Automation Lab IIT Bombay An Introduction to Model ... techniques/ppt... · An Introduction to Model Predictive Control TEQIP Workshop, ... Advanced Control ... robotics, fuel cells,

1

An Introduction to Model Predictive Control

TEQIP Workshop, IIT Kanpur

22’nd Sept., 2016

Sachin C. Patwardhan

Dept. of Chemical Engineering

I.I.T. Bombay

Email: [email protected]

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


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


44

Hierarchy of control system functions

2


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


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


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


8

Development of MPC Relevant Linear Perturbation Models

3


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


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


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


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


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


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)

γ1,γ2 (0.7,0.6) (0.43,0.34)

Model Parameters


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)


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)

γ1,γ2 (0.7,0.6)

5


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


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


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


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


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


22

Linear Discrete Time Model

Discrete time linear model under Assumption 1:

ddT

kk

kkkk

T

d

T

d

HAΓBAΓAΦ

Cxy

dΓΓuΦxx

)exp(;)exp(;)exp(

)()(

)()()()1(

00

dTkkk

k

Qddd

d

)()()(

:)(

ECov

covariance known with process noise white mean zero


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

Notation

uuyyxx


23


Rvvvv T(k)(k)E(k)Cov with (k)

covariance known with process noise white mean zero with

corrupted are tsMeasuremen


T

dddTd

Td

d

d

kkEkCov

kEkE

kk

ΓQΓΓddΓw

0dΓw

dΓw

)()()(

)()(

)()( define Now,

Tddd

k

ΓQΓQ

w

matrix covariance

with process stochastic mean zero a is Thus, )(


24


)()()(

)()()()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


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


9/18/2016 State Estimation 26

On-line State Estimator

)()()1|(ˆ)|1(ˆ

)1|(ˆ)()(

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


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


4 Tank Experimental Setup

28

Quadruple Tank Experimental Setup at the Automation Lab, IIT Bombay

8


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


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


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


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

LeΓuΦxx

(Model developed using System Identification Toolbox of MATLAB)

9


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


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


35

Brief Review of Linear Quadratic Optimal Control


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


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


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


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


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


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


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.


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|(ˆ)(

)1()1()2|1(ˆ)1|(ˆ

)2|1(ˆ)1()1(

kkk

kkkkkk

kkkk

xGu

eLuxx

xCye

Predictor Kalman using tionimplementa Controller

Is the closed loop stable under the nominal conditions?


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

ε

x

CL0

ΓGΓGΦ

ε

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

1CLΓGΦ

w

0detdet

][det

CLIΓGΦICLI0

ΓGΓGΦI

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


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


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


47

Linear Model Predictive Control


4848


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


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


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]


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


5252

Moving Horizon Formulation

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

Output

Constraints

14


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


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


5555

MPC with State Estimation

DynamicModel

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

On-line Optimizer


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


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|(ˆ)()1|(ˆ)()(

kE

kkkkkkk

pjkkk

jkjk

kjkjkjk

,...,2,1)1|(ˆ)(ˆ

)(ˆ)(ˆ

)|()(ˆ)1(ˆ

for with xz

zCy

ΓuzΦz


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


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


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


6161


Future instant (k+2)

Future instant (k+1)


6262


Future instant (k+p)


6363


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


6464


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


6565

Constraints on Inputs

q is called the Control Horizon

In a practical implementation control horizon (q) << prediction horizon (p)

To reduce the degree dimension of the on-line optimization problem, degrees of freedom available for shaping the future trajectory are often restricted to first q moves

by imposing input constraints of the form


6666

Constraints on 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

Bounds on rate of change of manipulated inputs

Bounds on the manipulated inputs


67

Control Horizon

Schematic Representation of Control Horizon and Input Bounds



Input Blocking Constraints

Alternatively, degree of freedom for shaping the future trajectory can be used through input blocking constraints

q is called the Control Horizon

In a practical implementation control horizon (q) << prediction horizon (p)

18



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


70

Input Blocking and Bounds

Schematic Representation of Input Blocking and Input Bounds


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


72

Future Setpoint Trajectory

19



Steady State Target Computation

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


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


75


75

Subject to following constraints(a) Model Prediction Equations

(b) Bounds on future inputs and predicted outputs


76


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 Φ are outside unit circle, the procedure for computing the terminal weighting matrix is given in

Muske and Rawlings (1993)

20


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.


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.


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


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


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


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


83

Unconstrained QP Formulation

Using these notations, unconstrained version of MPC problem

can be stated as follows


84


22


85


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

The optimum solution to above minimization problem is


86


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


87

Constrained QP Formulation

The constrained MPC formulation at k’th sampling instant

can be re-cast as


88

Constrained QP Formulation

23


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.


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


Nominal Stability

91

Let us formulate MPC in terms of a generalized loss function


Nominal Stability

92

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

24


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


Nominal Stability

94

Thus, it follows that


Nominal Stability

95


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


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


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


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


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


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


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


103

Comparison of Servo Responses

Controlled Outputs


104

Comparison of Servo Responses

Manipulated Inputs

27


105

Comparison of Regulatory Responses

Controlled Outputs

(Open Loop Observer Based MPC Formulation)


106


Manipulated Inputs


107


Unmeasured Disturbances


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


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)



Commercial Products

(Ref.: Qin and Badgwell, 2003)



Linear MPC Applications (2003)




Industrial Application: Ammonia Plant


29


113

Adaptive and Non-Linear Model Predictive Control


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


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)|(ˆ0

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



Example: Control of Tennessee Eastman Problem

Primary controlled variables: Product concentration of GProduct Flow rate



TE Problem: Objective Function



TE Problem: Operating Constraints



TE Problem: Transition Control

Primary Controlled Outputs

Managing large

setpoint transitions

needs either

on-line model

adaptations or

use of nonlinear

prediction models

32


125125


Secondary Controlled Outputs

Adaptive Model

Predictive Control

is still an open

research area. No

commercial

adaptive MPC is

available yet


126126


Manipulated Inputs




Manipulated Inputs


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


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


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


131131

Nonlinear MPC: Vendors



132132

NMPC: Applications (2003)


34


133133

Summary


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


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


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


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


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.


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.


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, 559–572, 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, 17–35.

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


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, 157–175.

Deshpande, A., Patwardhan, S. C., Narasimhan, S. Intelligent State Estimation for Fault Tolerant Nonlinear Model Predictive Control, Journal of Process Control, 19, 187–204, 2009.

140

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