Multivariable Lec6

Embed Size (px)

Citation preview

  • 7/30/2019 Multivariable Lec6

    1/54

    Multivariable ControlMultivariable Control

    Ali Karim our

    Assistant Professor

  • 7/30/2019 Multivariable Lec6

    2/54

    Chapter 6

    Chapter 6

    Introduction to Decou lin Control and Uncertaint

    Topics to be covered include:

    Decoupling

    Pre and post compensators and the SVD controller

    ecoup ng y a e ee ac

    Ali Karimpour July 2012

    2

  • 7/30/2019 Multivariable Lec6

    3/54

    Chapter 6

    Introduction

    BuAxx +=&BAsICsG 1)()( =

    )()(.....)()()()()( 12121111 susgsusgsusgsy pp+++=

    ....................................................................................

    ....................................................................................

    ..... 22221212 pp

    )()(.....)()()()()( 2211 susgsusgsusgsy pppppp +++=

    output is controlled by more than one input.

    Ali Karimpour July 2012

    3

    , ,

    very difficult to control a multivariable system.

  • 7/30/2019 Multivariable Lec6

    4/54

    Chapter 6

    Topics to be covered

    Pre and post compensators and the SVD controller

    Decoupling by State Feedback

    Diagonal controller (decentralized control)

    Uncertainty in MIMO Systems

    Ali Karimpour July 2012

    4

  • 7/30/2019 Multivariable Lec6

    5/54

    Chapter 6

    Decoupling

    Definition 6-1

    A multivariable system is said to be decoupled if its transfer-function matrix is diagonal

    and nonsingular.

    A conceptually simple approach to multivariable control is given by a two-steps

    procedure in which

    1. We first design a compensator to deal with the interactions in G(s) and

    )()()( sWsGsG ss =Decoupling

    Ali Karimpour July 2012

    5

    . .

    )()()( sKsWsK ss=)(sKs

  • 7/30/2019 Multivariable Lec6

    6/54

    Chapter 6

    Decoupling

    1. We first design a compensator to deal with the interactions in G(s) and

    sWsGsG = Decou lin

    Dynamic decoupling s.frequencieallatdiagonalis)(sGs

    1 ss ==

    (s)l(s)GK(s)IslsK -s1havewe)()(byThen ==

    It usuall refers to an inverse-based controller.

    Steady-state decoupling diagonal.is)0(sG

    1=

    Approximate decoupling at frequency 0

    s

    possible.asdiagonalasis)( 0jGs

    1

    Ali Karimpour July 2012

    6

    0s

    )(ofionapproximatrealais 00 jGG s forselectiongoodaisfrequency 0BW

  • 7/30/2019 Multivariable Lec6

    7/54

    Chapter 6

    Decoupling

    The idea of using a decoupling controller is appealing, but there are several difficulties.

    a. We cannot in general choose Gs freely. For example, Ws(s) must not cancel any

    RHP-zeros and RHP poles in G(s)

    b. As we might expect, decoupling may be very sensitive to modeling errors and

    uncertainties.

    c. The requirement of decoupling may not be desirable for disturbance rejection.

    One popular design method, which essentially yields a decoupling controller, is the

    internal model control (IMC) approach (Morari and Zafiriou).

    Ali Karimpour July 2012

    7

    Anot er common strategy, w c avo s most o t e pro ems ust ment one , s to

    use partial (one-way) decoupling where Gs(s) is upper or lower triangular.

  • 7/30/2019 Multivariable Lec6

    8/54

    Chapter 6

    Topics to be covered

    Pre and post compensators and the SVD controller

    Decoupling by State Feedback

    Diagonal controller (decentralized control)

    Uncertainty in MIMO Systems

    Ali Karimpour July 2012

    8

  • 7/30/2019 Multivariable Lec6

    9/54

    Chapter 6

    Pre and post compensators and the SVD controller

    The pre compensator approach may be extended by introducing a post compensator

    )()()()( sWsGsWsGssps

    =

    The overall controller is then

    )()()()( sWsKsWsK spss=

    Ali Karimpour July 2012

    9

  • 7/30/2019 Multivariable Lec6

    10/54

    Chapter 6

    Topics to be covered

    Pre and post compensators and the SVD controller

    Decoupling by State Feedback

    Diagonal controller (decentralized control)

    Uncertainty in MIMO Systems

    Ali Karimpour July 2012

    10

  • 7/30/2019 Multivariable Lec6

    11/54

    Chapter 6

    Decoupling by State Feedback

    In this section we consider the decoupling of a control system in state space

    representation.

    DuCx

    BuAxx

    +=

    +=&Let DBAsICsG += 1)()(Suppose 0|D|ifThen

    diagonalsGsG 1)()(

    ( ) 11111111 )()()( ++=+= DBDCBDAsICDDBAsICsG

    u n e case o =

    )()()( tTrtKxtu +=Static state feedback

    Ali Karimpour July 2012

    11

    ryu +=a c ou pu ee ac

    Dynamic output feedback

  • 7/30/2019 Multivariable Lec6

    12/54

    Chapter 6

    Decoupling by State Feedback

    Decoupling through state feedback

    Cxy

    BuAxx

    =

    +=&( ))()()(Suppose 1 trtFxEtu =

    Cx

    rBExFBEAx

    =

    += 11 )(&haveThen we

    The transfer function matrix is 111 )()( += BEFBEAsICsG)

    We s a er ve n t e o ow ng t e con t on on G s un er w c t e system can e

    decoupled by state feedback.

    Ali Karimpour July 2012

    12

  • 7/30/2019 Multivariable Lec6

    13/54

    Chapter 6

    Decoupling by State Feedback

    Theorem 6-1 A system represented by

    Cxy

    BuAxx

    =

    +=&

    with the transfer function matrix G(s) can be decoupled by state feedback of the form

    ( ))()()( 1 trtFxEtu =

    if and only if the constant matrix E is nonsingular.

    =

    d

    new

    s

    sG

    0

    )(

    1

    OFurthermore the new system is in the form:

    1E

    mds0

    dAC 11

    =

    =

    )(

    0

    0lim

    .

    .

    12

    sG

    s

    sEE

    pd

    d

    sO

    =F

    .

    .

    2

    2

    Ali Karimpour July 2012

    13

    Ep

    Proof: See Linear system theory and design Chi-Tsong Chen

    pd

    pAC

  • 7/30/2019 Multivariable Lec6

    14/54

    Chapter 6

    Decoupling by State Feedback

    Example 6-1 Use state feedback to decouple the following system.

    01010

    xyuxx

    =

    +

    =110

    00

    10

    6116

    100&

    Solution: Transfer function of the system is

    +++

    +

    +++

    ++

    ==

    66

    6116

    6

    6116

    116

    )()(

    2323

    2

    1

    s

    sss

    s

    sss

    ss

    BAsICsG

    ++++ 6565 22 ssss

    The differences in degree of the first row ofG(s) are 1 and 2, hence d1=1 and

    ]01[6116

    66116

    116lim23231 =

    +++

    ++++

    ++= sss

    ssss

    sssEs

    The differences in de ree of the second row ofG s are 2 and 1 hence d=1 and

    Ali Karimpour July 2012

    14]10[

    65

    6

    65

    6lim

    222=

    ++

    ++

    =

    ss

    s

    sssE

    s

  • 7/30/2019 Multivariable Lec6

    15/54

    Chapter 6

    Decoupling by State Feedback

    =

    10

    01E

    Solution (continue):

    Now E s un tary matr x an c ear y nons ngu ar so ecoup ng y state ee ac s

    possible and

    =

    =01011

    d

    dACF

    2

    ( )

    ==

    )()(01001

    )()()( 1 trtxtrtFxEtu

    The decoupled system is

    rxrBExFBEAx

    +

    =+= 10

    01

    6116

    000

    )( 11&

    xCxy

    ==

    001

    006116

    Ali Karimpour July 2012

    15Exercise 1: Derive the corresponding decoupled transfer function matrix.

  • 7/30/2019 Multivariable Lec6

    16/54

    Chapter 6

    Property of Decoupling by State Feedback

    1- All poles of decoupled are on origin.

    2- Decoupled system is:

    ndd

    decouple ssdiagsG= ...,,)( 1

    3- No transmission zero in decoupled system.

    4- Transmission zero of the system are deleted .

    5- Unstable transmission zero is the main limitation of method.

    Ali Karimpour July 2012

    16

  • 7/30/2019 Multivariable Lec6

    17/54

    Chapter 6

    Decoupling by State Feedback

    Exercise 2: Decouple following system and find the decoupled transfer function.

    100000

    xyuxx

    =

    +

    =1000

    00

    11

    0100

    0000&

    Exercise 3: Use state feedback to decouple the following system and put the

    poles of new system on s=-3.

    xyuxx

    =

    +

    =110

    00110

    01

    100

    010

    &

    Ali Karimpour July 2012

    17

  • 7/30/2019 Multivariable Lec6

    18/54

    Chapter 6

    Topics to be covered

    Pre and post compensators and the SVD controller

    Decoupling by State Feedback

    Diagonal controller (decentralized control)

    Uncertainty in MIMO Systems

    Ali Karimpour July 2012

    18

  • 7/30/2019 Multivariable Lec6

    19/54

    Chapter 6

    Diagonal controller (decentralized control)

    Another simple approach to multivariable controller design is to use a diagonal or

    . .

    Clearly, this works well ifG(s) is close to diagonal, because then the plant to be

    ,

    K(s) may be designed independently.

    However if off dia onal elements in G s are lar e then the erformance with

    Ali Karimpour July 2012

    19

    decentralized diagonal control may be poor because no attempt is made to counteract

    the interactions.

  • 7/30/2019 Multivariable Lec6

    20/54

    Chapter 6

    Diagonal controller (decentralized control)

    e es gn o ecen ra ze con ro sys ems nvo ves wo s eps:

    _

    2_ The design (tuning) of each controllerki(s)

    Ali Karimpour July 2012

    20

  • 7/30/2019 Multivariable Lec6

    21/54

    Chapter 6

    Input-Output Pairing

    Definition of RGA (Relative Gain Array)

    uu +=Physical Meaning of RGA: Let

    ijijij hg /=2221212 ugugy +=

    Relative gain?

    i

    jiij u

    yuyg

    = or0inputsotherifandbetweenrelation

    uk= ,

    ijiij

    u

    yuyh

    = or0outputsotherifandbetweenrelation

    2121111 ugugy

    =

    +=1

    212 u

    gu =

    ikyk

    = ,0

    121

    12111 )( ug

    ggy

    +=

    Ali Karimpour July 2012

    21TGGGRGAG == )()(

    22

  • 7/30/2019 Multivariable Lec6

    22/54

    Chapter 6

    Input-Output Pairing

    Example: Let

    ==

    1

    1)( TGGG

    2221212

    2121111

    ugugy

    ugugy

    +=

    +=

    =1 Open loop and closed loop gains are the same,so interactions has no effect.

    =0 g11=0 so u1 has no effect on y1.

    0

  • 7/30/2019 Multivariable Lec6

    23/54

    Chapter 6

    Input-Output Pairing

    RGA property:

    - .

    2- Its rows and columns sum to 1.

    3- The RGA is identity matrix if G is upper or lower triangular.

    4- Plant with large RGA elements are ill conditioned.

    5- Suppose G(s) has no zeros or poles at s=0. Ifij() (0) exist andhave different signs then one of the following must be true.

    * G(s) has an RHP zeros. * Gij(s) has an RHP zeros.* gij(s) has an RHP zeros.

    Ali Karimpour July 2012

    23

    6- If gijgij(1-1/ij) then the perturbed system is singular.

    7- Changing two columns/rows of G leads to same changes to its RGA

  • 7/30/2019 Multivariable Lec6

    24/54

    Chapter 6

    Diagonal controller (decentralized control)

    Example 6-2 Select suitable pairing for the following plant

    = 7.04.85.154.16.52.10

    )0(G

    ...

    Solution: RGA of the system is

    = 43.037.094.0

    41.145.196.0

    )0(

    ...

    Ali Karimpour July 2012

    24

  • 7/30/2019 Multivariable Lec6

    25/54

    Chapter 6

    Diagonal controller (decentralized control)

    The RGA based techniques have many important advantages, such as very simple in

    calculation as it only uses process steady-state gain matrix and scaling independent.

    Moreover, using steady-state gain alone may result in incorrect interaction measures and

    consequently loop pairing decisions, since no dynamic information of the process istaken into consideration.

    Many improved approaches, RGA-like, have been proposed and described in all

    process control textbooks, for defining different measures of dynamic loop

    .

    [1] D.Q. Mayne, The design of linear multivariable systems,Automatica, vol. 9, no. 2, pp.

    Relative Omega Array (ROmA),

    , . .

    [2] ARGA Loop Pairing Criteria for Multivariable Systems

    A. Balestrino, E. Crisostomi, A. Landi, and A. Menicagli ,2008

    Absolute Relative Gain Array (ARGA),

    Ali Karimpour July 2012

    25

    ,

    [3] RNGA based control system configuration for multivariable processes

    Mao-Jun He, Wen-Jian Cai *, Wei Ni, Li-Hua XieJournal of Process Control 19 (2009) 10361042

  • 7/30/2019 Multivariable Lec6

    26/54

    Chapter 6

    Diagonal controller (decentralized control)

    Next example, for which the RGA based loop pairing criterion gives an

    inaccurate interaction assessment, are employed to demonstrate the

    e ect veness o t e propose nteract on measure an oop pa r ng cr ter on.

    Example 6-3:

    Consider the two-input two-output process:

    RGA=Diagonal pairing= - agona pa r ng

    To illustrate the validity of above results, decentralized controllers

    forboth diagonal and off-diagonal pairings are designed respectively based on

    the IMC-PID controller tuning rules.To evaluate the output control performance, we consider a unit step set-point

    Change of all control loops one-by-one and the integral square error (ISE) is

    Ali Karimpour July 2012

    26

    used to evaluate the control performance.

  • 7/30/2019 Multivariable Lec6

    27/54

    Chapter 6

    Diagonal controller (decentralized control)

    The simulation results and ISE values are given in Fig. 3. The results show

    that the off-diagonal pairing gives better overall control system performance.

    off-diagonal

    diagonal

    Ali Karimpour July 2012

    27

  • 7/30/2019 Multivariable Lec6

    28/54

    Chapter 6

    Topics to be covered

    Pre and post compensators and the SVD controller

    Decoupling by State Feedback

    Diagonal controller (decentralized control)

    Uncertainty in MIMO Systems

    Ali Karimpour July 201228

  • 7/30/2019 Multivariable Lec6

    29/54

    Chapter 6

    Uncertainty in MIMO Systems

    LQG Control: Optimal state feedback

    ( )

    +=

    0

    dtRuuQzzJ TT

    r

    0and0,where >=== TTxz

    The optimal solution for any initial state is

    txtu r=

    where

    BRK T1=

    Where X=XT0 is the unique positive-semidefinite solution of the algebraic

    Riccati equation

    Ali Karimpour July 2012290

    1 =++

    QMMXBXBRXAXA

    TTT

  • 7/30/2019 Multivariable Lec6

    30/54

    Chapter 6

    Uncertainty in MIMO Systems

    Robustness Properties

    For an L R-controlled s stem if the wei htR is chosen to be dia onal then

    ( )( )11 += BAsIKIS r satisfies ( ) ,1)( jS

    mikand ii ,...,2,1,5.00 =

  • 7/30/2019 Multivariable Lec6

    31/54

    Chapter 6

    Uncertainty in MIMO Systems

    Example 6-4: LQR design of a first order process.

    32 +=

    ssG uxx

    101

    +

    =&23 ++ ss

    The cost function to be minimized is

    [ ]xy 11=

    dtRuyr +=0

    .=

    75.3816]-86.7008[1

    ==

    XBRKT

    r

    ( ) ibkA 9828.61596.7 =

    ( ) 0 allforstablebkAmikand ii ,...,2,1,5.00 =

  • 7/30/2019 Multivariable Lec6

    32/54

    Chapter 6

    Uncertainty in MIMO Systems

    Example 6-5: Decoupling controller

    =564721

    )(2

    sssG

    ss

    1

    The pre compensator approach may be extended by introducing a post compensator

    )(

    2

    20

    176

    87)(

    76

    87sG

    s

    ssG d=

    +

    +=

    )()()()( sWsGsWsG ssps =

    The overall controller is then

    )()()()( sWsKsWsK spss=

    =

    =k

    k

    k

    ksK

    0

    0

    76

    87

    0

    0

    76

    87)(

    We have good stability margin in both channel.

    k 0

    Ali Karimpour July 201232

    xerc se : er ve s a y marg n or eren va ue o

    +

    =k

    sK0

    )(

    For k=1 so find the smallest that lead to instability. Repeat for k=2.

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    33/54

    Chapter 6

    Uncertainty in MIMO Systems

    Type of uncertainty

    .

    Model structure and order are known, but (some)arameter values are uncertain.

    )(

    )(

    ass

    ksG

    +

    =

    Dynamic (frequency-dependent) uncertainty or nonparametricuncertainty. unstructured uncertainty

    There exists (some) erroneous or missing

    dynamics. Usually unmodeled dynamics is in

    Ali Karimpour July 201233

    high frequencies.

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    34/54

    Chapter 6

    Uncertainty in MIMO Systems

    Type of unstructured uncertainty

    Ali Karimpour July 201234

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    35/54

    Chapter 6

    Uncertainty in MIMO Systems

    Parametric uncertaintyNonparametric uncertainty

    Example 6_6: Consider a plant with parametric uncertainty

    maxmin0 )(1

    )( = sGsG +psNow let

    1,

    2/)(

    )(

    1

    )1(maxmin

    maxmin

  • 7/30/2019 Multivariable Lec6

    36/54

    p

    Uncertainty in MIMO Systems

    Parametric uncertaintyNonparametric uncertainty

    Example 6_7: Consider a plant with two parametric uncertainty

    3,,2 = kek

    sG s

    + s

    Ali Karimpour July 201236

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    37/54

    p

    Uncertainty in MIMO Systems

    3,,21

    )( +

    =

    kes

    ksG sp

    Consider additive uncertainty as:

    += ,1)();()()()( jsswsGsG Ap

    Additive uncertainty can be representBy multiplicative one:

    ( ) ,1)(;)()(1)()( jsswsGsG Mp +=

    Ali Karimpour July 201237

    )()(

    jGjwM =

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    38/54

    Uncertainty in MIMO Systems

    System without uncertainty

    =

    wPPz 1211

    Ku =

    NwwPKPIKPPz =+= 211

    221211 )(

    2221

    ),( KPFN l=

    System with uncertaintystructure

    u ou

    uncertainty

    Ali Karimpour July 20123838

    Suitable forrobust

    performance analysis

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    39/54

    Uncertainty in MIMO Systems

    System without uncertainty

    =

    wPPz 1211

    Ku =

    NwwPKPIKPPz =+= 211

    221211 )(

    2221

    ),( KPFN l=

    S stem with uncertaint N structure

    =

    w

    u

    NN

    NN

    z

    y

    2221

    1211

    = yu

    FwwNNINNz =+= 121

    112122 )(

    Ali Karimpour July 201239

    ),( = FF u

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    40/54

    Robust Stability of Parametric Uncertain Systems

    Robust stability in parametric uncertainty

    Ali Karimpour July 201240

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    41/54

    Robust Stability of Unstructured Uncertain Systems

    structureNSystem with uncertaintySystem without uncertainty

    Suitable forrobust performance analysis

    FwIz =+= 1

    Suitable fornominal performance analysis

    NwwPKPIKPPz =+= 1

    ionConfigurat

    on roenera

    Checking robuststability?

    Ali Karimpour July 201241Suitable forcontroller design

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    42/54

    Robust Stability of Unstructured Uncertain Systems

    structureNSystem with uncertaintySystem without uncertainty

    Suitable forrobust performance analysisSuitable fornominal performance analysis

    FwIz =+= 1

    If there is no uncertainty we

    structureM

    N11, N12, N21 and N22 are

    stable

    Ali Karimpour July 2012

    42

    Suitable forcontroller design

    Suitable forrobust stability analysis11=

    Chapter 6

  • 7/30/2019 Multivariable Lec6

    43/54

    Robust Stability of Unstructured Uncertain Systems

    structureM

    Suitable forrobust stability analysisNS:Nis internally stable

    RS: NS and F=F u(N,) is stable for any ||||1

    Theorem: RS for unstructured(full) perturbation.

    Assume that the nominal systemM(s) is stable (NS) and that the

    perturbations (s) are stable. ThenThe M-structure is stable

    ( )

  • 7/30/2019 Multivariable Lec6

    44/54

    Robust Stability of Unstructured Uncertain Systems

    Suitable forrobust stability analysisSystem without uncertainty

    12 wwp +=

    = MuySystem with additive uncertainty

    2

    1

    1 )( wGKIKwM+=

    Robust stabilit condition: In

    the case of |||| 1

    11

  • 7/30/2019 Multivariable Lec6

    45/54

    Robust Stability of Unstructured Uncertain Systems

    System without uncertaintySuitable forrobust stability analysis

    )( 12 wwIGGp +=

    = MuySystem with multiplicativeinput uncertainty

    21 ww +=

    Robust stability condition: In

    the case of |||| 1

    1)( 1

  • 7/30/2019 Multivariable Lec6

    46/54

    Uncertainty in MIMO Systems

    Suitable forrobust stability analysisSystem without uncertainty

    p 12

    = MuySystem with multiplicative

    2

    1

    1 )( wGKIGKwM+=

    ou pu uncer a n y

    Robust stability condition: In

    the case of |||| 1

    11

  • 7/30/2019 Multivariable Lec6

    47/54

    Robust Stability of Unstructured Uncertain Systems

    System without uncertaintySuitable forrobust stability analysis

    1

    12 )(= GwwIGGp

    System with inverse additive uncertainty = Muy

    21 ww +=

    Robust stability condition: In

    1)( 21

    1

  • 7/30/2019 Multivariable Lec6

    48/54

    Robust Stability of Unstructured Uncertain Systems

    System without uncertaintySuitable forrobust stability analysis

    112 )( = wwIGGp 112

  • 7/30/2019 Multivariable Lec6

    49/54

    Robust Stability of Unstructured Uncertain Systems

    System without uncertaintySuitable forrobust stability analysis

    GwwIGp1

    12 )( = 112

  • 7/30/2019 Multivariable Lec6

    50/54

    Robust Stability of Unstructured Uncertain Systems

    Suitable forrobust stability analysis

    Perturbed PlantUncertainty M in M-structure

    Additive uncertainty2

    1

    1 )( wGKIKwM+=12 wwGGp +=

    Multiplicative input uncertainty2

    1

    1 )( GwGKIKwM+=)( 12 wwIGGp +=

    Multiplicative output uncertainty2

    1

    1 )( wGKIGKw+=GwwIGp )( 12+=

    Inverse additive uncertaint 1=1

    =p

    Inverse multiplicative input uncertainty2

    1

    1 )( wKGIwM+=( ) 112

    = wwIGGp

    Ali Karimpour July 2012

    50

    Inverse multiplicative output uncertainty2

    1

    1 )( wGKIwM+=( ) GwwIGp

    1

    12

    =

    Chapter 6structureM

  • 7/30/2019 Multivariable Lec6

    51/54

    Robust Stability of Unstructured Uncertain Systems

    System with coprime factor uncertainty Suitable forrobust stability analysis

    ll NMG1=

    )()( 1 NlMlp NMG ++=

    [ ]MN = 11)( +

    = lMGKII

    M

    Since there is no weight for uncertainty so the theorem is

    Ali Karimpour July 2012

    51

    :

  • 7/30/2019 Multivariable Lec6

    52/54

    Robust Stability of Unstructured Uncertain Systems

    Remind Example 6-5: Decoupling controller

    564721 ss kk 087087

    +++ 25042232 ssss kk 076076

    Exercise 4: Derive stability margin for different value of if

    =k

    sK0

    )(

    For k=1 so find the smallest that lead to instability.

    )( += IGGpConsider system with multiplicative input uncertainty

    ) 1)( 1

  • 7/30/2019 Multivariable Lec6

    53/54

    Robust Stability of Unstructured Uncertain Systems

    Suitable forrobust stability analysis)GGKIK 1)(/1)( +

  • 7/30/2019 Multivariable Lec6

    54/54

    Robust Stability of Unstructured Uncertain Systems

    Exercise 5: Consider following block diagram. We have both input and output uncertainty.

    a) Find the set of possible plants(G )

    b) Find M and derive robust stability condition. ( )1and,1

    io

    Exercise 6: Assume we have derived the following detailed model:

    Ali Karimpour July 2012

    54

    Suppose we chose G(s)=3/(2s+1) with multiplicative uncertainty. Derive suitable scaling

    Matrix.