This document describes applying active disturbance rejection control (ADRC) to a Toyota hybrid synergy drive system. It provides background on PID controllers and introduces ADRC as an alternative. ADRC incorporates a transient profile generator, nonlinear feedback, and an extended state observer. The document simulates an ADRC controller for a Toyota hybrid powertrain model. Simulation results show the extended state observer can accurately estimate the state of charge over time, with the error decreasing to near zero. In conclusion, ADRC shows potential for controlling unknown parameters in hybrid electric vehicles.

1. MICHIGAN TECHNOLOGICAL UNIVERSITY
Active Disturbance Rejection Control
Final Report
Amol Galande
4/21/2014

2. 1
Contents
Introduction ..................................................................................................................................................2
PID & ADRC ...............................................................................................................................................2
Application of ADRC:.....................................................................................................................................9
Toyota Hybrid Synergy Drive: ...................................................................................................................9
Simulation and Results:...............................................................................................................................10
Conclusion:..................................................................................................................................................12
References: .................................................................................................................................................13
Glossary:......................................................................................................................................................13

3. 2
Introduction
Active disturbance rejection control was first described by Dr Jingqing Han and was motivated
by undesirable transient response features of PID control. The Active disturbance rejection
control is a control system that estimates the disturbance entering into the system and filters it
out from the output. This feature of the ADRC can be attributed to the non-linear feedback
incorporated in it that allows the plant output to reach steady state in a finite amount of time
as compared to the conventional PID controller. The PID controller although one of the most
dominant types of controller that is currently being used in the industrial scenario, has been
unable to cope with the increasing speed, accuracy and efficiency demands. The main aspect
where the ADRC is different from the PID controller is that it is error driven rather than being
plant or model based. Another important feature, the speed of control input or the rate of
change of control input to the plant model can be varied based on the physical limitations of
the physical components of system. This makes the ADRC compatible with a host of plant
model requiring only the speed of the controller to be adjusted. Its adaptability and flexibility
and the fact that it is model independent could make it viable for mass production and mass
installation. Although the ADRC is a very practical and convenient product it hasn’t been able to
break through the dominance of PID controllers in the industry. This could be ascribed to the
fact that its principle and its application hasn’t been understood well and this paper attempts to
apply the concept ADRC in a Toyota Hybrid Synergy Drive control system.
PID & ADRC
The control equation for the PID controller is based on the error signal .i.e. the difference
between the reference input and the plant output. The basic control law can be given as:
Here the coefficients of the proportional, integral and derivative control are adjusted based
on the plant model, the more precise the model the better the plant output for a given
control input and the corresponding error will be smaller. But the problem with this control
law is the magnitude of control output from the controller is based on the magnitude of the
error. As the plant output gets closer to the steady state the control magnitude reduces and
the plant approaches steady state at infinite time. The PID control does not handle high
frequency changes in the error signal as the derivative control amplifies the disturbance and
makes the system unstable, this could also lead into failure of physical components of the
system due to rapid increase in control input i.e. supplied voltage. For this reason the D is
neglected in some cases when using a PID controller.
(1)

4. 3
The reputation of PID controller is that it can be mass produced and has simple control law
which is easy to tune based on the model of the plant, but its flaws are being evident with
the increasing demands from this controller. The development of ADRC is a step by step
improvement over the flaws of PID controller. In the plot 1 below step input is used with a
PID controller, the problem here is when the input signal jumps from a positive to a
negative value the controller in an effort to reduce the error changes the control input
abruptly which can affect the physical components of the system.
Plot 2: PID controller output with a Transient profile generator
The solution to this using a transient profile generator that smoothens the slope i.e. rate of
change of the reference input provided to the system based on the physical limitations. Thus
just by using a TPG the abrupt change in the control input is avoided. As mentioned above the
magnitude of the control output of the PID controller is based on the magnitude of the error
signal thus the plant output reaches steady state at infinite time. The remedy to this problem is
using Non-linear feedback functions of the form ‘fal’ and ‘fhan’ as controllers for a given plant.
2 4 6 8 10 12 14 16 18 20
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time (sec)
magnitudeofinputsignal
reference input
plant output
2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (sec)
Magnitudeofinputsignal
input signal
output signal
Plot 1: PID closed loop control system output

5. 4
Here the e is the error signal, δ is the set threshold defined by the user and α helps regulate the
speed of the control output. As seen from the above equation the controller regulates the
output not only based on the error signal but also a threshold value set for the state variable
being controlled. When α is set as 1 the controller becomes linear and is only dependent on the
error signal but when it is changed to 0 it becomes a bang-bang controller that takes the plant
to steady state instantaneously. Thus for values between 1 and 0 the plant output reaches
steady state within finite time.
plot 3: PID and NFC response to a step input.
Plot 3 displays the plant response using a PID and a NFC controller. The NFC controller get the
plant output to steady state at time t=5.8 sec when α=0.1, while the PID controller has a steady
state error of 0.005 even at the time t=10sec. The NFC plant output has a steeper slope and
reaches the final value 1 faster than the PID controller. The controller response can be speed up
by changing the value of α. At α=0.05 the plant reaches steady state at t=4.5 sec.
The feature that makes ADRC interesting is the extended state observer. The ESO makes the
ADRC model independent i.e. a model with unknown states and disturbances can be controlled.
This is possible because the unknown states and disturbances are set as state variables by the
ESO, are observed, estimated and feedback to the controller. Thus the output of the plant can
be manipulated via the controller even with an imprecise system model.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time (sec)
magnitudeofstepinput
PID response
Step input
NFC response a=0.1
NFC response a=0.05
U= (2)

6. 5
Considering a double integrator plant model with unknown plant states and disturbance, it can
be represented as:
Here the represents the unknown plant variable and disturbance, u is
the control input and y is the plant output. In an ESO the unknown function is set as a state
variable (x3) that can be observed, thus enabling the output to be controlled with a rough or
imprecise model.
The observer uses forward Euler’s method to estimate the state variables. The differential
equations for the extended state observer can be described as:
Here ‘e’ represents the error between the estimate of y and the plant output y and β01, β02 and
β03 are observer gains. For simplicity the observer gains can be made linear, considering h as
the sampling period the gain can be taken as:
1
The extended state observer estimates the disturbance and rejects it while the plant states are
feedback to the controller.
1
Jing Qing Han, "From PID to Active Disturbance Rejection Control," Industrial Electronics, IEEE Transactions on , vol.56, no.3,
pp.900,906, March 2009 [equation (1)-(6)]
(3)
(5)
(4)
(6)

7. 6
Figure 1: Simulink model for extended state model
In the above model all the three elements have been incorporated i.e. the transient profile
generator, the non-linear feedback combination and the extended state observer which serves
as an alternative to the PID controller. The plant states are x1, x2 and x3 while the estimated
states are z1, z2, z3.
Plot 4: plant state x1 and estimated state z1
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
time (secs)
plantstatex1andestimatez1
)plant state x1
estimated state z1

8. 7
Plot 5: error between plant state x1 and estimate z1
In plot 5 the error signal is limited to 0.2 and does not go beyond that value, thus the estimator works
well for plant state x1.
Plot 6: Plant state x2 and estimated state z2
Plot 7: error plant state x2 and estimate z2
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
time (secs)
errorbetweenplantstatex1&estimatez1
0 1 2 3 4 5 6 7 8 9 10
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time (secs)
plantstatex2andestimatez2
plant state x2
estimated state z2
0 1 2 3 4 5 6 7 8 9 10
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
time (secs)
errorbetweenplantstatex2&estimatez2

9. 8
For the estimate for plant state x2 and plant state x3 i.e. z2 and z3 respectively a lag is introduced in the
estimate for these two states. This can be attributed to the integrators being used to calculate the
estimates in the extended state observer model. Although the lag is introduced the estimate error is
restricted between the ranges -0.4 to 0.4.
Plot 8: Plant state x3 and Estimated state z3
Plot 9: error plant state x3 and estimated state z3
It can be seen that the plant states are tracked well by the extended state observer, plus as the
transient profile generator and the non-linear feedback is being used as the controller the
speed of the control output can be controlled according to the system requirements. Similarly
the plant states x2 in figure 9 and plant state x3 are estimated fairly well by the extended state
observer.
0 1 2 3 4 5 6 7 8 9 10
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
time (secs)
plantstatex3andestimatez3
plant state x3
estimated state z3
0 1 2 3 4 5 6 7 8 9 10
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time (secs)
errorbetweenplantstatex3&estimatez3

10. 9
Application of ADRC:
Toyota Hybrid Synergy Drive:
Toyota Prius was launched in US in the year 2000 and since then has been one of the top selling
models in the hybrid class. The main reason for success of Prius is the ingenious powertrain
design. Most hybrids either have a series or parallel configuration to deliver power to the
vehicle drive, but the Prius has both series and parallel configuration allowing it to take
advantage of both the configurations. Toyota hybrid synergy drive (THS) consists of a planetary
gear system consisting of the sun gear, carrier gear and the pinion gear connecting with the
generator, the engine and the motor. The motor is also connected to the vehicle differential
directly enabling it to recapture energy that is lost during breaking. This energy is fed back to
the battery and can be used to assist the engine during high torque demands or can be used in
electric mode during start stop situation in the city. It helps improve the city and highway fuel
economy of Prius.
Figure 2: Schematic for Toyota Power Train Configuration
2
In figure 1 the Engine can directly transfer torque to the reduction gears via the carrier gear,
while the motor can directly transfer torque to the reduction gears and the wheels this forms
the parallel mode of power transfer. In the series mode the engine power is supplied to the
generator connected to the sun gear, the generator can either supply it to the battery for use
later or directly to the motor. The series and parallel mode of power transfer can take place
simultaneously because of the THS.
2
http://www.ae.pwr.wroc.pl/filez/20110606092430_HEV_Toyota.pdf

11. 10
Simulation and Results:
The control system for a hybrid is linear and rule based since there are multiple parameters
depending on each other, this requires precise modeling of the various systems incorporated in
the hybrid power train. Implementing the ADRC in a hybrid electric, with its ability to estimate
unknown parameter can have potential benefits. This paper is an attempt to check the
feasibility of using a non-linear controller in Hybrid.
In the THS the engine, motor and the generator are connected to carrier, ring and the sun gear
via mechanical gears thus it is important to develop a dynamic equation connecting the
torques, inertia and rotational speed of these components. The friction and viscous forces are
neglected in the model. The final dynamic model consists of three differential equation for
engine speed (ωe), ring gear speed (ωr) and state of charge (SOC). Ideal assumptions have been
made in the derivation of the differential equations.
The ring gear and the engine relate to the vehicle speed as they are directly connected to the
vehicle differential. In the above equations
The non-linear differential equation for SOC is represented as:
(7)
(8)

12. 11
3
Thus the equation (7)-(9) form the governing equations for the Toyota Hybrid Synergy Drive
powertrain system. Precise calculation of state of charge is required in a hybrid since the
actuation of the motor and generator depends on it but precise model of the vehicle battery
system is difficult and hence it presents an opportunity to implement the ADRC. The equation
(9) is a non-linear equation and is selected as the state that will be observed by the extended
state observer.
Figure 3: Closed loop THS control system.
In the above model the state variables ωe and ωr are not observed by the extended state
observer but are controlled by the non-linear feedback controller. The blocks E, G and M are
the engine, generator and the motor torque gains. The error signal to the controller is the
difference between the reference signal and the state estimates from the extended state
observer.
3
Jinming Liu; Huei Peng; Filipi, Z., "Modeling and Control Analysis of Toyota Hybrid System," Advanced Intelligent Mechatronics.
Proceedings, 2005 IEEE/ASME International Conference on , vol., no., pp.134,139, 2005 [equations (7)-(9)]
Te
Tg
Tm
z1
y
z3d
e
fcn
Z3 dot
u
z3
e
z2d fcn
Z2 dot
z1
z2
y
z1d fcn
Z1 dot
Te
Tg
Tm
wr
wedot
wrdot
fcn
Wrdot & Wedot
simout
To Workspace
Engine Rpm required
Ring gear RPM required
SOC
Rate of Soc Change
Signal Builder
Scope
Tm
Tg
wr
SOCfcn
SOC
1
Ms
1
s
Integrator4
1
s
Integrator3
1
s
Integrator2
1
s
Integrator1
1
s
Integrator
1
Gs
0.2
Es
ee
we
er
wr
e1
z1
e2
z2
ue
ug
um
fcn
Controler
(9)
Controller
Extended State
observer
THS modelE
G
M

13. 12
Plot 8: SOC and Estimate of SOC using ESO
Plot 9: Error signal of SOC estimate
Plot 8 and 9 are related to the performance of the extended state observer. In plot 9 the error
signal is initially high but steadily decreases with increment in time and stabilizes to after
25seconds. This is because initial the estimate will be a default value input into the controller,
based on the error the controller springs into action reducing the error.
Conclusion:
The ADRC is a brilliant solution for the current needs of a high performance controller that can
overcome the modeling errors of the plant model. It not only inherits the advantages of a PID
control but also overcomes the issues related to a PID controller. The fact that is allows
0 2 4 6 8 10 12 14 16 18 20
-0.05
0
0.05
0.1
0.15
0.2
Time in (sec)
StateofCharge
SOC
Estimated SOC
0 5 10 15 20 25 30
0.5
0.55
0.6
0.65
0.7
0.75
time (sec)
errorinSOCestimation

14. 13
uncertainties and imperfection in the plant model is one most important features of the ADRC.
This makes the ADRC plant independent i.e. one controller can be used for any plant model
with minor gain tuning. The non-linear feedback combination is quick in tracking the reference
inputs and can manipulate speed and the smoothness of the control output based on the
physical limitations which was not possible using the PID controller.
References:
1. https://techinfo.toyota.com/techInfoPortal/staticcontent/en/techinfo/html/prelogin/docs/prius
phvdisman.pdf
2. http://www.ae.pwr.wroc.pl/filez/20110606092430_HEV_Toyota.pdf
3. www.toyota.com
4. https://www1.eere.energy.gov/vehiclesandfuels/avta/pdfs/hev/batterygenIIIprius0462.pdf
5. http://web.ornl.gov/~webworks/cppr/y2001/rpt/121813.pdf
Glossary:
Te – Engine Torque
Tm – Motor Torque
Tg – Generator Torque
Ie – Engine Inertia
Im – Motor Inertia
Ig – Generator Inertia
ωe - Engine RPM
ωr – Ring gear RPM
ωm – Motor RPM
ωg – Generator RPM
m – Mass of the tire
fr – Rolling friction
Cd – Air Drag coefficient
A – Vehicle frontal area
K – Final drive ratio
ρ – Air density
All units are in MKS.