Predictive control algorithms in constrained control systems

Predictive control algorithms in constrained control

systems tolerating sensor faults

Piotr Marusak

Institute of Control and Computation Engineering

Warsaw University of Technology

Plan of presentation

1. Introduction

2. Predictive algorithms

2.1. Analytical controllers

2.2. Numerical controllers

3. Including information about sensor faults in predictive algorithms,

in control systems with constrained outputs

3.1. Shift of constraints

3.2. Change of the set–point value

3.3. The case of operation in the multi–layer control system

structure

4. Summary

Introduction

• The fault tolerant control systems can offer the possibility of

maintaining continuous control system operation despite a fault

that occurred (till the failure is fixed)

• The loss of measurement means the interruption of the feedback

loop

— unstable operating point: guide the process to the region of

safe operation

— stable operating point: continue operation in the acceptable

way

• Taking failures into consideration is relatively easy in predictive

control algorithms

The idea of the predictive control

ucalculated

k k+1 k+p timek–1

yset–point

ypredicted

k+s

past future

∆uk

Fig. 1. Idea of predictive control; p – prediction horizon, s – control

horizon, ∆uk – control signal change at current iteration

Analytical predictive control algorithms

The basic idea of the predictive algorithms is to minimize the

following performance index:

A unique solution:

where

Only the elements of the vector ∆∆∆∆u are used at each time step.

The control law can be obtained.

( ) ( )∑∑∑∑=

−

=

+

= =

+ ∆⋅+−⋅=io l

j

s

i

j

kikj

l

j

p

i

j

kik

j

kj uyyJ1

1

0

2

|

1 1

2

| λκ

( ) ( )yyκAλAκAu ~1−⋅⋅⋅+⋅⋅=

− TT∆∆∆∆

[ ]T

||1

2

|

2

|1

1

|

1

|1 ,,,,,,,,,~ oo l

kpk

l

kkkpkkkkpkkk yyyyyy++++++

=⋅+= KKKK∆uAyy

j

kku |∆

Analytical predictive control algorithms

The basic idea of the predictive algorithms is to minimize the

following performance index:

A unique solution:

where

Only the elements of the vector ∆∆∆∆u are used at each time step.

The control law can be obtained.

( ) ( )yyκAλAκAu ~1−⋅⋅⋅+⋅⋅=

− TT∆∆∆∆

[ ]T

||1

2

|

2

|1

1

|

1

|1 ,,,,,,,,,~ oo l

kpk

l

kkkpkkkkpkkk yyyyyy++++++

=⋅+= KKKK∆uAyy

j

kku |∆

( ) ( )

⋅⋅+−⋅⋅−= ∆uλ∆uyyκyy TT J

Fuzzy analytical algorithms

• The controller is a combination of many sub–controllers

• Parameters of sub–controllers are derived beforehand

• Output value of the whole controller is a sum of weighted

outputs of local controllers

Fig. 3. Block diagram of the analytical fuzzy predictive controller

~

~

~Predictivecontroller no 1.

∆ukPredictivecontroller no i.

Predictivecontroller no l.

w1

wi

wl

+

∆uk–j, yk–j, ek

uk–j, yk–j

lku~∆

u

fuzzy reasoning

iku~∆

u

1ku~∆

u

Taking constraints into consideration in analytical algorithms

Fig.4 . Block diagram of the control system with analytical predictive controller

and constraints included in the controller

Constraints put on:

ykky

y+

–

+∆uk uk

Plant∆u u∆u u

Analyticalcontroller

+

z–1

actuatoractuatormodel

• control changes

If ∆uk < ∆umin, then

∆uk = ∆umin.

If ∆uk > ∆umax, then

∆uk = ∆umax.

• control values

If uk–1 + ∆uk < umin, then

∆uk = umin – uk–1.

If uk–1 + ∆uk > umax, then

∆uk = umax – uk–1.

∆∆∆∆umin ≤ ∆∆∆∆u ≤ ∆∆∆∆umax, umin ≤ u ≤ umax, ymin ≤ y ≤ ymax,

• In a nonlinear case, in order to avoid problems connected with

general nonlinear optimization, effective algorithms with model

linearization and quadratic optimization are used

• A few such algorithms are available, so the algorithm most

suitable for a given nonlinear plant can be selected and

a compromise between control performance and computation

demand can be achieved

Numerical predictive control algorithms

Following problem is solved at each iteration:

subject to the constraints:

( ) ( )

⋅⋅+−⋅⋅− uλuyyκyyu

∆∆∆∆∆∆∆∆∆∆∆∆

TT min

Basic approach to sensor fault accommodation

• Control the loop in which the fault occurred in the open–loop

configuration (feedforward control)

— in practice: calculation of the free response using predicted

instead of measured value of the output with damaged

measurement

— the problem: the disturbances acting on the control plant

will not be compensated on the output with broken

measurement

• Use of the disturbance measurements is crucial

Control plant I (rectification column)

y1 – methanol concentration in a distillate,

y2 – methanol concentration in an effluent,

u1 – flow of a reflux,

u2 – flow of the steam into a boiler,

u3 – feed flow

),(

12,13

9,419,14

8,3

)(

)(

14,14

4,19

19,10

6,6

10,21

9,18

17,16

8,12

)(

)(3

4

2

1

48

4

2

1sU

s

s

e

sU

sU

s

e

s

e

s

e

s

sY

sY

s

ss

s

⋅

+

++

⋅

+

−

+

+

−

+=

−

−−

−

Fig. 5. Responses of the control system to the set–point change and the change of the

disturbance u3; continuous line – failure of the y1 sensor in 80th minute,

dotted line – normal control system operation

Constraints put on output variable values

• Change (shift) of constraints in the predictive controller

ymin + rmin ≤ y ≤ ymax – rmax

• Change of the set–point value

• Modification of the set–point values by the optimization layer

Fig. 6. Responses of the control system to the set–point change and the change of the

disturbance u3; failure of the y1 sensor in 80th minute; dotted line – before,

solid line – after the shift of constraints

Fig. 7. Responses of the control system to the set–point change and the change of the

disturbance u3; failure of the y1 sensor in 80th minute; continuous line – set–point

change to 0.95, dotted line – shift of constraints after detection of the sensor failure

Operation in multi–layer control system structure

Economic optimization problem

subject to

• Exact nonlinear plant static model

• Desired approach when an analytical predictive controller is used

),(min uyy

Je

maxmin uuu ≤≤

maxmaxminmin ryyry −≤≤+

Operation in multi–layer control system structure

Economic optimization problem

subject to

• Possible modification after sensor fault detection:

),(min uyy

Je

maxmin uuu ≤≤

maxmaxminmin ryyry −≤≤+

,minminmin

jjjcrr +=

jjjcrr maxmaxmax +=

* R.B. Newell, P.L. Lee: Applied process control – a case study; Prentice Hall, 1989

Fig. 8. Evaporator system

Output Variables

L2 – separator level,

X2 – product composition,

P2 – operating pressure

Manipulated variables

F2 – product flowrate,

P100 – steam pressure,

F200 – cooling water flowrate

Control plant II (evaporator system*)

ProductF2, X2, T2

Evaporator

Steam

F100 P100T100

P2

Condensate

FeedF1, X1, T1

SeparatorL2

Coolingwater

F200, T200

Condenser

T201

F4, T3

F5

F3

LC

• Economic performance index (cost of production):

• Constraints:

P100 ≤ 400 kPa, F200 ≤ 400 kg/min,

25 % ≤ X2

• Measured disturbance F1 (feed flow)

F1 = F10 + F1a ⋅sin(2⋅π⋅t/To),

F10 = 10 kg/min, F1a = 0.4 kg/min, To = 400 min

2100 21 FcPcJe ⋅−⋅=

Fuzzy predictive controller

Fig. 9. Membership functions of the FDMC controller

P2k

fp(P2k)

40 60

0

1

X2

k

fp (X2

k )15

35

0 1

Area 4

Area 2Area 1

Area 3

ZP1 ZP2

ZX

1Z

X2

Fig. 10. Responses to the change of F1 by 0.3 kg/min of the control system

without and with the optimization layer employed;

failure of the X2 sensor occurred in the 150th minute of simulation;

above: output signals, below: control signals

Fig. 11. Responses of the control system to the changes of the disturbance F1 signal;

constraints in the optimization layer not changed, changed once, changed according to

the needs after detection of the X2 sensor failure that occurred in the 150th minute;

above: output signals, below: control signals;

Je=1353.5552, Je=−932.9755, Je=−946.7594

Summary

• Effective and relatively little complicated methods of sensor

fault toleration in control systems with predictive controllers

were introduced

• Described approaches are simple and easy to implement

• The methods can be used in cases of both: analytical and

numerical algorithms based on linear and nonlinear plant models

• The presented examples of methods application in the control

systems of MIMO plants illustrate their efficiency