active disturbance rejection control

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  • 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 hasnt 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 hasnt 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, is the set threshold defined by the user and 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 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 =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 . At =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 Eulers 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 01, 02 and

    03 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

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