274
Asian Journal of Control, Vol. 8, No. 3, pp. 274-280, September 2006
－Brief Paper－
OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC
VEHICLES BASED ON THEORY OF SWITCHED SYSTEM
Licun Fang and Shiyin Qin
ABSTRACT
The focus of this paper is the control strategy used to control
general parallel hybrid electric vehicles (HEV). The torque split control problem of HEV is formulated as the optimal control of a
switched system. A model-based strategy for fuel-optimal control is
presented. The optimal control problem of such a switched system is
formulated as a two-stage optimization problem. Dynamic programming is utilized to determine the optimal control action that minimizes
the cost function. Simulated results indicate that this method is effective.
KeyWords: Hybrid electric vehicles (HEV), optimal control, dynamic programming, switched system.
I. INTRODUCTION
The most challenging goals facing the automotive industry are increasing fuel economy and reducing emissions.
There are a series of constraints that are imposed not only
by our society but also by cooperative agreements and legislative efforts [1]. HEVs are beginning to demonstrate
their capability to satisfy these requirements, and they will
become a viable alternative to conventional vehicles in the
future [2,3].
The hybrid powertrain is an integrated system that
may consist of the following components: an internal
combustion engine (IC engine), a battery pack, and an
electric machine (EM) which can be utilized as a traction
motor or generator. In such a system, each sub-system is
also a complex system which has its own functionality and
desired performance. Moreover, all of the sub-systems
need to be coordinated in an optimal manner to achieve the
desired objectives. Consequently, an integrated vehiclelevel controller is required to accomplish such tasks [4].
Recent reports on HEVS have placed great emphasis on the
control strategy, and several approaches to it have been
Manuscript received July 18, 2005; revised January 25,
2006, accepted May 15, 2006.
The authors are with the School of Automation Science and
Electrical Engineering, Beihang University, Beijing 100083,
China (e-mail: licun_fang@buaa.edu.cn, qsy@buaa.edu.cn).
proposed. The power management strategies can be
roughly classified into three types: static optimization
methods [5,6], global optimization methods based on dynamic programming [7,8], and rule-based control approaches [9,10].
HEV systems simultaneously exhibit several kinds of
dynamic behavior, such as continuous-time dynamics,
discrete-time dynamics, jump phenomena, switching and
the like. Therefore, an HEV is a typical hybrid dynamical
system, and many challenges exist in the design of a vehicle system controller for an HEV [4]. The focus of this
paper is the control strategy used to control general parallel
HEVs. The remainder of this paper is organized as follows.
Section 2 will describe the hierarchical architecture of the
HEV system. The different powertrain components models
used for simulation and control are presented in section 3.
The optimal control problem is solved in section 4. In section 5, a case study investigated through simulation is presented, and the results obtained in optimizing the control of
fuel consumption based on the optimal strategy proposed in
this paper are given. Conclusions are drawn in section 6.
II. CONTROL SYSTEM ARCHITECTURE
An HEV is a typical hybrid dynamical system [4].
Most hybrid dynamical systems have been designed using
the concept of centralized and distributed control. A
L. Fang and S. Qin: Optimal Control of Parallel Hybrid Electric Vehicles Based on Theory of Switched Systems
two-level hierarchical control architecture [11] is used in
the synthesis of the control law for the hybrid powertrain
shown in Fig. 1. The supervisory powertrain controller
(SPC) represents a high-level vehicle control system that
can coordinate the overall powertrain to satisfy certain
performance indices. Based on the driver’s demand and
operating parameter feedback signals of the vehicle and its
components, the SPC must determine the desired propelling torque distribution between the IC engine and EM, or
the desired brake torque distribution between the EM and
friction brake system, and the gear position; therefore, the
main function of the SPC is to determine the desired output
of various sub-systems according to driver’s commands,
system parameter feedback signals and a pre-programmed
control strategy. Different control strategies will result in
different fuel economy and emission characteristics. The
focus of this paper is the control strategy used to control
general parallel HEVs.
In order to narrow our research scope, only a double
shaft parallel powertrain is considered. In this configuration,
the engine power and electric motor power can provide
torque to the wheels separately through the transmission. It
should be pointed out that the strategies for dealing with
this situation can also be applied to other powertrain arrangements by making appropriate amendments. Figure 2
shows a schematic diagram of the double shaft parallel
HEV system considered in this paper.
Driver
Supervisory powertrain
control
CAN bus
Engine
controller
Engine
Motor /Battery
controller
Clutch
Motor
Trans.
controller
Brake
controller
Trans.
Vehicle
The following components are utilized in the configuration:
• spark ignition direct injection engine: 50kW;
• permanent magnet motor: 10kW;
• NiMh battery: 20kW, 2kWh;
• 5speed automated manual transmission.
III. SYSTEM MODEL
A quality dynamic model is paramount for the development of an effective control strategy, especially when
dealing with a complex coordinated control problem such
as a parallel HEV. An HEV powertrain dynamic model
describing key system components can be a significant aid
to the development of the control law and provides insight
into the dominant dynamic effects in the control law synthesis process. To this end, the components of a parallel
HEV are modeled. However, developing a vehicle controller is the key objective of this paper, so simplified models
are presented. Dynamic models of various primary HEV
components, such as the ICE, motor, battery, transmission
and vehicle, are described below.
3.1 Engine
To avoid the dependence on the availability of specific
efficiency maps for the IC engine and the electric motor
while optimizing and formulating the HEV optimal control
strategy, a universal representation, the Willans lines model
of an IC engine and motor, is utilized [12]. A brief summary of the parametrization is given in the following paragraphs. The scalability of an IC engine is achieved through
the use of two engine sizing parameters, the displacement
Vd and the stroke S. Here, some concepts and corresponding variables, such as the mean piston speed cm, the available mean effective pressure pma and the mean effective
pressure pme, are introduced as follows:
cm =
Sωe
,
π
Battery
Fig. 1. Hierarchical control architecture of an HEV.
Pma =
Reduction
gears
Motor
Gear
box
Wheel
Mechanical coupling
Pme =
Wheel
Clutch
Battery
Fig. 2. Block diagram of the parallel HEV.
ICE
275
4π QLHV m f
Vd ωe
4π Te
,
Vd
(1)
,
(2)
(3)
where ωe is the ICE angular speed, Te is the ICE effective
f is the fuel mass flow rate, QLHV is the fuel
torque, m
low heating value, and e is the “inner” ability to transform
chemical to mechanical energy.
The engine’s efficiency can be approximated by an affine relation between the torque and fuel mass available in
one cycle:
Asian Journal of Control, Vol. 8, No. 3, September 2006
276
Te = e ⋅ QLHV
m f
ωe
− Tloss ,
(4)
where Tloss represents the inner losses. The resulting affine
function shows the dominant effect observed in all ICEs,
i.e., the dramatic decrease in efficiency as the torque output
decreases.
Based on Eqs. (1) ~ (3) and some necessary substitution in Eq. (4), a dimensionless definition of engine efficiency η can be obtained:
pme
,
pma
(5)
where Pme = ePma − Pml ,
(6)
η=
and pml denotes the inner loss (see [12] for more details).
3.2 Motor
In this section, the same concepts that were introduced
above for an IC engine are developed for an EM. Let pem be
the available electrical power. Then, for each time instant,
the following equation defines the efficiency of the EM:
ωem Tem, e = η (⋅) pem = η (⋅) Vi ,
(7)
where ωem and Tem, e are the EM angular speed and effective
torque, respectively; V and i are the voltage and current,
respectively. Again, as a first approach, the motor’s efficiency can be approximated by assuming that an affine
relationship between the effective torque and input energy
exists:
Tem, e = eem ⋅ V ⋅
i
− Tem, loss
ωem
= eem Tem, a − Tem, loss ,
ωw (k ) =
ωe (k ) ωem (k )
=
.
ρ
R(k )
(9)
The second one is the torque relation:
Tw (k ) = R (k ) ⋅ Te (k ) ⋅ η gb + ρ Tm (k ) η red ,
(10)
where ωw is the wheel angular speed, R(k) is the gear ratio,
ρ is the reduction gear ratio between the EM and the
wheels, Tw(k) is the torque of the wheels, and Te(k) and
Tm(k) are the output torque from the engine and the motor,
respectively. η gb denotes the efficiency of the gear box, and
η red is that of the reduction gears.
3.5 Vehicle
The modeling of the vehicle dynamics is derived from
the basic equation for solid-body motions (Newton’s Second Law):
1
Dair CD A f V 2
2
dV
+ Mi
+ Mg sin (θ) ,
dt
F = MgCr +
(11)
where F is the force required at the vehicle wheels to reach
a certain acceleration at speed V ; Mg, Cr , Dair, CD , Af , Mi ,
and θ are the mass of the vehicle, the local acceleration of
gravity, the coefficient of the rolling resistance between the
tires and the road surface, the air density, the coefficient of
the aerodynamic drag for the vehicle in the traveling direction, the frontal area of the vehicle, the inertial mass of the
vehicle, including the rotational inertia contribution, and
the gradient of the road, respectively.
The vehicle acceleration a and speed V are given by
(8)
where eem, Tem, a and Tem, loss are the “inner” ability to transform electrical energy into mechanical energy, based on the
available mean effective torque of the EM and inner loss
(see [12] for more details).
3.3 Battery
The practical implementation of this control strategy is
dependent on the availability of an on-line SOC (State Of
Charge) estimator. The battery performance, such as the
voltage, current and efficiency, from a purely electric
viewpoint, is the outcome of thermally dependent electrochemical processes that are quite complicated. A model
based upon open circuit voltage is used in this paper. Further details about this method can be found in [13].
3.4 Transmission
Two main relations describe the parallel double-shaft
arrangement of an HEV. The first one is the speed relation:
1
η gb R(k ) T / r − MgCr − Dair CD Af V 2 − mg sin(θ)
2
a=
,
Mi
(12)
V = ∫ adt ,
(13)
where r is the radius of the wheels.
IV. OPTIMAL CONTROL
4.1 Problem Formulation
In the operation process of a powertrain, the transmission controller determines the gear shifting sequence and
selects the appropriate transmission gear based on the
transmission output speed, the position of the acceleration
pedal and braking pedal, the current gear position and the
clutch state. The powertrain system can be formulated as a
switched system. As is well-known, a discrete-time
switched system can be described as a set of piece-wise
L. Fang and S. Qin: Optimal Control of Parallel Hybrid Electric Vehicles Based on Theory of Switched Systems
difference equations [14]:
Te _ min ( we (k )) ≤ Te ≤ Te _ max ( we ( k )) ,
x( k + 1) = f ( x(k ), u ( k ), m(k ))
⎧ f1 ( x( k ), u (k )), if m( k ) = 1
⎪
=⎨
#
⎪ f ( x (k ), u (k )), if m(k ) = I ,
⎩ I
(14)
(15)
where Twh_req is the required torque of the vehicle.
For a vehicle control system, once a driving cycle is
given, the wheel speed profile wwh_req is known, and the
torque Twh_req required to follow the speed profile can be
determined in each time step by inversely solving the dynamic equations (11) ~ (13). The practical wheel speed wwh
can be detected by a speed sensor.
Problem 1. The power control of an HEV can be formulated as an optimal control problem, in which the optimization goal is to find the control input Te(k) and switching
sequence of gear shifting g(k) needed to minimize a cost
function in a given driving cycle:
⎧ N −1
J = ψ( x ( N )) + ⎨ ∑ L( x( k ), Te (k ), m(k ))
⎩ k =0
N −1
⎫
+ ∑ M k ( xi , xi' ) ⎬ ,
k =0
⎭
(17)
where Te_min (we(k)) and Te_max (we(k)) are the minimal and
maximal output torque of the ICE at speed we(k);
where x(k) is a state vector of the system, u(k) is a vector of
the continuous-time control variables, and m(k) ∈{1, 2, …I}
is a finite integer set in which every element corresponds to
a kind of operating mode with event attributes. An HEV
can be considered as a special case of this model. For a
vehicle control system, the continuous-time state vector is
(V, SOC); the control variables u(k) are Te(k) and Tm(k);
m(k) is the gear position, which is a discrete-event state
variable. If the state variables and control variables are
inserted into Eq. (14), a set of piece-wise difference equations for an HEV can be obtained. It should be pointed out
that the motor torque becomes a dependent variable instead
of a control variable due to the equality constraint on the
driveline torque:
Twh _ req = Te ( k ) + Tm (k ) ,
277
(16)
where N is the duration of the driving cycle, ψ (x (N)) is the
cost function associated with the error in the terminal SOC,
L(x (k), Te(k), m (k)) is the instantaneous fuel consumption
rate, and xi, xi represent different gear positions. Mk(xi, xi) is
the cost function of mode switching between xi and xi; in
order to simplify the problem, the cost function of mode
switching is assumed to be a constant set and to take values
from this set.
In order to ensure drivability of the vehicle and
safe/smooth operation of each subsystem, it is necessary to
satisfy the following constraints:
we _ min ≤ we (k ) ≤ we _ max ,
(18)
where we_min and we_max are the minimal and maximal
speeds of the ICE;
SOCmin ≤ SOC(k ) ≤ SOCmax ,
(19)
where SOCmin and SOCmax are the lower limit and upper
limit of the SOC of the battery;
Tm _ min ( wm (k ), SOC( k )) ≤ Tm (k )
≤ Tm _ max ( wm (k ), SOC(k )) ,
(20)
where Tm_min (wm(k), SOC(k)) and Tm_max (wm(k), SOC(k))
are the minimal and maximal output torque of the EM at
speed wm(k) for the battery SOC(k).
4.2 Solution
For the optimal control problem of a switched system,
the following conditions must be satisfied [15].
Condition 1. An optimal solution (σ*, u*) exists, where σ*
is the optimal switching sequence and u* is the optimal
input control.
Condition 2. For any given switching sequence σ, there
exists a corresponding u* = u* (σ) such that Jσ(u) = J(σ, u)
is minimized.
If the above conditions are satisfied, then, the following
equation holds:
min J (σ, u ) = min min J (σ, u ) .
σ∈∑ , u∈U
σ∈∑
u∈U
(21)
Therefore, the optimal control problem of a switched system can be formulated as a two-stage optimization problem.
Stage 1. (a) Fix the total number of switches to be K, fix
the order of the active subsystem, let the minimum value of
J with respect to u be a function of the K switching instants,
i.e., J1 = J1(t1, t2, …, tk) for K ≥ 0, and then find J1. (b)
Minimize J1 with respect to t1, t2, …, tk.
Stage 2. (a) Vary the order of the active subsystem to find
an optimal solution under K switches. (b) Vary the number
of switches K to find an optimal solution.
Dynamic programming is a powerful tool for solving
general dynamic optimization problems. The main advantage is that it can easily deal with the constraints and
nonlinearity of such problems while obtaining a globally
optimal solution. The dynamic programming technique is
based on the Bellman’s Principle of Optimality, which
Asian Journal of Control, Vol. 8, No. 3, September 2006
states that the optimal policy can be obtained if we first
solve a one stage problem involving only the last stage and
then gradually extend it to the first stage.
As a matter of fact, dynamic programming has been
used to solve the optimal control of a switched system [16].
i
By denoting Σ [k, N] as a subset of Σ[k, N] which contains all
σ∈Σ[k, N], starting with subsystem i at instant k, one can
i
compute the cost functions V (x(k), k), i ∈ I by solving Eq.
i
'
(22), where i, i represent a subsystem and V (x (k), k) is the
minimum value of J if the system starts at time k with state
x(k) and subsystem i:
i
V ( x( k ), k ) = min
min J (σ, u )
i
σ∈∑ k , N u∈U
[
]
⎧ ψ(( N )) if k = N
⎪
⎪ min{L( x(k ), u (k ))
= ⎨ u(k )
(22)
j
⎪
+ min
V ( x(k + 1), k + 1)} if 0 ≤ k < N .
⎪⎩
j∈{i ,i ' }
Consequently, the optimal solution (σ, u) of a discretetime switched system can be constructed by solving
i
V (x(k), k) backwards and finally finding (x(0), 0).
In a double-shaft parallel hybrid configuration, conditions 1 and 2 are satisfied, and the gear ratio of the transmission only affects the torque contribution of the IC engine; consequently, we can divide the development of the
overall optimal control strategy into two stages:
(1) implement a gear shifting strategy, which selects the
switching sequence of gears in a discrete set Σ to optimize the operation of the engine;
(2) implement a power split strategy, which defines the
best power split between the ICE and EM.
With the above two-stage optimization process, the
optimal control policy (Te(k), g(k)) can be constructed.
4.3 Problem Simplification
The big challenge in utilizing the above algorithm to
find the optimal feedback control policy is overcoming the
difficulty posed by the huge computation burden. It is, thus,
necessary to adopt some approaches to accelerate the
computation speed. In this paper, we simplify the above
two steps as described below.
(1) Shifting Sequence of Gear Position
Determining the gear shift strategy is crucial to improving the fuel economy of hybrid electric vehicles [17].
The gear shifting sequence of an automatic transmission
can be modeled as a discrete-time dynamic system:
⎧ M , m( k ) + g ( k ) ≥ M
⎪
m(k + 1) = ⎨ 1, m( k ) + g (k ) ≤ 1
⎪m(k ) + g ( k ), otherwise,
⎩
(23)
where M is the number of gear positions, and g(k) is constrained to take on values from among −1, 0, and 1, representing downshift, hold and upshift, respectively.
From the speed profile of a given driving cycle, the
optimal gear position can be constructed through static
optimization [9], the total switching numbers of gear position can be fixed and the shifting sequence can be determined; consequently, the problem can be transformed into
an optimal control problem for a switched system with a
fixed switch number and fixed switching sequence.
(2) The Optimal Torque Input
The state and control variables are first quantized into
a finite grid. For a given driving cycle, the vehicle model
can be replaced by a finite set of operating points parameterized by the required torque and speed. Pre-computed
look-up tables can be constructed for recording the next
states and cost function as a function of the quantized states,
control inputs and operating points. Once these tables are
built, they can be used to solve the problem in a very efficient manner.
The dynamic programming procedure produces an
optimal, time-varying, state-feedback control policy (Te*(k),
g*(k)) for a driving cycle. This control policy can then be
used as a state feedback controller.
V. CASE STUDY
To verify the control policy determined by the dynamic programming algorithm, a case was studied and will
be discussed in this section. The UDDS (Urban Dynamometer Driving Schedule) driving cycle is a standard test
driving cycle utilized by the Environmental Protection
Agency to certify the fuel economy and emission performance of vehicles which are driven in urban areas.
The optimal control policy was found through DP
(Dynamic Programming) in a UDDS driving cycle. In order to evaluate the final fuel economy, the optimal control
law was applied to the full-order parallel HEV model. The
terminal SOC constraint was selected as 0.65, and the initial SOC in the simulation was chosen as 0.65 for the purpose of calculating the fuel economy.
The dynamic trajectories of the vehicle under the optimal control policy for the UDDS driving cycle are shown
in Fig. 3 and Fig. 4.
Difference of Speed(km/h)
278
0.04
0.03
0.02
0.01
0
-0.005
0
200
400
600
Time(s)
800
1000
1200
Fig. 3. Difference between the required and achieved speeds.
L. Fang and S. Qin: Optimal Control of Parallel Hybrid Electric Vehicles Based on Theory of Switched Systems
Speed(km/h)
100
omy is achieved by optimizing the gear shifting policy and
torque splitting strategy. In general, torque split algorithms
developed through DP are more accurate, so the approach
proposed in this paper can be utilized to assess other control strategies.
A
50
0
0.70
0
200
400
600
800
1000
1200
B
REFERENCES
SOC
0.68
0.66
0.64
0
200
400
600
Overall ratio
10
800
1000
1200
800
1000
1200
800
1000
1200
C
5
0
0
200
400
600
0.6
D
Fuel(L)
0.4
0.2
0
279
0
200
400
600
Time(s)
Fig. 4. Dynamic trajectories of the actual speed profile, battery SOC, fuel consumption and overall ratio.
The largest difference between the desired vehicle
speed and the actual speed is within 0.04km/h, which
means that the control strategy can trace the objective
speed correctly. The SOC trajectory starts at 0.65 and ends
at around 0.65 with a small quantization error. The fuel
consumption of the control policy is 0.4502L, and the average fuel consumption is 3.8 L/100Km. The results were
compared with the default control strategy in ADVISOR
2002, that is, the electrical peaking energy management
strategy. The fuel consumption of the control policy is
found to be 0.4695L, and the average fuel consumption is
3.92 L/100Km, so the optimal control law is effective in
improving fuel economy.
VI. CONCLUSIONS
The optimal torque split problem of a parallel HEV
has been formulated as the optimal control problem of a
switched system. A control strategy for general parallel
HEVs has been developed in this paper. The proposed optimal torque split strategy for parallel HEVs, designed with
the aid of dynamic programming, can improve the overall
system efficiency. A two-stage dynamic programming algorithm has been used to solve the minimum fuel optimal
control problem for a parallel hybrid electric vehicle. In
order to reduce the computation complexity, several approaches have been utilized to accelerate the computing
speed. A dynamic optimal solution to the energy management problem over a driving cycle has been obtained. The
simulation results indicate that improvement in fuel econ-
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