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  • 8/17/2019 Journal on Lyapunov Stability Theory Based Control of Uncertain Dynamical Systems

    1/10

    Sl M J.

     ONTROL

     N OPTIMIZATION

    Vol.

    21, No. 2,

    March

    1983

    1983

    Society

    for Industrial and

    Applied Mathematics

    0363-0129/83/2102-0006

     01.25/0

     

    NEW CLASS

    OF

    STABILIZING

    CONTROLLERS

    FOR

    UN ERT IN

    DYN MI L

    SYSTEMS

    B. R.

    BARMISH?,

    M.

    CORLESS

     ND

    G.

    LEITMANN

    Abstract.

    This

    paper

    is

    concerned

    with the

    problem

    of

    designing

    a

    stabilizing controller

    for a class

    of

    uncertain

    dynamical

    systems. The vector of

    uncertain

    parameters

    q .) is

    time-varying,

    and

    its

    values

    q t

    li e

    within a

    prespecified bounding

    set

    Q

    in

    R

    p

    Furthermore,

    no statistical

    description

    of

    q .)

    is

    assumed,

    and

    the controller

    is

    shown to

    render

    the

    closed

    loop

    system

     practically

    stable

    in

    a

    so-called guaranteed

    sense;

    that

    is,

    the desired

    stability

    properties

    are

    assured

    no

    matter what

    admissible

    uncertainty q .)

    is

    realized. Within

    the

    perspective

    of

    previous research in

    this

    area,

    this

    paper contains

    one

    salient feature:

    the

    class of

    stabilizing

    controllers which

    we

    characterize

    is shown

    to

    include linear

    controllers when the

    nominal

    system happens

    to

    be linear and

    time-invariant. In

    contrast,

    in

    much

    of

    the

    previous

    literature

     see, for

    example,

    [1], [2], [7],

    and

    [9] ,

    a

    linear

    system

    is stabilized

    via

    nonlinear

    control. Another

    feature

    of

    this paper is

    the

    fact that

    the methods of

    analysis

    and

    design

    do

    not

    rely

    on

    transforming

    the

    system

    into

    a more

    convenient canonical

    form; e.g., see

    [3].

    It

    is also

    interesting

    to

    note that

    a

    linear

    stabilizing

    controller

    can

    sometimes be

    constructed even

    when

    the

    system

    dynamics

    are nonlinear.

    This

    is

    illustrated

    with

    an

    example.

    Key

    words,

    stability,

    uncertain

    dynamical systems,

    guaranteed

    performance

    1.

    Introduction.

    During

    recent

    years,

    a

    number

    of

    papers

    have

    appeared

    which

    deal

    with

    the

    design

    of

    stabilizing

    controllers for

    uncertain

    dynamical

    systems;

    e.g.,

    see

    [1]-[7].

    In

    these

    papers

    the

    uncertain

    quantities

    are

    described

    only

    in

    terms

    of

    bounds on their

    possible

    sizes;

    that

    is

    no

    statistical

    description

    is assumed.

    Within

    this

    framework,

    the

    objective

    is

    to find

    a

    class

    of

    controllers

    which

    guarantee

     stable

    operation

    for

    all

    possible

    variations

    of

    the uncertain

    quantities.

    Roughly speaking,

    the

    results to date

    fall into

    two

    categories.

    There are

    those

    results which

    might appropriately

    be termed

    structural in

    nature;

    e.g.,

    see

    [1]-[3],

    [6].

    By

    this

    we

    mean

    that

    the

    uncertainty

    cannot

    enter

    arbitrarily

    into

    the

    state

    equations;

    certain

    preconditions

    must

    be met

    regarding

    the locations of the

    uncertainty

    within

    the

    system

    description.

    Such conditions are often referred

    to

    as matching

    assumptions.

    We

    note

    that

    in

    this situation

    uncertainties

    can be

    tolerated

    with

    an

    arbitrarily

    large

    prescribed

    bound.

      second

    body

    of

    results

    might

    appropriately

    be

    termed

    nonstruc-

    rural

    in

    nature;

    e.g.,

    see

    [4]

    and

    [5].

    Instead

    of

    imposing matching assumptions

    on

    the

    system,

    these

    authors

    permit

    more

    general

    uncertainties at

    the

    expense

    of

     sufficient smallness

    assumptions

    on

    the allowable sizes

    of

    the

    uncertainties.

    This

    work falls within the class of structural

    results

    mentioned above.

    Our

    motivation comes

    from a

    simple

    observation.

    Namely, given

    a

    theory

    which

    yields

    stabilizing

    controllers for

    a

    class

    of

    uncertain

    nonlinear

    systems,

    it

    is

    often desirable

    for

    this

    theory

    to have the

    following

    property:

    upon

    specializing

    the

     recipe

    for

    controller

    construction

    from

    nonlinear

    to linear

    systems,

    one of the

    possible

    stabilizing

    control laws should

    be linear

    in

    form. It

    is

    of

    importance

    to note

    that

    existing

    results

    do

    not have this

    property.

    Upon

    specialization

    to

    the

    linear

    case,

    one

    typically

    obtains

    controllers

    of

    the

    discontinuous

     bang-bang variety; e.g.,

    see

    [1]

    and

    [2].

    One

    can

    often

    approximate

    these

    controllers

    using

    a

    so-called saturation

    nonlinearity; e.g.,

    see

    *

    Received

    by

    the

    editors

    March

    4,

    1981, and

    in

    revised

    form January

    15 ,

    1982.

     

    Department of Electrical

    Engineering,

    University

    of

    Rochester,

    Rochester,

    New

    York 14627.

    The

    work

    of

    this

    author

    was

    supported

    by

    the

    U.S. Department

    of

    Energy

    under

    contract no. ET-78-S-01-3390.

    Department

    of

    Mechanical

    Engineering,

    University

    of

    California

    at

    Berkeley,

    Berkeley, California

    94720.

    The

    work of

    these

    authors

    was

    supported

    by

    the National

    Science

    Foundation

    under

    grant

    ENG

    78-13931.

    246

      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

      I  A  M   l  i  c  e  n  s  e  o  r  c  o  p  y  r  i  g  h  t  ;  s  e  e

      h  t  t  p  :  /  /  w  w  w .  s  i  a  m .  o  r  g  /  j  o  u  r  n  a  l  s  /  o  j  s  a .  p  h  p

  • 8/17/2019 Journal on Lyapunov Stability Theory Based Control of Uncertain Dynamical Systems

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    STABILIZING CONTROLLERS FOR

    UNCERTAIN

    DYNAMICAL

    SYSTEMS

    247

    [7].

    Such

    an

    approach

    leads to

    uniform

    ultimate boundedness of the

    state

    to

    an

    arbitrarily

    small

    neighborhood

    of

    the

    origin;

    this

    type

    of behavior

    might

    be termed

    practical

    stability.

    Our

    desire in

    this

    paper

    is

    to

    develop

    a controller

    which is linear when

    the

    system

    dynamics

    are

    linear.

    By

    taking

    known

    results

     such

    as

    in

    [3])

    which were

    developed

    exclusively

    for

    linear

    systems,

    one

    encounters a

    fundamental

    difficulty

    when

    attempting

    to

    generalize

    2

    to a

    class

    of

    nonlinear

    systems; namely,

    it is

    no

    longer possible

    to

    transform

    the

    system dynamics

    to

    a

    more convenient

    canonical form. The

    subsequent

    analysis

    is free of

    such

    transformations.

    2.

    Systems,

    assumptions

    and

    the

    concept

    of

    practical stability.

    We

    consider

    an

    uncertain

    dynamical

    system

    described

    by

    the

    state

    equation

    2 t)

    =f x t),

    t)+

    Af x

     t ,

    q t),

    t

    +[B x t),

    t)+ AB x t),

    q t),

    t)]u t),

    where

    x(t)R

    is the

    state,

    u(t)R

    is

    the

    control,

    q(t)R

    p

    is the

    uncertainty

    and

    f x,

    t ,

    Af x,

    q,

    t ,

    B

     x,

    t

    and

    AB

     x,

    q,

    t

    are

    matrices

    of

    appropriate

    dimensions

    which

    depend

    on

    the

    structure

    of the

    system.

    Furthermore,

    it

    is assumed that the

    uncertainty,

    q .):R-->R

    p

    is

    Lebesgue

    measurable and its values q t)

    lie within

    a

    prespecified

    bounding set

    Q

    cR for all R We denote this

    by writing

    q(.)M(Q).

    As mentioned

    in the

    introduction,

    given

    that

     stabilization is

    the

    goal,

    we

    must

    impose

    additional

    conditions

    on the

    manner in which

    q t)

    enters

    structurally

    into the

    state

    equations.

    We

    refer

    to

    such

    conditions

    as

    matching

    assumptions.

    Assumption

    1.

    There

    are

    mappings

    h(.):R xRPxRR

    and

    E( ):R xRPxR-R

     

    such

    that

    Af x,

    q,

    t B  x,

    t)h

     x, q,

    t ,

    AB x,

    q,

    t)=

    B x, t)E x, q,

    t ,

    liE

     x, q,

    t ll

    <

    1

    for

    all

    x

    R

     

    q

    O

    and R.

    We

    note that this

    assumption

    can

    sometimes be

    weakened. For

    example,

    in

    [9]

    a

    certain

    measure

    of

    mis-match is introduced and

    results

    are

    obtained

    under

    the

    proviso

    that

    this

    measure

    does

    not

    exceed

    a

    certain

    critical

    level

    termed

    the

    mis-match

    threshold.

    Our

    second

    assumption

    reflects

    the

    fact

    that the uncertainties

    must

    be

    bounded

    in

    order to

    permit

    one

    to

    guarantee

    stability.

    Assumption

    2.

    The set

    O

    c

    R

    is

    compact.

    Our next

    assumption

    is introduced

    to

    guarantee

    the

    existence of solutions

    of

    the

    state

    equations.

    Assumption

    3.

    The

    mappings

    f .

     :

    R

    x

    R

    R

    n

    B

      ):

    R

    x

    R R

    mxn,

    h

     .)

    and

    E .)  see Assumption 1)

    are continuous.

    3

    This

    notion

    is

    no t

    to

    be interpreted

    in the

    sense

    of

    Lasalle

    and

    Lefschetz

    [12]

    but

    as defined

    subsequently.

    That

    is ,

    one

    begins

    with

    a

    linear

    control

    law for

    a

    linear

    system

    and

    generalizes

    the

    controller in

    such

    a

    way

    that

    is

    applies

    to a

    class

    of nonlinear

    systems.

    In fact,

    one

    can

    modify

    the

    analysis

    to

    follow

    so

    as

    to allow

    mappings which

    are

    Carath6odory

    and

    satisfy

    certain

    integrability

    conditions.

    See,

    for

    example,

    Corless and

    Leitmann

    [7].

    All the

    results

    of

    this

    paper

    still

    hold

    under

    this

    weakening

    of

    hypotheses.

      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

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    248

    B.

    R.

    BARMISH

    M.

    CORLESS

    AND G.

    LEITMANN

    In

    order

    to

    satisfy

    our

    final

    assumption,

    one

    may

    need

    to

     precompensate

    the

    so-called

    nominal

    system,

    that

    is ,

    the

    system

    with

    Af x,

    q,

    t)=--O

    and

    AB(x,

    q,

    t)----0;

    e.g.,

    see

    [2].

    Thus,

    prior

    to

    controlling

    the

    effects

    of

    the

    uncertainty,

    it

    may

    be

    necessary

    to

    employ

    a

    portion

    of

    the control

    to

    obtain

    an

    uncontrolled

    nominal

    system

    (UC)

    (t)=f(x(t),t)

    that has

    certain

    stability

    properties

    embodied in

    the

    next

    assumption.

    Assumption

    4.

    f 0,

    t)-

    0

    for

    all

    R

    and,

    moreover,

    there

    exist a C function

    V(.):R

    xR

    [0, o0)

    and

    strictly

    increasing

    continuous

    functions

    y(.),

    y2 ),

    y3 ):

    [0,

    00)

     

    [0, 00) satisfying

    4

    yl(0)

    y2(0)--y3(0)

    0 and limr_.

    yl r)

    limr_.

    TE r)

    lim_.oo

    ya r) o, such

    that for

    all

     x, t)

    e R

    R,

     2.2)

    v (llxll)

    Moreover,

    defining

    the

    Lyapunov

    derivative

    o( ):

    R

     

    R

     

    R

    by

     2.3)

    o(X,

    t)

    a

    0

    V(x,

    t)

    =+V xV(x,t)f(x,t),

    at

    where

    V

    denotes the

    transpose

    of the

    gradient operation,

    we also

    require

    that

    t)

     

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    STABILIZING

    CONTROLLERS

    FOR

    UNCERTAIN DYNAMICAL SYSTEMS 249

    (ii)

    Given

    any

    r>0

    and

    any

    solution

    x(.):[to, tl]-->R

    n,

    X to =Xo,

    of

    (2.4)

    with

    Ilxoll

    0

    such

    that

    IIx

     t l l

    R

    can

    be

    continued over

    [to,

    oo).

    (iv)

    Given

    any

    d

    >_-

    _d,

    any

    r >

    0 and

    any

    solution

    x

    (.)

    [to,

    oo)

    -- >

    R

     ,

    x

    (to)

    Xo,

    of

    (2.4)

    with

    Ilxoll--_d

    and

    any

    solution

    x( ): [to, oo)-->R , X(to)

    =Xo,

    of

    (2.4),

    there

    is

    a

    constant 6

     a >

    0

    such

    that

    Ilxoll-=

    to .

    3.

    Controller construction.

    We

    take

    _d

    >

    0 as

    given

    and

    proceed

    to

    construct

    a

    control

    law

    p

    _e(

    which

    will

    later

    be shown

    to

    satisfy

    conditions

    (i)-(v)

    in

    the

    definition

    of practical

    stabilizability.

    Construction

    of

    pa_( ).

    The first

    step

    is to

    select

    functions

    AI .

    and

    A2(.) R

    R

    -->

    R

    satisfying

    (3.1) A x,

    t)-->_max

    Ilh(x,

    q,

    t ll,

    (3.2)

    1

    > A).(x,

    t)

    >-max

    liE(x,

    q,

    t l[.

    q

    The

    standing Assumptions

    1-4

    assure that

    there

    is

    a

    A2(x,

    t)

    such that

    1)

    A2(x, t)

    [0,

    oo) satisfying

    a(x,t)

    (3.3)

     g(x,

    t)>-4[1_

    AE x t ][C2-Cl.o X,

    t ]

    where

     

    and

    C2

    are

    any

     designer chosen

    nonnegative

    constants such

    that

    a)

    C1<

    1;

    b)

    either

    C1

     

    0

    or

    C2

     

    0;

    (3.4)

    c)

    C2

     

    0

    whenever

    limx_,0

    [A(x,

    t /o X,

    t) ]

    does

    not

    exist;

     

    v

    v _d .

    )

    1

     C

    Note that these conditions can indeed be satisfied because

    of

    continuity

    of

    the

    3 .

    and

    the fact

    that lim_,o

    y(r)=

    0

    for

    1, 2,

    3.

    This

    construction

    then

    enables

    one to let

    (3.5)

    Pa(X,

    t)a---y(x,

    t)B (x, t VV x,

    t) .

    Remark.

    In fact,

    (3.3)

    and

    (3.5)

    describe

    a

    class

    of

    controllers

    yielding

    practical

    stability.

    It

    will

    be

    shown

    in

    5

    that

    this

    class

    includes

    linear

    controllers

    when

    the

    nominal

    system

    happens

    to

    be linear

    and

    time-invariant.

    4.

    Main

    result

    and

    stability

    estimates.

    The theorem

    below and its

    proof

    differ

    from

    existing

    results

    (see

    [1],

    [2]

    and

    [6])

    in

    one

    fundamental

    way:

    The

    control

    p_a( )

    which

    leads

    to the

    satisfaction

    of

    the conditions

    for

    practical

    stabilizability

    degenerates

      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

      I  A  M   l  i  c  e  n  s  e  o  r  c  o  p  y  r  i  g  h  t  ;  s  e  e

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  • 8/17/2019 Journal on Lyapunov Stability Theory Based Control of Uncertain Dynamical Systems

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    250 B.

    R.

    BARMISH,

    M.

    CORLESS

    AND

    G. LEITMANN

    into

    a linear

    controller whenever the

    nominal

    system,

    obtained

    by

    setting

    Af(x(t),

    q(t),

    t)=--O

    and

    AB(x(t),

    q(t),

    t)=-O

    in

    (2.1),

    is linear

    and time-invariant.

    This

    will be demonstrated

    in

    the

    sequel.

    In

    fact,

    even

    for

    certain

    nonlinear nominal

    systems,

    the

    controller

    turns out

    to

    be linear. This

    phenomenon

    will

    be

    illustrated

    with an

    example

    of a nonlinear

    pendulum.

    Central

    to

    the

    proof

    of

    the

    theorem below

    is

    one

    fundamental

    concept

    a

    system satisfying

    Assumptions

    1-4

    admits

    a

    control such

    that

    the

    Lyapunov

    function for the

    nominal

    system (UC)

    is also

    a

    Lyapunov

    function

    for

    the

    uncertain

    system (2.1).

    THEOREM 1.

    Subfect

    to

    Assumptions 1-4,

    the

    uncertain

    dynamical system

    (2.1)

    is

    practically

    stabilizable.

    Proof.

    For a

    given

    _d>0

    and

    a

    given

    uncertainty

    q . M Q ,

    the

    Lyapunov

    derivative

     .) Rn

    R

    R

    for

    the closed

    loop

    system

    obtained

    with the feedback

    control

    (3.5)

    is

    given by

     x,

    t)a_

    o X,

    t)+

    V’V(x,

    t){Af(x,

    q(t),

    t)

    (4.1)

    +[B

    (x, t)+

    aB

    (x,

    q(t),

    t)]pa_(x,

    t)}.

    By

    using

    the

    matching

    assumptions

    in

    conjunction

    with

    (3.5), (4.1)

    becomes

    q’(x,

    t)=

    f 0(x,

    t)-3/(x,

    t)llB x,

    t)VxV(x,

    t ll

    +

    V’V(x, t)B(x,

    t)[h(x,

    q(t),

    t)

    -y(x,

    t)E(x,

    q(t),

    t)B (x,

    t)VxV(x,

    t)].

    Letting

    (.)

    R R

    -- >

    R be

    given

    by

     (x,

    t)

    a---B x,

    t)VxV(x,

    t) ,

    and

    recalling

    the

    definition

    of

    AI(.

    and

    Az(. ),

    a

    straightforward computation

    yields

     x, t)_-< 0 x, t)-[1-A2(x,

    t)]y(x,

    t)ll

    (x,

    t l[

    =

    +

    +/-(x,

    t)ll

    (x,

    t [I

    Now

    there are

    two

    cases

    to

    consider.

    Case

    1.

    The

    pair

    (x,

    t)

    is such

    that

    Al(x, t)

    0.

    It

    then

    follows

    from

    the

    preceding

    inequality

    that

     x,

    t)

    0.

    Moreover, in

    view

    of

    (3.3)

    and

    the conditions

    on

    the

    Ci ,

    o (x, t)_

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    STABILIZING

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    251

    Combining

    Cases

    1 and

    2,

    and

    noting

    that

    C1

    < 1,

    we conclude

    (as

    a

    consequence

    of

    Assumption 4)

    that

    (4.2)

     x, t) _-_d,

    using

    the

    estimates

    provided

    in

    [7],

    one can

    define

    0

    if r

    _-<

    (yl

    yl)(a),

    (4.3)

    T(d-,

    r)

    A

    y2(r)

    Z(’]/_l

    yl

     y1)(d)

    otherwise,

    (1

    C1)( y3

     -

     

    1) =

    C2

    and

    in accordance with

    [7],

    the

    desired

    uniform

    ultimate

    boundedness

    condition

    (iv)

    holds with

    the

    proviso

    that

    (4.4)

    (1

    C1)(y3

    yl

    yl)(a)-

    C2

    >0.

    Note

    that

    this

    requirement

    is

    implied

    by

    the satisfaction

    of

    condition

    (d)

    of

    (3.4)

    which

    entered

    into

    the

    construction

    of

    the

    controller.

    Finally,

    to

    complete

    the

    proof,

    it

    remains to

    establish the

    desired

    uniform

    stability

    property. Indeed,

    le t

    d->_d

    be

    specified

    and

    notice that if

    6(d)=

    R,

    the

    following

    property

    will

    hold

    Given

    any

    solution

    x( ) [t0,

    )R ,

    X(to)=Xo

    of

    (2.4)

    with

    tlx011=

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    252

    B.

    R. BARMISH

    M.

    CORLESS

    AND

    G.

    LEITMANN

    where D

    (.),

    E

    (.)

    and

    v

    (.)

    have

    appropriate

    dimensions

    and

    depend continuously

    on

    their

    arguments.

    In

    accordance with

    Assumption

    4,

    the

    matrix

    A

    must

    be

    asymptoti-

    cally

    stable.

    To

    obtain

    a

    Lyapunov

    function

    for the

    uncontrolled

    nominal

    system,

    we

    select

    simply

    an

    n

    x

    n positive-definite

    symmetric

    matrix

    H

    and solve

    the

    equation

    (5.3)

    A P

    +

    PA  H

    for P which

    is

    positive-definite;

    see

    [11].

    Then

    we have

    (5.4)

    V(x, t) x’Px

    and

    (5.5) 0 x, t)=-x’Hx.

    It is clear from

    (5.4)

    and

    (5.5)

    that one

    can

    take

    the

    bounding

    functions

    yi(’)

    to

    be

    (5.6)

    yl(r) h

    min[e]r

    2,

     y2

    (r)

    -

    h

    max[.P]r

    2,

     y3

    (r)

    h

    min[H]r

    2,

    where

    h

    max(min)[

    denotes

    the

    operation

    of

    taking

    the largest

    (smallest)

    eigenvalue.

    Construction

    of

    the

    controller. We

    take

    _d

    >0 as

    prescribed

    and construct

    the

    controller

    p_d( given

    in

    3.

    Using

    the

    notation

    above,

    we

    define first

    6

    (5.7)

    po

    a--maxl[O(q)[I,

    pz

    a-maxl[E(q)ll-

    4(1

    -pE)Chmin[H]

    Case

    2.

    po

    O, p

    >

    O.

    Clearly,

    it suffices

    to

    take

    C

     0 and

    (5.12)

    y(x,

    t)_=

    y0

     

    02

    4(1

    p)C2

    where

    C2

    is

    required

    to

    satisfy

    condition

    d)

    of

    (3.4). Using

    the

    descriptions

    of the

    yi(.

    given

    in

    (5.6),

    this

    amounts

    to

    restricting

    C2

    by

    C2

    X

    min[e]

    <

    Amin[H]_d

    2

    (5.13)

    1

    C1

    Amax[P]

    with

    C1

    0

    in

    the

    above.

    6

    One

    ca n

    in

    fact

    us e

    overestimates

    to

    and

    tSE

    for

    Po

    and

    pE

    as long as

    the

    inequality

    tSz

    < is

    satisfied.

      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

      I  A  M   l  i  c  e  n  s  e  o  r  c  o  p  y  r  i  g  h  t  ;  s  e  e

      h  t  t  p  :  /  /  w  w  w .  s  i  a  m .  o  r  g  /  j  o  u  r  n  a  l  s  /  o  j  s  a .  p  h  p

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    STABILIZING

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    FOR UNCERT IN

    DYN MIC L

    SYSTEMS 253

    Case 3.

    Oo

    >0,

    p

    >0.

    Now,

    in order to

    satisfy

    (5.9),

    we select

    Ca

    (0,

    1),

    C2

    satisfying

    (5.13)

    and

    (5.14) y(x,

    t

    =-

    yo

    >max

     

    (por

    +p)2

    }

     

    4(1-pE)[Clhmin[n]r

    2-t-

    C2]

    Letting

    f(r)

    denote the bracketed

    quantity

    in

    (5.14)

    above,

    a

    straightforward

    but

    lengthy

    differentiation

    yields

    (5.15)

    maxf(r)=

    1

     

    020

     P }

    _->0

    4(1-Oz)

    Clhmin[H]

    Hence,

    any

    3 o

    equal

    to or

    exceeding

    this

    maximum value

    will

    be

    appropriate

    in

    (5.14).

    6. Illustrative

    example.

    We

    consider

    now

    the

    simple

    pendulum

    which was

    analyzed in

    [7]. However,

    here

    it

    will

    be

    shown

    that

    the desired

    practical

    stability

    can

    actually

    be

    achieved

    via a

    linear

    control. This

    may

    seem

    somewhat

    surprising

    in

    light

    of

    the

    fact

    that

    the

    nominal

    system

    dynamics

    are nonlinear.

     

    pendulum

    of

    length

    is

    subjected

    to

    a

    control

    moment u

    (.) (per

    unit

    mass).

    The

    point

    of

    support

    is

    subject

    to an uncertain

    acceleration

    q(.),

    with

    [q t [

    0

    is

    a

    given

    constant. In

    order

    to

    satisfy

    the

    assumptions

    of

    2 one must

    assure

    a

    uniformly

    asymptotically

    stable

    equilibrium

    for

    (UC),

    the

    uncontrolled

    nominal

    system.

    Hence,

    for

    a

    given _d

    >

    0,

    we

    propose

    a

    controller

    of the

    form

    (6.2)

    u(t)

    =-bx(t)-CXE(t)+pa_(x(t), t ,

    where b and

    c are

    positive

    constants

    and

    p_a(.)

    will

    be

    specified

    later

    in

    accordance

    with

    the

    results

    of 3. The

    linear

    portion

    of

    the controller

    (6.2)

    is

    used

    to

    obtain

    a

    stable nominal

    system.

    Substitution

    of

    (6.2)

    into

    (6.1)

    now

    yields

    the state

    equation

    2

     t

    f

    (x

     t , t

    +

    B [p

    (X  t ,

    t +

    h (x

     t ,

    q(t),

    t ],

    6.3)

    where

    (6.4)

    B

    1

    f(x,

    t

    -bx-cxE-a

    sinx

    h(x,

    q, t

    -q

    cos

    xl

     

    suitable Lyapunov function

    for the

    uncontrolled

    nominal

    system

    (with

    x

    0

    as

    equilibrium)

    is

    2\

    2

    (6.5)

    V x,t = b+gc

     x

    +cxxE+x

    +2a 1-cosx ,

    and, provided

    b is

    sufficiently

    large,

    the

    associated

    3 (.

    are

    given by

    3 (r)=Alr

    2,

    2r

    +

    4a if

    r >

    /3(r)

    X3r

    2

      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

      I  A  M   l  i  c  e  n  s  e  o  r  c  o  p  y  r  i  g  h  t  ;  s  e  e

      h  t  t  p  :  /  /  w  w  w .  s  i  a  m .  o  r  g  /  j  o  u  r  n  a  l  s  /  o  j  s  a .  p  h  p

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    254

    B.

    R. BARMISH,

    M.

    CORLESS

    AND

    G.

    LEITMANN

    A

    ,5

    Xmax[P],

     

    3

    min

    {/,

    c

    }

    6.7)

     6.8)

    P=[

    b+c2

    ]

     

    b+amin

    si

    x

    >0.

    C

    x

    Following

    the

    procedure

    described

    in

    3 for

    the

    construction of

    the

    controller

    p

     .),

    we select

    first

     6.9) l(X,t)=lcosxl,

    =(x, t

    0.

    Inequality  3.3)

    can then

    be

    assured by requiring

     

    2

    p

    COS

    X

     6.

    0)

    e x,

    t

    4[

    Given our desire

    for a linear

    feedback,

    one can select

    C

    0 and

    satisfy  6.10) by

    choosing

     

    6.11)

     x,

    t)v0>

    =4C

    To

    complete

    the

    design,

    C

    must be selected to

    satisfy

    condition

    d)

    of

     3.4).

    The

    analysis

    must

    account for two

    cases,

    depending

    on

    the

    size

    of the

    given

    radius

    >

    0.

    Case 1.

    a>

    h

    4a. The

    required

    conditions

    on

    C

    are

    _

    h3

    d

    6.12) 0

    0

    is

    chosen

    suNciently

    small

    so that

     6.13)

    h2C2

     a

    1-cos

    0. As the

    radius

    decreases,

    C

    decreases,

    which

    in

    turn implies that

    T0

    increases.

    In

    contrast,

    the

    nonlinear

    saturation

    controller

    of

    [7]

    remains

    bounded

    by

    the

    bound

    of the

    uncertainty,

    and

    the radius

    can

    be

    decreased

    by

    increasing

    the nonlinear

    gain; i.e.,

    by

    approaching

    a discontinuous

    control.

    .

    eels This

    paper

    addresses

    the so-called problem

    of

    practical

    stabiliza-

    bility

    for

    a

    class

    of uncertain

    dynamical

    systems.

    In

    contrast

    to

    previous

    work

    on

    problems

    of

    this

    sort,

    the

    main

    emphasis

    here

    is

    on the

    structure

    of

    the controller.

    It

    is

    shown

    that

    by choosing

    the function

    T

    in

    a

    special

    way,

    the

    resultant

    control

    law

    can

    often

    be

    realized as

    a

    linear time-invariant

    feedback.

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    [1]

    G.

    LEITMANN,

    Guaranteed

    asymptotic

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    GUTMAN, Uncertain

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      D  o  w  n  l  o  a  d  e  d  0  1  /  0  3  /  1  3  t  o  1  2  8 .  1  4  8 .  2  5  2 .  3  5 .  R  e  d  i  s  t  r  i  b  u  t  i  o  n  s  u  b  j  e  c  t  t  o  S

      I  A  M   l  i  c  e  n  s  e  o  r  c  o  p  y  r  i  g  h  t  ;  s  e  e

      h  t  t  p  :  /  /  w  w  w .  s  i  a  m .  o  r  g  /  j  o  u  r  n  a  l  s  /  o  j  s  a .  p  h  p

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    STABILIZING

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    FOR

    UN ERT IN DYN MI L

    SYSTEMS

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    BARMISH

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    A.

    VINKLER

     N J

    WOOD Multistep Guaranteed

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    Press New York,

     1961 .

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