LPV Model Identification of an EVVT System

Transcription

1 015 American Control Conference Palmer House Hilton July 1-3, 015. Chicago, IL, USA LPV Model Identification of an EVVT System Jie J. Yang, Shupeng Zhang, Ruitao Song, and Guoming G. Zhu Abstract In this paper, a family of discrete-time system models of an Electric Variable Valve Timing (EVVT) actuator for internal combustion engines under different operational conditions were obtained through the closed-loop system identification, where the complicated EVVT actuating system was treated as a black box. Since it is almost impossible to hold the EVVT cam phasing system at the desired operational condition under open-loop control, closed-loop system identification was adopted. Closed-loop EVVT system models were obtained using the PRBS q-markov Cover system identification, and with the known closed-loop controller the open-loop system models can be obtained under the given system operational condition such as engine speed, oil viscosity, and battery voltage. The LPV (Linear Parameter Varying) system model was formed based on the obtained family of open-loop discrete-time EVVT models, and the resulting LPV model was further validated by the experimental data. The obtained LPV model is intended to be used for designing the gain-scheduling LPV controllers using the LMI (Linear Matrix Inequality) convex optimization. I. INTRODUCTION In recent years strict emission regulations and increasing fuel economy requirement lead to the development of new internal combustion (IC) engine techniques including VVT (Variable Valve Timing), turbocharger, cooled EGR (exhaust gas recirculation), etc. The optimal timing of both intake and exhaust valves could be quite different over different engine operational conditions. Without the VVT system the engine valve timings have to be compromised. Continuously variable valve timing systems have been used in engine industry since early nineties. With the flexibility of adjusting valve timing under different engine operational conditions the engine combustion process can be optimized with minimal pumping loss and emissions under partial engine load and the engine maximal torque and power output can also be improved. The most commonly used VVT systems are hydraulic vane type VVT and electric cam phasing system. Hydraulic vane type VVT uses a solenoid to change the oil pressure ratio between the pair of chambers located at both sides of the vane to push or pull the cam shaft at either advanced or retarded direction to the desired cam phase angle. This type of VVT systems is widely used due to its low cost and minor modification requirement to the existing engine. However, J. Yang is with Shanghai Jiaotong University, Shanghai, China. This work was completed while he was visiting MSU (yangjiejt@gmail.com) S. Zhang and R. Song are with the Department of Mechanical Engineering at the Michigan State University (MSU), E. Lansing, MI 4884, USA (zhangs30@msu.edu and songrui1@msu.edu). G. Zhu is with the Department of Mechanical Engineering and Electrical/Computer Engineering at MSU, East Lansing, MI 4884, USA (phone: ; fax: ; zhug@egr.msu.edu). the hydraulic VVT also has its limitations. The VVT system only works after the engine oil is warmed up and therefore the cam timing has to remain in the locked position during the cold start, which prevents using the VVT from reducing cold start emissions. Comparing with the hydraulic VVT, the Electric VVT (EVVT) is independent of the engine oil pressure and temperature, which makes it feasible to be operated during the cold start. Also, the EVVT response time is much faster than that of the hydraulic VVT. Therefore, the EVVT has its potential for mode transition between SI (Spark Ignited) and HCCI (Homogeneous Charge Compression Ignition) combustion. An electric motor driven VVT was studied in this paper. The EVVT system (see Fig. 1) consists of a ring gear driven by the engine timing belt through a pulley; four planetary gears connected to the carrier that is driven by an electric motor; and the sun gear is connected to the cam shaft directly. Cam phase change is a function of the integrated speed difference between the VVT pulley and electric motor. In this modeling work, crankshaft speed is considered one of varying parameters; and the vehicle battery voltage, that fluctuates due to the vehicle electrical load variation, is the second varying parameter. Figure 1. Electric planetary gear VVT system In reference [1], the physical model of the EVVT system was developed and a controller was designed based on presumed parameters. And reference [] presents a discrete-time LPV (Linear Parameter Varying) model with oil viscosity as the measurable varying parameter along with a l to l LPV controller. The simulation results showed that the LPV controller is efficient under the targeted engine operational conditions. To develop a comprehensive discrete-time LPV model, in this paper the EVVT system was treated as a black box and a family of discrete-time system models were obtained through closed-loop system identification. The LPV modeling and control techniques have been widely used in automotive industry. Reference [3] proposes a methodology of identifying LPV models for nonlinear systems and the proposed methodology was applied to nonlinear turbocharged diesel engines. In reference [4] a data-based grey-box LPV model was /$ AACC 473

2 developed and the gain-scheduling H controller was designed. The LPV control technique was also used in [5] to compensate the time delay of the air-fuel ratio control loop for a lean burn SI engine. In [5] an LPV throttle model was established by converting the highly nonlinear system into an LPV system, and an H static output feedback controller was designed to guarantee the robust performance. This paper presents an approach to use closed-loop system identification to obtain a family of open-loop discrete-time EVVT models and then form the corresponding LPV system model based upon the identified models. The identified LPV model is intended to be used for designing LPV controllers in future work. To obtain the linear discrete-time EVVT model under a specific operational condition, the indirect closed-loop system identification approach discussed in [6] and [11] was used to obtain the open-loop system models based upon the identified closed-loop models, and the q-markov COVariance Equivalent Realization (q-markov Cover) system identification method [7]-[10] using the PRBS (Pseudo-Random Binary Signals) was used to obtain the closed-loop discrete-time models. The q-markov cover theory was initially used for model reduction, and it guarantees that the reduced order system preserves the first q-markov parameters of the original system. The q-markov realization based upon the input and output data can be used for system identification. It can be used for identifying the system models when the input and output data has the same sampling rate or different sampling rate (multi-rate system identification). The system excitation used for the system identification can be pulse, white noise, or PRBS, and in this paper the PRBS excitation was used. The paper is organized as follows. The following section reviews the framework of closed-loop system identification and theory of the q-markov Cover using PRBS; Section III describes the architecture of the EVVT system; and Section IV provides the bench testing results and discussions. Conclusions are drawn in the last section. II. SYSTEM IDENTIFICATION FRAME WORK Consider a general form of a linear time-invariant closed-loop system shown in Fig., where r is the reference signal, n is the measurement noise, u and y are system input and output. As discussed in reference [11], there are many approaches for closed-loop system identification, categorized as direct, indirect, and joint input-output approaches. In this paper, we utilize the knowledge of the given controller dynamics to calculate the open-loop plant model from identified closed-loop system model and this approach is called the indirect approach. To ensure the quality of the identified plant model, a proportional controller is used to form the closed-loop system to improve the system identification accuracy [11] - [1]. Figure. Closed-loop system identification framework Note that the input and output relationship of the generalized closed-loop system, shown in Fig., can be expressed below: -1 y H r GK(I + GK) r (1) Let ˆΗ be the identified closed-loop transfer function from r to y. The open-loop system model G ID can be calculated based upon the identified closed-loop transfer function ˆΗ, assuming that the inverse of (I - ˆΗ ) -1 exists. Note that the closed-loop controller transfer function is used to solve for the open-loop system models as follows ˆ ˆ ID ) -1-1 G Η(I -Η K () As discussed in the Introduction section PRBS was used as the excitation signal for system identification. The PRBS is a switch digital signal and can be easily generated for engineering practice. The most commonly used PRBS is based on maximum length sequences (called m-sequences) [13] for which the length of the PRBS signal is m (equal to n -1), where n is an integer, called the order of PRBS. Let z -1 represent a delay operator, and define ˆp (z -1 ) and p (z -1 ) to be polynomials -1 -n p(z ) a z a z a p( ˆ z ) (3) n 1 1 where a i is either 1 or 0, and obeys binary addition, i.e., & (4) and the non-zero coefficients a i of the polynomial are defined in the following table that can be found in [13]. TABLE I. NONZERO COEFFICIENTS OF PRBS POLYNOMIAL Polynomial order(n) Period of seq. (m) Nonzero coefficients 6 63 a5,a a4,a a,a3, a4,a a5,a a7,a a9,a11 Then the PRBS can be generated by the following formula 1 uˆ ( k+ 1) pˆ ( z ) uˆ ( k ), k 0,1,..., (5) where u( ˆ 0) 1 and uˆ ( 1) uˆ ( ) uˆ ( n) 0. Defined the following sequence a ; If k is even; s( k) -a; If k is odd. (6) Then, the signal u( k) s( k) [-a+ auˆ ( k) ] (7) is called the inversed 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 of m and u( k) u( k+m). The mean of the inverse PRBS is m-1 1 m uemu( k) u( k) 0 (9) m k 0 And the autocorrelation ( Ruu ( τ ) Emu( k + τ) u( k)) of u is a, τ 0; m-1 1 -a, τ m; Ruu ( τ ) u( k+ τ ) u( k) τ a /m, τ is odd. m k 0 -a /m, is even; (10) 474

3 The inversed PRBS is used for the q-markov Cover identification algorithm conducted in this paper. For convenience, in the rest of this paper, term PRBS is used to represent the inversed PRBS. Also in this paper, closed-loop system models were identified in discrete-time domain using PRBS-GUI [10] developed for the multi-rate PRBS q-markov Cover. The advantage of using the GUI (graphic user interface) in [10] is that the number of the Markov parameters and the order of the identified model can be adjusted easily based upon identification results shown on the GUI. III. EVVT SYSTEM ARCHITECTURE In this paper an electric motor driven VVT was studied. The EVVT system, shown in Fig. 1, is based upon a planetary gear set. The ring gear, running at the half speed of the crankshaft, is driven by the crankshaft through the engine timing belt. Note that for the traditional valve-train system the ring gear is connected to the camshaft directly, resulting in a fixed valve timing. While for the EVVT system the ring gear and the camshaft is connected through the planetary gear set. The planetary gear carrier is driven by an electric motor and four plant gears engage both ring and sun gears at the same time, where the sun gear is connected to the camshaft. The speed of the camshaft is determined by the ring gear speed together with the VVT motor speed, which provides the engine with the flexible valve opening timing. The modeling work of the EVVT system can be found in [1], and the relation between cam phase and the speed of VVT motor and ring gear is: Φ n ( s+nr )( c r ) s n Ω Ω (11) s Equation (11) indicates that the cam phase is the integration of the speed difference between the VVT motor and ring gear. Therefore the cam phase adjustment can be achieved by controlling the VVT motor speed with respect to the engine speed. To hold the cam phase at a constant value the VVT motor speed should be held to match the ring gear speed; to advance the cam phase the VVT motor speed shall be faster than the ring gear speed; and to retard cam phase, the VVT motor speed shall be slower than the ring gear speed. Tests were conducted to investigate the step response deviations under various vehicle battery voltage and engine speed and the results can be found in Fig. 3. It can be found that when the vehicle battery voltage, supplied to the EVVT motor, drops from 14.0V to 9.5V at 800 RPM engine speed the settling time of step response is almost doubled due to the reduced torque available for the EVVT motor to drive the ring gear. However, when the engine speed reduced from 1400 RPM down to 800 RPM with a fixed battery voltage of 9.5V, the settling time is cut by half because at low motor speed the available motor torque increases. This study indicates that the EVVT response time is heavily dependent on both engine speed and battery voltage and these two were selected as the LPV model parameters. Figure 3. Step response under different speed and voltage IV. BENCH SYSTEM IDENTIFICATION This section discusses the closed-loop system identification process using the PRBS input signal. A family of the EVVT system models were obtained under different engine speeds and vehicle battery voltages. The time responses of the identified EVVT models were compared with those obtained from the bench tests. Finally the obtained family of EVVT models were converted to a single LPV model with engine speed and vehicle battery voltage as the varying parameters. A. Test Bench Setup A test bench shown in Fig. 4 was set up for identifying the EVVT system. A Ford 5.4L V8 engine cylinder head was modified to mount the EVVT actuator on the test bench. The cylinder head has a single camshaft driving two intake valves and one exhaust valve. The crankshaft, simulated by an AC motor, is connected to the camshaft through a timing belt. An encoder was installed on the AC motor to generate crank position signal every crank degree along with a gate signal (one pulse per 360 crank degrees). Both encoder signals were used to obtain engine position and speed. A cam position sensor with four pulses per engine cycle, was installed on the back of the camshaft to detect the engine firing Top Dead Center (TDC) and to calculate the cam phase. An electrical oil pump was used to supply pressurized oil for lubricating both EVVT actuator and cylinder head assembly. Intake and exhaust valves were installed to provide the cyclic camshaft torque load. Opal-RT real-time engine prototype controller was used to control the EVVT and collect the experimental data for the test bench. The engine crank and cam position signals were used to calculate the engine crank angle and cam phase in real-time. For the closed-loop system identification a well-tuned proportional controller was used due to relatively high open-loop gain caused by the integral relationship of the speed difference between the ring gear and VVT motor. The Opal-RT controller output is the VVT motor speed command. The identified closed-loop EVVT system model is a transfer function between the reference cam phase and the calculated cam phase signal; see Fig

4 Figure 4. EVVT test bench B. Bench Test Results The EVVT has an adjustable cam phase range of 0 to 40 degree with the most retarded position at 0 degree and the most advanced position at 40 degree. The nominal operational condition is centered at 5 and the magnitude is selected to be 7.5 to avoid saturation which will results in high system identification error. Other PRBS q-markov cover system identification parameters can be found in Table II. experiment one. The chart in Fig. 6 indicates a dominant 1 st order dynamic due to the large gap between the first and second singular values. The singular value drops sharply for the remaining singular values, indicating that selecting high model order would not improve the identified model accuracy significantly. Therefore, the nd and 4th order open-loop models were selected for investigating the effect of model order to model accuracy. Test condition at 800 RPM with 14.0V battery voltage is used for illustration. TABLE II. CLOSED-LOOP IDENTIFICATION PARAMETERS Engine Speed (RPM) VVT Motor Power (V) Input Sample Rate (ms) Output Sample Rate (ms) PRBS Order 9 Number of Markov Parameters 30 Identified Open-loop Model Order The engine speed of the test bench is limited to 1750 RPM due to the speed limitation of the AC motor used to simulate the crankshaft speed. As a result, closed-loop system identification tests were conducted at three engine speeds (800, 1100, and 1400 RPM). The vehicle battery voltage was selected to be 14V, 11.5V, and 9.5V for three engine speeds. There are total of nine test points. The reference signal (PRBS signal) and the system response signal were logged and analyzed using the PRBS GUI [10] in Matlab. Number of Markov parameters was selected to optimize the system identification accuracy in terms of the step responses, and the identified model order is determined by the order index associated with the singular values of the PRBS response correlation matrix; see Fig. 6. Figure 6. Order selection diagram in the PRBS system ID GUI The open-loop system model for the EVVT operated at 800 RPM with 14.0V battery voltage is obtained by substituting the identified EVVT closed-loop system model to equation (), along with the known proportional controller. The Bode plot of the open-loop system model is shown in Fig. 8. The small spike near 4.5 rad/sec of the 4 th order open-loop model is related to the cyclic disturbance introduced by timing belt vibrations since this vibration mode was also observed in other conditions and the vibration frequency increases as the engine speed goes up. The Bode plot of the second order system, shown in Fig. 8 with the green line, has almost the same frequency response as the 4 th order model except that the dynamics related to the timing belt variation was eliminated. Other operational conditions were investigated in the same way and a family of nd order models (9 test conditions as discussed before) was obtained; see Table III. It is worth noting that in the continuous identified model, there is one pole very close to zero for all the conditions. From system model [1] it is believe that this is corresponding to the integration action of the system and was manually set it to zero. Figure 5. Model order selection Fig. 6 shows the order selection chart from the PRBS system ID GUI, where the EVVT was operated with 14V battery voltage at 800 RPM. The time response of the 4th order identified model were compared with experimental results using the same PRBS excitation in Fig. 7. From Fig. 7 it can be found that the response of the identified model using the same PRBS excitation is fairly close to the Figure 7. PRBS response of model and experiments at 800 RPM 14.0V 476

5 C. Model Validation To evaluate the accuracy of the identified system models, the model step responses were compared with the experimental responses using the same proportional controller. The closed-loop step responses were used for the evaluation purpose due to the fact that the cam phase cannot be maintained at the desired constant position under open-loop control. During the step experiments the cam phase reference was increased from 0 to 30 crank degrees and then decreased back to 0 crank degrees so that the EVVT operates in its unsaturated region. For the simulation study, the reference signal varies from 0 to 10 crank degrees and then down to 0 crank degree since the identified models are linear. The corresponding step responses are compared in Fig. 9. α( N( t ),V ( t)) G( N,V ) (1) s( s+ β ( N( t ),V ( t))) where α and β are varying parameters that belong to the following regions α( N(t), V (t)) 0.595, (13) [ ] 800 rpm at 9.5V 1400 rpm at 14.0V Figure 8. Bode plot of identified open-loop plant at 1400 RPM,9.5V Step responses at both conditions are fairly close to the original response. It can be observed that for the test results the advancing responses are slightly slower than the retarding responses. This is mainly due to the extra torque required by the EVVT motor for acceleration to achieve the advanced valve time. The step-up response of the model is very close to the experiment while the step-down model response is slower than the experiment response. The identified model compromises both advancing and retarding processes. This explains the larger overshoot of the model during advancing operation and slower response during retarding operation compared with the experiment results. The noise found in experimental responses is believed to be related to the backlash effect between gears and timing belt since the magnitude of the spikes remains around 1 crank degree (resolution of cam phase calculation). D. Construct the LPV Model In this section the identified family of linear discrete-time models were converted into a single LPV model with engine speed and vehicle battery voltage as the varying parameters. The LPV controller design based on the identified LPV model will be the subject of the future work. The open-loop transfer functions of the EVVT system under various conditions were listed in Table III. Note that the continuous time models of the discrete-time identified EVVT system models are obtained by mapping it back to continuous-time domain and the final LPV model is in the following form Figure 9. Closed-loop step response comparison at two conditons [ ] β ( N(t), V (t)) 6.975, (14) Parametersα and β are plotted in Fig. 10. Note that α is associated with the DC gain of the transfer function for LTI (Linear Time Invariant) system and -β is the pole location of the first order dynamics. From Fig. 10 it can be observed that, the vehicle battery voltage has significant influence on both parametersα and β.the higher the battery voltage is, the larger both α and β. This is mainly due to the fact that the motor available torque increases as the battery voltage goes up. Note that the motor output satisfies the following equation Km Tm Kmi ( Vm K ɺ aθ ) (15) R where T m is motor torque, V m is voltage, ɺ θ is the motor speed. The effects of the engine speed to both parameters with a given voltage are mixed due to the following facts. At high voltage (14.0V), the AC motor is torque saturated and as the engine (motor) speed increases the motor output power increases leading to increased β and α. However, when the battery voltage drops (to 9.5V) the motor output 477

6 power saturates at high engine (motor) speed, leading to slow response (small β) and reduced DC gain (α ). As a summary, the engine speed also has less influence on α and β than the vehicle battery voltage. V. CONCLUSION Closed-loop PRBS (Pseudo-Random Binary Signal) q-markov Cover system identification was used to obtain a family of the electric VVT (variable valve timing) system models under different engine speeds and vehicle battery voltages. Since the crank based cam position sampling rate is different from the time-based control input, multi-rate q-markov Cover system identification was adopted. The identified system models provide fairly close responses to experiment results under both PRBS and step excitations. The effect of model order to model accuracy was investigated and the nd order model was selected based upon model accuracy. The identified models were also validated using experimental step responses. The identified family of LTI (linear time invariant) models were converted into a single LPV (linear parameter varying) model and it will be used for LPV controller design in future. Figure 10. α and β Plots under different engine speeds and battery voltages TABLE III. IDENTIFIED MODELS Voltage (V) Speed (RPM) s( s ) s( s ) s( s ) s( s+ 8.47) s( s ) 0.45 s( s ) s( s+ 10.6) s( s+ 1.4) s( s ) REFERENCES [1] Zhen Ren and Guoming G. Zhu, Modeling and Control of an Electric Variable Valve Timing System for SI and HCCI Combustion Mode Transition American Control Conference on San Francisco, CA, USA 011 [] Andrew White, Jongeun Choi, and Guoming Zhu, Dynamic, Output-Feedback, Gain-Scheduling Control of an Electric Variable Valve Timing System. Washington, DC, USA, 013. [3] X. Wei and L. del Re, Gain Scheduled H Control for Air Path Systems of Diesel Engines Using LPV Techniques, IEEE Transactions on Control Systems Technology, vol. 15, no. 3, pp , 007. [4] J. Salcedo and M. Martnez, LPV identification of a turbocharged diesel engine Applied Numerical Mathematics, vol. 58, pp , 008. [5] S. Zhang, A. P. White, J. Yang, and G. Zhu, LPV Control of an Electronic Throttle, 013 ASME Dynamic Systems and Control Conference, San Francisco, CA, October, 013. [6] U. Forssel and L. Ljung, Closed-loop identification revisited, Automatica, 35, pp , 1999 [7] R. E. Skelton and B.D.O. Anderson, Q-Markov covariance equivalent realization, International Journal of Control, Vol. 53, No. 1, 1986 [8] K. Liu and R. E. Skelton, Identification and control of NASA s ACES structure, Proceedings of American Control Conference, Boston, Massachusetts, USA, [9] G. G. Zhu, R. E. Skelton, and P. Li, Q-Markov Cover identification using pseudo-random binary signals, International Journal of Control, Vol. 6, No. 1, 1995, pp [10] G. Zhu, Weighted multirate q-markov Cover identification using PRBS an application to engine systems, Mathematical Problems in Engineering, Vol. 6. pp. 01-4, 000. [11] Z. Ren, G. G. Zhu, Pseudo-random binary sequence closed-loop system identification error withintegration control, Journal Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 33, in Press, 009 [1] B. Codrons, B. D. O. Anderson, M.Gevers, Closed-loop identification with an unstable or nonminimum phase controller, Automatica38, pp , 00 [13] W. W. Peterson, Error Correcting Coding, MIT Technical Press, Cambridge, Massachusetts, USA,