Predictive Controller Tuning

1
Predictive Controller Tuning Steady State
Target Selection A Covariance Assignment
Approach

• Donald J Chmielewski
• Illinois Institute of Technology
• Annual Meeting of the AIChE
• November 2000, Los Angeles, CA

2
Previous Work

• Tuning of Predictive Controllers
• - Cutler (1983)
• - Shridhar Cooper (1998)
• - Loeblein Perkins (1999)
• Steady State Target Selection
• - Muske Rawlings (1994)
• - de Hennin et. al. (1994)
• - Rao Rawlings (1999)
• - Loeblein Perkins (1999)

3
Output Envelope Prediction
-------------------
z
-------------------
w
---------------------
-------------------
t
t
Process
Outputs
Disturbance Inputs
Measurements
Sensor Noise
Controller
State Estimator
--------------------

u
---------------------
---------------------
v
--------------------
t
t
4
Example Pre-Heated Reactor
O2 out

• Manipulated Variables
• Reactant Feed Rate (F)
• Fuel Feed Rate (Ff)
• Vent Position (V)

CO out
Vent Position
Furnace
To , F
TR
TF
CSTR

Fuel Feed

• Control Variables
• Reactor Temperature (TR)
• Furnace Temperature (TF)
• Furnace Oxygen (O2)
• Furnace CO (CO)

• Disturbance Input
• Feed Temperature (To)

5
Infinite Horizon Predictive Control
8
S

• Min xT(k)Qx(k) uT(k)Ru(k)
• s.t. x(k1) A x(k) B u(k)
• xi (k) lt xi
• ui (k) lt ui

k 0
_
_
T
T
Unconstrained Solution
P (A-BL) P(A-BL) L RL Q L (B PB R) B
PA
T
T
-1
6
Open Loop Response
7
Closed Loop Response
8
Closed Loop Response
9
Covariance Analysis

• Closed Loop Process
• x(k1) (A BL)x(k)
  Gw(k)

• Covariance of the State
• Sx (A-BL) Sx (A-
  BL)

T
T
G SwG

• Covariance of the Control Input

T
Su L Sx L
T
10
Steady State Covariance of Scalar Signals

• State Variables
• lim E ( xi (k) )2 ei Sx ei
• Input Variables
• lim E ( ui (k) )2 ei Su ei ei LSx Lei
• where e i is the ith Row of the Identity
  Matrix

T
8
k
T
T
T
8
k
11
Covariance Bounded Design
There Exists L s.t. ei ?x eiT lt xi2 and eiL
?x LTeiT lt ui2 If and Only If There Exits X gt
0 and Y s.t.
eiXeiT lt xi2
X-AXATBYAT BY AYTB-G ?wGT BTYT X
ui2 eiY YTeiT X
gt 0 and
gt 0
-1
One such L is given by YX
12
Selection of Covariance Bounds
xi
Xi
max
ui
Ui
s u
--------------------------------------------------
--------------------------------------------------
-
--------------------------------------------------
-------
s x
min
ui
xi
max
xi
min

• xi min( xi - xi ), ( xi - xi
  )
• ui min( ui - ui ), ( ui - ui
  )

max
min
SS
SS
max
min
SS
SS
13
Selection of Covariance Bounds
xi
Xi
max
ui
s u / 2
Ui
--------------------------------------------------
--------------------------------------------------
-
--------------------------------------------------
-------
s x
2
min
ui
xi
max
xi
min

• xi min( xi - xi ), ( xi - xi
  )
• ui min( ui - ui ), ( ui - ui
  )

max
min
SS
SS
max
min
SS
SS
14
Covariance Bounded Tuning

• Covariance Bounded Synthesis
• Given Sw , Dxi s Dui s
  L
• Tuning of Predictive Controller
• Given Sw , Dxi s Dui s
  Q R

15
Covariance Bounded LQR Design

• There Exists Q gt 0 R gt 0 s.t.
• ei Sx ei lt xi i 1n
• ei LSx Lei lt ui i 1m
• where Sx (A-BL) Sx (A- BL) G Sw G
• P (A-BL) P(A-BL) L
  RL Q
• L (B PB R) B PA

2
T
T
T
2
T
T
T
T
- 1
T
T
If . . .
16
There Exists X gt 0 Y gt 0 s.t. ei
A X (A ) ei lt xi i 1 .. n
ui eiYB BYei
X X - BYB gt 0 X - AXA
ABYBA gt 0 X - AXA - AGSwG A 2ABYB A
ABYB
BYB A
X
- 1
-1
T
T
2
T
2
gt
0
i 1 .. m
T
T
T
T
T
T
T
T
T
T
T
gt
0
T
T
17

• Give a pair (X, Y) that satisfy the previous
  Linear Matrix Inequalities (LMIs).
• Then,
• Q (X - BYB ) - A X A
• R Y
• will yield the Covariance Bounded LQR

- 1
- 1
T
T
-1
18
Minimum Covariance Design
ei A Y (A ) ei lt sxi
sxi lt Dxi2 sui
eiYB sui lt Dui2 BYei
X X - BYB gt 0 X - AXA
ABYBA gt 0 X - AXA - AGSwG A 2ABYB
A ABYB
BYB A
X
min S
cxi sxi cui sui
sxi sui
- 1
-1
T
T
T
gt
0
T
T
T
T
T
T
T
T
T
T
T
gt
0
T
T
19
Closed Loop Response ( Minimized Temperature
Covariance )
20
Closed Loop Response ( Minimized Reactant Feed
Covariance )
21
Steady State Target Selection

• Minimum Covariance Design suggests that
• Profit a ( xi ui )
• Real Time Optimization Employs
• Profit a ( xi ui )

2
2
22
Covariance Based Target Selection
ei A Y (A ) ei lt sxi
sxi lt Dxi2 sui
eiYB sui lt Dui2 BYei
X X - BYB gt 0 X - AXA
ABYBA gt 0 X - AXA - AGSwG A 2ABYB
A ABYB
BYB A
X
min S
1/2
1/2
cxi (sxi ) cui ( sui )
sxi sui
- 1
-1
T
T
T
gt
0
T
T
T
T
T
T
T
T
T
T
T
gt
0
T
T
23
Acknowledgments

• Michael J.K. Peng
• Armor College of Engineering, IIT
• Department of Chemical Environmental
  Engineering, IIT

24
Covariance Bounded LQR Design

• There Exists Q gt 0 R gt 0 s.t.
• ei Sx ei lt xi i 1n
• ei LSx Lei lt ui i 1m
• where Sx (A-BL) Sx (A- BL) G Sw G
• P (A-BL) P(A-BL) L
  RL Q
• L (B PB R) B PA

2
T
T
T
2
T
T
T
T
- 1
T
T
If and Only If. . .
25
There Exists X gt 0 , Y gt 0 and Z s.t.
ei Xei xi i 1 ..
n ui ei Z Z ei
X X - GSwG
AX -BZ (AX-BZ) X
X (AX - BZ)
Z AX - BZ X
0 Z 0
Y
T
2
gt
2
gt
0
i 1 .. m
T
T
T
gt
0
T
T
gt
0
T
26

• Give a triple (X,Y, Z) that satisfy the previous
  Linear Matrix Inequalities (LMIs).
• Then,
• Q X (AX-BZ) X (AX-BZ) - X ZYZX
• R Y
• will yield the Covariance Bounded LQR

- 1
- 1
T
T
-1
-1
-1