FAULT-TOLERANT CONTROL OF CHEMICAL PROCESS

SYSTEMS USING COMMUNICATION NETWORKS

Nael H. El-Farra, Adiwinata Gani

& Panagiotis D. Christofides

Department of Chemical Engineering

University of California, Los Angeles

2003 AIChE Annual Meeting

San Francisco, CA.

November 20, 2003

INTRODUCTION

• Process control system failure:

¦ Typical sources:

? Failure in control algorithm? Faults in control actuators and/or measurement sensors

¦ Induce discrete transitions in continuous dynamics

• Motivation for fault-tolerant control:

¦ Preserve process integrity & dependability

¦ Minimize negative economic & environmental impact:? Raw materials waste, production losses, personnel safety, · · ·, etc.

• Dynamics of chemical processes:

¦ Nonlinear behavior? Complex reaction mechanisms ? Arrhenius reaction rates

¦ Input constraints

? Finite capacity of control actuators

ISSUES IN FAULT-TOLERANT CONTROLOF CHEMICAL PROCESSES

• Availability of multiple control configurations:

¦ Actuator/sensor redundancy

¦ Different manipulated variables

• Illustrative example

Coolant

Out

Controller

Coolant

In

F0, T

0, C

A0

F1, T

1, C

A1

Temperature

Sensor

CSTR 1

Q1

Composition

Analyzer

= Automatic Control Valve

Coolant

Out

Controller

Temperature

Sensor

CSTR 2

Q2

Composition

Analyzer

F2, T

2, C

A2

Coolant

In

ISSUES IN FAULT-TOLERANT CONTROLOF CHEMICAL PROCESSES

• Distributed interconnected nature of process units:

¦ Propagation of failure effects

¦ Large numbers of distributed sensors & actuators involved

¦ Efficient means of communication required

• Types of control system architecture:

Plant

Sensor 1 Sensor n

Computer

Actuator 1 Actuator m... ...

? Point-to-point connections

Sensor

Process 2

Sensor Actuator Actuator

Computer Computer

Monitor

Process 1

Plant

? Distributed control system

INTEGRATING COMMUNICATION NETWORKS IN CONTROL

• Defining feature:

Information exchanged using a network among control system components

• Networked control structure

01101001010101101001010101000101111000101011101010100010101010000011111101011010010100010101000101

Network Channel

a1

a2

s1

s2

sR

vM

v2

v1 w

1

w2

yR

y2

aM

u2

uM

wR

Controller

Plant

x(t)

y1

u1

sensor

delay

actuator

delay

reference

input

actuator

input

sensor

output

... ...

... ...

...

...

...

...

¦ Economic & operational benefits:

. Reduced system wiring

. Ease of diagnosis & maintenance

. Enhanced fault-tolerance:? Rerouting signals? Activation of redundant components

¦ Implementation issues:

? Bandwidth limitations

? Network delays

INTEGRATING COMMUNICATION NETWORKS IN CONTROL

Process 1

Sensor

1

Sensor

n

Computer

Actuator

1

Actuator

m... ...

Plant

Supervisor

Process 2

Sensor

1

Sensor

n

Computer

Actuator

1

Actuator

m... ...

Process 3

Sensor

1

Sensor

n

Computer

Actuator

1

Actuator

m... ...

Local

Network

Local

NetworkLocal

Network

Plant Network... ...

... ...

• Practical implementation considerations:

¦ Size and complexity of the plant

¦ Number of sensors & actuators

¦ Bandwidth limitations in data transmission

¦ Network scheduling and communication delays

PRESENT WORK

• Scope:

¦ Nonlinear process systems

? Input constraints ? Control system failures

• Objectives:

¦ Integrated approach for fault-tolerant control system design

. Design of nonlinear feedback controllers

? Nonlinear dynamics ? Input constraints

. Design of supervisory switching laws

? Orchestrate transition between control configurations

. Use of communication networks

¦ Application to two chemical reactors in series

• Approach:¦ Lyapunov-based nonlinear control ¦ Hybrid systems theory

A HYBRID SYSTEMS FRAMEWORK FOR FAULT-TOLERANTPROCESS CONTROL

• State-space description:

x(t) = fi(x(t)) +m∑

l=1

gli(x(t))uli(t)

i(t) ∈ I = 1, 2, · · · , N <∞uli,min ≤ uli(t) ≤ uli,max

¦ x(t) ∈ IRn : continuous process state variables

¦ ui(t) ∈ IRm : manipulated inputs for i-th mode

¦ i(t) ∈ I : discrete variable controlled by supervisor

¦ N : total number of control configurations

¦ fi(x), g(l)i (x) : sufficiently smooth nonlinear functions

.x = f (x) + g (x)u3 3 3 Discrete Events

.x = f (x) + g (x)u4 4 4

Dis

cre

te E

ven

ts

Discrete Events.

Discrete Events

1 1 1x = f (x) + g (x)u

mode 4

Discre

te E

ven

ts

.x = f (x) + g (x)u2 2 2

mode 1 mode 2

mode 3

• Faults induce discrete events superimposed on continuous dynamics

FAULT-TOLERANT CONTROL PROBLEM FORMULATION

• Coordinating feedback & switching over networks:

¦ Synthesis of a family of stabilizing feedback controllers

? Model for each mode of the hybrid plant: x = fi(x) +Gi(x)ui

? Magnitude of input constraints: |ui| ≤ ui,max

? Family of Lyapunov functions: Vi, i = 1, · · · , N¦ Design of supervisory switching laws that orchestrate mode transitions

i(t) = φ(x(t), i(t−), t)

¦ Design of network communication logic

? Handling bandwidth limitations? Handling transmission delays

• Objective:

¦ Maintain closed-loop stability under failure situations

FEEDBACK CONTROLLER DESIGN

• Lyapunov-based nonlinear control law:

ui = −ki(x, ui,max)(LgiVi)T

¦ Example: bounded robust controller(El-Farra & Christofides, Chem. Eng. Sci., 2001; 2003)

. Controller design accounts for constraints.

• Explicit characterization of stability region:

®

­

©

ªΩi(ui,max) = x ∈ IRn : Vi(x) ≤ cmaxi & Vi(x) < 0

¦ Explicit guidelines for mode switchings

¦ Larger estimates using a combination of several Lyapunov functions

MODEL PREDICTIVE CONTROL

• Control problem formulation

¦ Finite-horizon optimal control:®

­

©

ªP (x, t) : minJ(x, t, u(·))| u(·) ∈ U∆, Vσ(x(t+ ∆)) < Vσ(x(t))

¦ Performance index:

J(x, t, u(·)) = F (x(t+ T )) +∫ t+T

t

[‖xu(s;x, t)‖2Q + ‖u(s)‖2R]ds

. ‖ · ‖Q : weighted norm. . Q, R > 0 : penalty weights.

. T : horizon length. . F (·) : terminal penalty.

¦ Same Vσ as that for boundedcontroller design.

¦ Bounded controller may pro-vide “good” initial guess.

k+i|k

futurepast

predicted state trajectory

computed manipulated inputtrajectory

prediction horizon

k k+1 k+p

...

x

HYBRID PREDICTIVE CONTROL(El-Farra et. al., Automatica, 2004; IJRNC, 2004; AIChE J., 2004)

maxu

Con

tol l

evel

x(t)

i

|u| < x(t) = Ax(t) + Bu (t).

Supe

rvis

ory

laye

r

Perofrmance objectives

MPCcontroller

"Optimality"

i=1i=2

stabilit

y region Ω (u )max

Bounded controller

"Constraints"

Switching logic

i = ?

Plan

t

• Switching logic:

ui(x(t)) =

Mi(x(t)), 0 ≤ t < T ∗

bi(x(t)), t ≥ T ∗

¨§

¥¦T ∗ = infT ∗ ≥ 0 : LfiVi(x) + LgiVi(x)Mi(x(T ∗)) ≥ 0

¦ Initially implement MPC, x(0) ∈ Ωσ(umax)

¦ Monitor temporal evolution of Vσ(xM (t))

¦ Switch to bounded controller only ifVσ(xM (t)) starts to increase

V(x)=Cmax

V(x)=C2

V(x)=C1

umaxΩ( )

x(T )

Bounded controller’s

Switching

x(0)

*

Active Bounded controlActive MPC

Stability region

SUPERVISORY SWITCHING LOGIC FOR FAULT-RECOVERY(El-Farra & Christofides, AIChE J., 2003)

• Basic mechanism for preserving closed-loop stability:

¦ Switching between failed & well-functioning configurations

• Limitations imposed by input constraints

¦ Stability regions of control configurations

• Switching policy: mode switching ensures fault-tolerance provided that

¦ State within the stability region of fall-back configuration at time of failure

x(Tf ) ∈ Ωj(uj,max)

Stability region for mode 1

x(0)

switching to mode 1"safe"

switching to mode 2"safe" (u )max1

Ω

Ω

Stability region for mode 2

max)(u2

t2-1

t1-2

Switching between stability regions

• Implications:

? Determines tolerable-failure times fora given reconfiguration strategy

? Determines selection of fall-back con-figuration for a given failure time

DESIGN & IMPLEMENTATION OF COMMUNICATION LOGIC

Unit 1 ...

Local

Controller

Plant

Supervisor

Other communication traffic

from other local controllers

Type of failure

Time of failure

Operating Conditions

Size of disturbance

Failure Information

Local

Controller

Unit 2 Unit n

Process

ModelProcess

Model

Plant

011010010101011010

01101001010101 01101001010101

Minimize unnecessary

communication costs (tradeoff

between network resources &

control performance)

Account for transmission delay

Objective:

Minimize effects of fault propagation

Fault-recovery Tasks

Design robust controller

Compute fault-recovery region

Incorporate delays

Select/activate fall-back config.

u = p(x,umax

,b)

:= (x,umax

,b, tfail)

APPLICATION TO CHEMICAL REACTORS

Coolant

Out

Coolant

In

F0, T

0, C

A0

F1, T

1, C

A1

Temperature

Sensor

CSTR 1

Composition

Analyzer

Coolant

Out

CSTR 2 F2, T

2, C

A2

Coolant

In

01101001010101101001010101000101111000101011101010100 01101001010101101001010101000101111000101011101010100

Temperature

Sensor Composition

Analyzer

01101001010101101001010101000101111000101011101010100010101010000011111101011010010100010101000101

Computer Computer

Plant

Supervisor

Local

Network

Local

Network

Plant Network

01101001010101101001010101000101111000101011101010100

F3, T

03, C

A03

• Process dynamic model:'

&

$

%

dCA1

dt=

F0

V1(CA0 − CA1 )− k0 exp

( −ERT1

)CA1

dT1

dt=

F0

V1(T0 − T1) +

(−∆Hr)

ρcpk0 exp

( −ERT1

)CA1 +

Q1(t)

ρcpV1

dCA2

dt=

F1

V2(CA1 − CA2 )− k0 exp

( −ERT2

)CA2 +

F3

V2(CA03 − CA2 )

dT2

dt=

F1

V2(T1 − T2) +

(−∆Hr)

ρcpk0 exp

( −ERT2

)CA2 +

Q2(t)

ρcpV2+F3

V2(T03 − T2)

FAULT-TOLERANT CONTROL PROBLEM FORMULATION

Coolant

Out

Coolant

In

F0, T

0, C

A0

F1, T

1, C

A1

Temperature

Sensor

CSTR 1

Composition

Analyzer

Coolant

Out

CSTR 2 F2, T

2, C

A2

Coolant

In

01101001010101101001010101000101111000101011101010100 01101001010101101001010101000101111000101011101010100

Temperature

Sensor Composition

Analyzer

01101001010101101001010101000101111000101011101010100010101010000011111101011010010100010101000101

Computer Computer

Plant

Supervisor

Local

Network

Local

Network

Plant Network

01101001010101101001010101000101111000101011101010100

F3, T

03, C

A03

• Control objective:

¦ Under normal operation: stabilize both reactors at unstable steady-states

¦ Under controller failure: preserve closed-loop stability of CSTR 2

• Candidate control configurations:

¦ Under normal operation: (Q1, Q2): |Q1| ≤ Qmax1 , |Q2| ≤ Qmax2

¦ Under failure conditions: (Q2, CA03): |Q2| ≤ Qmax2 , |CA03 − CA03s | ≤ CmaxA03

CLOSED-LOOP SIMULATION RESULTS? Closed-loop state & input profiles under well-functioning & failed controllers

(failure occurs at t = 5 min)

0 5 10 15 20 25 30

300

320

340

360

380

Time (hr)

Temp

eratu

re, T

1 (K)

0 5 10 15 20 25 303.5

3.6

3.7

3.8

3.9

4

Time (hr)

Reac

tant c

once

ntrati

on, C

A1 (k

mol/m

3 )

0 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8x 10

5

Time (hr)

Rate

of he

at inp

ut, Q

1 (KJ/h

r)

CSTR 1

0 50 100 150 200 250 300300

320

340

360

380

400

420

440

Time (hr)

Temp

eratu

re, T

2 (K)

0 50 100 150 200 250 3002.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Time (hr)

Reac

tant c

once

ntrati

on, C

A2 (k

mol/m

3 )

0 50 100 150 200 250 300−12

−10

−8

−6

−4

−2

0

2

4x 10

5

Time (hr)

Rate

of he

at inp

ut, Q

2 (KJ/h

r)

CSTR 2

CLOSED-LOOP SIMULATION RESULTS¦ Closed-loop state & input profiles for CSTR 2 when

? (Q1, Q2) configuration fails at t = 5 min

? (Q2, CA03) configuration activated (transmission of “disturbance bounds”over network – no delays)

0 0.2 0.4 0.6 0.8 1400

500

600

700

800

900

1000

1100

Time (hr)

Tem

pera

ture

, T2 (K

)

Activating fall−back controller

Failure (no delays)

0 0.2 0.4 0.6 0.8 1−12

−10

−8

−6

−4

−2

0

2x 10

5

Time (hr)

Rate

of h

eat i

nput

, Q2 (K

J/hr

)

Failure

Activating fall−back controller(no delays)

0 0.2 0.4 0.6 0.8 10

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (hr)

Rea

ctan

t con

cent

ratio

n, C

A2 (k

mol

/m3 )

Failure

Activating fall−back controller

(no delays)

0 0.2 0.4 0.6 0.8 11.6

1.8

2

2.2

2.4

2.6

Time (hr)

Activating fall−back controller (no delays)

Failure

Inle

t rea

ctan

t con

cent

ratio

n, C

A03 (k

mol

/m3 )

CLOSED-LOOP SIMULATION RESULTS

? Implementation of fault-tolerant control strategy over network with total delay(fault-detection/communication/actuator activation) of τD = 2 min & τD = 3.3 min

0 0.2 0.4 0.6 0.8 1200

300

400

500

600

700

800

900

1000

1100

Time (hr)

Tem

pera

ture

, T2 (K

)

τDelay

= 3.3 min

τDelay

= 2 min

Failure

0 0.2 0.4 0.6 0.8 1−3

−2

−1

00

x 106

Time (hr)

Rat

e of

hea

t inp

ut, Q

2 (KJ/

hr)

Failure

τDelay

= 2 min

τDelay

= 3.3 min

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

Time (hr)

Rea

ctan

t con

cent

ratio

n, C

A2 (k

mol

/m3 )

Failure

τDelay

= 2 min

τDelay

= 3.3 min

0 0.2 0.4 0.6 0.8 11.6

1.8

2

2.2

2.4

2.6

Time (hr)

Inle

t rea

ctan

t con

cent

ratio

n, C

A03

(km

ol/m

3 )

Failure

τDelay

= 2 min

τDelay

= 3.3 min

EFFECT OF DELAYS ON FAULT-TOLERANCE

¦ Tradeoff between:

? Network design (communication logic & delays)? Control system design (fault-recovery region, switching logic)

420 440 460 480 500 520 540 5601

2

3

4

Temperature, T2 (K)

Reac

tant

Con

cent

ratio

n, C

A2 (k

mol

/m3 )

Steady−state

(T2(0),C

A2(0))

Failure

Switching after τdelay

= 2 min

Fault−recovery region, CSTR 2

EFFECT OF DELAYS ON FAULT-TOLERANCE

¦ Tradeoff between:

? Network design (communication logic & delays)? Control system design (fault-recovery region, switching logic)

420 440 460 480 500 520 540 5601

2

3

4

Temperature, T2 (K)

Reac

tant

Con

cent

ratio

n, C

A2 (k

mol

/m3 )

Steady−state

(T2(0),C

A2(0))

Failure

Switching after τdelay

= 3.3 min

Fault−recovery region, CSTR 2

CONCLUSIONS

• Chemical process systems with:? Nonlinear dynamics ? Input constraints ? Control system failures

• Integrated approach for fault-tolerant control over communication networks:

¦ Design of constrained nonlinear feedback controllers:

¦ Design of fault-tolerant supervisory switching laws:

? Stability regions of control configurations

¦ Design of communication logic:

? Accounting for network resource limitations

? Effects of delays on fault-tolerance

• Approach brings together tools from Lyapunov and hybrid systems theory

• Application to two chemical reactors in series

ACKNOWLEDGMENT

• Financial support from NSF, CTS-0129571, is gratefully acknowledged