Date post: | 13-Apr-2017 |

Category: |
## Documents |

Upload: | amol-galande |

View: | 97 times |

Download: | 7 times |

Share this document with a friend

Embed Size (px)

of 14
/14

Transcript

MICHIGAN TECHNOLOGICAL UNIVERSITY

Active Disturbance Rejection Control

Final Report

Amol Galande

4/21/2014

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

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)

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)

ma

gn

itud

e o

f in

pu

t sig

na

l

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)

Ma

gn

itud

e o

f in

pu

t sig

na

l

input signal

output signal

Plot 1: PID closed loop control system output

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 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

time (sec)

magnitu

de o

f st

ep in

put

PID response

Step input

NFC response a=0.1

NFC response a=0.05

U= (2)

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)

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 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

time (secs)

pla

nt sta

te x

1 a

nd e

stim

ate

z1

)plant state x1

estimated state z1

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)

err

or

be

twe

en

pla

nt sta

te x

1 &

estim

ate

z1

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)

plan

t sta

te x

2 an

d es

timat

e z2

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)

err

or

be

twe

en

pla

nt sta

te x

2 &

estim

ate

z2

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)

pla

nt sta

te x

3 a

nd e

stim

ate

z3

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)

err

or

be

twe

en

pla

nt sta

te x

3 &

estim

ate

z3

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 Configuration2

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

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)

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 model E

G

M

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)

Sta

te o

f C

ha

rge

SOC

Estimated SOC

0 5 10 15 20 25 300.5

0.55

0.6

0.65

0.7

0.75

time (sec)

err

or

in S

OC

estim

atio

n

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.

Recommended