Robust Nonlinear Observer for a
Non-collocated Flexible System
Mohsin Waqar
M.S.Thesis Presentation
Friday, March 28, 2008
Intelligent Machine Dynamics Lab
Georgia Institute of Technology
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
2
Problem Statement
•Examine the usefulness of the Sliding Mode Observer as part
of a closed-loop system in the presence of non-collocation and
model uncertainty.
3
Non-Minimum Phase Behavior
Causes:
Combination of non-collocation of actuators and sensors and the flexible
nature of robot links
Detection:
•System transfer function has positive zeros.
Effects:
•Limited speed of response.
•Initial undershoot (only if odd number of pos. zeros).
•Multiple pos. zeros means multiple direction reversal in step
response.
•PID control based on tip position fails.
•Limited gain margin (limited robustness of closed-loop system)
•Model inaccuracy (parameter variation) becomes more troubling.
4
Control Overview
Noise
V
+
Commanded
Tip Position
u
Feedforward
Gain F
Linear
Motor
F
Flexible
Link
δ
y
Sensors
-
Feedback
Gain K
Control objective:
x̂
Observer
Accuracy, repeatability and steadiness of
the link tip.
5
Test-Bed Overview
R
PCB 352a
Accelerometer
PCB
Power Supply
Anorad Encoder
Readhead
Anorad
Interface Module
LV Real Time 8.5
Target PC
w/
NI-6052E DAQ
Board
C
LS7084
Quadrature
Clock Converter
NI SCB-68
Terminal
Board
+
+
-
Anorad DC Servo 160VDC
Amplifier
PWM
Linear
Motor
-10 to +10VDC
6
Flexible Link Modeling – Assumed Modes Method
c
E, I, ρ, A, L
m
F
w(x,t)
x
A Few Key Assumptions:
•3 flexible modes + 1 rigid-body mode
•Undergoes flexure only (no axial or torsional displacement)
•Horizontal Plane (zero g)
•Light damping (ζ << 1)
•Only viscous friction at slider
7
Flexible Link Modeling – Assumed Modes Method
Mq  Cq  Kq  Q
E, I, ρ, A, L
F
 K  M   0

q  
  T Q  T C  [diag ( 2 )]


T
w(x,t)
x
EI

 AL4

m
M 
1
2
x  Ax  Bu
 w( x  0, t ) 
y
   x    Cx  Du

 w( x  L, t ) 
 x1  1 
 x   
 2  2
 x3  3 
   
 x4    4 
 x5  1 
   
 x6   2 
 x   
 7  3
 x8   4 
8
Flexible Link Model vs Experimental
Experimental Data
AMM Model Data
Tip Mass (kg)
0.110
0.25
Length (m)
0.32
0.48
Width (m)
0.035 (1 3/8”)
0.04
Thickness (m)
.003175 (1/8”)
0.0024
Material
AISI 1018 Steel
Not Applicable
Density (kg/m3)
7870
9838
Young’s Modulus (GPa)
205
205
First Mode (Hz)
5.5
5.7
Second Mode (Hz)
49.5
49.0
Third Mode (Hz)
130.5
219.3
9
Flexible Link Modeling – Lumped Parameter Model
 0
 k

 m
x 1
 0
 3k

 m2
c
m2


1
0
c
m1
k
m1
0
0
3c
m2

3k
m2
0 
 0 

c
 0 
 
 
m1 
x 0 F
1 
 
 3 
3c 
 m2 
 
m2 
m1
J2
F
Model Data
k
y2
y1
Tip Mass (kg)
0.110
Base Mass (kg)
20
Stiffness (N/m)
131.4
Damping (N-s/m)
0.04
Resulting First Mode (Hz)
5.5
Resulting Positive Zero
3.06e3
10
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
11
Steady State Kalman Filter - Overview
Why Use?
•Needed when internal states are not measurable directly (or costly).
•Sensors do not provide perfect and complete data due to noise.
•No system model is perfect.
Notable Aspects:
•Optimal estimates (Minimizes mean square estimate error)
•Predictor-Corrector Nature
•Designed off-line (constant gain matrix) and reduced computational burden
•Design is well-known and systematic
12
SteadyHow
Stateit Kalman
works - Kalman
Filter – How
Filterit works
Plant Dynamics
Measurement & State Relationships
Noise Statistics
Kalman
Filter
State Estimates
with minimum
square of error
Initial Conditions
Filter Parameters:
Noise Covariance Matrix Q
– measure of uncertainty in plant. Directly
tunable.
Noise Covariance Matrix R
– measure of uncertainty in measurements.
Fixed.
Error Covariance Matrix P
– measure of uncertainty in state estimates.
Depends on Q.
Kalman Gain Matrix K
– determines how much to weight model
prediction and fresh measurement. Depends
on P.
13
Steady State Kalman Filter – How it works
r
u
F
-
x
B
+
Filter Design:
1/s
y
v
+
C
A
1. Find R and Q
1a) For each measurement,
find μ and σ2 to get R
+
K
-
1b) Set Q small, non-zero
2. Find P using Matlab CARE fcn
3. Find K=P*C'*inv(R)
4. Observer poles given by
eig(~A-LC)
5. Tune Q as needed
x̂
+
B
+
1/s
ŷ
C
~A
Kc
14
Steady State Kalman Filter – How it works
r
Observer dynamic equation:
ˆ ˆ  Bu  K ( yˆ  y )
xˆ  Ax
F
-
u
B
+
1/s
xC
v
y +
A
L
B
+
+
1/s
x̂
C
ŷ
~A
K
Closed-loop system with observer:
 xˆ   A  BK C  KC
 
 BK C
 x 
 u    KC
 yˆ   C  DK
C
  
 y    DKC
KC   xˆ   BK C
 
A   x   BK C
0
 KC
ˆ
 x
0      DKC
x

 DKC
C 
K  r 
0  v 
0
r 
0  
v

I 
15
+
-
Steady State Kalman Filter – A Limitation
Example:
Given a second order dynamic system with a single
measurement,
 x1   x2 
x    f 
 2  
y  x1
Then the Kalman filter in presence of parametric uncertainty is given
by
 xˆ1   xˆ2  K1 x1 
  ˆ

 xˆ2   f  K 2 x1 
And the observer error dynamics are given by
 x1   x2  K1 x1 
 


f

K
x
x
2 1
 2 
f  f  fˆ
16
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
17
Sliding Mode Observer – Lit. Review
•Walcott and Zak (1986) and Slotine et al. (1987) – Suggest a
general design procedure based on variable structure systems (VSS)
theory approach. Simulations show superior robustness properties.
•Chalhoub and Kfoury (2004) – Use VSS theory approach.
Simulations of a single flexible link with observer in closed-loop show
superior robustness properties.
•Kim and Inman (2001) – Use Lyapunov equation approach. Superior
robustness properties shown by simulations and experimental results of
closed-loop active vibration suppression of cantilevered beam (not a
motion system).
•Zaki et al. (2003) – Use Lyapunov approach. Experimental results.
18
Observer in open loop.
Sliding Mode Observer – Definitions
• Sliding Surface – A line or hyperplane in state-space
which is designed to accommodate a sliding motion.
• Sliding Mode – The behavior of a dynamic system
while confined to the sliding surface.
1
• Signum function (Sgn(s))   
1
s  0
if 

s  0
• Reaching phase – The initial phase of the closed loop
behaviour of the state variables as they are being
driven towards the surface.
19
Sliding Mode Observer – Overview
Example:
n  2, y  x1
Sliding
Surface
If Single Sliding Surface:
Error
Vector
Trajectory
s1  xˆ1  x1  x
Then Dynamics on
Sliding Surface:
x   x  0,   0
x
Sliding Condition:
s1s1   s1
x
(0,0)
20
Sliding Mode Observer – Form
xˆ  Axˆ  Bu  K L ( y  yˆ )  K s (sgn( y  yˆ ))
Example:
Given a second order dynamics system with a
single measurement,
 x1   x2 
x    f 
 2  
y  x1
The error dynamics in the presence of parametric
uncertainty are given by
 x1   x2  L1 x1  k1 sgn( x1 ) 
 


f

L
x

k
sgn(
x
)
x
2 1
2
1 
 2 
f  f  fˆ
21
Sliding Mode Observer – VSS Theory Approach
Notable Aspects:
•Sliding mode gains are selected individually one gain at a time.
•Gains are dependent on one another.
•Must select upper bounds on parametric uncertainties.
•Must select upper bounds on estimate errors.
Limitations:
•As number of measurements increase, higher likelihood of more
unknowns than constraint equations. Some gains must be set to
zero.
•If measurements are not directly states, approach becomes
unmanageable.
•Sliding mode gain Ks is time-varying.
22
Sliding Mode Observer – Lyapunov Approach
Given the SMO error dynamics
x  ( A  K LC) x  Ks (sgn( y  yˆ ))  Ax
Walcott and Zak show that the following implementation assures
stable error dynamics:
K s   P 1C T
( A  K LC) P  P( A  K LC )T  Qp
 
Depends on
Ax
Formally, the Lyapunov function candidate V  e Pe can be used to
show that V is negative definite and so error dynamics are stable.
T
23
Boundary Layer Sliding Mode Observer
  P 1C T sgn( y  yˆ )

S 
y  yˆ
1 T
 P C


IF
y  yˆ   

y  yˆ   
Notable Aspects:
•As width of B.L. decreases, BLSMO becomes SMO.
•As estimate error tends to zero, so does S.
24
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
25
Simulation Studies - Overview
w
G
v
+
D
r
u
F
+
B

+
-
+
x
C
y
•Noise statistics inherited
from experimental test-bed.
•Feedback gain designed to
keep control signal u < 62
N.
A
ρ
ε
Ks
+
KL
+
B
+
+

A
x̂
ŷ
C
+
Parameter Variation
Studies:
•Vary tip mass.
•Observer design
parameters: ρ, Qp , and λ.
•Parameter variation from
+60% to -60%.
KC
D
26
Simulation Studies - Overview
w
G
Performance Metric:
v
+
D
(For lumped-parameter models)
•Position Mean Square Estimate
Error:
Norm of vector
 MSE ( x1 ) 
 MSE ( x ) 
3 

r
u
F
B
-
+
+

+
x C
y
A
ρ
ε
Ks
+
KL
B
+
+
+
-

x̂
ŷ
C
+
A
KC
•Velocity Mean Square Estimate
Error:
Norm of vector
D
 MSE ( x2 ) 
 MSE ( x ) 

4 
Similar approach for assumed modes
method model.
27
Simulation Studies – Results
•Sliding mode behavior seen in error space.
•SMO (Qp = 4, ρ = 1) and BLSMO (Qp = 4, ρ = 1, λ = 0.005).
28
Simulation Studies – Results
•Discontinuous state
function for SMO.
•Smoothed state
function for BLSMO.
29
Simulation Studies – Results
Tip Position:
•Kalman Filter vs. BLSMO
(Qp = 2.2e3, ρ = 2.5, λ =
150)
•30% parameter variation.
•Lumped parameter model.
Tip Velocity:
•Result:
Reduced error estimates
from BLSMO.
30
Simulation Studies – Results
•Lumped parameter model.
•Result:
Position Mean Square
Estimate Error (m)
Larger variation in
performance between
different SMO designs.
BLSMO (roe=1,Q=4,lambda=0.005)
BLSMO (roe=1,Q=7.5,lambda=0.003)
BLSMO (roe=1,Q=19,lambda=0.001)
Kalman Filter
Little variation in
performance between
different BLSMO designs.
BLSMO estimate errors are
lower than SMO.
1.E-04
1.E-05
BLSMO estimate errors are
lower than Kalman filter.
1.E-06
1.E-07
1.E-08
1.E-09
-60
-40
-20
0
Parameter Variation (%)
20
40
60
31
Velocity Mean Square Estimate Error (m/s)
Simulation Studies – Results
3.5E-05
3.0E-05
2.5E-05
2.0E-05
1.5E-05
1.0E-05
5.0E-06
BLSMO
Kalman Filter
1.0E-08
-60
-40
-20
0
20
40
60
Parameter Variation (%)
•Lumped parameter model.
•Result:
With Gaussian white measurement noise, BLSMO (Qp = 2.2e3, ρ =
0.01, λ = 5) outperforms Kalman filter.
32
Simulation Studies – Results
•Modified inertia lumped
parameter model.
•Result:
Unstable error dynamics
for Kalman filter in
presence of 21%
parameter variation.
Stable error dynamics for
BLSMO (Qp = 3.65e6, ρ
= 60, λ = 1) under same
conditions, up to 32%
parameter variation.
33
Simulation Studies – Results
Closed-Loop Tip Response:
•Lumped parameter model with 30%
parameter variation.
•BLSMO (Qp = 2e3, ρ = 2.5, λ = 150).
•Result:
Due to improved estimation, commanded
tip excitation decreases.
•Modified inertia lumped parameter model
with 25% parameter variation.
•BLSMO (Qp = 3.65e6, ρ = 60, λ = 1).
•Result:
Due to improved estimation,
Unstable tip response is stabilized.
34
Simulation Studies – Results
•Assumed modes method model.
•Result:
BLSMO (Qp = 2.5e11, ρ = 5, λ = 37) offers no estimation advantage.
Closed-loop tip response could not be improved.
•Why?
-No state directly measured.
-Parameter variation effects A, B, C and D.
-According to Matlab, observability depends on link parameters.
35
Simulation Studies – Summary of Results
The Good:
•SMO estimates are superior to Kalman filter.
•BLSMO estimates are superior to SMO.
•In presence of Gaussian white noise, BLSMO
estimates remain superior to Kalman filter.
•Improved estimation can stabilize an unstable
tip response or at the very least reduce closedloop tip tracking error.
36
Simulation Studies – Summary of Results
The Bad:
•Robust observer with assumed mode method
model not any more robust than Kalman filter.
•Anomaly at +60% parameter variation in many
results.
•All parameters selected by trial and error
manner.
37
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
38
Experimental Studies – Overview
•Controller and observer based on lumped parameter model.
•Model outputs tip acceleration. (accelerometer signal not
integrated)
•Noise covariance matrix for Kalman filter reflects:
A standard deviation of 1.97e-5 meters in the position
measurement.
A standard deviation of 0.0161 m/s2 in the acceleration
measurement.
•Tip position is commanded in closed-loop control by
penalizing state x1 in the method of symmetric root locus and
in design of the feed-forward gain F.
39
Experimental Studies – Overview
LabVIEW GUI
•Allows direct
control over
hardware at runtime.
•Relays status
information to
developer.
•Updates at 10hz to
minimize overhead.
40
Experimental Studies – Results
Tip Acceleration:
•Loop rate 1khz.
•Kalman filter.
Base Position:
•First mode suppressed by
state-feedback in 1.5
seconds.
•A filtered square wave
trajectory is tracked by link
tip.
41
Experimental Studies – Results
•Tip acceleration displayed.
•Loop rate 1khz.
•Tracking filtered square wave.
Tip mass increased by 426%
Tip mass decreased by 70%
42
Experimental Studies – Results
•Link base position displayed.
•Tracking filtered square wave
trajectory for link tip.
•Parameter variation of 91%
in link length.
•SMO (Qp=1.5e7, ρ=10)
shows estimate chatter.
•BLSMO (Qp=1.5e7, ρ=10,
λ=5) shows no estimate
chatter.
•Damping effect on base
motion apparent.
43
Experimental Studies – Results
•Link tip acceleration displayed.
•Tracking filtered square wave
trajectory for link tip.
•Parameter variation of 91% in
link length.
•SMO (Qp=1.5e7, ρ=10) shows
estimate chatter.
•BLSMO (Qp=1.5e7, ρ=10, λ=5)
shows no estimate chatter.
•Damping effect on tip motion
44
apparent.
Experimental Studies – Results
•Control signal is displayed.
•Tracking filtered square wave
trajectory for link tip.
•Parameter variation of 91% in
link length.
•SMO (Qp=1.5e7, ρ=10) shows
very high control activity.
•BLSMO (Qp=1.5e7, ρ=10, λ=5)
shows reduced control activity.
45
Experimental Studies – Results
Base Position:
•Studies could not be
completed because of
restrictive bounds
placed on observer
design parameters ρ
and λ.
•The structure of the output matrix C in combination with
large sliding mode gain Ks and large feedback gain Kc can
lead to discontinuities in the estimates which can cause
discontinuities in the control signal:
For ρ > 50
For λ < 1
46
Experimental Studies – Summary of Results
•Robust observer parameter Qp fixed off-line while ρ and λ can
be tuned on-line.
•Small computational over-head.
•SMO and BLSMO have an apparent damping effect on motor
when tracking a time-varying reference signal in presence of
parametric uncertainty.
•Kalman filter is surprisingly robust to parameter variation.
Although room for estimate improvement does exist.
•Marginal stability resulting for parameter variation appears to
be caused more by degraded performance of controller than of
the Kalman filter.
•Estimation chatter lead to chatter in control signal and
overheated motor.
47
Agenda
1.
•Background:
Problem Statement
Non-collocation and Non-minimum Phase Behavior
Observer and Controller Overview
Test-bed Overview
Plant Model
2.
•Optimal Observer – The Kalman Filter
3.
•Robust Observer – Sliding Mode
4.
•Results:
Simulation Studies
Experimental Studies
5.
•Conclusions
48
Scoring the Sliding Mode Observer
What is a useful observer anyway?
•Robust (works most of the time)
•Accuracy not far off from optimal estimates
•Not computationally intensive
•Straightforward design
•Straightforward implementation
49
Scoring the Sliding Mode Observer
Strong points:
•Simulations indicate optimality is not sacrificed for
robustness.
•Simulations show that improving estimates alone can
improve closed-loop tip tracking errors significantly.
•On physical system, operates at fast control rates and is
applicable to real-time control of fast motion systems.
•On physical system, offers high tunability at run-time. (can
even revert to Kalman filter on-the-fly)
•In simulations and on physical system, easy to design.
50
Scoring the Sliding Mode Observer
Weak points:
•In simulations and on physical system, more particular about
linear system model than Kalman filter.
•On physical system, more difficult to implement than Kalman
filter. Significantly more trial and error tuning needed.
•On physical system, without boundary layer, can harm
hardware.
51
Robust Nonlinear Observer for a
Non-collocated Flexible System
Mohsin Waqar
M.S.Thesis Presentation
Friday, March 28, 2008
Intelligent Machine Dynamics Lab
Georgia Institute of Technology
u  Fr  K c xˆ
F = 2.24e4
Kc   1.01e4 0.056e4 3.25e6 0.066e4
0.34 
 1.2e  4
 3.2e  8 4.5e  5 

Ks  
 2.8e  4
0.48 


 1.1e  7 1.9e  4 
xˆ  Axˆ  Bu  K L ( y  yˆ )   K s (sgn( y  yˆ ))
 528 0.863
 713 0.184 

KL  
1.08e3 0.983 


1.06e3 0.965 
0
0
1
0 

C

 1.195e3 0.391 1.195e3 0.391
53