Robust%20Nonlinear%20Observer%20for%20a%20Non-collocated%20Flexible%20System

1
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
2
Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
3
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.

4
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.

5
Control Overview
Noise V

Commanded Tip Position
y
F
u
d
Linear Motor
Flexible Link
Sensors
Feedforward Gain F
-
Observer
Feedback Gain K
Control objective Accuracy, repeatability and
steadiness of the link tip.
6
Test-Bed Overview
R
PCB 352a Accelerometer
PCB Power Supply
C
LS7084 Quadrature Clock Converter
Anorad Encoder Readhead
Anorad Interface Module

-
LV Real Time 8.5 Target PC w/ NI-6052E DAQ Board

NI SCB-68 Terminal Board
160VDC
Anorad DC Servo Amplifier
Linear Motor
-
PWM
-10 to 10VDC
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Flexible Link Modeling Assumed Modes Method
c
E, I, ?, 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 (? ltlt 1)
• Only viscous friction at slider

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Flexible Link Modeling Assumed Modes Method
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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
Youngs 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
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Flexible Link Modeling Lumped Parameter Model
c
Model Data
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
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Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
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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

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How it works - Kalman Filter
Steady State Kalman Filter How it works
Plant Dynamics
Kalman Filter
State Estimates with minimum square of error
Measurement State Relationships
Noise Statistics
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.
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Steady State Kalman Filter How it works
v

• Filter Design
• Find R and Q
• 1a) For each measurement, find µ and s2 to get
  R
• 1b) Set Q small, non-zero
• 2. Find P using Matlab CARE fcn
• Find KPC'inv(R)
• Observer poles given by eig(A-LC)
• 5. Tune Q as needed

-
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Steady State Kalman Filter How it works
Observer dynamic equation
Closed-loop system with observer
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Steady State Kalman Filter A Limitation
Example Given a second order dynamic system with
a single measurement,
Then the Kalman filter in presence of parametric
uncertainty is given by
And the observer error dynamics are given by
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Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
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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. Observer in open loop.

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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.
• Signum function (Sgn(s)) if
• Reaching phase The initial phase of the closed
  loop behaviour of the state variables as they are
  being driven towards the surface.

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Sliding Mode Observer Overview
Example
Sliding Surface
If Single Sliding Surface Then Dynamics on
Sliding Surface Sliding Condition
Error Vector Trajectory
(0,0)
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Sliding Mode Observer Form
Example Given a second order dynamics system
with a single measurement,
The error dynamics in the presence of parametric
uncertainty are given by
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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.

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Sliding Mode Observer Lyapunov Approach
Given the SMO error dynamics
Walcott and Zak show that the following
implementation assures stable error dynamics
Depends on
Formally, the Lyapunov function candidate
can be used to show that is
negative definite and so error dynamics are
stable.
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Boundary Layer Sliding Mode Observer
IF

• Notable Aspects
• As width of B.L. decreases, BLSMO becomes SMO.
• As estimate error tends to zero, so does S.

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Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
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Simulation Studies - Overview

• Noise statistics inherited from experimental
  test-bed.
• Feedback gain designed to keep control signal u lt
  62 N.
• Parameter Variation Studies
• Vary tip mass.
• Observer design parameters ?, Qp , and ?.
• Parameter variation from 60 to -60.

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Simulation Studies - Overview

• Performance Metric
• (For lumped-parameter models)
• Position Mean Square Estimate Error
• Norm of vector
• Velocity Mean Square Estimate Error
• Norm of vector
• Similar approach for assumed modes method model.

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Simulation Studies Results

• Sliding mode behavior seen in error space.
• SMO (Qp 4, ? 1) and BLSMO (Qp 4, ? 1, ?
  0.005).

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Simulation Studies Results

• Discontinuous state function for SMO.
• Smoothed state function for BLSMO.

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Simulation Studies Results
Tip Position

• Kalman Filter vs. BLSMO (Qp 2.2e3, ? 2.5, ?
  150)
• 30 parameter variation.
• Lumped parameter model.
• Result
• Reduced error estimates from BLSMO.

Tip Velocity
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Simulation Studies Results

• Lumped parameter model.
• Result
• Larger variation in performance between different
  SMO designs.
• Little variation in performance between different
  BLSMO designs.
• BLSMO estimate errors are lower than SMO.
• BLSMO estimate errors are lower than Kalman
  filter.

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Simulation Studies Results

• Lumped parameter model.
• Result
• With Gaussian white measurement noise, BLSMO (Qp
  2.2e3, ? 0.01, ? 5) outperforms Kalman
  filter.

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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, ?
  60, ? 1) under same conditions, up to 32
  parameter variation.

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Simulation Studies Results

• Closed-Loop Tip Response
• Lumped parameter model with 30 parameter
  variation.
• BLSMO (Qp 2e3, ? 2.5, ? 150).
• Result
• Due to improved estimation, commanded tip
  excitation decreases.
• Modified inertia lumped parameter model with 25
  parameter variation.
• BLSMO (Qp 3.65e6, ? 60, ? 1).
• Result
• Due to improved estimation,
• Unstable tip response is stabilized.

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Simulation Studies Results

• Assumed modes method model.
• Result
• BLSMO (Qp 2.5e11, ? 5, ? 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.

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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 closed-loop
  tip tracking error.

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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.

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Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
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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.

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Experimental Studies Overview
LabVIEW GUI

• Allows direct control over hardware at run-time.
• Relays status information to developer.
• Updates at 10hz to minimize overhead.

41
Experimental Studies Results
Tip Acceleration

• Loop rate 1khz.
• Kalman filter.
• First mode suppressed by state-feedback in 1.5
  seconds.
• A filtered square wave trajectory is tracked by
  link tip.

Base Position
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Experimental Studies Results

• Tip acceleration displayed.
• Loop rate 1khz.
• Tracking filtered square wave.
• Tip mass increased by 426
• Tip mass decreased by 70

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Experimental Studies Results

• Link base position displayed.
• Tracking filtered square wave trajectory for link
  tip.
• Parameter variation of 91 in link length.
• SMO (Qp1.5e7, ?10) shows estimate chatter.
• BLSMO (Qp1.5e7, ?10, ?5) shows no estimate
  chatter.
• Damping effect on base motion apparent.

44
Experimental Studies Results

• Link tip acceleration displayed.
• Tracking filtered square wave trajectory for link
  tip.
• Parameter variation of 91 in link length.
• SMO (Qp1.5e7, ?10) shows estimate chatter.
• BLSMO (Qp1.5e7, ?10, ?5) shows no estimate
  chatter.
• Damping effect on tip motion apparent.

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Experimental Studies Results

• Control signal is displayed.
• Tracking filtered square wave trajectory for link
  tip.
• Parameter variation of 91 in link length.
• SMO (Qp1.5e7, ?10) shows very high control
  activity.
• BLSMO (Qp1.5e7, ?10, ?5) shows reduced control
  activity.

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Experimental Studies Results
Base Position

• Studies could not be completed because of
  restrictive bounds placed on observer design
  parameters ? and ?.

• 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 ? gt 50 For ? lt 1

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Experimental Studies Summary of Results

• Robust observer parameter Qp fixed off-line while
  ? and ? 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.

48
Agenda
1.

• Background
• Problem Statement
• Non-collocation and Non-minimum Phase Behavior
• Observer and Controller Overview
• Test-bed Overview
• Plant Model
• Optimal Observer The Kalman Filter
• Robust Observer Sliding Mode
• Results
• Simulation Studies
• Experimental Studies
• Conclusions

2.
3.
4.
5.
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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

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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.

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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.

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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
53
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