVT phaser actuator is working against engine oil pressure, while when camshaft is moving toward retard side, oil pressure is working with the actuator. Therefore, the system has a slower step up response than step down one. It looks like that the linear identified mode chosen to approximate the fast step down V. CONCLUSION Closed-loop system identification method, using PRBS q-markov Cover, was applied to obtain open-loop system models of a VVT cam actuator system. Constrained by the sample rate of the crank-based cam position sensor (a function of engine speed) and time based control scheme, the actuator control sample rate is different from cam position sensor one. The multi-rate system identification is a necessity. The system identification results, based upon the HIL simulation, show that the closed loop system identification is capable of providing accurate open-loop system models around a desired cam phase; and the identification accuracy were not affected by the engine speed that changes cam phase sample rate. The same technique is also applied to a cam phase actuator bench system, where open-loop system identification is not feasible. The proposed closed-loop system identification approach provides both nd and 4 th order models whose time responses are fairly close to bench responses. The next step is to evaluate the closed loop performance with these controllers design based upon the identified models. APPENDIX Table 4 Identified plant models Engine Speed(rpm) Identified Open-loop Plant (4 th Order) Identified Open-loop Plant ( nd Order) s 5.7s 585.6s.8 s s 5.7s s.63s 78s 3. s 4.6 s.63s s 3.9s 354s.88 s 9.3.4s.4s s 4.54s 597s 8.54 s.38 s 6.78s 34.8 REFERENCES [] Y. Moriya, A.Watanabe, H. Uda, H. Kawamura, M. Yoshiuka, M. Adachi, A Newly Developed Intelligent Variable Valve Timing System - Continuously Controlled Cam Phasing as Applied to New 3 Liter Inline 6 Engine, SAE paper 96579, 996. [] P. H. Dugdale, R. J. Rademacher, B. R. Price, J. W. Subhedar, R. L. Duguay, Ecotec.4L VVT: A Variant of M s lobal 4-Cylinder Engine, SAE paper 5--94, 5 [3] R. Simpson, Worm gear driven variable cam phaser, US Patent 66667, 3. [4] J. Poole, J. Patton, B. oodwin, Modeling and Simulating a VVT System for Robust Design, SAE paper 8--9, 8 [5]. I. ustavsson, L. Ljung, and T. Soderstorm, Identification of process in closed-loop identifiability and accuracy aspects, Automatica, Vol. 3, pp [6] M. Leskens, L. B. M. Van Kessel, and P. M. J. Van den Hof, MIMO closed-loop identification of an MSW incinerator, Control Engineering Practice, Vol., pp [7] U. Forssel and L. Ljung, Closed-loop identification revisited, Automatica, 35, pp. 5-4, 999 [8] P. M. J. Van Den Hof, and R. J. P. Schrama, Identification and control - closed-loop issues, Automatica, Vol. 3, No., pp , 995. [9] R. E. Skelton and B.D.O. Anderson, Q-Markov covariance equivalent realization, International Journal of Control, Vol. 53, No., 986 [] K. Liu and R. E. Skelton, Identification and control of NASA s ACES structure, Proceedings of American Control Conference, Boston, Massachusetts, USA, 99. [].. Zhu, R. E. Skelton, and P. Li, Q-Markov Cover identification using pseudo-random binary signals, International Journal of Control, Vol. 6, No., 995, pp []. Zhu, Weighted multirate q-markov Cover identification using PRBS an application to engine systems, Mathematical Problems in Engineering, Vol. 6, pp. -4,. [3] Z. Ren,.. Zhu, Pseudo-random binary sequence closed-loop system identification error with integration control, Journal Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 33, in Press, 9 [4] B. Codrons, B. D. O. Anderson, M.evers, Closed-loop identification with an unstable or nonminimum phase controller, Automatica 38, pp. 7-37, [5] W. W. Peterson, Error Correcting Coding, MIT Technical Press, Cambridge, Massachusetts, USA, 96 [6] A.. Stefanopoulou, J. S. Freudenberg, J. W. rizzle, Variable Camshaft Timing Engine Control, IEEE Transactions on Control Systems Technology, Vol. 8, No., pp. 3-34, 669