Home >Documents >Advanced Digital Controls - Digital Control MBC500 Luong.pdf · PDF fileAdvanced Digital...

Advanced Digital Controls - Digital Control MBC500 Luong.pdf · PDF fileAdvanced Digital...

Date post:29-Aug-2018
Category:
View:215 times
Download:0 times
Share this document with a friend
Transcript:
  • Advanced Digital Controls

    University of California, Los Angeles

    Department of Mechanical and Aerospace Engineering

    Report Author David Luong Winter 2008

  • Luong

    Professor T.C. Tsao Page 2 3/21/2008

    Table of Contents

    Abstract4 Introduction..5 Physical Plant Description System Connection..6 System Identification

    Analytical Modeling7 State Space Representation....11 Transfer Function Form.11 Plant Decoupling12 Experimental Modeling using Digital Signal Analyzer.14 Frequency Characterization...15 System Isolation.15 System Decoupling16 Curve Fitting..17 Analytical and Experimental Comparison.18 Controller Designs.20

    Methodology..20 Internal Model Principle21 Robust Stability Analysis Framework...22 Selection of Sampling Time..26

    Direct and Indirect Lead-Lag Controller...27 Design

    Continuous-time.27 Discrete-time..32 Sensitivity and Complementary Sensitivity Analysis34 Robust Stability Analysis...35 Simulation..36 Implementation..39 Sinusoidal Reference Tracking..41

    State Estimation Feedback.42 Formulation42 Design47

    Sensitivity and Complementary Sensitivity Analysis49 Robust Stability Analysis...50 Simulation..51 Implementation..53 Summary54

  • Luong

    Professor T.C. Tsao Page 3 3/21/2008

    Pole Placement and Model Matching (RST) Design.55 Formulation55 Design57 Sensitivity and Complementary Sensitivity Analysis59 Robust Stability Analysis...61 Simulation..63 Implementation..66 Summary68 2H and H Norm69 Formulation69 Design71 Sensitivity and Complementary Sensitivity Analysis72 Robust Stability Analysis...73 Simulation..74 Implementation..75 Summary76 Zero-Phase Feed-forward Error Tracking..77 Formulation77 Design79 Sensitivity and Complementary Sensitivity Analysis79 Robust Stability Analysis...79 Simulation..80 Implementation..82 Summary84 Repetitive Control..85 Formulation85 Design88 Robust Stability Analysis...89 Simulation..90 Implementation..92

    Summary94 Appendix95 MATLAB m-files..95 system_id...96 lead_lag_design104 state_feedback_observer_full..110 state_feedback_observer_integrator_full.112 modelmatch..115 dioph120 RST..121 Youla_example1..122 H2ModelMatching...125 designZeroPhase..126 zeroPhase.128 designRepControl129 Augmented State Observer Feedback Loop Gain Derivation..132

  • Luong

    Professor T.C. Tsao Page 4 3/21/2008

    ABSTRACT

    Figure 1: Magnetic Bearing MBC 500 This report explores various digital controller designs on a magnetic bearing system. Models of the translational and rotational dynamics in the y-direction are obtained analytically and validated against experimental frequency response data. The report illustrates the theory, design, and implementation of several digital controllers with considerations given to stability, robustness, performance, and reference tracking of step and periodic external signals. The framework of this report starts with an understanding of the magnetic bearing system from a dynamics point of view. With a model of the system in hand, the controllers were motivated and theorized, designed in MATLAB and Simulink environments, and implemented on a xPC setup connected to the MBC500. From considerations beginning with the classical lead-lag compensator to more modern control designs in repetitive control, the reader should note the improvements, as well as the tradeoffs, as the methodologies progress. The MATLAB m-files used to conduct analysis and simulations are included in the Appendix. A description of their function appears on the first page in that section.

  • Luong

    Professor T.C. Tsao Page 5 3/21/2008

    INTRODUCTION Physical Plant The magnetic bearing is a shaft suspended by 4 electromagnetic actuators, two on each end. The actuators are oriented in the horizontal x and vertical directions y. There are four Hall Effect sensors placed in a similar manner. The manufacturer provides an optional analog controller programmed to stabilize the plant. In identifying the plant, these controllers are turned on by loop switches on the left. Twelve user defined signals exist on the MBC500. For each electromagnet and Hall Effect sensor combination are three signals: input reference r, control voltage u, and position voltage y. Figure X highlights the I/O configuration.

    Figure 1: Inputs and Outputs to the Magnetic Bearing

  • Luong

    Professor T.C. Tsao Page 6 3/21/2008

    System Connection The connection layouts of the coupled and decoupled magnetic bearing system are given in Figures 2 and 3, respectively.

    Figure 2: Bearing System seen by Controller Cx1 x and y directions coupled

    Figure 3: Bearing System seen by Controller Cx1 x and y directions decoupled

  • Luong

    Professor T.C. Tsao Page 7 3/21/2008

    ANALYTICAL MODELING The modeling of the magnetic bearing is performed separately on the plant and the controller. The signal flow for the plant is D/A Voltage Amplifier Current Electromagnets Force Mechanical Dynamics Motion Sensor Voltage A/D

    And the signal flow for the controller is

    A/D Control D/A A mathematical description of the system is next needed to determine its dynamics. The forces and measurements are shown in Figures 1 and 2.

    Figure 1: MBC500 System Configuration

    Figure 2: Rotor Configuration

    The rigid body dynamics are investigated assuming the shaft does not rotate. This allows for the decoupling of the x and y system states, and their individual input/output descriptions. The table below describes the symbols used and their descriptions. Note that the analysis is exactly the same for the y-direction dynamics.

  • Luong

    Professor T.C. Tsao Page 8 3/21/2008

    Symbol Description

    0x The horizontal displacement of the center of the rotors mass. 1 2,x x The horizontal displacements of the rotor at the left and right bearing positions, respectively. 1 2,X X The horizontal displacements of the rotor at left and right Hall Effect sensor positions, respectively. The angles that the long axis of the rotor makes with

    the z-axis.

    1 2,F F The forces exerted on the rotor by left and right bearings, respectively.

    Table 1: Modeling Definitions

    The equations of motions governing the magnetic bearing system are given as

    0 1 2

    0 2 12 2

    1 0 2

    2 0 2

    1 0 22

    2 0 22

    ( ) cos ( ) cos

    ( )sin( )sin( )sin( )sin

    L L

    L

    L

    L

    L

    F mx F F

    M I F l d F l d

    x x l dx x l dX x l dX x l d

    = = +

    = = +

    =

    = + +

    =

    = + +

    The set of equations can be compactly represented in state-space form. Applying the small angle approximationssin , cos 1 to linearize the system, we get

    0 0

    0 01 1

    10 0

    21 1

    2 2

    0

    21 02

    22 2

    0 00 1 0 00 0 0 0

    0 00 0 0 1( ) ( )0 0 0 0

    1 0 ( ) 01 0 ( ) 0

    m m

    L LI I

    L

    L

    x xFx xF

    l d l d

    xl dX x

    l dX

    = + +

    = +

    The equation for the magnetic forces can be described by

    2 2

    2 2

    ( 0.5) ( 0.5)( 0.0004) ( 0.0004)

    i icontrol controli

    i i

    i iF k k

    x x+

    = +

    Since we assume small displacements in these forces, we can linearize the above equation about its equilibrium point ( , ) (0,0)

    ii controlx i = . The Taylor series approximation at this

    point is

  • Luong

    Professor T.C. Tsao Page 9 3/21/2008

    _ __

    ( , ) (0,0) (0,0) ( 0) (0,0) ( 0)i ii i c i i i c ii c i

    F FF x i F x ix i

    = + +

    with 2 2

    _ _3 3

    (0,0)

    _ _2 2

    _ _ (0,0)

    2( 0.5) 2( 0.5)4375

    ( 0.0004) ( 0.0004)

    2( 0.5) 2( 0.5)3.5

    ( 0.0004) ( 0.0004)

    c i c ii i

    i i i i

    c i c ii i

    c i i i c i

    i iF Fkx x x x

    i iF Fki x x i

    + = = +

    + = = +

    Thus, the linearized magnetic force is

    _4375 3.5i i c iF x i= + With the small angle approximation of

    1 0 02 2

    2 0 02 2

    ( )sin ( )( )sin ( )

    L L

    L L

    x x l d x l dx x l d x l d

    =

    = + + +

    the solution of the magnetic forces is

    1 0 _2

    2 0 _2

    4375 4375( ) 3.5

    4375 4375( ) 3.5

    Lc i

    Lc i

    F x l d i

    F x l d i

    = +

    = + + +

    The state-space representation can now be written as

    0 0

    0 0

    1 12

    21 1

    2 2

    1 1_1

    _ 21 1

    2 2

    0 00 1 0 04375 0 4375( ) 00 0 0 0

    0 0 4375 0 4375( ) 00 0 0 1( ) ( )0 0 0 0

    0 03.5 0

    0 0 0 3.5( ) ( )

    Lm m

    r r rL

    L LI I

    m m c

    cL L

    I I

    l dx x x

    l dl d l d

    ii

    l d l d

    = + + +

    + +

    2

    20 00

    8750 3.5 3.5_1

    _ 28750( ) 3.5 3.5

    2 2

    21 2

    22 2

    0 1 0 0 0 00 0 0

    0 0 0 1 0 0( ) ( )0 0 0

    1 0 ( ) 01 0 ( ) 0

    L

    m m m cr r

    cl d L L

    I II

    L

    rL

    ix x

    il d l d

    l dXx

    l dX

    +

    = + +

    = +

  • Luong

    Professor T.C. Tsao Page 10 3/21/2008

    Current Amplifier Dynamics The setup includes a dual-channel current amplifier that is governed by the following differential equation:

    4 4

    1 0.25( )2.2 10 2.2 10i i icontrol control control