Integration of design and control robust approach using mpc and pi controllers
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Integration of Design and Control : Robust approach using MPC and PI controllers. N. Chawankul, H. M. Budman and P. L. Douglas Department of Chemical Engineering University of Waterloo. Outline. Introduction Objectives Methodology Case study Results Conclusions. Introduction.

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Integration of design and control robust approach using mpc and pi controllers

Integration of Design and Control :Robust approach using MPC and PI controllers

N. Chawankul, H. M. Budman and P. L. Douglas

Department of Chemical Engineering

University of Waterloo


Outline

Outline

Introduction

Objectives

Methodology

Case study

Results

Conclusions


Introduction

Introduction

Integrated Approach

1- Design+Control

min (capital +operating

+variability costs)

st

stability

actuator constraints

Traditional Approach

1- Design:

min (capital + operating costs)

2- Control for designed plant

-stability

-actuator constraints

-performance specs:

Small Overshoot

Short Settling Time

Large closed loop bandwidth


Variability cost

disturbance

output

Variability Cost

  • Variability Cost = Cost of Imperfect Control

    For a disturbance d (green):

    What is the cost due to off-spec product (blue)?


Robust control approach

Robust Control Approach

  • To test stability and calculate variability cost we need a model.

    Nonlinear model: stability (Lyapunov-difficult)

    variability (numerically- difficult).

    “Robust” linear model:

    nonlinear model= family of linear models

    family of linear models= nominal model

    +model uncertainty (error)


Integration of design and control robust approach using mpc and pi controllers

Introduction

Previous Approaches

Our approach

• Linear Dynamic Nominal Model +

Model Uncertainty

(Simple optimization problem)

• Variability cost into cost function :

One objective function

• Centralized Control : MPC

• Nonlinear Dynamic Model

(difficult optimization problem)

• Variability cost not into cost function:

Multi-objective optimization

• Decentralized Control : PI /PID


Objectives of the current work

Objectives of the current work

  • Variability using MPC based on a nominal model and model error.

  • Cost of variability in one objective function together with the design cost.

  • Model uncertainty (as a function of design variables) into the objective function.

  • The robust stability criteria as a process constraint.

  • Compare the traditional method to integrated method.

  • Preliminary study on SISO system (distillation column) with MPC.


Methodology

Methodology

  • Model Predictive Control (MPC)

  • Nominal Step Response and Uncertainty

  • Process Variability

  • Optimization

    - Objective Function

    - Constraints


Mpc controller

MPC Controller

d

Simplified MPC block diagram

W

Sd

r

u

y

+

+

MPC

Process

+

+

+

-

u(k)

past

future

target



y(k)



y(k+1/k)

k

k-1

k+1

k+n

k+3

k+2


Nominal step response and uncertainty

Nominal Step Response and Uncertainty

Step Response Model, Sn

y

u

Sn

S6

S5

S4

1

S3

S2

S1

t

0

t

u

Nominal step response model, Sn,nom

1

t

0

-1

Uncertainty,

Upper bound

Nominal step response

Actual Sn

Lower bound

t

Sn-Sn,nom


Process variability 1 y f w

Process variability-1y = f(W)

W

(Sinusoid unmeasured disturbance)

r=0

y

u

+

MPC

Process

+

+

-

Substitute (k), u(k-1) into the first equation and apply Z-transform


Process variability 2

disturbance

output

Process variability-2

Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids)

Amplitude of output,y

With phase lag

Amplitude of disturbance,W

Consider worst case variability :


Optimization

Optimization

MinimizeCost(u,c) = Capital Cost + Operating Cost + Variability Cost

u,c

Such thath(u,c) = 0(equality constraints)

g(u,c)  0(inequality constraints)

u is a vector of design variables.

c is a vector of control variables.


Constraints

Constraints

  • h(u,c) = 0 (equality constraints)

    • steady state empirical correlations

  • g(u,c)  0 (inequality constraints)

    • manipulated variable constraint

    • robust stability


Inequality constraints 1

Inequality Constraints- 1

Consider the MPC controller gain, KMPC:

Manipulated variable constraint

where

 is a manipulated variable weight

The infinity norm of

A is the amplitude of the disturbance.


Inequality constraints 2

Inequality Constraints- 2

Block diagram of the MPC and the interconnection M-

M

Z(k)

w(k)

U(k)

W1

W2

Z-1I

(k+1/k)

u(k)

U(k-1)

+

T2

N1

+

Li

Kmpc

T1

+

+

H

+

-

2. Robust stability constraint (Zanovello and Budman, 1999)

H

N1

+

+

Mp

H

-

+

N2

Z-1I

U(k+1)

U(k)

M

z

w


Case study distillation column preliminary study siso system

Case study- Distillation ColumnPreliminary study: SISO system

 Depropanizer column from Lee, 1994

 adjust reflux ratioto control the mole fraction of propane in distillate

RR

A

+

Feed

-

MPC

= 0.783 (propane)

XD*

Ethane

Propane

Isobutane

N-Butane

N-Pentane

N-Hexane

Q


Process model

(Equality Constraints)

Process Model

The mathematic expressions of the process variables (N, D, Q) as functions of RR

 RadFrac model in ASPEN PLUS

 different column design, 19 – 59 stages

 design variables (number of stages and column diameter) are functions of nominal RR

Number of stages VS. RR


Input output model

Input/Output Model

(Equality Constraints)

y

Sn

S3

  • First Order Model

S2

S1

t

Dynamic simulation using ASPEN DYNAMICS

Step change on RR by  10 % (19- 59 stages)

Upper bound

Nominal step response

Lower bound

63.2 %


Cost cc u oc u vc u c annualized capital cost cc luyben and floudas 1994 day operating cost oc day

Objective Function

Cost = CC(u) + OC(u) + VC(u,c)

Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day)

Operating cost, OC ($/day)

where Q = reboiler duty (GJ/hr)

OP = operating period (hrs)

UC = Utility cost ($/GJ)


Variability cost inventory cost 1

Variability Cost (Inventory cost) - 1

V1

Variability cost, VC ($/day)

- assume sinusoid unmeasured disturbance, W

disturbance induces process variability

consider a holding tank to attenuate the product variation

calculate the volume of the holding tank

- calculate the loss due to the product held in the tank

V2

where P = product price, N = payoff period (10 years),

i = interest rate (10%) and V = volume of the holding tank


Integration of design and control robust approach using mpc and pi controllers

Variability Cost (Inventory cost) - 2

 The worst case variability:

A simple

mass balance

Feed disturbance

Distillation

Column

Cin

Cout

Holding

V

F out

F in

The required volume of the holding tank

spec


Integration of design and control robust approach using mpc and pi controllers

Two different approaches

Traditional Method

  • Integrated Method

Robust Performance

(Morari, 1989)


Results 1

W

RR

D (m)

N

Capital cost

($/day)

Operating cost

($/day)

Variability cost

($/day)

Total cost ($/day)

1

2.4

0.1849

5.8

31

551

654

68

1273

3

2.8

0.1705

6.23

28

530

703

168

1401

5

3.0

0.1653

6.5

27

529

726

257

1512

7

3.4

0.1357

6.9

25

538

774

284

1596

Results-1

Results from Integrated Method: W is a product price multiplier.


Results 2

W

Total Cost of integrated method ($/day)

Total Cost of traditional method ($/day)

Saving ($/day)

% Saving

1

1273

1297

24

1.8

3

1401

1481

80

5.4

5

1512

1665

153

9.2

7

1596

1849

253

13.7

Results-2

Comparison using Traditional and Integrated methods


Imc control

IMC Control

(Morari and Zafiriou, 1989) is used.

  • Internal Model Control, IMC

Gd(s)

d

+

r

y

+

+

C(s)

Gp(s)

-

+

y’

-

Gp(s)

F(s)

 IMC-based PID parameters for


Results

Results

Comparison using both methods


Conclusions

Conclusions

  • single objective function

  • linear dynamic model + model uncertainty

  • MPC variability cost is explicitly incorporated in the objectivefunction

  • integrated approach results in lower costs

  • - savings can be significant; >13% for high value products

  • On-going work: Formulate the MIMO problem with MPC


    Acknowledgement

    Acknowledgement

    Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)


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