ANALYSIS AND CONTROL DESIGN OF A HYDRO-MECHANICAL HYDRAULIC HYBRID PASSENGER VEHICLE
Teck Ping, Sim Mechatronics and Intelligent Machines Lab Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455 Email:
[email protected] ABSTRACT This paper gives the dynamic analysis of a hydromechanical transmission (HMT) drive train with regeneration and independent wheel torque control of a hydraulic hybrid passenger vehicle. From this analysis, we formulate the HMT control system, which is made up of high, mid and low-level control systems. The high-level consists of a state of charge management and the mid-level translates the storage requirement specified by the high-level into desired internal speed and gear ratio to be executed by the low-level. In this paper we focus on the lowlevel control analysis and design, where the actuation authority to regulate the internal speed variable comes from either the engine (mode 1) or the hydraulic system (mode 2). Experimental studies show good tracking performance of the proposed control systems and enable our vehicle system to be driven in the proposed HMT architecture.
INTRODUCTION Transportation by far is the largest consumer of petroleum (69% in 2007) in the United States and this consumption has been a growing trend (6 million barrels per day in 1954 to 14.3 million barrels per day in 2007) [1]. Reversing this trend requires the development of more energy efficient vehicles. Since 2006 the Engineering Research Center for Compact and Efficient Fluid Power at the University of Minnesota has been developing a hydraulic hybrid passenger vehicle with a hydro-mechanical transmission (HMT) drive train with regeneration and independent wheel torque control [2]. Hydraulic hybrid vehicle research has mainly been centered on two architectures - the parallel and series hybrid architecture.
Perry Y. Li Center for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455 Email:
[email protected] The parallel architecture uses a mechanical drive train with a hydraulic pump/motor in-line between the transmission and the axle. The benefits of this system are; 1) it allows for regeneration and 2) it enables efficient power transfer from the engine to the wheels. However, with this architecture the engine speed is determined by the vehicle speed and the available gear ratios, preventing optimal engine management. This can be prevented using the series architecture. In the series architecture, the engine and the wheel are not mechanically connected but are coupled by a hydrostatic transmission [3] so that full engine management is possible. Specifically, in this architecture, the engine is directly coupled to a hydraulic pump and the wheels are coupled to one or more hydraulic pump/motors. This means that the power transmission suffers heavily from the efficiency of the hydraulic systems, which is a downside to this architecture. The HMT drive train with regeneration and independent wheel torque control proposed in [2] exhibits the benefits of both the parallel and series architectures. It enables efficient power transmission through the mechanical transmission and allows for optimal engine management. The hydro-mechanical drive train splits the engine power through two paths, mechanical and hydraulic, which are recombined before the wheel. The power split is achieved by using a planetary differential system. One of the key elements in realizing the fuel efficiency gain of this hydraulic hybrid vehicle is the development of an effective control system layer to regulate the numerous subsystems (lowlevel) and an energy management layer (mid and high-level) to optimize the fuel usage based on the drive requirements. In this paper, we focus on the analysis and control design of the former layer.
HYDRO-MECHANICAL TRANSMISSION (HMT) DRIVE TRAIN WITH REGENERATION AND INDEPENDENT WHEEL TORQUE CONTROL
accumulator HPA
TL1 , ωL1
ωR2 HR
uc
Te
Je , be , ce
pump/motor E
uB
−TL1 , ωL1
1 clucth on uc = 0 clutch off
uL
pump/motor B clutch
uL JLE , bLE , cLE
HR −TLc , ωc
G ωe
RR TLc , ωc
differential
−TL2 , ωL2
RB
TL2 , ωL2 TB , ωB
ICE
E(ωe , ue )
ωc
G ωL2
mechanical transmission
Jveh , bveh , cveh
HR
JB , bB , cB dveh uB
uR pump/motor F
Figure 2. Free body diagram of one wheel HMT system. (TLc , TL1 , TL2 ) give the internal torque variables.
HPA pump/motor E/F
ICE
pump/motor B Figure 1. Proposed HMT drive train with independent wheel torque control of a passenger vehicle system
displacement of the hydraulic pump/motors. This enables direct regeneration and hence, enables realization of traction control (instead of the normal friction brakes which are energy wasteful). HMT DRIVE TRAIN DYNAMICS In developing the HMT drive train dynamics, for simplicity and without loss of generality, consider a one wheel HMT configuration. The free body diagram of this system is given in Figure 2. The dynamics of this system is given by: Back pump dynamics ˙ L1 + bLE ωL1 + cLE = TL1 + uL JLE ω
The proposed HMT drive train with regeneration and independent wheel torque control of a passenger vehicle is shown in Figure 1. This architecture consists of a small Perkins 403C11, 19.5kW, 3 cylinder diesel engine combined with the input power of two variable displacement hydraulic pump/motors (28cc Sauer-Danfoss Series 42 Axial Piston Pump/Motor) connected in tandem. The combined output power is channeled to a mechanical transmission system (low or high gear options). The output of the transmission is recombined in two differentials with the input power of two back variable displacement pump/motors (28cc Sauer-Danfoss Series 42 Axial Piston Pump/Motor). The output of the differential powers each wheel. All the pump/motors are connected to a high pressure accumulator (HPA) for energy storage and a low pressure accumulator. This architecture enables: 1. an extra degree of freedom to allow for operation of the system to operate at its highest efficiency through complementary use of the engine and hydraulic systems. 2. independent control of torque at each wheel by varying the
(1)
where (JLE , bLE , cLE ) give the inertia, damping and frictional torque of pump/motor E, with ωL1 being the speed of pump/motor E. uL gives the torque output of pump/motor E (for ωL1 positive, ‘-’ uL value indicates pumping and ‘+’ uL value indicates motoring and vice-versa for ωL1 negative). Wheel and vehicle dynamics ˙ L2 + bveh ωL2 + cveh = TL2 + dveh Jveh ω
(2)
where (Jveh , bveh , cveh ) give the inertia, damping and frictional torque of the vehicle, with ωL2 gives the wheel speed. dveh gives the equivalent torque on wheel due to drag on the vehicle. Mechanical transmission dynamics ˙ c + bmech ωc + cmech = TLc + Tmech Jmech ω
(3)
where ωc is the carrier speed and (Tmech , Jmech , bmech , cmech ) give the output mechanical torque (from the hydraulic pump/motor B
and the engine), inertia, damping and frictional torque of the mechanical system. (Tmech , Jmech , bmech , cmech ) depend on the clutch state, uc (i.e. uc = 0 indicates clutch is disengaged and uc = 1 indicates clutch is engaged) and are defined as follows: ( RR GRB uB + RR GTe Tmech = RR GRB uB
if uc = 1 otherwise
2 2 2 (Je + JB RB )G RR (be + bB R2B )G2 R2R Jmech )GRR bmech = (ce + cB RB 2 2 JB RB G RR cmech 2 2 b B RB G RR cB RB GRR
Eliminating the internal state TL from (9)-(11) and substituting (12) we have: Ã |
(4)
JLE HR
!µ ¶ + HR Jveh −2 JHLER ˙ L2 ω = ˙c ω −2 JHLER 4 JHLER + HR Jmech {z } J
µ
where d¯1 = HR d¯veh − d¯LE and d¯2 = 2d¯LE + HR d¯mech . if uc = 1 (5) otherwise
where RB is the transmission ratio of the belt-sprocket assembly (pump/motor B to the mechanical transmission), RR is the transmission ratio of the chain-sprocket assembly(mechanical transmission to the differential) and G is the gear ratio. Te gives the torque output of the engine and uB gives the torque of pump/motor B. Also, (Je , be , ce ) give the inertia, damping ratio and frictional torque of the engine and (JB , bB , cB ) give the inertia, damping ratio and frictional torque of pump/motor B. Differential system 2ωc = HR ωL1 + ωL2 TL1 /HR = TL2 := −TL TLc = 2TL
(6) (7) (8)
where HR is the transmission ratio of the belt-chain-sprocket assembly (pump/motor E to the differential). Substituting (7)-(8) into (1)-(3) we have:
Internal/External Coordinate Transformation In this section we look into decomposing the drive-train dynamics into external and internal coordinates. External coordinate being those that corresponds to the vehicle motion and internal coordinate being the complementary coordinate in the Riemannian sense. The advantage of such a transformation is that it decouples the system into meaningful quantities in such a way that concepts like energy conservation, passivity, power flow are preserved and clearly exhibited [4–6]. Consider the following form of coordinate transformation: µ
ωL2 ωc
¶
˙ L1 = −HR TL + uL + (−bLE ωL1 − cLE ) JLE ω | {z } ˙ L2 = −TL + (dveh − bveh ωL2 − cveh ) Jveh ω | {z }
(10)
d¯veh
˙ c = 2TL + Tmech + (−bmech ωc − cmech ) Jmech ω {z } |
(11)
d¯mech
¶µ ¶ 10 ωL a1 ωint | {z }
(14)
where ωL is the external coordinate and ωint is the internal coordinate. Coefficient a in S is to be determined such that µ T
S JS =
Jext 0 0 Jint
¶ (15)
meaning that the internal coordinate is orthogonal to the external coordinate. Using (15), a can be solved as:
(9)
d¯LE
µ =
S
a=
2JLE 4JLE + HR2 Jmech
(16)
Therefore the internal coordinate(or speed), ωint is given by: ωint = ωc −
2JLE ωL2 4JLE + HR2 Jmech
(17)
Also, Jext and Jint can now be given by:
Differentiating (6) and rearranging we have: ˙ L1 = (2ω ˙c −ω ˙ L2 )/HR ω
−uL + d¯1 2uL + HR Tmech + d¯2
(13) ¶
(12)
JLE Jmech Jint = 4JLE /HR + HR Jmech
Jext = HR Jveh +
(18)
Jint
(19)
In the new coordinates we have:
Table 1. Values of the inertia terms in equation (22) for our HMT hybrid vehicle system at all possible mechanical system configurations
2JLE 2JLE ¯ (HR Tmech + 2uL ) + d¯1 + d2(20) HR Jint HR Jint = 2uL + HR Tmech + d¯2 (21)
˙ L = −uL + Jext ω ˙ int Jint ω
Let Text := −uL + H2JR LE Jint (HR Tmech + 2uL ) and Tint := 2uL + HR Tmech , then we have: µ
Text Tint
¶
µ =
4JLE HR Jint
2
−1
2JLE HR Jint
1
¶µ
uL HR Tmech
¶ (22)
˙ int = Tint + d¯2 Jint ω
2JLE ¯ d2 HR Jint
(23) (24)
In this formulation Text can now be commanded by the driver, but we have a freedom to choose Tint to control ωint such that the fuel economy can be optimized. HMT CONTROL SYSTEM DESIGN The HMT control system is separated into three distinct hierarchies as follows: High-Level Control: The high-level control consists of a state of charge management of the hydraulic system to optimize the fuel economy. Mid-Level Control: The mid-level control translates the storage requirement specified by the high-level into desired internal speed, ωint and gear ratio, G. Low-Level Control: The low-level control realizes the internal speed, ωint requirement as specified by the previous level. In this paper we focused on the implementation of the low-level control on our prototype vehicle system. In this control we use Tint to control the internal speed, ωint to the desired value with a tracking bandwidth of 1Hz or approximately 6.3rad/s. This specification is arbitrary but at this stage of the hybrid vehicle development we feel that it would be sufficient for our testing needs. The same control methodologies as proposed here can be redesigned for different bandwidth specifications. From (22) and since the H4JR LE Jint − 1 term largely contribute to Text (from table 1), hence we can realize Text using the torque of the pump/motor E (by simply changing the displacement of the pump/motor E through a sensed foot-pedal command), i.e.: Text ≈ −uL
Clutch, uc
4JLE HR Jint
Jmech
Jint
[kgm2 ]
[kgm2 ]
2JLE HR Jint
−1
Low
1
100.28
100.37
-1.00
0.0005
Low
0
5.19
5.29
-0.98
0.0104
Hi
1
14.86
14.96
-0.99
0.0037
Hi
0
0.77
0.86
-0.87
0.0635
To regulate ωint , consider the internal system dynamics. From (21) and (4) we have:
with this definition, from (20) and (21) we have: ˙ L = Text + d¯1 + Jext ω
Gear
(25)
( RR GRB HR uB + RR GHR Te + 2uL + d¯2 = RR GRB HR uB + 2uL + d¯2
if uc = 1 otherwise (26) Hence, to regulate the internal speed,ωint , we use the engine torque, Te when the clutch is engaged (uc = 1). The hydraulic torque of the front pump/motor,uB is an extra degree of freedom for the high level control to optimize the fuel economy. When the system is declutch then the hydraulic torque, uB alone is used to regulate ωint . The following subsections present the design of the engine speed and hydraulic speed regulation control modes. Since 2JLE /HR Jint ≈ 0 by table 1, from equation (17), ˙ int Jint ω
ωint ≈ ωc
(27)
Henceforth, the problem of regulating ωint is stated here as a problem of regulating ωc . Mode 1: Engine speed regulation The off-the shelf Perkins 403C-11 engine is already speed regulated with an in-built mechanical governor. Hence, the throttle input, ue ∈ [0, 1] gives a reference speed command to the engine’s governor. In our setup, this throttle input, ue is actuated by a stepper motor. This allows the engine to be controlled. From equations (17) and (26) the dynamics of the system regulated by the engine’s governor in this control mode is given by: ˙ c = RR GHR Te + RR GRB HR uB + 2uL + d¯2 Jint ω | {z }
(28)
disturbances
This engine’s governor control system is shown in Figure 3, where Kgov denotes the engine governor control map and E is the engine map. While, this could have simplified the implementation of the engine speed regulation, testing has shown that this control system is incapable of rejecting the disturbances on the
RR GRB uB + 2uL ωc∗
Kgov
Te
E
1 Jint s
RR GHR
dωc
ωc∗∗
ωc
ue
Gf f (s)
K(s)
f −1 (·)
f (·)
GE (s)
ωc
-
Gc (s)
d¯2 Figure 5. Figure 3.
designed using the plant model given by (30). To enable tracking of the desired speed value with a bandwidth of 1Hz, a reference feed-forward control is designed. The closed loop transfer function with the PI controller is re-identified from the actual closedloop system i.e.:
ωc[rad/s]
11 10 9 8
0
10
20
30
40
50
30
40
50
+ (s − 18.04)(s + 17.90) B− c (s)Bc (s) := K c s3 + 10.77s2 + 106.5s + 366.9 Ac (s) (31) Since, this is a non-minimum phase system, the design of the reference feed-forward control takes the form of the zero-phase error tracking controller (ZPETC) [7] i.e.
Gc (s) = −1.142
t[s] 20 uB[Nm]
Engine speed regulation control system
Engine’s governor control system
0 −20 −40
0
10
20
G f f (s) =
t[s]
Figure 4. The carrier speed, ωc under a constant throttle input to the engine (top figure). The drop in the carrier speed is due to loading on the drive train by activating pump B.
HMT system (see Figure 4). To address this issue a control system is designed to reject the disturbances and enable tracking of the desired speed value with a bandwidth of 6.3 rad/s. To design this control system, consider first the model for the mapping from ω∗c → ωc . With both pump/motor B and E deactivated (i.e. uB = 0 and uL = 0), this map is found to be best modeled by: ωc = GE (s) f [g(ω∗c )]
(29)
where GE (s) is given by a second order system with a zero and uˆe = f (ue ) is a nonlinear static map and ue = g(ω∗c ) maps the engine reference to the throttle position at nominal. In this configuration, GE (s) is identified in the actual system to be: GE (s) = −1319.9
s − 18.42 s2 + 10.81s + 62.52
(30)
The designed control system is shown in Figure 5. To compensate for the change of speed (see an example in Figure 4) due to the activation of pump/motor B and/or E and other disturbances (collectively given by dωc in Figure 5), a PI controller is
±Ac (s)B− 1 c (−s) − n 2 Kc B+ c (s)Bc (0) (s/ω f + 1)
Setting the cutoff frequency of the low-pass filter, ω f to be sufficiently high (we take ω f = 50rad/s), pick n = 3 so that G f f (s) is proper, and from (31) we have:
G f f (s) =
0.002691s4 + 0.07752s3 + 0.8094s2 + 6.158s + 17.81 8e − 006s4 + 0.001343s3 + 0.08148s2 + 2.074s + 17.9
Implementing this control in our vehicle prototype as shown in Figure 1, enables rejection of disturbances (compared to the results in Figure 4 with the engine’s speed governor). In this case the disturbance source is introduced by the loading with pump B (i.e. uB in (28)). The controller also enables good tracking of the reference input and achieves the required bandwidth of 6.3 rad/s. This is shown in the bode plot given in Figure 7. The sinusoidal tracking results of the PI controller with and without the ZPETC are shown in Figure 8. Mode 2: Hydraulic speed regulation In this control mode, the clutch is disengaged and regulation of ωint ≈ ωc (see equation (27)) is achieved solely by the tandem pump/motor B. From (26) we have: ˙ c = RR GRB uB HR + 2uL + d¯2 Jint ω {z } | {z } | uˆB
disturbances
(32)
sinusoidal ref. speed with frequency of 1.5rad/s 15 14
10 ω** ,ω [rad/s] c c
ωc[rad/s]
11
9 8
0
10
20
30
40
50
t[s]
13 12 11 10 9
20 uB[Nm]
8
without ZPETC 0
with ZPETC
5
10
0
15
time[s] sinusoidal ref. speed with frequency of 6.0rad/s 15
−20
14
0
10
20
30
40
50
ω** ,ω [rad/s] c c
−40
t[s]
Figure 6. The carrier speed, ωc with ω∗∗ c set to 10 rad/s (top figure). There is no drop in the carrier speed despite the loading on the drive train by activating pump B(bottom figure).
13 12 11 10 9 8
without ZPETC 0
with ZPETC
5
10
15
time[s] 15
Figure 8. The carrier speed, ωc (blue line), tracking the sinusoidal reference carrier speed, ω∗∗ c command(red line) through mode 1
Magnitude (dB)
10 5
Gff (s)Gc (s)
0
2uL + d¯2
-5
ωc∗∗
Gc (s)
Phase (deg)
-10 360
Gf f (s)
K(s)
uˆ
+
-
Gff (s)Gc (s)
270
180
Ka s(s+a)
ωc
Gc (s)
Figure 9. Hydraulic speed regulation control system. with integral controller and G f f (s) is a ZPETC.
Gc (s)
K(s) is a LQR
90 0
10
1
10
2
10
Frequency (rad/s)
Figure 7. Bode plot of the engine speed regulation control system with (red line) and without (blue line) reference feed-forward. The ’o’ and ’x’ denote the experimental results of the magnitude and phase shift with and without reference feed-forward respectively.
where the driving torque to regulate ωc is given by the hydraulic torque, uB of pump/motor B. uB is given by: uB = PDm xB
(33)
P is the hydraulic system pressure, Dm is the maximum displacement of the pump/motor and xB ∈ [−1, 1] is the displacement factor that we can electronically change. uL is the hydraulic torque
of pump/motor E (under driver’s command) and is used here to ˙ L2 are treated as unknown disturdrive the wheel. d¯2 and 2JHLE ω R bances. By equation (32), the open-loop system, GuˆB →ωc (s) is an integrator. However closed loop identification shows that the openloop system is best described by a cascade of a first order system and an integrator i.e. GuˆB →ωc (s) = Ka /s(s + p). The first order dynamics maybe attributed to the swash-plate control dynamics of the axial piston pump/motor. To regulate the carrier speed, the control given in Figure 9 is developed. First, a LQR with integral controller (a Kalman estimator is used to enable full state feedback) is designed for the identified open loop plant, GuˆB →ωc (s). Figure 10 shows the step responses of this design. However, as shown in the bode plot in Figure 11, the resulting closed-loop system has a bandwidth lower than 6.3rad/s. The closed-loop system given in the bode
16
sinusoidal ref. speed with frequency of 1.5rad/s 8
ωc
7
c
12
,ω [rad/s] ω** c c
ω** ,ω [rad/s] c c
14
ω**
10 8
6 5 4
6 3
4
0
5
10 time[s]
15
with ZPETC
without ZPETC 0
5
10
20
15 time[s]
20
25
30
sinusoidal ref. speed with frequency of 6.0rad/s 8
20
7 ,ω [rad/s] ω** c c
Figure 10. LQR with integral control response to step inputs. The system achieves steady state in approximately 1.5 second.
6 5 4
Magnitude (dB)
0
Gff (s)Gc (s)
3
5
10
15 time[s]
20
25
30
-20
Gc (s)
Figure 12. The carrier speed, ωc (blue line), tracking the sinusoidal reference carrier speed, ω∗∗ c command(red line) through mode 2
-40
-60 360
Gff (s)Gc (s)
270 Phase (deg)
with ZPETC
without ZPETC 0
180
Gc (s)
90 0 0
10
1
10
2
10
Frequency (rad/s)
Figure 11. Bode plot of the hydraulic speed regulation control system with (red line) and without (blue line) reference feed-forward. The ’o’ and ’x’ denote the experimental results of the magnitude and phase shift with and without reference feed-forward respectively.
plot is based on a reduced order form and is identified here to be: Gc (s) =
−43.98s + 1769 s3 + 60.98s2 + 980.1s + 1757
To obtain the desired bandwidth of 6.3rad/s, a reference feedforward (ZPETC) is designed and is given as follows: 1.4e − 005s4 + 0.0014s3 + 0.048s2 + 0.579s + 0.993 G f f (s) = (1/200s + 1)4 The improvement in the tracking performance is shown in the
bode plot given in Figure 11. The sinusoidal tracking results of the LQR with integral controller with and without ZPETC are shown in Figure 14. In all the results presented so far, uL has been kept at 0. From (32), uL can be treated as a disturbance source. Figure 13 demonstrate that disturbances can be rejected using the designed control system. However, for a more effective rejection of the disturbance on the carrier speed due to uL , we can introduce disturbance feed-forward. For our system the feed-forward strategy requires a non-trivial modeling of the frictional effects and torque losses of pump/motor E. This will be a topic for future work.
DRIVE RESULTS The prototype vehicle is tested and driven using the proposed control system. Figure 14 shows the HMT drive result using the proposed low-level control system. In this result the carrier speed, ωc or equivalently the internal speed, ωint is regulated at a constant value. The wheel speed, ωL2 , is driven by the torque applied on hydraulic motor E, uL ≈ Text . uL is commanded by the driver.
CONCLUSION This paper presents the analysis and design of the low-level control for a prototype hydro-mechanical transmission (HMT) drive train with independent wheel torque control. The low-level control enables regulation of the internal speed variable, ωint .
(a)
6 ω [rad/s]
10 9
c
ωc[rad/s]
5.5 5
7
4.5
0
2
4 time[s]
6
L2
0
0
20
40
60 t[s] (b)
80
100
120
0
20
40
60 t[s] (c)
80
100
120
0
20
40
60 t[s]
80
100
120
50 0 −50
−10 40
−20
20
L
−15
u [Nm]
uL[Nm]
−5
Figure 13.
c **
ωc
100
8 ω [rpm]
4
ω
8
0
2
4 time[s]
6
8
Rejecting the effect of the external torque, uL applied to hy-
0 −20
draulic motor E on the carrier speed, ωc . Figure 14. (a) The regulated carrier speed; (b) the wheel speed; (c) the torque supplied by hydraulic motor E
ωint is an essential degree of freedom to optimize the fuel usage in the high and mid-level control. Good regulation of this variable is therefore critical for the operation of this hybrid vehicle, which has been demonstrated here. Future efforts include investigating the bump-less transfer of the different modes (i.e. from mode 1 to mode 2 or vice versa). This is non-trivial due to the sudden dynamical change as the system declutches. In addition, we are investigating other possible modes with this architecture such as locking up hydraulic pump/motor E/F (making the system to be a pure parallel system), putting the gear in neutral and locking up the differential carriers(making the system to be a pure series system), etc. This work lays the foundation for implementing the high-level and mid-level control systems for maximizing fuel usage and hence improves fuel mileage of a passenger vehicle. These will be presented in future publications.
ACKNOWLEDGMENT This material is based upon work performed within the ERC for Compact and Efficient Fluid Power, supported by the National Science Foundation under Grant No. EEC-0540834. Thanks go to the team members of the Hydraulic Hybrid Passenger Vehicle (HHPV) project - Stephen Sedler, Felicitas Mensing, Jonathan Meyer, Robert Ertel, David Grandall, and Justin Lapp for their help in carrying out these experiments.
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