MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator (EVPS)
group seminar on Oct. 19th, 2000
[presenter]
[advisor]
[team partner]
[project sponsor]
ARG ALLEYNE RESEARCH GROUP
Rong Zhang
Prof.. Andrew Alleyne
Eko Prasetiawan
Caterpillar
Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Overview
❑1.
❑2.
❑3.
❑4.
❑5.
ARG
Problem statement
Introduction to LQG/LTR control
EVPS LQG/LTR design
EVPS LQG/LTR performance
Conclusions
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
1. Problem statement
◼
❑
❑
❑
❑
1. Problem statement
2. Introduction to LQG/LTR control
3. EVPS LQG/LTR design
4. EVPS LQG/LTR performance
5. Conclusions
❑Introduction to the Earthmoving Vehicle
Powertrain
An analogy between passenger vehicle
powertrain and EVP
❑EVPS schematic and I/O list
❑Need for coordination
A tracking example
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
An analogy
ARG
Passenger Vehicle
Earthmoving Vehicle
•Prime mover:
Usually Spark-Ignition type engine
(gas)
•Prime mover:
Usually Compression-Ignition
type engine (diesel)
•Torque Converter:
Mechanical gearbox
•Torque→pressure converter:
Hydraulic pump
•Resistance speed control:
Brake
•Resistance speed control:
Flow valve
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
EVPS schematic...
4
3
1
5
2
3
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
… and I/O list
A MIMO control system
Controlled outputs: load speeds (3)
ARG
Components Engine
Pump
Loads
5 Inputs
throttle(1)
disp.( 1)
flow valve (3)
9 Outputs
speed (1)
disp. (1)
pressure(1)
speeds(3)
pressures (3)
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Need for coordination!
❑A tracking example
Tracking references
Speed Response of open loop system to Flow Valve#1 variation
100
50
dnm1(rpm)
Node 1: A rising step
Node 2: 0
Node 3: 0
nm1(rpm)
0
-50
-100
5
6
7
8
9
10
11
12
13
14
15
10
11
12
13
14
15
10
time(sec)
11
12
13
14
15
100
dnm2(rpm)
Using only one input:
Flow Valve 1...
nm2(rpm)
50
0
-50
-100
5
6
7
9
nm3(rpm
)
100
50
dnm3(rpm)
8
0
-50
-100
5
6
7
8
9
data0724_plot1018.m@18-Oct-2000 14:35:10
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
❑5 control inputs without coordination...
Step Mission
1
2
3
4
To control nm1  to
reference
To bring nm2 and
nm3 back to 0
Actions
Results
uvalve1 
But nm2 and nm3
uvalve2  , uvalve3

But uvalve1  because of
shared flow
uengine  , upump  But nm2 and nm3 as
To increase flow
supply, forcing nm1
well
back to reference
......
To bring nm2 and A big mess!
nm3 back to 0? ... ......
[Q] How to take actions at the right time, right direction and right amount?
[A] Coordination needed !
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
2. Introduction to LQG/LTR control
❑Pole placement?
“Performance” vs. “Cost”
◼
◼
❑
❑
❑
1. Problem statement
2. Introduction to LQG/LTR control
3. EVPS LQG/LTR design
4. EVPS LQG/LTR performance
5. Conclusions
❑LQR controller -- “optimal” feedback law
❑LQR estimator -- “optimal” filter by Kalman
❑LQG controller design
Optimal controller + Optimal estimator
❑LQG/LTR controller design
Optimal + Optimal  Robustness
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Pole placement?
x = Ax + Bu 
y = Cx + Du   eig ( A − BK )

u = − Kx 
Poles will be here!
[Q1] Where should the target poles be placed?
Too slow? poor performance!
Too fast? expensive controller and surprising power bill!
[Q2] Is there an “optimal” controller balancing both
Performance and Cost?→ “Punishment philosophy”
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
LQR controller
In this method, pole locations are not designed directly. Instead, find a good
u=-Kx that minimizes:

(
)
J =  x T Qx + u T Ru dt , Q  0, R  0
0
Q and R are Performance Index or “Punishment Matrices”
•Want a quicker state convergence? make Q bigger to punish large states!
•Want to keep control efforts within saturation range or at a lower cost?
make R bigger to punish overacting inputs!
[Solution]
Theoretical: ARE equation finds us a good K
Practical: Matlab command ‘lqr’
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
LQR estimator
Not all the states are available, how to construct them from y? An estimator
xˆ = Axˆ + Bu + L( y − Cxˆ )

x = Ax + Bu
  e = ( A − LC )e

e = xˆ − x

Find a good L(ue=-LCe) that minimizes:

(
)
J =  e T Qe e + u e Re u e dt , Qe  0, Re  0
T
0
If Qe and Re are determined by process and
measurement noise level...
A Kalman Filter!
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[Solution]
Theoretical: ARE equation
Practical: Matlab command
‘lqr’ ‘kalman’
Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
LQG = LQR control + Kalman filter
❑ It’s a Optimal + Optimal design, but is it “optimal” in
the sense of robustness? No!
LQG/LTR
= LQG + Robustness recovery
❑ Using a recovery procedure (r=0 to inf), to make the
LQG closed-loop closer to that of the Target Loop:
the ideal LQR loop with full-state feedback.
[Solution]
Theoretical: Loop Transfer Recovery procedure
Practical: Matlab command ‘ltru’ ‘ltry’
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Optimal + Optimal  Robustness
LQR with Full-state feedback
LQG with measurements feedback
S in g u la r V a lu e s
SV BODE PLOT --- LQG/LTR (recov. gain --->0)
100
80
Singular Value Bode Plot
100
Singular Value Bode Plot
☺
80

40
60
SV - db
Singular Values (dB)
60
20
40
0
-2 0
20
-4 0
0
-6 0
-20
-4
10
-8 0
10
-4
10
-2
10
0
10
2
10
4
F re q u e n c y (ra d / s e c )
-3
10
-2
10
-1
10
0
1
10
10
Frequency - Rad/Sec
2
3
10
10
4
10
r = 0 (no recovery)
ARG
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Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
S in g u la r V a lu e s
100
80
Loop transfer recovery...
60
Singular Values (dB)
40
20
0
-2 0
-4 0
-6 0
S V B O D E P L O T --- L Q G / L TR (re c o v. g a in ---> 1 )
Closer to the target loop
100
100
Singular Value Bode Plot
80
SV - db
SV - db
10
-2
10
0
10
2
10
4
☺
40
20
20
0
0
-2 0
-2 0
-4 0
-4 0
-6 0
-6 0
-8 0
-8 0
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
F re q u e n c y - R a d / S e c
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-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
F re q u e n c y - R a d / S e c
r = 1 (small recovery)
ARG
-4
F re q u e n c y (ra d / s e c )
60
40
10
10
Singular Value Bode Plot
80

60
-8 0
S V B O D E P L O T --- L Q G / L TR (re c o v. g a in ---> 1 0 0 0 0 0 )
r = 105 (large recovery)
15
Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
3. EVPS LQG/LTR design
◼
◼
◼
❑
❑
1. Problem statement
2. Introduction to LQG/LTR control
3. EVPS LQG/LTR design
4. EVPS LQG/LTR performance
5. Conclusions
❑Plant Model (14 states) to Design Plant
Model (17 states)
To insure 0 tracking errors to step inputs, the PM
is augmented by 3 free integrators.
❑LQG design
Good “Punishment Matrices” are found and tested
❑LQG/LTR design
Robustness or the ideal LQR is recovered
ARG
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
uv1
F lo w V a lv e
 r ef
D p = K p
S w as h - pl ate
Lo a d Uni t # 1
D y n a m ic s
Qv
D y n a m ic s
 Po
Pu

Pd
Lo ad
T m, l
D yn am i c s
Te ,l

+
P rim e
M o ve r
ne
Qp
-
U ps tr e a m

H o se
-
Flo w
P
+
Qv +
V al v e
-
T m, i
D o w n s tr ea m
H o se
-
Dm
+
H y d ra ul ic
M o to r
n m1
n
 Q vi
i =1
+
+
...
...
Q vi
...
Qv n
Qm
Dm
nm
+ Q v1
Pu
uvi
L o a d U n it # i
nmi
...
uv n
Pu
Lo a d Uni t # n
nm n
Plant Model
A14x14
EVPS System
Design Plant Model
A17x17
Three 1/s’ added to insure 0 tracking error
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Kc
+
nm,ref
nm,ref
0143 
I 
 33 
179
xI

-
C LTR
Ki
+
x̂
+
BLTR
y
-
LQG/LTR
Controller
1
s
LQG Controller
Kp
u
+
Plant
y
u
517
+
n̂m
Estimator
ALTR
1717
nm,ref
(3)
LQG/LTR
Controller
(17 states)
(5)
u
Plant
(14 states)
LQG/LTR Controller
(9)
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Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
4. EVPS LQG/LTR performance
❑Simultaneous tracking
◼
◼
◼
◼
❑
1. Problem statement
2. Introduction to LQG/LTR control
3. EVPS LQG/LTR design
4. EVPS LQG/LTR performance
5. Conclusions
• Different nodes track different speed references
• The total flow demand changes
❑Disturbance rejection
• One of the 3 nodes is subject to a pressure disturbance
• The TOTAL flow demand does not change
• The distribution of pressures among the 3 nodes is changed
ARG
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Simultaneous speed-tracking
Speed Response of LQG/LTR Tracking, r=1000
dnm1(rpm)
100
0
nm1(rpm)
-50
-100
15
10
5
0
nm2(rpm)
100
dnm2(rpm)
•nm1
tracking +/- 100rpm reference
50
50
•nm2
being regulated
0
-50
-100
nm3(rpm
)
100
dnm3(rpm)
15
10
5
0
50
0
-50
-100
0
15
10
5
•nm3
tracking - 60rpm reference
time(sec)
data1016 tr1000track@18-Oct-2000 16:36:32
Opposite direction
Same direction
l
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Pressures of simu-tracking
Pressure Response of LQG/LTR Tracking, r=1000
dp1(MPa)
1
•pd1
increased to push through more flow
0.5
0
pd1(MPa)
-0.5
-1
0
15
10
5
dp2(MPa)
1
0.5
•pd2
unchanged to maintain the same flow
0
pd2(MPa)
-0.5
-1
0
5
10
15
10
•pd3
decreases to push through less flow
15
dp3(MPa)
1
0.5
0
pd3(MPa)
-0.5
-1
0
5
time(sec)
data1016ltr1000track@18-Oct-2000 17:13:46
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Control inputs of simu-tracking
throttle(deg)
Control Inputs of LQG/LTR Tracking, r=1000
50
•Throttle when total flow demand
45
40
0
5
10
15
Dp(V)
5.5
•Pump when total flow demand
5
4.5
0
5
10
15
uv1(V)
5
•Flow 1 when speed reference 1
4
3
0
5
10
15
uv2(V)
5
4
3
0
5
10
15
0
5
10
15
•Flow 2 compensates for pressure
resulted from total flow
uv3(V)
5
4
3
•Flow 3 when speed reference 3
time(sec)
data1016ltr1000track@18-Oct-2000 16:15:17
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Pressure disturbance at node 1
Pressure Response of LQG/LTR Disturbance Rejection, r=1000
dp1(MPa)
1
0.5
Pressure step as disturbance
is applied at node 1 only
0
pd1(MPa)
-0.5
-1
0
5
10
15
10
15
10
15
dp2(MPa)
1
0.5
0
pd2(MPa)
-0.5
-1
0
5
dp3(MPa)
1
0.5
Neighbor node pressure
doesn’t change significantly
0
pd3(MPa)
-0.5
-1
0
5
time(sec)
data1016ltr1000dist@18-Oct-2000 17:14:14
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Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Speeds of disturb. rejection
Speed Response of LQG/LTR Disturbance Rejection, r=1000
15
•nm1
decreases when disturbance pressure
squeezes out some flow; then
regulated by the controller
15
•nm2
increases by pressure disturbance
squeezes in some flow from neighbor
node; then regulated by the controller
dnm1(rpm)
100
50
0
nm1(rpm)
-50
-100
0
5
10
dnm2(rpm)
100
50
0
nm2(rpm)
-50
-100
0
5
10
dnm3(rpm)
100
50
0
-50
-100
0
5
nm3(rpm
)
10
15
•nm3
increases by pressure disturbance
squeezes in some flow from neighbor
node; then regulated by the controller
time(sec)
data1016ltr1000dist@18-Oct-2000 17:14:08
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Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
Control inputs of disturb. rejection
total flow demand not changed!
throttle(deg)
Control Inputs of LQG/LTR Disturbance Rejection, r=1000
50
45
40
0
15
10
5
Dp(V)
5.5
•Pump doesn’t need to change much
5
4.5
•Throttle compensates for small total
pressure
0
15
10
5
uv1(V)
5
•Flow 1 to fight disturbance pressure
4
3
0
5
10
15
0
5
10
15
0
5
10
15
uv2(V)
5
4
3
uv3(V)
5
4
3
•Flow 2 compensates for upstream
pressure caused by load 1
•Flow 3 compensates for upstream
pressure caused by load 1
time(sec)
data1016ltr1000dist@18-Oct-2000 17:13:59
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Rong Zhang
MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
◼
◼
◼
◼
◼
5. Conclusions
1. Problem statement
2. Introduction to LQG/LTR control
3. EVPS LQG/LTR design
4. EVPS LQG/LTR performance
5. Conclusions
❑ An LQG/LTR MIMO controller is successfully designed
and implemented
 The system: 14 states, 9 measurements, 5 inputs
 The design plant model with free integrators: 17 states
 The LQG/LTR controller: 17 states, 9 inputs, 5 outputs
❑ It has satisfying tracking and disturbance rejecting
performance
❑ It’s robustness and working range are subject to
further validation
❑ Model reduction technique will be used to simplify the
controller
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
References
 M. Athans, "A tutorial on the LQG/LTR method," presented at
American Control Conference, Seattle, WA, 1986. A quick start.
 B. D. O. Anderson and J. B. Moore, Optimal Control, Linear
Quadratic Methods. Eaglewood Cliffs, New Jersey: Prentice-Hall,
1990. A textbook.
 J. C. Doyle and G. Stein, "Multivariable Feedback Design: Concepts
for a Classical/Modern Synthesis," IEEE Trans. Automat. Contr., vol.
AC-26, pp. 4-16, 1982. A classic.
 A. Saberi, B. M. Chen, and P. Sannuti, Loop Transfer Recovery:
Analysis and Design. London: Springer-Verlag, 1993. A monograph.
 Matlab manual online “Robust Control Toolbox” at:
http://www.mathworks.com/access/helpdesk/help/pdf_doc/robust/
robust.pdf A useful tool.
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MIMO LQG/LTR Control
for the Earthmoving Vehicle Powertrain Simulator
An earthmoving vehicle powertrain
Control
2
Hydr.
Pump
5
4
Steering
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1
Engine
3
Drive
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