Automation Lab IIT Bombay An Introduction to Model ... techniques/ppt...An Introduction to Model Predictive Control TEQIP Workshop, ... Advanced Control ... robotics, fuel cells, internet search engines, planning and scheduling, ...

  • Published on
    23-May-2018

  • View
    212

  • Download
    0

Transcript

  • 1

    An Introduction to Model Predictive Control

    TEQIP Workshop, IIT Kanpur

    22nd Sept., 2016

    Sachin C. Patwardhan

    Dept. of Chemical Engineering

    I.I.T. Bombay

    Email: sachinp@iitb.ac.in

    Automation LabIIT Bombay

    22

    Outline

    Motivation

    Development of MPC Relevant Linear Models

    Review of Linear Quadratic Optimal Control

    Linear Model Predictive Control Formulation

    Adaptive MPC

    Nonlinear Model Predictive Control Formulation

    Summary and Research Directions

    Automation LabIIT Bombay

    33

    Long Term Scheduling and Planning

    On-line Optimization

    Multivariable / Nonlinear Control

    Regulatory (PID) Control

    Plant

    Slow Parameter drifts

    MarketDemands /Raw materialavailability

    MVFast Load Disturbances

    PV

    Advanced

    Control

    Setpoints PV, MV

    Plant Wide Control Framework

    Automation LabIIT Bombay

    44

    Hierarchy of control system functions

  • 2

    Automation LabIIT Bombay

    55

    Why Multi-Variable Control?

    Most rear systems have multiple inputs and

    multiple controlled outputs

    Systems exhibit complex and multi-variable

    interactions between inputs and outputs variables

    Need to operate a system within operating

    constraints

    Safety limits

    Input saturation constraints

    Product quality constraints

    Automation LabIIT Bombay

    66

    Why Model Predictive Control?

    Need to control over wide operating range

    Process nonlinearities

    Changing process parameters / conditions

    Conventional approach : Multi-loop PI - difficult to tune

    Ad-hoc constraint handling using logic programming

    (PLCs): lack of coordination

    MPC deals with multivariable interactions,

    operating constraints, and process nonlinearity

    systematically

    Automation LabIIT Bombay

    77

    Model Predictive Control

    Most widely used multivariable control scheme in

    the process industries over last 35 years

    Used for controlling critical unit operations (such as

    reactors) in refineries world over

    With increasing computing power, MPC is

    increasingly being applied in diverse application

    areas: robotics, fuel cells, internet search engines,

    planning and scheduling, control of drives, bio-

    medical applications

    Automation LabIIT Bombay

    8

    Development of MPC Relevant Linear Perturbation Models

  • 3

    Automation LabIIT Bombay

    MPC Relevant Linear Model

    Critical Step: Development of a control relevant linear

    perturbation model for developing MPC scheme

    Approach 1: If a reliable mechanistic or grey-box

    dynamic model is available, then a linear perturbation

    model can be developed using local linearization

    Approach 2: Alternatively, a linear perturbation model

    can be developed using input-output data generated by

    deliberately exciting the system for a short period and

    using the system identification tools

    9

    Automation LabIIT Bombay

    10

    Local Linearization

    modelon perturbatilinear a develop to

    )( of odneighborhoin expansion seriesTaylor apply we

    ),(point operating statesteady and

    ,,,

    :Modelt Measuremen

    dt

    d :Dynamics State

    modelparameter lumped aGiven

    D,U,X

    D,U,X

    DYUX

    G(X)Y

    D)U,F(X,X

    drmn RRRR

    0D,U,XFX

    X

    D,U

    )(dt

    d

    for solvingby found becan StateSteady ingCorrespond

    )( inputs operating statesteady Given

    Automation LabIIT Bombay

    11

    Local Linearization

    modelon perturbati derive to(1)Equation from (2)Equation Subsract

    )2()(dt

    d

    StateSteady At

    )1()()(

    )()(

    )( of odneighborho in theexpansion seriesTaylor

    )()(

    )(

    D,U,XFX

    DDD

    FUU

    U

    F

    XXX

    FD,U,XF

    X

    D,U,X

    D,U,XD,U,X

    D,U,X

    tt

    tdt

    d

    XXX

    GYY

    XXX

    GYXX

    X

    GXGY

    X

    XX

    )()()(

    )()()(

    Modelt Measuremen

    )(

    )()(

    ttty

    ttt

    Automation LabIIT Bombay

    12

    Local Linearization

    D-Dd ;U-Uu

    Y-Yy;X-Xx

    Cxy

    HdBuAxx

    (t)(t)(t)(t)

    (t)(t)(t)(t)

    eson variablPerturbati

    Modelon PerturbatiLinear Time Continuous

    dt

    d

    ),,(at computed

    ;;;

    matrices Define

    DUX

    X

    GC

    D

    FH

    U

    FB

    X

    FA

  • 4

    Automation LabIIT Bombay

    13

    Example: Quadruple Tank System

    21

    21

    h and h :Outputs Measured

    v and v :Inputs dManipulate

    1

    4

    114

    4

    44

    2

    3

    223

    3

    33

    2

    2

    224

    2

    42

    2

    22

    1

    1

    113

    1

    31

    1

    11

    )1(2

    )1(2

    22

    22

    vA

    kgh

    A

    a

    dt

    dh

    vA

    kgh

    A

    a

    dt

    dh

    vA

    kgh

    A

    agh

    A

    a

    dt

    dh

    vA

    kgh

    A

    agh

    A

    a

    dt

    dh

    Pump 2V2

    Pump1V1

    Tank3

    Tank 2

    Tank 1

    Tank 4

    Automation LabIIT Bombay

    Model Parameters

    14

    Steady state Operating ConditionsP- : Minimum Phase P+ : Non-minimum Phase

    A1, A3 [cm2] 28

    A2, A4 [cm2] 32

    a1, a3 [cm2] 0.071

    a2, a4 [cm2] 0.057

    kc [V/cm] 0.5

    g [cm/s2] 981

    P- P+

    h1, h2 [cm] (12.4,12.7) (12.6,13)

    h3, h4 [cm] (1.8,1.4) (4.8,4.9)

    v1,,v2 [V] (3,3) (3.15,3.15)

    k1,k2, [cm3/V] (3.33,3.35) (3.14,3.29)

    1,2 (0.7,0.6) (0.43,0.34)

    Model Parameters

    Automation LabIIT Bombay

    Linearization of Quadruple Tank Model

    15

    uxx

    0)1(

    )1(0

    0

    0

    1000

    01

    00

    01

    0

    001

    4

    11

    3

    22

    2

    22

    1

    11

    4

    3

    42

    4

    2

    31

    3

    1

    A

    kA

    kA

    kA

    k

    T

    T

    TA

    A

    T

    TA

    A

    T

    dt

    d

    xy

    000

    000

    c

    c

    k

    k4,3,2,1for

    2 i

    g

    h

    a

    AT i

    i

    ii

    P- P+

    (T1,T2) (62,90) (63,91)

    (T3,T4) (23,30) (39,56)

    Automation LabIIT Bombay

    Linearization of Quadruple Tank Model

    16

    000.50

    0000.5

    00.03122

    0.047860

    0.062810

    00.08325

    0.03334-000

    00.04186-00

    0.0333400.01107-0

    00.0418600.01595-

    CB

    A

    Quadruple Tank System

    Continuous Time State Space Model Matrices

    Developed at Steady State

    Operating Point

    )( U,X

    P-

    h1, h2 [cm] (12.4,12.7)

    h3, h4 [cm] (1.8,1.4)

    v1,,v2 [V] (3,3)

    k1,k2, [cm3/V] (3.33,3.35)

    1,2 (0.7,0.6)

  • 5

    Automation LabIIT Bombay

    Discrete Dynamic Models

    17

    ..0,1,2,....k:kTt

    instant at available are (k), ts,measuremen Thus,

    sec T of interval sampling uniform

    and rate constant a at sampled are tsMeasuremen

    :Sampling tMeasuremen 1.

    sAssumption

    k

    y

    TktkTtfor )1( u(k)u(t)

    interval sampling the during

    constant piecewise are inputs dManipulate

    :hold order zero with tionReconstruc Input 2.

    Development of computer oriented discrete dynamic models

    Automation LabIIT Bombay

    18

    Digital Control: Measured Outputs

    0 5 10 15 201.8

    2

    2.2

    2.4

    2.6

    2.8

    3

    3.2

    Sampling Instant

    Measu

    red

    Ou

    tpu

    t

    0 5 10 15 201.8

    2

    2.2

    2.4

    2.6

    2.8

    3

    3.2

    Sampling Instant

    Measu

    red

    Ou

    tpu

    t

    ADC

    Continuous Measurement

    from process

    Sampled measurement

    sequence to computer

    Output measurements are available onlyat discrete sampling instant Where T represents sampling interval

    ,....,,: 210 kkTtk

    Automation LabIIT Bombay

    19

    Digital Control: Manipulated Inputs

    In computer controlled (digital) systems Manipulated inputs implemented through DACare piecewise constant

    0 2 4 6 8 10 12 14 16 18 202

    2.1

    2.2

    2.3

    2.4

    2.5

    2.6

    2.7

    2.8

    2.9

    Sampling Instant

    Man

    ipu

    late

    d In

    pu

    t

    0 2 4 6 8 10 12 14 16 18 202

    2.1

    2.2

    2.3

    2.4

    2.5

    2.6

    2.7

    2.8

    2.9

    Sampling Instant

    Ma

    nip

    ula

    ted

    In

    pu

    t S

    eq

    uen

    ce

    DAC

    Input Sequence

    Generated by computerContinuous input profile generated by DAC

    TkttkTtforkututu kkk )1()()()( 1

    Automation LabIIT Bombay

    20

    Linear DiscreteTime Model

    TktkTtfor

    kt

    )1(

    )()(

    uu

    constant piecewise are inputs dManipulate

    Systems Controlled Computer

    DifficultyDisturbance inputs d(t) are NOT piecewise constant functions!

    How to develop a discrete time model?

    TktkTtforkt )1()()( dd

    interval sampling the during functions constant piecewise

    as edapproximat be can inputs edisturbanc the

    that so enough small is (T) interval Sampling

    :1 Assumption gSimplifyin

  • 6

    Automation LabIIT Bombay

    Unmeasured Disturbances

    21

    0 20 40 60 80 100-2.5

    -2

    -1.5

    -1

    -0.5

    0

    0.5

    1

    1.5

    2

    2.5

    samples

    d(k

    )Typical

    piecewise constant

    unmeasured disturbance generated using zero

    mean Gaussian random process

    with unit variance

    Automation LabIIT Bombay

    22

    Linear Discrete Time Model

    Discrete time linear model under Assumption 1:

    ddT

    kk

    kkkk

    T

    d

    T

    d

    HABAA

    Cxy

    duxx

    )exp(;)exp(;)exp(

    )()(

    )()()()1(

    00

    dTkkkk

    Qddd

    d

    )()()(

    :)(

    ECov

    covariance known with process noise white mean zero

    :2 Assumption gSimplifyin

    kT)(t(k) and kT)(t(k)kT),(t(k)

    Notation

    uuyyxx

    Automation LabIIT Bombay

    23

    Linear Discrete Time Model

    Rvvvv T(k)(k)E(k)Cov with (k)covariance known with process noise white mean zero with

    corrupted are tsMeasuremen

    :3 Assumption gSimplifyin

    TdddTdTd

    d

    d

    kkEkCov

    kEkE

    kk

    Qddw

    0dw

    dw

    )()()(

    )()(

    )()( define Now,

    Tddd

    k

    QQ

    w

    matrix covariance

    with process stochastic mean zero a is Thus, )(

    Automation LabIIT Bombay

    24

    Linear Discrete Time Model

    )()()(

    )()()()1(

    kkk

    kkkk

    vCxy

    wuxx

    Combining all the simplifying assumptions, we arrive at a linear discrete time computer controlrelevant dynamic model of the form

    where w(k) and v(k) are assumed to be uncorrelated random sequences with zero mean and know variances

    RvvQww TT kkEkkE )()(;)()(

    Q quantify uncertainties in state dynamicsand/or modeling errors

    R quantifies variability of measurement errors

  • 7

    Automation LabIIT Bombay

    Quadruple Tank System

    25

    2244 05.005.0

    00

    0

    IRIQ

    C 0.50

    0000.5

    00.1438

    0.21590

    0.30550.01209

    0.022760.4001

    0.8465000

    0 0.81120

    0.149300.94620

    00.181300.9233

    Discrete Time State Space Model Matrices Sampling Time T = 5 sec

    Automation LabIIT Bombay

    9/18/2016 State Estimation 26

    On-line State Estimator

    )()()1|()|1(

    )1|()()(

    kkkkkk

    kkkk

    Leuxx

    xCye

    Estimator Prediction Recursive

    1

    TT

    TT

    CCPRCPL

    LCPQPP

    Equation Riccati Algebraic using Computed

    Matrix Gain Observer

    For example, steady state Kalman predictor

    A suitable state estimator can be developed using the linear perturbation model

    Automation LabIIT Bombay

    Data Driven Model Development

    2727

    0 2 4 6 8 10 12 14 16 18 202

    2.1

    2.2

    2.3

    2.4

    2.5

    2.6

    2.7

    2.8

    2.9

    Sampling Instant

    Man

    ipu

    late

    d In

    pu

    t

    Physical System

    0 5 10 15 201.8

    2

    2.2

    2.4

    2.6

    2.8

    3

    3.2

    Sampling Instant

    Measu

    red

    Ou

    tpu

    t

    Man. Input Perturbations

    System ResponseUnmeasured Disturbances

    ARX/ARMAX/ State Space Model

    Identification

    Data Driven Linear State Space Model

    Automation LabIIT Bombay

    4 Tank Experimental Setup

    28

    Quadruple Tank Experimental Setup at the Automation Lab, IIT Bombay

  • 8

    Automation LabIIT Bombay

    29

    Splitting Data for Identification and Validation

    0 500 1000

    -5

    0

    5

    y1

    Input and output signals

    0 500 1000

    -0.5

    0

    0.5

    1

    Samples

    u1

    Identification Data Validation data

    Automation LabIIT Bombay

    30

    Identification Data: Inputs

    0 1000 2000 3000 4000 5000-0.5

    0

    0.5

    u1(k

    ), v

    olts

    Identification Data: Manipulated Inputs

    0 1000 2000 3000 4000 5000-0.5

    0

    0.5

    Time (sec)

    u2(k

    ), v

    olts

    Automation LabIIT Bombay

    31

    Identification Data: Outputs

    0 1000 2000 3000 4000 5000

    -10

    0

    10

    y 1(k

    ), c

    m

    Identification Data: Output Perturbations

    0 1000 2000 3000 4000 5000-10

    0

    10

    Time (sec)

    y 2(k

    ), c

    m

    Automation LabIIT Bombay

    32329/18/2016 32

    Data Driven State Space Model

    9/18/2016 32

    17.81-9.2753.95101.85

    15.3716.0249.27-121.57

    0.4667-0.1522

    0.13840.1326-

    0.7090.322-

    0.45430.5758

    10

    0.0099-0.0288

    0.0113-0.0525-

    0.00210.0018

    0.00280.0046

    0.86260.04200.0426-0.0316-

    0.0333-0.73420.0032-0.1180

    0.10510.03800.96030.0108-

    0.0404-0.0522-0.0145-0.9521

    3-

    CL

    ;

    Innovation form of State Space Model (or observer)Sampling Time T = 5 sec

    )()()(

    )()()()1(

    kkk

    kkkk

    eCxy

    Leuxx

    (Model developed using System Identification Toolbox of MATLAB)

  • 9

    Automation LabIIT Bombay

    Model Validation: Inputs

    33

    0 500 1000 1500 2000 2500 3000-0.5

    0

    0.5

    u1(k

    ), v

    olts

    Validation Data: Manipulated Input Perturbations

    0 500 1000 1500 2000 2500 3000-0.5

    0

    0.5

    Time (sec)

    u2(k

    ), v

    olts

    Automation LabIIT Bombay

    Model Validation: Outputs

    34

    0 500 1000 1500 2000 2500 3000

    -10

    -5

    0

    5

    10

    Time

    Pe

    rturb

    atio

    n L

    eve

    l 1 (

    cm)

    Model Validation: Output 1

    Model Simulation

    Measured Output

    0 500 1000 1500 2000 2500 3000-10

    -5

    0

    5

    10

    Time

    Pe

    rtu

    rba

    tion

    Le

    vel 2

    (cm

    )

    Model Validation: Output 2

    Model Simulation

    Measured Output

    Identified models have reasonably accurate predictions

    Comparison of simulated model output with the

    measured outputs in the validation data set

    Automation LabIIT Bombay

    35

    Brief Review of Linear Quadratic Optimal Control

    Automation LabIIT Bombay

    3636

    Linear Quadratic Regulator

    )()(

    )()()1(

    kCxky

    kBukAxkx

    Model:

    ObjectiveRegulate the process at origin of the state space in the face of sudden impulse like, disturbances,

    which result in non-zero initial conditions

    matrices seighting Definite PositiveSymmetric :W,WW

    Cx(k)y(k)

    )(x(k)1)x(k

    u(k)Wu(k))()(

    )()(

    )1(),....,0(

    min

    that such)1(),....,0( sequence input Determine

    N x,

    1

    0u

    T

    u

    N

    kx

    T

    NT

    ku

    toSubject

    kxWkx

    NxWNx

    Nuu

    Nuu

    Square of distance

    from Origin

    PenalizeLarge

    manipulated inputs

    Final State

  • 10

    Automation LabIIT Bombay

    3737

    Summary: Quadratic Optimal Control

    (k)-G(k)(k)

    law control feedback state varying Time

    xu

    stage each at solution of optimality ensures which

    k each for definite ve and Symmetric : s(k)

    WS(N)

    with ....1 1,-N N, from starting

    time in backward solved Equation

    N

    )()()]()[1()]([)(

    )1()1(1

    kGWkGWkGkSkGkS

    kSkS

    uT

    xTT

    TT

    uWG(k)

    Equation Riccati time Discrete solving by computed matrix Gain

    N should be known a-priori and gain matrices have to be saved : not quite practical in many situations

    Automation LabIIT Bombay

    3838

    Algebraic Riccati Equation

    GGGSGS

    SSWG

    as (ARE) Equation Riccati Algebraic

    solving by computed be can which

    GG(k)SS(k) large, becomes N When

    u

    u

    T

    xT

    TT

    WW][][

    ;

    1

    ARE. to solution definite

    negative non and symmetric unique a exists there then

    W

    where observable is ),( if and

    lecontrollab is ),( if However, solutions. many has ARE

    u

    T

    x(k)-Gu(k)

    form assumes law control and

    Automation LabIIT Bombay

    3939

    Nominal Stability Analysis

    Theorem 1: Consider the time invariant dynamic model together with the LQ loss function. Assume that a positive-definite steady state solution exists for the algebraic Riccati equations. Then the steady state optimal strategy

    gives an asymptotically stable closed-loop system

    Proof: Define Lyapunov function

    matrix definite ve a is S :Note

    Automation LabIIT Bombay

    4040

    Nominal Stability Analysis

    Thus, the closed loop system is asymptotically stable for any choice of positive definite Wx and positive semi-definite Wumatrices

    Simultaneously guarantees closed loop stability and good closed loop performance

    By selecting Wx and Wu appropriately, it is easy to compromise between speed of recovery and magnitude of control signals.

  • 11

    Automation LabIIT Bombay

    4141

    Closed Loop Poles

    The poles of the closed loop system obtained

    by solving the characteristic equation

    It can be shown that the poles are the n stable eigenvalues of the generalized eigenvalue problem

    This equation is called the Euler equation of the LQ problem.

    circle. unit the inside are G- of poles all i.e. system, loop closed stable a gives controller LQ that shows 1 Theorem

    Automation LabIIT Bombay

    4242

    Linear Quadratic Optimal Output Regulator

    In many situations we are only interested in controlling certain outputs of a system

    The above modified objective function can be rearranged as follows

    and by setting

    we can use the Riccati equations derived above for controller design.

    Automation LabIIT Bombay

    4343

    Linear Quadratic Gaussian Regulator

    Linear Quadratic Gaussian (LQG) Regulator

    Design optimal state estimator (Kalman Predictor / Kalman Filter)

    Implement control law using estimated states

    )()()(

    )()()()1(

    kkk

    kkkk

    vCxy

    wuxx

    Dynamics Process

    )1|()(

    )1()1()2|1()1|(

    )2|1()1()1(

    kkk

    kkkkkk

    kkkk

    xGu

    eLuxx

    xCye

    Predictor Kalman using tionimplementa Controller

    Is the closed loop stable under the nominal conditions?

    Automation LabIIT Bombay

    449/18/2016 State Feedback Control 44

    Nominal Closed Loop Stability

    )(

    )(]0[

    )1|(

    )(

    ][)|1(

    )1(

    k

    k

    kk

    k

    kk

    k

    v

    w

    LI

    I

    x

    CL0

    GG

    x

    Dynamics Loop Closed Combined

    circle unit the inside equation loop closed the of sEigenvalue

    and 1

    that destablishe have e arguments, stability Lyapunov Through :Note

    1CLG

    w

    0detdet][det

    CLIGICLI0

    GGI

    Equation sticCharacteri Loop Closed

    Thus, even though the observer and controller are designed separately to be a-stable, the nominal closedloop system, implemented using the observer based

    feedback controller, is asymptotically stable

  • 12

    Automation LabIIT Bombay

    4545

    LQOC Formulation

    LQ Optimal Controller: Linear quadratic regulator

    can be further modified to

    Reject non-stationary (drifting) unmeasured

    disturbances

    Tracking arbitrarily changing setpoints for the

    controlled outputs

    Achieve robustness in the face of mismatch

    between the plant and the model

    Automation LabIIT Bombay

    4646

    Limitations of LQOC

    Difficult to incorporate or handle operating

    constraints explicitly

    Limits/constraints on the manipulated inputs

    Constraints on process outputs (arising from

    product quality, safety considerations)

    Algebraic Riccati Equations: AREs is notoriously

    difficult to solve for large dimensional systems

    Automation LabIIT Bombay

    47

    Linear Model Predictive Control

    Automation LabIIT Bombay

    4848

    Model Predictive Control

    Multivariable control based on on-line use of

    dynamic model and constrained optimization

    Developed by industrial researchers

    Dynamic Matrix Control (DMC) developed by Shell in

    U.S.A. (Cutler and Ramaker, 1979)

    Model Algorithmic Control developed by Richalet et. al.

    (1978) in France

    Can be used for controlling complex, large

    dimensional and non-square systems

  • 13

    Automation LabIIT Bombay

    4949

    Advantages of MPC

    Can be viewed as a modified version of the classical optimal

    control problem

    Can systematically and optimally handle

    Multivariable interactions

    Operating input and output constraints

    Basic Idea: Given a reasonably accurate model for plant

    dynamics, possible consequences of the current and future

    input moves on the future plant behavior can be forecasted

    on-line and used while deciding the input moves

    Automation LabIIT Bombay

    5050

    MPC: Basic Idea

    Finite Horizon formulation: Optimization

    problem is formulated over a finite window of time

    starting from current instant, i.e. over [k, k+p]

    (unlike over [k,) in the classical optimal control)

    Pro-active constraint management: Using the

    dynamic model, on-line forecasting is carried out

    foresee and avoid any possible constraint violations

    over the time window [k,k+p]

    Automation LabIIT Bombay

    5151

    MPC: Basic Idea

    On-line Constrained Optimization: At each

    sampling instant, a constrained optimization

    problem is formulated over the window and solved

    online to determine the current input u(k)

    Moving horizon implementation: The time window

    for control keeps moving or receding

    From [k, k+p] To [k+1, k+p+1]

    . and so on

    Automation LabIIT Bombay

    5252

    Moving Horizon Formulation

    (Kothare et al, (2000), IEEE Control Systems Technology)

    Output

    Constraints

  • 14

    Automation LabIIT Bombay

    5353

    MPC: Schematic Diagram

    Process

    Dynamic Model

    Dynamic Prediction

    Model

    Optimization

    MPC

    Set point Trajectory

    Disturbances

    Dynamic Model: used for on-line forecasting over a moving time horizon (window)

    Plant-model mismatch

    Inputs Outputs

    Automation LabIIT Bombay

    5454

    Components of MPC

    Internal model and state estimator

    Discrete Linear State Space Model developed from

    mechanistic approach or time series modeling

    (FIR or Finite Step Response models were used initially)

    State Estimator: Open loop observer / Kalman Predictor/

    Kalman Filter / Luenberger Observer / Innovation form of

    state observer developed from ARX / ARMAX / BJ model

    Prediction of Future Plant Behavior

    Key issue: Handling unmeasured drifting disturbances and

    plant model mismatch

    On-line constrained optimization strategy

    Quadratic programming

    Linear programming

    Automation LabIIT Bombay

    5555

    MPC with State Estimation

    DynamicModel

    (Kothare et al, (2000), IEEE Control Systems Technology)

    On-line Optimizer

    Automation LabIIT Bombay

    5656

    State Estimation and Prediction

    Consider state estimation and prediction using prediction form of observer

    Such a observer can be developed using any of the

    following approaches

    Kalman predictor

    Luenberger predictor

    State realization of ARX / ARMAX / BJ model

    Prediction estimate of the current state and innovation

  • 15

    Automation LabIIT Bombay

    5757

    Future Prediction

    in absence of model plant mismatch (MPM), model predictions over future time window [k + 1; k+p] can be generated as follows

    observer as used is predictor Kalman state steady the when

    i.e. noise, white mean zero a is

    sinnivation the mismatch, plant model of absence In

    0e

    xCyyye

    )(

    )1|()()1|()()(

    kE

    kkkkkkk

    pjkkk

    jkjk

    kjkjkjk

    ,...,2,1)1|()(

    )()(

    )|()()1(

    for with xz

    zCy

    uzz

    Automation LabIIT Bombay

    Future Trajectory Prediction

    In practice, Model Plant Mismatch (MPM) arises due to

    Changes in the steady state operating conditions

    Abrupt step changes/drifts in the unmeasured

    disturbances

    In the presence of MPM, the innovation sequence is no

    longer a zero mean white noise. The mean starts drifting

    and the sequence becomes a colored noise.

    Thus, the model predictions have to be corrected to

    account for for MPM

    5858

    Automation LabIIT Bombay

    Compensation for MPM

    The innovation signal contains a signature of MPM,

    which is typically a low frequency signal

    However, the innovation signal also contains the

    measurement noise, which is typically in the high

    frequency range

    Thus, a filtered version of the innovation sequence

    can be used as a proxy for MPM and can be used

    for correcting the future predictions

    59

    Automation LabIIT Bombay

    6060

    State Estimation and Prediction

    Innovation Bias Approach: Effect of model plant mismatch and /or unmeasured

    disturbance signal is extracted by filtering the innovationthrough a unity gain low pass filter

    model predictions over future time window [k + 1; k + p] with compensation for MPM are generated as follows

  • 16

    Automation LabIIT Bombay

    6161

    Future Trajectory Prediction

    Future instant (k+2)

    Future instant (k+1)

    Automation LabIIT Bombay

    6262

    Future Trajectory Prediction

    Future instant (k+p)

    Automation LabIIT Bombay

    6363

    Future Trajectory Prediction

    outputs future on esDisturbanc

    Unmeasured and Mismatch Model Plant of Effect

    outputs future on inputs

    future of Effect

    outputs future on state

    past the of Effect

    prediction

    output Future

    Interpretation of p step output prediction equation

    p is called as Prediction Horizon

    Automation LabIIT Bombay

    6464

    Future Trajectory Prediction

    Note: The predictions generated using the innovation bias approach is equivalent to carrying out predictions

    using the observer augmented with an artificially introduced integrated white noise model, i.e. prediction

    Generated using the following dynamic system

    Introduction of integrated white noise in predictionshelps in achieving offset free closed loop behavior

  • 17

    Automation LabIIT Bombay

    6565

    Constraints on Inputs

    q is called the Control Horizon

    In a practical implementation control horizon (q)

  • 18

    Automation LabIIT Bombay

    699/18/2016 State Feedback Control 69

    Input Blocking Constraints

    Bounds on the manipulated inputs

    Bounds on rate of change of manipulated inputs

    Since predictions are carried out online at each control instant, it is possible to choose future inputs moves such

    That the above constraints are respected

    Automation LabIIT Bombay

    70

    Input Blocking and Bounds

    Schematic Representation of Input Blocking and Input Bounds

    Automation LabIIT Bombay

    7171

    Future Setpoint Trajectory

    In addition to predicting the future output trajectory, at each instant, a filtered future setpoint trajectory is generated using a reference system of the form

    Automation LabIIT Bombay

    72

    Future Setpoint Trajectory

  • 19

    Automation LabIIT Bombay

    739/18/2016 State Feedback Control 73

    Steady State Target Computation

    Case: Number of manipulated inputs equals the number of controlled outputs and unconstrained solution exists

    Automation LabIIT Bombay

    74

    Constrained MPC formulation

    74

    Given the prediction model, input constraints and desired set point trajectory, the MPC problem at sampling instant k is formulated as follows

    Automation LabIIT Bombay

    75

    Constrained MPC formulation

    75

    Subject to following constraints(a) Model Prediction Equations

    (b) Bounds on future inputs and predicted outputs

    Automation LabIIT Bombay

    76

    Constrained MPC formulation

    76

    behavior output and input loop closed the shape to used are which

    ,parameters tunig as treated are matrices These

    matrix weighting input tesemidefini positive symmetric is and

    matrix weighting error definite positive symmetric a is

    U

    E

    W

    W

    equation Lyapunov discrete solving by found be can circle,

    unit the inside are of poles When equation. Lyapunov discrete

    solving by found be can matrix weighting state terminal The

    W

    W

    When some poles of are outside unit circle, the procedure for computing the terminal weighting matrix is given in

    Muske and Rawlings (1993)

  • 20

    Automation LabIIT Bombay

    77

    Moving Horizon Implementation

    77

    The resulting constrained optimization problem is solved on-line each sampling instant using any standard constrained optimization method.

    The controller is implemented in a moving horizon framework.

    Thus, after solving the optimization problem over window [k,k+p], only the first optimal move

    is implemented on the plant, i.e.

    The optimization problem is reformulated at the next sampling instant over time windows [k+1, k+p+1] based on the updated

    information from the plant and resolved.

    Automation LabIIT Bombay

    7878

    Moving Horizon Formulation

    Optimization problem transformed to Quadratic

    Programming (QP) problem for improving computing

    efficiency on-line and solved using efficient QP

    solvers available commercially.

    MPC formulation can control Non-square multi-variable

    systems i.e. systems with number of controlled

    outputs not equal to the number of manipulated

    inputs.

    In many practical situations, not all outputs have to be

    controlled at fixed setpoints but need to be

    maintained in some zone. Such zones can be easily

    defined using constraints on predicted outputs.

    Automation LabIIT Bombay

    79

    Quadratic Programming (QP)

    A constrained optimization problem is called as Quadratic Programming (QP) formulation if it Has following standard form

    bAU

    UHUUU

    to Subject

    TT FMin

    2

    1

    A large dimensional QP formulation can be solved very quickly using an efficient search method

    Through a series of algebraic manipulations, the Constrained MPC formulation can be transformed

    to a Quadratic Programming (QP) Problem

    Automation LabIIT Bombay

    80

    QP Formulation

    To understand how the MPC optimization problem can be

    transformed to a quadratic programming problem, considerMPC formulation without terminal state weighting

    (Note: QP formulation can be carried out with terminal state weighting also. It has been neglected here to keep the expressions relatively simple)

    the prediction model can be expressed as follows

  • 21

    Automation LabIIT Bombay

    81

    QP Formulation

    Matrix relating the effect of past statesto future predictions

    Matrix relating the effect ofpast unmeasured disturbancesand model plant mismatch on

    the future predictions

    Automation LabIIT Bombay

    82

    QP Formulation

    Matrix relating the effect of future manipulated inputsOn future predictionsConsists of impulse response coefficients of the modelReferred to as Dynamic Matrix in MPC literature

    Automation LabIIT Bombay

    83

    Unconstrained QP Formulation

    Using these notations, unconstrained version of MPC problem

    can be stated as follows

    Automation LabIIT Bombay

    84

    Unconstrained QP Formulation

  • 22

    Automation LabIIT Bombay

    85

    Unconstrained QP Formulation

    The unconstrained optimization problem can be reformulated as a Quadratic Programming problem as follows

    The optimum solution to above minimization problem is

    Automation LabIIT Bombay

    86

    Unconstrained QP Formulation

    Since only the first input move is implemented on the process

    With some algebraic manipulations, the above control law can be rearranged as follows

    From the above expression, it is easy to see that unconstrained MPC is a form of state feedback control law

    Advantage of unconstrained formulation: closed form control law can be obtained and, as a consequence,

    on-line computation time is small

    Automation LabIIT Bombay

    87

    Constrained QP Formulation

    The constrained MPC formulation at kth sampling instant

    can be re-cast as

    Automation LabIIT Bombay

    88

    Constrained QP Formulation

  • 23

    Automation LabIIT Bombay

    89

    Alternate Formulations

    To achieve offset free control, it is possible to develop MPC

    formulation based on the augmented state space model (see

    Muske and Rawlings, 1993; Yu et al., 1994).

    Early formulations of MPC, such as Dynamic Matrix Control

    (DMC), were based on open loop observer and were meant

    for open loop stable systems. These formulations can be

    derived by setting L = [0] in the innovation bias formulation.

    MPC formulation in this presentation has been developed

    using Kalman predictor. It is straightforward to develop a

    similar formulation based on the Kalman filter.

    Automation LabIIT Bombay

    Nominal Stability

    90

    Proved for the deterministic version of MPC undercertain simplifying assumptions

    Assumption 2: The true states are perfectly measurable

    Assumption 3: It is desired to control the system at the origin

    Assumption 1: There is no model plant mismatchor unmeasured disturbances are absent andboth internal model (i.e. observer) and plantevolve according to

    Automation LabIIT Bombay

    Nominal Stability

    91

    Let us formulate MPC in terms of a generalized loss function

    Automation LabIIT Bombay

    Nominal Stability

    92

    Let us denote the optimal solution of the resulting constrained optimization problem at instant k as

  • 24

    Automation LabIIT Bombay

    Nominal Stability

    93

    Let optimum solution of the MPC problem over the window [k + 1, k + p + 1] be denoted as

    We want to examine

    A non-optimal but feasible solution for the optimization problem over window [k + 1, k + p + 1] is

    For this feasible solution, the following inequality holds

    Automation LabIIT Bombay

    Nominal Stability

    94

    Thus, it follows that

    Automation LabIIT Bombay

    Nominal Stability

    95

    Automation LabIIT Bombay

    Nominal Stability

    96

    Thus, it follows that

    and the nominal closed loop system is globally asymptotically stable.

    Thus, under the nominal conditions, MPC guarantees global asymptotic stability and optimal performance.

    It is remarkable that we are able to construct a Lyapunov function using the MPC objective function.

  • 25

    Automation LabIIT Bombay

    97

    Tuning of MPC

    Process

    Dynamic Model

    Unknown DisturbancesSet Point Filter

    MPC

    Plant-model mismatch

    Inputs Outputs

    Robustness Filter

    Set Point

    Facilitates Performance specification

    Guard again plant-model mismatch

    Automation LabIIT Bombay

    98

    Tuning of MPC

    Prediction Horizon: Typically chosen close to open

    loop settling time (60 to 100 samples)

    Control Horizon: Typically chosen small (5 to 10)

    to avoid model inversion problems

    Input rate constraints

    Zone / Range Control: Not necessary to specify

    set points on each output. Instead, high and

    low limits can be defined within which output

    should be maintained

    Automation LabIIT Bombay

    9999

    Example: Shell Control Problem

    Controlled Outputs :(y1) Top End Point (y2) Side Endpoint (y3) Bottom Reflux Temperature

    Manipulated Inputs :(u1) Top Draw (u2) Side Draw (u3) Bottom Reflux Duty

    Unmeasured Disturbances:(d1) Upper reflux

    (d2) Intermediate reflux

    Automation LabIIT Bombay

    100100

    119

    2.7

    144

    42.4

    133

    38.4140

    9.6

    160

    72.5

    150

    39.5150

    88.5

    160

    77.1

    150

    05.4

    )(

    192220

    151418

    272827

    s

    e

    s

    e

    s

    es

    e

    s

    e

    s

    es

    e

    s

    e

    s

    e

    sG

    sss

    sss

    sss

    u

    )()()()()( sdsGsusGsy du

    `

    132

    26.1

    127

    14.1120

    83.1

    125

    52.1140

    44.1

    145

    2.1

    )(1515

    2727

    ss

    s

    e

    s

    es

    e

    s

    e

    sGss

    ss

    d

    Characteristics Large time delays High degree of

    multivariable interactions

    Shell Control Problem (SCP)

  • 26

    Automation LabIIT Bombay

    101101

    SCP: MPC Tuning Parameters

    Operating Constraints

    3

    1

    5.0

    5.05.0

    y

    y

    Input Limits

    2,15.05.0

    3,2,15.05.0

    iford

    iforu

    i

    i

    3,2,105.005.0 iforuiRate Limits

    Output Constraints

    Prediction Horizon : 40 Control Horizon : 5

    11.015.1

    011

    diagW

    diagW

    u

    e

    Sampling interval(T) = 2 min

    Automation LabIIT Bombay

    102

    SCP: PID Tuning Parameters

    Multi-loop PID control: Three independent PID controllers with no coordination among them

    PID Pairing and Tuning (y1) Top End Point - (u1) Top Draw

    Kc = 0.3 , Ti = 13 min, Td = 0

    (y2) Side Endpoint - (u2) Side Draw

    Kc = 0.23 , Ti = 30 min, Td = 0

    (y3) Bottom Reflux - (u3) Bottom Reflux Duty

    Kc = 0.28 , Ti = 9 min, Td = 0

    Automation LabIIT Bombay

    103

    Comparison of Servo Responses

    Controlled Outputs

    Automation LabIIT Bombay

    104

    Comparison of Servo Responses

    Manipulated Inputs

  • 27

    Automation LabIIT Bombay

    105

    Comparison of Regulatory Responses

    Controlled Outputs

    (Open Loop Observer Based MPC Formulation)

    Automation LabIIT Bombay

    106

    Comparison of Regulatory Responses

    Manipulated Inputs

    Automation LabIIT Bombay

    107

    Comparison of Regulatory Responses

    Unmeasured Disturbances

    Automation LabIIT Bombay

    108108

    SCP: Sequential Servo Changes

    0 50 100 150 200 250-0.5

    -0.4

    -0.3

    -0.2

    -0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    Sampling instant

    Mea

    sure

    d O

    utp

    uts

    Shell Control Problem

    y(1)

    y(2)

    y(3) Note:DecoupledServoResponse. Change in oneSetpoint Does not affect the other outputs

    With driftingUnmeasured

    disturbances

  • 28

    Automation LabIIT Bombay

    109109

    SCP: Sequential Servo Changes

    0 50 100 150 200 250-0.5

    -0.4

    -0.3

    -0.2

    -0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    Sampling instant

    Man

    ipu

    late

    d I

    np

    uts

    Shell Control Problem

    u(1)

    u(2)

    u(3)

    Automation LabIIT Bombay

    1109/18/2016 State Feedback Control 110

    Commercial Products

    (Ref.: Qin and Badgwell, 2003)

    Automation LabIIT Bombay

    1119/18/2016 State Feedback Control 111

    Linear MPC Applications (2003)

    (Ref.: Qin and Badgwell, 2003)

    Automation LabIIT Bombay

    1129/18/2016 State Feedback Control 112

    Industrial Application: Ammonia Plant

    (Ref.: Qin and Badgwell, 2003)

  • 29

    Automation LabIIT Bombay

    113

    Adaptive and Non-Linear Model Predictive Control

    Automation LabIIT Bombay

    Dealing with Model-Plant Mismatch

    Adaptive Model Predictive Control: Active

    approach

    On-line Model Maintenance: Identify model

    parameters on-line, either intermittently using a

    batch of data, or, on-line using recursive

    parameter estimation

    Robust Model Predictive Control: Passive approach

    Incorporate robustness at the design stage

    114

    Automation LabIIT Bombay

    115

    Adaptive MPC

    Process

    System

    Identification

    Model Predictive Controller

    Inputs

    Disturbances

    Outputs

    Faults

    Identified Model Parameters

    Set point

    Online model parameter estimation: using Recursive Least Squares/

    Pseudo-linear Regression

    Automation LabIIT Bombay4 Tank Experimental Setup

    116

    Quadruple Tank Experimental Setup at the Automation Lab, IIT Bombay

  • 30

    Automation LabIIT BombayAMPC of Quadruple Tank System

    Parameter Value

    Prediction Horizon(p) 100

    Control Horizon(q) 5

    10

    Forgetting Factor 0.999 (in recursive least squares)

    Filter Coefficient 0.95

    117

    40cm)|(0

    3%)|u(3%

    /0.95mu0 3

    jkky

    jkk

    hr

    AMPC Tuning Parameters

    Automation LabIIT BombayTracking Performance

    ACODS 2016 118

    Automation LabIIT BombayRelative Sensitivity Index

    119

    100)0(

    )0()()(%

    Index Sensitvity Relative

    ]-I)[()(y Matrix Sensitivit Model u1

    ij

    ijijij

    G

    GkGkS

    kCkG

    Automation LabIIT BombayTracking Performance

    120ACODS 2016

  • 31

    Automation LabIIT Bombay

    1219/18/2016 State Feedback Control 121

    Example: Control of Tennessee Eastman Problem

    Primary controlled variables: Product concentration of GProduct Flow rate

    Automation LabIIT Bombay

    1229/18/2016 State Feedback Control 122

    TE Problem: Objective Function

    Automation LabIIT Bombay

    1239/18/2016 State Feedback Control 123

    TE Problem: Operating Constraints

    Automation LabIIT Bombay

    1249/18/2016 State Feedback Control 124

    TE Problem: Transition Control

    Primary Controlled Outputs

    Managing large

    setpoint transitions

    needs either

    on-line model

    adaptations or

    use of nonlinear

    prediction models

  • 32

    Automation LabIIT Bombay

    125125

    TE Problem: Transition Control

    Secondary Controlled Outputs

    Adaptive Model

    Predictive Control

    is still an open

    research area. No

    commercial

    adaptive MPC is

    available yet

    Automation LabIIT Bombay

    126126

    TE Problem: Transition Control

    Manipulated Inputs

    Automation LabIIT Bombay

    1279/18/2016 State Feedback Control 127

    TE Problem: Transition Control

    Manipulated Inputs

    Automation LabIIT Bombay

    128128

    Need for Nonlinear Control

    Linear prediction model based MPC:

    limits applicability to small regions around

    operating point

    Real systems are nonlinear: use of linear controllers can generate sub-optimal performance

    Nonlinear MPC Need to achieve tight control of highly nonlinear

    systems

    Control of time varying (batch / semi-batch) systems

    Grade transition problems in polymer processing

  • 33

    Automation LabIIT Bombay

    129129

    Models for Nonlinear MPC (NMPC)

    First Principles / Phenomenological

    / Mechanistic / Grey Box

    Based on physics of the problem

    Energy and material balances

    Thermodynamic models

    Conservation laws: conservation of charge

    Valid over wide operating range

    Provide insight in the internal working of systems

    Development and validation process: difficult and

    time consuming, requires a domain expert for

    development

    Automation LabIIT Bombay

    130130

    Models for Nonlinear MPC

    Data Driven / Black Box Models

    Dynamic models developed directly from input-output data

    Model Forms Nonlinear Difference Equations (NARX, NARMAX etc.)

    Artificial Neural Networks

    Limitations Valid over limited operating range

    Provide no insight into internal working of systems

    Development process: much less time consuming

    and comparatively easy

    Automation LabIIT Bombay

    131131

    Nonlinear MPC: Vendors

    (Ref.: Qin and Badgwell, 2003)

    Automation LabIIT Bombay

    132132

    NMPC: Applications (2003)

    (Ref.: Qin and Badgwell, 2003)

  • 34

    Automation LabIIT Bombay

    133133

    Summary

    Model Predictive Control

    provides a coordinated approach to handling of multi-

    variable interactions and operating constraints

    deal with control problems of non-square systems

    transparent way of tuning controller through objective

    function weights and rate limits to achieve desirable

    closed loop performance

    can handle nonlinear systems effectively

    Very flexible control tool for addressing wide

    variety of control problems

    Automation LabIIT Bombay

    134134

    Current Research Directions

    Developing reliable nonlinear models capturing

    effects of unmeasured disturbances

    Incorporating robustness at design stage

    Integrating fault diagnosis with MPC/NMPC

    formulations

    Development of improved state estimation

    schemes

    Embedding MPC / NMPC on a chip

    Automation LabIIT Bombay

    135135

    Current Research Directions

    Fast NMPC for robotic and other fast

    applications like automobiles

    Improving MPC relevant optimization schemes:

    guaranteed convergence

    Coordinated MPC: Developing multiple MPC that

    cooperate and control a large system

    Stochastic MPC: Handling uncertainty in

    unmeasured disturbances and parameters

    Automation LabIIT Bombay

    136

    References

    Books with excellent material on LQOC and MPC

    Astrom, K. J. and B. Wittenmark, Computer Controlled Systems, Prentice Hall, 1990.

    Camacho, E. C. and C. Bourdons, 1999, "Model Predictive Control", Springer Verlag, London.

    Franklin, G. F. and J. D. Powell, Digital Control of Dynamic Systems, Addison-Wesley, 1989.

    Goodwin, G., Graebe, S. F., Salgado, M. E., Control System Design, Phi Learning, 2009.

    Glad, T., Ljung, L. Control Theory: Multivariable and Nonlinear Methods, Taylor and Francis, 2000.

    Sodderstrom, T. Discrete Time Stochstic Systems. Springer, 2003.

    Rawlings, J. B., Mayne, D. Q., Model Predictive Control: Theory and Design, Nob Hill Publishing, 2015.

  • 35

    Automation LabIIT Bombay

    137

    References

    MPC and Related Important Review Articles

    Garcia, C. E., Prett, D. M. , Morari, M. Model predictive control: Theory and practice - A survey. Automatica, 25 (1989), 335-348.

    Morari, M. , Lee, J.H., Model Predictive Control: Past, Present and Future, Comp. Chem. Engg., 23 (1999), 667-682.

    Henson, M.A. (1998). Nonlinear Model Predictive Control : Current status and future directions. Computers and Chemical Engg,23 , 187- 202.

    Lee, J.H. (1998). Modeling and Identification for Nonlinear Model Predictive control:Requirements present status and future needs. International Symposium on Nonlinear Model Predictive control,Ascona, Switzerland.

    Meadows, E.S. , Rawlings, J. B. Nonlinear Process Control, ( M.A. Henson and D.E. Seborg (eds.), New Jersey: Prentice Hall, Chapter 5.(1997).

    Qin, S.J., Badgwell, T.A. A servey of industrial model predictive control technology, Control Engineering Practice 11 (2003) 733-764.

    Automation LabIIT Bombay

    138

    References

    Useful / Important Papers

    Muske, K. R. , Rawlings, J. B. ; Model Predictive control with linear models, AIChE J., 39 (1993), 262-287.

    Muske, K. R. ;Badgwell, T. A. Disturbance modeling for offset-free linear model predictive control. Journal of Process Control, 12 (2002), 617-632.

    Ricker, N. L., Model Predictive Control with State Estimation, Ind. Eng. Chem. Res., 29 (1990), 374-382.

    Yu, Z. H. , Li , W., Lee, J.H. , Morari, M. State Estimation Based Model Predictive Control applied to Shell Control Problem: A Case Study, Chem. Eng. Sci., (1994), 14-22.

    Patwardhan S.C. and S.L. Shah (2005) From data to diagnosis and control using generalized orthonormal basis filters. Part I: Development of state observers, Journal of Process Control,15,7, 819-835.

    Automation LabIIT Bombay

    References

    Srinivas, K., Shaw, R., Patwardhan, S. C., Noronha, S. Adaptive model predictive control of multivariable time-varying systems. Ind. Eng. Chem. Res., 2008, 47, 2708-2720.

    Badwe, A., Singh, A., Patwardhan, S. C., Gudi. R. D., A Constrained Recursive Pseudo-linear Regression Scheme for On-line Parameter Estimation in Adaptive Control. Journal of Process Control, 20, 559572, 2010.

    Srinivasarao,M.; Patwardhan,S. C.; Gudi, R. D. Nonlinear predictive control of irregularly sampled multi-rate systems using nonlinear black box observers. Journal of Process Control, 2007, 17, 1735.

    Srinivasrao, M.; Patwardhan, S.C. ; Gudi, R. D. From data to nonlinear predictive control. 2.. Improving regulatory performance using identified observers. Ind. Eng. Chem. Res., 2006, 45, 3593-3603.

    139

    Automation LabIIT Bombay

    References

    Prakash, J.; Patwardhan, S. C.;Narasimhan, S. Integrating model based fault diagnosis with model predictive control. Ind. Eng. Chem. Res., 2005, 44, 4344-4360.

    Patwardhan, S.C. ; Manuja, S.; Narasimhan, S.; Shah, S. L From data to diagnosis and control using generalized orthonormal basis filters. Part II: Model predictive and fault tolerant control. Journal of Process Control, 2006, 16, 157175.

    Deshpande, A., Patwardhan, S. C., Narasimhan, S. Intelligent State Estimation for Fault Tolerant Nonlinear Model Predictive Control, Journal of Process Control, 19, 187204, 2009.

    140

Recommended

View more >