CHINESE JOURNAL OF MECHANICAL ENGINEERING
Vol. 28,aNo. 5,a2015 ·971·
DOI: 10.3901/CJME.2015.0512.070, available online at www.springerlink.com; www.cjmenet.com; www.cjme.com.cn
Constant Speed Control of Four-stroke Micro Internal Combustion Swing Engine
GAO Dedong1, 2, LEI Yong1, *, ZHU Honghai3, and NI Jun3
1 State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China 2 School of Mechanical Engineering, Qinghai University, Xining 810016, China
3 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105-2125, USA
Received November 7, 2014; revised March 4, 2015; accepted May 12, 2015
Abstract: The increasing demands on safety, emission and fuel consumption require more accurate control models of micro internal combustion swing engine (MICSE). The objective of this paper is to investigate the constant speed control models of four-stroke MICSE. The operation principle of the four-stroke MICSE is presented based on the description of MICSE prototype. A two-level Petri net based hybrid model is proposed to model the four-stroke MICSE engine cycle. The Petri net subsystem at the upper level controls and synchronizes the four Petri net subsystems at the lower level. The continuous sub-models, including breathing dynamics of intake manifold, thermodynamics of the chamber and dynamics of the torque generation, are investigated and integrated with the discrete model in MATLAB Simulink. Through the comparison of experimental data and simulated DC voltage output, it is demonstrated that the hybrid model is valid for the four-stroke MICSE system. A nonlinear model is obtained from the cycle average data via the regression method, and it is linearized around a given nominal equilibrium point for the controller design. The feedback controller of the spark timing and valve duration timing is designed with a sequential loop closing design approach. The simulation of the sequential loop closure control design applied to the hybrid model is implemented in MATLAB. The simulation results show that the system is able to reach its desired operating point within 0.2 s, and the designed controller shows good MICSE engine performance with a constant speed. This paper presents the constant speed control models of four-stroke MICSE and carries out the simulation tests, the models and the simulation results can be used for further study on the precision control of four-stroke MICSE.
Keywords: MICSE, constant speed control, hybrid model, spark timing, valve duration
1 Introduction
Alternative micro power generation technologies have been extensively studied in both industry and universities in the past few years. Numerous new concepts, designs, and systems have been proposed and explored to solve the micro power generation problems, such as micro combustion engines[1], micro fuel cells[2], solar cells[3], thermoelectric converters[4–5], nuclear batteries[6], etc. Various micro power generation systems using combustion have been proposed and investigated to realize high energy conversion efficiency from hydrocarbon fuel to electrical power[7–10]. Problems reported in the literatures appear to concentrate on fabrication, fluid flow and thermal management at the micro-scale size.
The concept of micro internal combustion swing engine (MICSE) was proposed and studied by MIJIT[7]. Typical applications of MICSE include laptop computers, handheld electronics, portable domestic robots, micro air vehicles
* Corresponding author. E-mail: [email protected]
Supported by National Natural Science Foundation of China (Grant No. 51475422), and Science Fund for Creative Research Groups of National Natural Science Foundation of China (Grant No. 51221004)
© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015
and micro satellites, etc. A swing engine in principle is a free-piston engine, where the oscillating movement of the swing arm (piston) is rotational instead of linear as in most cases. Combustion occurs in four chambers separated by a single rotating swing-arm. A swing engine can operate both in two-stroke mode and four-stroke mode. The differences between these two operational modes basically lie in the configurations of the intake and exhaust valves, as well as spark and valve timing sequences. The paper focuses on the four-stroke MICSE. The challenges of developing MICSE systems come not only from design and manufacturing, but also from the control. One of the major difficulties of developing MICSE control systems is due to its nature as a free piston engine. Because of the lack of reliable control systems, the first generation free-piston engines were given up in the 1960s[11–13]. With the development of electronic control technology (sensor, actuator and computer technology), the control system is not a limiting factor anymore.
However, the hybrid nature of the MICSE requires a new modeling methodology. In the engine control literature, there are three major approaches used for different engine modeling tasks: 1) thermodynamic models, 2) input-output models, specific to each application or engine configuration,
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3) nonlinear models, physically based (generalized) for wide range of operation[14–16]. All of these models share a common feature, i.e., they are average-value models. However, the increasing demands on safety, emission, and fuel consumption require the use of more accurate models. From a control engineering point of view, internal combustion engines are “hybrid system”, i.e., they include discrete-event and continuous-time subsystems[17–18]. The combustion chambers in a MICSE have distinctive states corresponding to the stroke they are experiencing, which can be regarded as a discrete-event system; while air flow dynamics and generator dynamics are in the continuous-time domain.
The swing engine has a natural frequency determined by the piston or swing arm and the gas enclosed in the combustion chamber. In literature, only a few papers addressed the dynamics and control problems on free piston engines. JOHANSEN, et al[13,19], addressed the simplest type of free piston engines, i.e., single piston and single combustion chamber. The scope of their studies do not address the control issues encountered in MICSE: 1) a 4-stroke, oscillating swing arm engine, 2) four combustion chambers, and 3) active exhaust valve and passive intake valve configuration. Some pioneering work in the MICSE timing control has been done by MIJIT[7], where the fundamental valve and sparking timing operation principles are explained. However, his work does not cover the constant speed control problems.
Constant speed control problems for internal combustion engines have been a very popular topic. One particular type of such problems is the idle speed control (ISC) problem. Idle speed operation in spark ignition, four-stroke, internal combustion engines is typically characterized by low engine speed cases (between 500 r/min to 1000 r/min), low to medium engine torque, and a closed primary throttle. Various control strategies have been proposed and applied to the ISC design[20]. These include linear controllers, such as PID[21–22], linear-quadratic optimization[23], H-infinity[24–25], nonlinear controllers, such as fuzzy-logical controls[26], and neural network based controls[27–28]. The goal for the MICSE control is to design a controller which regulates the MICSE voltage output at a desired constant level in a fast and smooth performance. The fluctuation in external load resistance is regarded as a disturbance. The constant speed control problem in a four-stroke MICSE application is different from the idle speed controls mentioned above. The main difference is how to control the air/fuel flow into the combustion cylinders or chambers.
The performance of a MICSE is evaluated by fuel economy and “drivability”. Here the “drivability” is defined as fast and smooth response of the MICSE, instead of the general qualitative evaluation of automotive powertrains’ operating qualities, including idle smoothness, cold and hot starting, throttle response, power delivery, and tolerance for altitude changes. It is desirable to keep the MICSE output voltage close to a constant level to respond
to the change in external load with good transient performance. The control objective of constant output voltage can be implemented through manipulating the two actuators, spark ignition and exhaust valves.
This paper is organized as follows: section 2 presents the prototype and operation principle of the four-stroke MICSE. In section 3, the hybrid model is constructed to understand the dynamics of MICSE system with a Petri net, and a nonlinear model and a linearized model is developed for the controller design. Section 4 designs a constant speed feedback controller with sequential loop closing design. In section 5, the simulation results are provided to illustrate some of the properties of the close loop control system. Finally, conclusions and future work are given in Section 6.
2 Prototype and Operation Principle
of Four-stroke MICSE
The configuration of the MICSE-base hybrid systems is shown in Fig. 1, where a rechargeable battery serves as the auxiliary power storage device. The MICSE systems presented here burns the methane as the fuel. The prototype and operation principle of four-stoke MICSE are described in this section.
Fig. 1. MICSE configuration with a battery
2.1 Description of four-stroke MICSE system
Fig. 2 provides a prototype of the four-stroke MICSE system.
Fig. 2. Prototype of the four-stroke MICSE system For a single combustion chamber, the mass flow is controlled by a passive intake valve and an active exhaust valve. A reed valve is constructed of flexible leaf springs,
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·973· and of cantilever design, i.e., firmly fixed at one end and
free to deflect substantially at the other end under load. The reed valve allows the intake flow to enter the combustion chamber from the manifold when the intake manifold pressure exceeds the combustion chamber pressure and block the intake flow under all other conditions.
The exhaust flow is controlled by an active poppet valve train, as shown in Fig. 3. A poppet valve is a mushroom- shaped valve that raises perpendicular from its seat. It is driven by a solenoid which is activated by the trigger signal generated from the engine control unit. The poppet valve allows the burned gases exhaust into the air through the exhaust port when it is open and blocks the exhaust flow under all other conditions. The combustion of the air/fuel mass in the chamber is initiated by a spark plug. A micro control unit controls the timings of the spark ignition and exhaust valve operation and a micro generator converts the mechanical power into electrical power.
Fig. 3. Exhaust valve assembly for the four-stroke MICSE prototype system
2.2 Four-stroke MICSE operation principle
Fig. 4 depicts the operation principle of MICSE four-stroke mode. Repeatedly and sequentially firing one of the four combustion chambers (A, B, C and D) at proper swing speed drives the swing arm from a cold start state into a steady state oscillation. Simultaneously the generator, coupled and driven by the swing arm, generates directly electrical power output.
Fig. 4. MICSE four-stroke mode operation principle
Fig. 5 gives a detailed description of the main events in a single operation cycle of the four-stroke mode. In Fig. 5, ωIGN is defined as the angular speed of the swing arm when the combustion starts; ωEVO and ωEVC are the angular speeds when the exhaust valve opens and closes, respectively; ωIVO and ωIVC are defined as the angular speeds when the intake valve opens and closes, respectively.
Fig. 5. Main events in a single operation cycle of the four-stroke mode
3 MICSE Modeling
The framework for modeling a four-stroke MICSE consists of four major interacting blocks, namely, the intake manifold, the actuators, the combustion chambers, and the micro generator. Fig. 6 shows a block diagram representation of a four-stroke MICSE system.
Fig. 6. Four-stroke MICSE block diagram representation The manifold pressure pm is controlled by its upstream air/fuel delivery system and experiences cyclic fluctuations due to the opening and closing of the intake reed valve. The mass flow rate of air/fuel loaded into the combustion chambers depends on the manifold pressure pm and the swing arm angular position θ. The torque Te generated by the MICSE is the sum of the torque Ti produced by the four combustion chambers (i=A–D). The torque Ti is determined by the mass mci of air/fuel in combustion chambers, spark advance timing τi, and valve closing timing φi. The spark timing and valve timing are synchronized with the evolution of combustion chambers,
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which is determined by the continuous but oscillatory motion of the swing arm. The micro generator dynamics are impacted by the engine torque Te, load torque TLoad and friction torque Tfriction. The micro generator converts the mechanical energy into the electrical energy, which is delivered to load devices.
The MICSE constant speed control uses different strategies in responding to different magnitudes of external load resistance change. For large changes in load resistance, the supervisory controller relies on dual mode switch (DMS), i.e., switching between two-stroke and four-stroke modes, to maintain constant speed/voltage output. For small changes in load resistance, it uses variable exhaust valve duration timing and spark timing to reach the control objective. There are various intake and exhaust strategies: late intake valve closing (LIVC), early intake valve closing (EIVC), late intake valve opening (LIVO), early intake valve opening (EIVO), late exhaust valve closing (LEVC), early exhaust valve closing (EEVC), late exhaust valve opening (LEVO) and early exhaust valve opening (EEVO). However, since the passive intake reed valve is not controllable in the case of the MICSE, only the following four strategies are feasible: LEVC+LIVC, EEVC+EIVC, LEVO+LIVO, and EEVO+EIVO. Here we focus on the case of small changes in load resistance based on exhaust valve closing timing, i.e., LEVC+LIVC and EEVC+EIVC.
The approach employed to solve the four-stroke MICSE speed control problem follows the following procedure, as shown in Fig. 7.
Fig. 7. Procedure of four-stroke MICSE speed control
3.1 MICSE hybrid modeling
3.1.1 Modeling MICSE engine cycle with a Petri net The intake, compression, expansion and exhaust strokes occurred sequentially in a single combustion chamber, while they happened concurrently in all the four combustion chambers. In order to capture these sequential and concurrent behaviors in a MICSE, a two-level Petri net is introduced, as shown in Fig. 8. The Petri net subsystem at the upper level controls and synchronizes the four Petri net subsystems at the lower level through the three places {P0, P1, P2}.
Fig. 8. A two-level Petri net describing MICSE sequential and concurrent behaviors
Fig. 9 shows the Petri net subsystem at upper level. In Fig. 9, the event of swing arm reaching maximum angle (MA) generates a token in place P3, which enables transition t3. Transition t3 fires spontaneously and adds a token to places {P0, P4, P5}. The tokens presented in place P0 are passed on to four subsystems at the lower level through spontaneous transition t0, which synchronize the four subsystems. Two timed transitions {t4, t5} are enabled as their input places {P4, P5} have the necessary number of tokens, but they fire only after firing delays set by the given spark advance timing and exhaust valve timing commands and then add a token to {P1, P2}. The tokens presented in places {P1, P2} at different times will be consumed by one of the four subsystems depending on the state of the subsystems, i.e., only the subsystems in the state of compression or exhaust will consume these tokens in place P1 or P2.
Fig. 9. Petri net subsystem at upper level
Fig. 10 gives the Petri net subsystem structure on the lower level for a single combustion chamber (The letter “A” in the subscription represents the chamber A). Transitions {tA1, tA2, tA3, tA4} fire spontaneously once they are enabled. The transitions' firing sequence is determined by the token circulating among places {PA1, PA2, PA3, PA4} and their firing time is controlled by the token appearing in one of the three places {P0, P1, P2}.
Fig. 10. Petri net subsystem at lower level
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·975· The above Petri net representation describes the
connections and interactions of the combustion chambers. The combustion chambers jointly impact the overall engine system through exerting torques on the swing arm. The overall torque generated by the combustion chambers can be expressed as Eq. (1):
A B C D
e .
T =T -T +T -T (1) The torque Ti generated by each chamber at each phase relies on the thermodynamics of the air/fuel mixture before and after combustion. The profile of Ti is mainly determined by the phases of the chamber and subjected to the influences from the spark ignition timing, and the exhaust valve timing which changes the mass of fresh air/fuel charge loaded into the chamber during the intake phase, and the residual gas fraction.
3.1.2 Breathing dynamics of intake manifold
The model of the breathing process is a critical part of the engine modeling task because it affects most engine performance indices: fuel economy and emissions. The dynamic equation for the manifold is based on the principles of conservation of mass, thermodynamic energy and momentum which are satisfied by assuming uniform pressure and temperature in the plenum between the air/fuel mixing system and the intake valves. The air/fuel mixture into the intake manifold is assumed to be homogeneous.
There is no throttle existing between the air/fuel mixing system and the intake manifold. A pipe with a cross-sectional area of Ap is used to directly connect the air/fuel mixing system and the manifold. The mass air/fuel flow rate into the manifold Mmuin is defined as
p u m u m
in mu
( , , ), ,
0, otherwise, M A p p p p M = íìïï
ïïî
≥ (2)
where pu and pm are the pressures of the air/fuel mixing system and the intake manifold, respectively. The reed valves located between the manifold and combustion chambers are assumed to have a constant opening area Ar
for simplicity. The dynamics of reed valves, driven by the pressure difference between the manifold and the combustion chamber, are not considered in this research. The mass air/fuel flow rate from the manifold into a combustion chamber Mcmout is defined as
r m c m c
out cm
( , , ), ,
0, otherwi e,s
M A p p p p
M = íìïï ïïî
≥ (3)
where pc is the pressures of the combustion chamber. After taking into account of the steady flow energy equation, ideal gas law, and isentropic relation, the mass flow rate M(A, p1, p2) that passes through a nozzle can be approximately calculated via Eq. (4)[29]:
( )
1 2
1 1
cr
1 2 11
2 cr
( , , )
( ) 2 1 ( ) , ( ) ,
1
2 , ( ) .
1
M A p p
pr pr pr pr
pr pr
-
+ -
=
ìï é ù
ï ê ú
ïï ê - ú >
ï -
ï ê ú
ï ë û
ïíïï
ï æ ö
ï ç ÷
ï ç ÷÷ >
ï çç + ÷
ï è ø
ïî
(4)
where γ is the ratio of specific heats of the air/fuel mixture, Λ and the pressure ratio pr are rewritten as:
1 2
D 0 1
, .
p p
C A pr
RT p
= = (5)
where R is the gas constant, T0 is the gas temperature, CD is the flow or discharge coefficient, A is the cross-sectional area of the nozzle, p1 and p2 are the upstream and downstream pressures, respectively. The critical pressure ratio (pr)cr is defined as Eq. (6):
2 1
cr cr
1
( ) 2 .
1 pr p
p
æ ö÷ æ ö÷ -
ç ÷ ç ÷
=ççç ÷÷ =ççç ÷÷ è + ø
è ø (6) Therefore, Mmuin and Mcmout can be calculated with Eq. (4). The manifold dynamics, described by the first order differential equation that relates the rate of change of the manifold pressure pm to the flow rates into the manifold
in
Mmu and out of the manifold Mcmout, is written as Eq. (7):
in out
m M
m mu cm m
m
d ( ), .
d
p RT
K M M K
t = - = V (7)
Based on the manifold temperature Tm, the manifold volume Vm and the specific gas constant R, the constant Km
can be calculated. This model is a representation of the intake manifold filling dynamics, which can be applied to the four-stroke MICSE.
3.1.3 Thermodynamics of the chamber
Related to the pressure of each chamber, Eq. (1) can be rewritten as Eq. (8):
e ( intakei compi expani exhausti ) S A,
T = p -p +p -p A L (8)
where pintakei , pcompi , pexpani and pexhausti are the chamber pressures, AS is the swing arm side surface area, and LA is the distance between the radial location of the center-of-pressure on the swing arm face and the central pivot point. The positive sign before the pressure terms in the equation indicates they accelerate the swing arm rotation, while the negative sign means they decelerate the swing arm rotation.
Each combustion chamber is modeled as an open thermodynamic system. The assumptions are 1) the gas
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inside the chamber is uniform in composition and state at each point in time, and 2) that state and composition vary with time due to heat transfer, work transfer, mass flow across the boundary, and boundary displacement[29]. Based on the first law of thermodynamics and the ideal gas law, an equation for pressure derivative p can be obtained as Eq. (9)[29]:
b u
b ,
R R T m V
p p x
R T m V
æ - ö÷
ç ÷
= çç + + - ÷÷
çè ø
(9)
where p(t) is the chamber pressure, xb is the fraction of burned gas, V(t) is the chamber volume, m(t) is total mass of gases in the chamber, T(t) is the temperature of the mixed gases in the chamber, Rb and Ru is the gas constants of burned gas and unburned gas, respectively. All above parameters except xb could be measured directly or indirectly. For the four different strokes, the fraction of burned gas xb has the form as Eq. (10):
cmin l
bcm b bl b
l bl b
b
l bl b bc
l bl b
( ) ( ), ITS,
( ), CMS,
( ) , EPS,
( ), EHS,
M M
x x x x
m m
M x x x m
M x x x
m
M x x m
ìïï - + -
ïïïï
ïïïï -
= íïïï
ïïï - +
ïïïï
ïïï - ïïïî
(10)
where xbcm and xbl are the burned gas fractions of the intake flow and leakage flow, respectively. Ml is the leakage flow rate through the clearance gaps. Mcmin is the flow rate into the intake chamber (Mcmin =Mcmout ). In the combustion/ expansion stroke, the mass burning rate xbc is derived from the Wiebe function and has the following form as Eq. (11)[29]:
1
bc u0 +1
b b
( 1) exp ,
m m m
t t
x a m x a
æ æ ö +ö÷
ç ç ÷ ÷
ç ÷ ÷
= + çç- çç ÷ ÷
÷ ÷
ç çè ø ÷
è ø
(11)
where xu0 is the initial mass fraction of fresh reactant charge in the chamber before combustion and τb is the burn duration. Variable parameters a and m can be adjusted based on the analysis of experimental data. Substituting the value of xb from Eq. (10) into Eq. (9), the pressures of four chambers could be obtained.
3.1.4 Torque generation
The oscillatory dynamics of the generator rotor shaft coupled with the swing arm is described as Eq. (12):
e Load friction,
T T T
= - J- (12)
where θ is the rotation angle of the swing arm, TLoad
represents the torque of the electromagnetic forces of the generator that is shaft coupled with the swing arm for electricity generation, Tfriction represents the friction torque and J denotes the sum of the swing arm inertia and the generator rotor inertia. The torque TLoad, resulted from electromagnetic force and current flow, is given as Eq. (13):
2 2 2 g
Load g
L g
( ) B L r , T BLr i t
R R
= =
+ (13)
where B is the magnetic field strength of the generator, L is the length of the generator coils, rg is the generator rotor radius, i(t) is the current, is the angular velocity of rotor, and RL and Rg are the external electrical load and the generator internal electrical resistance in the output circuit of the system.
The rotational friction arises when the swing arm moves relatively to the chamber wall. The damping torque due to friction opposes the motion, and depends on many factors, such as the surface properties, the nature of fluid flow between the surfaces and the relative velocity of sliding. The relationship between these factors and the generated damping torque is quite complex. We approximate this complex relation with a linear relationship, i.e., the damping friction torque is proportional to the engine speed, as shown in Eq. (14):
friction f ,
T =k (14)
where kf is an adjustable coefficient. 3.1.5 Model validation
Based on the above analysis, the hybrid models for the four-stroke MICSE systems are constructed using MATLAB Simulink. The block diagram for MATLAB realization is shown in Fig. 11.
The parameters need to be identified through the model validation process include burning time and friction coefficient. A trial and error approach is employed to find the values for these parameters. The data obtained from the model simulation is compared to experimental data. Fig. 12 shows the experimental and simulated results for DC voltage output obtained at five exhaust valve closing timing settings (ranging from 1 ms to 5 ms) when the spark timing advance is equal to zero.
In this study, the exhaust valves open right after the swing arm reaches an extreme position and close after its opening duration reaches a specified value, i.e. an integer value is selected from 1 ms to 5 ms. It is shown that the simulation results reasonably agree with the experiment results. Both of them demonstrate a similar trend that the voltage output increases along with the increase in exhaust valve opening duration.
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·977·
Fig. 11. Block diagram for MATLAB Simulink realization of the four stroke MICSE system
Fig. 12. A comparison of experimental and simulated DC voltage output for a four-stroke MICSE system
3.2 Nonlinear modeling and its linearization
The nonlinear model is built on the regression results of the states and output data collected from the hybrid model. The data collection strategy employed here is to 1) modify the hybrid model by adding an integral controller on the engine speed error and using the load as an actuator to drive the system to the desired speed, and 2) run this setup for several values of spark advance, exhaust valve duration, and desired speed, which are listed in Table 1.
Table 1. Four-stroke MICSE operating range
Parameter Start Stop Step
Spark advance Ts 0.3τ 0.5τ 0.05τ
Value duration D/ms 2 4 0.2
Desired speed ω/(rad • s–1) 50 70 2.5
3.2.1 Nonlinear modeling
In the four-stroke MICSE application, we utilize the exhaust valve closing timing for chamber air/fuel flow
regulation instead of the throttle in traditional throttled engine or the active intake valves in camless unthrottled engine. To simplify the control problem, only the exhaust valve closing strategies (i.e., LEVC and EEVC) are considered for the constant voltage control problem. Fig. 13 shows the block diagram of a four-stroke MICSE.
Fig. 13. Schematic block diagram of nonlinear MICSE system model
In the four-stroke MICSE application, intake manifold pressure experiences little variation and thus can be assumed as constant. In other words, the engine pumping rate doesn’t depend on intake manifold pressure. Moreover, the exhaust valve closing timing σ affects both the engine pumping process and air/fuel charge process through changing the residual gas fraction in the combustion chambers. However, the dynamics of exhaust valve open/close actuation is ignored in this study.
The system model for the four-stroke MICSE in Fig. 12 can be further expressed by the following continuous-time nonlinear 1st order ordinary differential equation (ODE) with delays:
s s
4 2 5 6 L
d 1 ( , , ) ( ) ( , ) ,
dt J f M T T f f R
= é - - ù
ê ú
ë û (15)
where ω denotes the engine speed, the subscript Ts refers to
Y GAO Dedong, et al: Constant Speed Control of Four-stroke Micro Internal Combustion Swing Engine
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the delay of a stroke in the time domain, and the engine pumping mass air/fuel charge M and air/fuel flow rate M are rewritten as Eq. (16):
3( , , ), 2( , ).
M = f M M = f (16) The regression results for the above nonlinear model are given in Appendix.
3.2.2 Linearized model and stability analysis
The above nonlinear model is linearized around a given nominal equilibrium point OP={ω0, RL0, σ0, τ0} as shown in Table 2.
Table 2. Target four stroke MICSE operating point
Parameter Value
Speed ω0/(rad • s–1) 60
Electrical load resistance RL0/Ω 3.2 Exhaust value closing timing τ0/ms 3
To simplify the following derivation, the delays are neglected. The linearization for each nonlinear block is illustrated in Fig. 14, which reveals model structure.
Fig. 14. Linearization for each nonlinear block
The coefficients kxx are obtained from the following equations:
p1 OP p2 OP
m1 OP m2 OP m3 OP
e e e
T1 OP 2 OP T3 OP
Load friction Load friction
LR OP LN OP
L
| , | ,
| , | , | ,
| , | , | ,
( ) ( )
| , | .
T
M M
k k
M M M
k k k
M
T T T
k k k
M
T T T T
k k
R
¶ ¶
= =
¶ ¶
¶ ¶ ¶
= = =
¶ ¶
¶
¶ ¶ ¶
= = =
¶ ¶ ¶
¶ + ¶ +
= =
¶ ¶
(17) From the linearized model, the transfer function between exhaust valve duration timing, spark advance timing, external electrical load and the engine speed is expressed as Eq. (18):
p1 m1 m3 T1 T3 LR L
LR T2 m2 p2 m1 T1
( )
( ) ( ) .
k k k k k K R
Js k k k k k k
= + + +
+ - - + (18)
Substitution of all the numerical values of the coefficients obtained at the nominal operating point result in a real root with a value of –9.904 8´102. The root lies in the open left half plane (OLHP) and therefore the uncontrolled system is stable.
In the above analysis, the time delays in the four-stroke MICSE system are ignored in the model linearization. These time delays mainly include induction-to-power (I-P delay), spark timing actuation delay and, valve duration timing actuation delay. Among them, the largest time delay is the I-P delay, which has a value of 2Ts or twice of the stroke duration. To describe the system more accurately, the effects of time delay must be considered in the engine modeling. Therefore, the new transfer function including delays can be written as:
p1 m1 m3 T1 s s
T3 s LR L LR T2
m2 p2 m1 T1 s
[( ) exp( ) exp( 2 )
exp( ) ] [ ( )
( ) exp( )].
k k k k T s T s
k T s K R Js k k
k k k k T s
= + - - +
- + / + - -
+ - (19)
To assess potential difficulties in the feedback controller design, the time delay is approximated as Eq. (20) using a first-order Padé approximation which has a non-minimum phase (NMP) zero in the open right half plane (ORHP) and a stable pole in the OLHP:
s s
s
1 ( 2)
exp( ) .
1 ( 2)
T T s
T s
- /
- =
+ / (20) Substitution of the time delay terms in the transfer function with the above delay approximation results in an uncontrolled system of three negative poles and two positive zeros. The numerical values for these poles and zeros are listed in Table 3.
Table 3. Numerical values for poles and zeros of the transfer function with time delays
Poles Value Zeros Value
p1 –1132.2 z1 200
p2 –200 z2 100
p3 –87.482 z3 –100
From Table 3, the two poles from the delay (p2 and p3) are much smaller than the pole of the uncontrolled system without any delay. Therefore, the uncontrolled four-stroke MICSE system is still a stable system even considering the effects of timing delays.
4 Feedback Controller Design
The goal for the MICSE control is to design a controller which regulates the MICSE voltage output at a desired constant level in a fast and smooth fashion and rejects the disturbance from external load resistance fluctuation. The exhaust valve duration and spark timing are used as two key control variables to realize the control objective.
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·979· A sequential loop closure design approach is employed
to design the controller for the constant speed regulation problem. Two steps are involved: 1) design the spark controller C2, and 2) design the exhaust valve controller C1. The approach is illustrated in Fig. 15.
Fig. 15. Controller design layout
In Fig. 15, G1, G2, G3 and G4 are written as the following equations:
1 3
1 2 3 T3
s 3
1 2
2 p1 m1 m3 T1 2 4 LR
3
, ,
( )
( )( )
2
( ) , .
( )
s z
G s z G k
J s p
JT s p s p
G k k k k s z G k
J s p
-
= - =
- - -
= + - =
-
(21)
4.1 Closing loop on spark timing
A proportional (P) controller is used to close the spark loop first. The reason for using a P controller for the spark loop is that the engine needs to operate with its optimal spark timing at steady state. The control of spark timing is to improve the transient response performance, but not to eliminate the steady state error. If the proportional control gain is too large, the closed-loop poles could move to the right half plane (RHP) and also the spark advance timing could saturate. The transfer function for the closed spark loop is written as Eq. (22):
2 1 3 P_Spark
2 1 3
1 . C G G
G = +C G G (22)
When the value for the proportional control gain is chosen as 3.75´10–5, the numerical values for the poles and zeros of the closed-loop transfer function are given in Table 4, where a comparison to the poles and zeros of the uncontrolled system is also made.
Table 4. Poles and zeros of closed-loop transfer function with P control of spark timing
Uncontrolled poles
Uncontrolled
zeros Closed-loop poles Closed-loop zeros
–1132.2 200 –415.44+375.76i 200
–200 –100 –415.44–375.76i –100
–87.482 – –94.63 –
A root locus for the closed loop system with proportional
control of spark is shown in Fig. 16. From the root locus, it is shown that the trajectory moves into RHP as the control gain increase. The proportional control increases both damping and natural frequency.
Fig. 16. Root locus for the closed-loop transfer function with P control of spark timing
4.2 Closing loop on valve duration timing
A proportional-integral (PI) controller is used to close the loop on valve duration timing. A PI controller for the valve loop will allow us to eliminate the steady state error in the speed output. The choice of the PI controller for the valve loop depends on the P controller for the spark loop. With the proportional control of spark timing given in the previous section, the design of the PI control of exhaust valve duration should meet the following requirements for transient response performance. The 90% rising time is less than 0.1 s, the settling time is less than 0.2 s and the overshot is less than 5%.
The PI control of the valve duration can be in the form of Eq. (23):
PI PI 1
( 1)
k s ,
C s
+
= (23)
where kPI is the control gain and τPI specifies the location of the zero added by the valve duration PI controller. Therefore, the closed-loop transfer function for PI control of valve duration can be written as Eq. (24):
1 1 2 PI_Value
2 1 3 1 1 2
1 .
C G G
G = +C G G +C G G (24)
The existence of a RHP zero in the above closed-loop transfer function implies that the control gain kPI can not be too large. A large value of the gain kPI would drive the closed-loop poles into the RHP and lead to an unstable closed-loop system. With the consideration for transient performance and constraints for stable system, the value for control gain kPI is selected as 0.001 6 while τPI is selected as 1/400. With these values, a root locus for the closed-loop system with PI control of valve duration is shown in Fig. 17. The numerical values for the poles and zeros in the closed-loop transfer function are given in Table 5.
Y GAO Dedong, et al: Constant Speed Control of Four-stroke Micro Internal Combustion Swing Engine
·980·
Fig. 17. Root locus for the closed-loop transfer function with PI control of valve duration
Table 5. Poles and zeros of closed-loop transfer function with PI control of valve duration
Poles Value Zeros Value
p1 –376.43+371.30i z1 –400
p2 –376.43–371.30i z2 –200
p3 –47.31 z3 100
p4 –47.312 – –
5 Simulation Results and Discussion
The purpose of the simulation is to illustrate some of the properties of the closed loop system with control of spark timing and valve duration. The simulation is accomplished through applying the above sequential loop closure control design to the hybrid four-stroke MICSE model developed. The simulation results for applying the controller to the hybrid model are shown in Fig. 18, which demonstrates the changes in the speed output of the engine, the spark timing actuation, and the valve duration timing actuation due to the change of external load resistance.
Fig. 18. Simulation results for applying closed loop controller (P control of spark timing and PI control of valve duration)
In the simulation example, a 20% step change in external load resistance is applied to the model. In the case of the uncontrolled system, the change in load resistance leads to an engine speed drop of 5 r/min and the system fails to return to its desired operating point. In the case of closed loop control, the system is able to quickly return to its desired operating point within 0.2 s, with the assistances from the P control of spark timing and PI control of valve duration. The sparking timing is intensively used during transient response and never used during steady state. However, a 15% step change in valve duration occurs corresponding to the step change in external load.
6 Conclusions
(1) Petri nets are used to describe the concurrency and interactions between the MICSE system modules. A two-level Petri net based hybrid model for four-stroke MICSE systems is developed.
(2) The MICSE hybrid models are beneficial for understanding the dynamics of the system and serves as a
“platform” for the development of control strategies to improve the engine performance.
(3) The developed nonlinear and linearized models address the impacts from the spark timing and valve duration on engine speed output. More importantly, it reveals how the exhaust valve duration significantly influences the engine breathing process which is a fundamental process for torque response, fuel economy, and emissions.
(4) It is shown from the simulation results that the control of spark timing and valve duration yields promising results towards realization of good engine performance. The simulation demonstrates the feasibility to replace throttles or camless intake valves with exhaust valves as actuators for a constant speed control problem.
Acknowledgments
We thank Dr. Rhett Mayor for his suggestions, and the reviewers and editor for the helpful comments.
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Biographical notes
GAO Dedong, born in 1980, is currently a PhD candidate at State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. He is a lecturer at School of Mechanical Engineering, Qinghai University, China. He received his master degree from Tsinghua University, China, in 2007. His research interests are computer simulation and bio-manufacturing. Tel: +86-135-19764535; E-mail: [email protected]
LEI Yong, born in 1976, is currently an associate professor at State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. He received his PhD degree from University of Michigan, USA, in 2007. His research interests include fault analysis, intelligent maintenance and precision control.
Tel: +86-153-06522727; E-mail: [email protected]
ZHU Honghai, born in 1977, received his PhD degree from The University of Michigan, Ann Arbor, USA, in 2006. His research interests include Modeling and control of combustion engines NI Jun, is currently a professor in University of Michigan, USA. He received his PhD degree from the University of Wisconsin-Madison, USA, in 1987. His research interests include precision machining, micro/meso systems and manufacturing processes, manufacturing process modeling and control, statistical quality design and improvement, etc.
Tel: +1-734-9362918; E-mail: [email protected]
Appendix
A1 Regression of function for nonlinear breathing process
The mass air/fuel flow from the manifold to the chambers is a complicated nonlinear function of engine characteristics including pressure losses in the intake valve and the wall friction[15]. It can be expressed by an empirical relationship assuming quasi-steady operating conditions, and averaging the mass air/fuel flow into the chambers over a stroke. The empirical relationship can be obtained by regarding the engine as a pump, omitting intake reed valve dynamics, and assuming constant values for intake temperature, and intake manifold pressure. The engine pumping mass air/fuel flow rate M is a function of the engine speed ω and exhaust valve closing timing σ. The resulting regression model can be obtained as Eq. (25):
Y GAO Dedong, et al: Constant Speed Control of Four-stroke Micro Internal Combustion Swing Engine
·982·
2
2 2 2 2 2 2
( , )
(1, , , , , , , , ).
M f F
= =
(25)
Fig. 19 and Fig. 20 show the regression results for manifold air/fuel flow rate, where the estimated flow rates using the regression model match their original values obtained from the hybrid model. The engine speed ω ranges from 50 rad/s to 70 rad/s and exhaust valve closing timing
varies between 2 ms and 4 ms.
Fig. 19. Regression results for M plotted against
Fig. 20. Regression results for M plotted against ω
A2 Regression of function for air/fuel charge
The air/fuel charge is an integration of mass air/fuel flow rate over a stroke. It is interesting to notice that the initial condition for the integration is the amount of the residual gas left in the chamber from the previous exhaust stroke, which is controlled by the exhaust valve closing timing. Meanwhile, the integration interval is defined as the duration of a stroke, which is strongly correlated with the engine speed. Therefore, the air/fuel charge M is a function of mass flow rate M , the exhaust valve duration , and the engine speed ω. The resulting regression model can be obtained as Eq. (26):
3
2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2
( , , ) (1, , , , , , ,
, , , , , , , ,
, , , , , ).
M f M F M M M
M M M M M
M M M M
= =
(26) The regression results for the air/fuel charge over a stroke are shown in Fig. 21, Fig. 22, and Fig. 23, where the estimated air/fuel charge using the regression model match their values obtained from the hybrid model.
Fig. 21. Regression results for M plotted against M
Fig. 22. Regression results for M plotted against ω
Fig. 23. Regression results for M plotted against
