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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
