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Control of a Solution Copolymerization Reactor using Piecewise Linear Models. Leyla Özkan APACT-03 York, UK April 30 th , 2003. Presentation Outline. Motivation Multi-Model Predictive Control Formulation Implementation on MMVA Solution Co-polymerization Reactor Stability Analysis

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Control of a solution copolymerization reactor using piecewise linear models

Control of a Solution Copolymerization Reactor using Piecewise Linear Models

Leyla Özkan

APACT-03 York, UK

April 30th , 2003


Presentation outline
Presentation Outline Piecewise Linear Models

  • Motivation

  • Multi-Model Predictive Control Formulation

  • Implementation on MMVA Solution Co-polymerization Reactor

  • Stability Analysis

  • Conclusion

APACT03-30 April, 2003


Motivation
Motivation Piecewise Linear Models

  • Polymerization reactors

    • Complex nonlinear kinetics and behavior

    • Difficult to specify control objectives

    • Global competition

      • Strict requirements on polymer properties

      • Grade transitions common

    • Control objective

      • Minimize grade transition time

Reduce Off-Specification Product

APACT03-30 April, 2003


Model predictive control

min Piecewise Linear Models

J

(

k

)

p

u

(

.

)

Past

Future

Set point

dx

Predicted y

=

f

(

x

,

u

,

t

)

dt

Closed loop y

Open loop u

£

£

u

u

u

min

max

£

£

y

y

y

min

max

Closed loop u

k

k+1

k+Hc

k+Hp

Hc

Hp

Model Predictive Control

  • Model Predictive Control

    • Class of control algorithms that solves optimization problem at every instant

APACT03-30 April, 2003


Multi model predictive control

[ Piecewise Linear Models

]

¥

T

=

+

+

+

J

x

(

k

m

|

k

)

Q

x

u

Ru

(

k

m

|

k

)

å

¥

I

T

+

+

(

k

)

(

k

m

|

k

)

(

k

m

|

k

)

finite horizon cost

infinite horizon cost

m

=

0

2

å

+

+

x

(

k

m

|

k

)

2

n

+

+

+

+

u

u

(

(

k

k

m

m

|

|

k

k

)

)

x

(

k

m

|

k

)

R

Q

2

2

=

m

n+1

I

å

+

Q

R

I

=

m

0

  • Effective in terminal region

  • Bounded by

  • State feedback controllaw

T

+

+

+

+

x

)

)

(

k

n

1

P x

(

k

n

1

Multi-Model Predictive Control

  • Infinite horizon objective function

  • Free input variables

  • Forces states towards

  • terminal region

APACT03-30 April, 2003


Quasi infinite horizon strategy

x Piecewise Linear Models(k+1|k), u(k+1|k)

x(k+2|k), u(k+2|k)

x(k|k), u(k|k)

n+1 m  

(x,u) = (0,0)

u(k+m|k)=Kx(k+m|k)

Terminal region

x(k+n+1|k)

Quasi-infinite horizon strategy

APACT03-30 April, 2003


Illustration of multi model predictive control

t

J

J

¥

n

OP2

x2

OP1

x1

Illustration of multi-model predictivecontrol

Control Recipe

If x terminal region

quasi-infinite horizon

If x terminal region

infinite horizon

APACT03-30 April, 2003


The resulting lmi problem

g Piecewise Linear Models

min

é

ù

1

*

*

*

*

*

L

ê

ú

0

.

5

g

Q

x

(

k

|

k

)

I

0

0

0

0

0

I

ê

ú

0

.

5

ê

ú

g

R

u

(

k

|

k

)

0

I

0

0

0

0

ê

ú

0

.

5

³

+

g

0

Q

x

(

k

1

|

k

)

0

0

I

0

0

0

ê

ú

I

0

.

5

ê

ú

+

g

R

u

(

k

1

|

k

)

0

0

0

I

0

0

ê

ú

M

O

ê

ú

ê

ú

+

+

x

(

k

n

1

|

k

)

0

0

0

0

0

Q

ë

û

é

ù

T

0

.

5

T

0

.

5

T

+

Q

(

A

Q

B

Y

)

QE

QQ

Y

R

t

t

t

t

I

t

ê

ú

T

T

+

+

x

x

(

A

Q

B

Y

)

Q

b

b

b

e

0

0

ê

ú

t

t

t

t

t

t

t

ê

ú

T

T

³

0

x

-

x

-

E

Q

e

b

(

I

e

e

)

0

0

t

t

t

t

t

ê

ú

0

.

5

ê

ú

g

Q

Q

0

0

I

0

I

ê

ú

0

.

5

g

R

Y

0

0

0

I

ë

û

t

The resulting LMI problem

Finite horizon

x(k) terminal region

Infinite horizon

APACT03-30 April, 2003


The resulting lmi problem1

g Piecewise Linear Models

min

x

Q

,

,

Y

i

s.t.

é

ù

T

1

x

(

k

|

k

)

³

0

ê

ú

ê

ú

x

(

k

|

k

)

Q

û

ë

and

é

ù

T

0

.

5

T

0

.

5

T

+

Q

(

A

Q

B

Y

)

QE

QQ

Y

R

i

i

i

i

I

i

ê

ú

T

T

+

+

x

x

(

A

Q

B

Y

)

Q

b

b

b

e

0

0

ê

ú

i

i

i

i

i

i

i

ê

ú

T

T

³

0

x

-

x

-

E

Q

e

b

(

I

e

e

)

0

0

i

i

i

i

i

ê

ú

0

.

5

ê

ú

g

Q

Q

0

0

I

0

I

ê

ú

0

.

5

g

R

Y

0

0

0

I

ë

û

i

-

-

-

1

1

1

=

g

x

=

 l

=

P

Q

K

Y

Q

i

i

Özkan, L. et.al. , Control of a solution copolymerization reactor using multi-model predictive control, Chem. E. Science, 58:1207-1221, 2003

The resulting LMI problem

Feasible

ONLY IF

bi=0

(x(k)  terminal region)

APACT03-30 April, 2003


Input constraints

j

=

1

,

2

,

,

n

£

u

(

k

)

u

u

k

0

j

j

,

max

³

Finite horizon cost:

Infinite horizon cost:

Impose directly

(

u

(

k

+

m

|

k

)

k

+

m

|

k

)

=

Kt x

+

£

Exists s.t.

X

u

(

k

m

|

k

)

é

ù

X

Y

j

2

t

³

£

0

X

u

ê

ú

jj

j

,

max

u

Y

Q

ë

û

j

t

,

max

m

=

0

, 1, …, n

j

j

=

=

1

1

,

,

2

2

,

,

,

,

n

n

u

u

m

=

n+1 ,…,

Input Constraints

APACT03-30 April, 2003


Mmva solution copolymerization reactor

Monomer (F Piecewise Linear ModelsA)

Monomer (FB)

Initiator (FI)

Coolant

Solvent (FS)

Coolant

Transfer Agent (FC)

Polymer

Solvent

Unreacted feed

Inhibitor (FZ)

Richards, J. R. et.al. , Feedforward and Feedback Control of a Solution Co-polymerization Reactor, AIChE Journal, 35(6):891-907, 1989

MMVA Solution Copolymerization Reactor

  • Characteristics:

    • Based on free radical mechanism

    • Realistic industrially

      • 12 states, 4 outputs

    • Dynamics depends on monomers’ feed ratio

Challenge: Optimization problem involves 200–300 variables

APACT03-30 April, 2003


Implementation on mmva copolymerization reactor

4

OP 2

3.8

Mw/E4 (kg/kmol)

3.6

OP 1

3.4

0

10

20

30

40

50

time (h)

Implementation on MMVA copolymerization reactor

  • Manipulated variables

    • FA, FB, FC, Tj

  • Obtaining multiple models

    • Select new desired operating point

    • Assume a trajectory

APACT03-30 April, 2003


Implementation on mmva cont d

( Piecewise Linear Modelsx,u) - (0,0)

  • Desired operating point

    • Driving force

    • Linear models are updated accordingly

  • Norm measure to define sequence of linear models

x3

x

x(k|k)

x2

x1

t

Implementation on MMVA (cont’d)

  • Simplifications

    • Only terminal region approximated as ellipsoid

APACT03-30 April, 2003


Implementation on mmva cont d1

Ap Piecewise Linear Models

Y

30

30

0

0

10

10

20

20

time (h)

time (h)

Implementation on MMVA (cont’d)

  • Effect of number of linear models

2 models

4

353.5

3 models

353.4

3.8

T (K)

353.3

4 models

Mw/E4 (kg/kmol)

353.2

3.6

353.1

3.4

Off–spec. (kg)

0.64

27

26

0.62

Open loop 302.4

25

0.6

2 models 134.8

Gpi (kg/h)

0.58

24

3 models 110.1

23

0.56

4 models 109.8

0.54

22

APACT03-30 April, 2003


Stability analysis

  • MPC Piecewise Linear Models

    • Lyapunov Approach

      • dx(t)/dt=f(x) and x=0 equilibrium

      • V(x)

        • V(x)>0, x 0 and V(x)=0  x=0

        • d V/dt<0, x 0

Asymptotically

stable origin

V(x)

Time

Stability Analysis

  • Conventional control

    • Check eigenvalues of closed-loop system

APACT03-30 April, 2003


Hybrid systems

Discrete Piecewise Linear Models

components

Continuous

dynamical systems

Logic commands (switches,automata)

Hybrid systems

Hybrid systems

  • Definition:

    • Dynamical systems with continuous and discrete state variables

APACT03-30 April, 2003


Multiple lyapunov functions

Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automatic Control, 43(4):475-482, 1998.

Multiple Lyapunov Functions

  • Consider Vi :i=1,2,…,N

  • Given S={x0: (i0,t0),(i1,t1),…,(iN,tN),..}

  • Denote

    • T: increasing sequence of times t0,t1,…tN

    • (T):even sequence of T:t0,t2,t4,

If Viis monotonically non-increasing on(T) then system is stable in the sense of Lyapunov

APACT03-30 April, 2003


Multiple lyapunov functions1

V1(x)

t0

t1

t2

t3

t4

t5

t6

V2(x)

Multiple Lyapunov Functions

  • Easy to interpret

  • N candidate Lyapunov functions

  • Requires stable subsystems

APACT03-30 April, 2003


Multi model predictive control with stability guarantee

Theorem analysis tools for switched and hybrid systems,

Ifx(k)  terminal region, the quasi–infinite horizon optimization problem is solvedincluding the contractive constraint. If x(k)  terminal regionthe infinite horizon optimization problem is solved. The closed loop system is stable if the feasible solutions of the control strategy defined are implemented in a receding horizon fashion

Multi-Model Predictive Control with Stability Guarantee

  • Contractive constraint

    •  (k) <  (k-1)  x  terminal region

  • Stability Analysis

APACT03-30 April, 2003


Multi model predictive control with stability guarantee1

  • Proof analysis tools for switched and hybrid systems,

    • Candidate Lyapunov function

terminal region

Ï

g

ì

(

k

)

x

(

k

)

=

V

í

a

(

(

x

))

T

terminal region

Î

x

(

k

)

Px

(

k

)

x

(

k

)

î

Multi-Model Predictive Control with Stability Guarantee

  • Remarks

    • Stability results depend on feasibility

    • States are measurable

    • @ k=ts

      • V(x(ts))-V(x(ts-1))<0; not required

APACT03-30 April, 2003


Mmva solution copolymerization revisited

1 analysis tools for switched and hybrid systems,

10

0

10

 (k)

xT(k)Px(k)

-1

10

log (V)

-2

10

-3

10

-4

10

0

5

10

15

t(h)

MMVA solution copolymerization revisited

  • Observations

    • V monotonically decreasing

    • ts=7.0 and 7.75 h

APACT03-30 April, 2003


Summary and conclusion
Summary and Conclusion analysis tools for switched and hybrid systems,

  • Multi-model control algorithm is developed

    • Hybrid structure

  • Implemented on a high dimensional problem

    • The effect of number of linear models

      • Decrease in transition time

    • Computational difficulty

      • Number of free input variables

    • Large size LMI’s solved on a realistic problem

  • Stability analysis

    • Contractive constraint

    • Multiple Lyapunov Functions

APACT03-30 April, 2003


Control of a solution copolymerization reactor using piecewise linear models

Thank you analysis tools for switched and hybrid systems,

for your attention

Leyla Özkan

APACT03-30 April, 2003