robust nonlinear observer for a non-collocated flexible system

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•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

•Project Roadmap

1.

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

Control objective: Accuracy, repeatability and steadiness of the link tip.

Linear Motor

Flexible Link Sensors

ObserverFeedback Gain K

FeedforwardGain F

Commanded

Tip Position

Noise

V

+

-

yδFu

x

6

Test-Bed Overview

NI SCB-68Terminal

Board

Anorad EncoderReadhead

Anorad Interface Module

LS7084Quadrature

Clock Converter

PCB 352aAccelerometer

PCB Power Supply

Anorad DC ServoAmplifier

Linear Motor

LV Real Time 8.5Target PC

w/NI-6052E DAQ

Board

R

C

+-+-

160VDC

PWM-10 to +10VDC

7

Flexible Link Modeling – Assumed Modes Method

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

m

w(x,t)

x

E, I, ρ, A, L

F

c

8

Flexible Link Modeling – Assumed Modes Method

Mq Cq Kq Q

0K M

12TM

m

x

E, I, ρ, A, L

F w(x,t)

4

EIAL

q 2[ ( )]T TQ C diag

1 1

2 2

3 3

4 4

5 1

6 2

7 3

8 4

xxxxxxxx

x Ax Bu

( 0, )( , )

w x ty x Cx Du

w x L t

9

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

10

c

y1

m1

m2

k

J2F

y2

Flexible Link Modeling – Lumped Parameter Model

1 1 1 1

22 2 2 2

0 1 0 0000

0 0 0 13

3 3 3 3

k c k cm m m m

x x F

k c k cm

m m m m

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

11

Agenda•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

•Project Roadmap

1.

2.

3.

4.

5.

12

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

13

How it works - Kalman Filter

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.

Kalman Filter

Plant Dynamics

Measurement & State Relationships

Noise Statistics

Initial Conditions

State Estimates with minimum square of error

Steady State Kalman Filter – How it works

14

+

-

+v

x

1/s

A

B C+

1/s

~A

B C+

K

+

Kc

F-

ur

x y

y

Filter Design:1. Find R and Q

1a) For each measurement, find μ and σ2 to get R

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

Steady State Kalman Filter – How it works

15

ˆˆ0

CC

CC

BK Kx rA BK KC KCxBKx vBK Ax

+-

+v

x

1/s

A

B C+

1/s

~A

B C+

L

+

K

F-

ur x y

y

0 0ˆ

ˆ 0 0C C

C C

C C

K Kux r

C DK DKyx v

y DK C DK I

ˆˆ ˆ ˆ( )x Ax Bu K y y

Steady State Kalman Filter – How it works

Observer dynamic equation:

Closed-loop system with observer:

16

Steady State Kalman Filter – A Limitation

1 2

2

x xx f

Example: Given a second order dynamic system with a single measurement,

1y x

2 1 11

2 12

ˆˆˆˆ

x K xx

f K xx

Then the Kalman filter in presence of parametric uncertainty is given by

And the observer error dynamics are given by

2 1 11

2 12

x K xxf K xx

ˆf f f

17

Agenda•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

•Project Roadmap

1.

2.

3.

4.

5.

18

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.

19

• 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.

11

Sliding Mode Observer – Definitions

00

ss

20

Sliding Mode Observer – Overview

(0,0)x

12,n y x

Error Vector Trajectory

Sliding Surface

1 1 1ˆs x x x

x0, 0x x

1 1 1s s s

Example:

If Single Sliding Surface:

Then Dynamics on Sliding Surface:

Sliding Condition:

21

Sliding Mode Observer – Form

ˆ ˆ ˆ ˆ( ) (sgn( ))L sx Ax Bu K y y K y y

Example: Given a second order dynamics system with a single measurement,

1 2

2

x xx f

1y x

The error dynamics in the presence of parametric uncertainty are given by

2 1 1 1 11

2 1 2 12

sgn( )sgn( )

x L x k xxf L x k xx

ˆf f f

22

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.

23

ˆ( ) (sgn( ))L sx A K C x K y y Ax Given the SMO error dynamics

Walcott and Zak show that the following implementation assures stable error dynamics:

1 TsK P C

( ) ( )TL L pA K C P P A K C Q

Sliding Mode Observer – Lyapunov Approach

Formally, the Lyapunov function candidate can be used to show that is negative definite and so error dynamics are stable.

TV e PeV

AxDepends on

24

1

1

ˆsgn( )

ˆ

T

T

P C y yS y y

P C

ˆ

ˆ

y y

y y

Boundary Layer Sliding Mode Observer

Notable Aspects:

•As width of B.L. decreases, BLSMO becomes SMO.

•As estimate error tends to zero, so does S.

IF

25

Agenda•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

•Project Roadmap

1.

2.

3.

4.

5.

26

B

A

C

KL

KC

F

D

v

r

G

w

u x

x y

y

+

+

+

-+

+

-

+

+

+

εKsρ

+

A

B C

D

Simulation Studies - Overview

•Noise statistics inherited from experimental test-bed.

•Feedback gain designed to keep control signal u < 62 N.

Parameter Variation Studies:

•Vary tip mass.

•Observer design parameters: ρ, Qp , and λ.

•Parameter variation from +60% to -60%.

27

B

A

C

KL

KC

F

Dv

r

Gw

u x

x y

y

+

+

+-

+

+

-++

+

εKsρ

+

A

B C

D

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.

1

3

( )( )

MSE xMSE x

2

4

( )( )

MSE xMSE x

28

Simulation Studies – Results

•Sliding mode behavior seen in error space.

•SMO (Qp = 4, ρ = 1) and BLSMO (Qp = 4, ρ = 1, λ = 0.005).

29

•Discontinuous state function for SMO.

•Smoothed state function for BLSMO.

Simulation Studies – Results

30

Simulation Studies – Results

•Kalman Filter vs. BLSMO (Qp = 2.2e3, ρ = 2.5, λ = 150)

•30% parameter variation.

•Lumped parameter model.

•Result:

Reduced error estimates from BLSMO.

Tip Position:

Tip Velocity:

31

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

-60 -40 -20 0 20 40 60Parameter Variation (%)

SMO (roe=0.5,Q=1e5) BLSMO (roe=0.5,Q=1e5,lambda=10)

SMO (roe=0.6,Q=5e3) BLSMO (roe=0.6,Q=5e3,lambda=195)

SMO (roe=0.25,Q=2.2e3) BLSMO (roe=0.25,Q=2.2e3,lambda=150)

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

-60 -40 -20 0 20 40 60

Parameter Variation (%)

Posi

tion

Mea

n Sq

uare

Es

timat

e Er

ror (

m)

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

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.

32

Simulation Studies – Results

1.0E-08

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

-60 -40 -20 0 20 40 60

Parameter Variation (%)

Velo

city

Mea

n Sq

uare

Est

imat

e Er

ror (

m/s

)

BLSMO

Kalman Filter

•Lumped parameter model.

•Result:

With Gaussian white measurement noise, BLSMO (Qp = 2.2e3, ρ = 0.01, λ = 5) outperforms Kalman filter.

33

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.

34

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.

35

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

-20 -10 0 10 20 30 40 50Parameter Variation (%)

BLSMO

Kalman Filter

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.

36

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.

37

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.

Simulation Studies – Summary of Results

38

Agenda•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

•Project Roadmap

1.

2.

3.

4.

5.

39

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.

40

Experimental Studies – Overview

•Allows direct control over hardware at run-time.

•Relays status information to developer.

•Updates at 10hz to minimize overhead.

LabVIEW GUI

41

Experimental Studies – Results

•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:

Tip Acceleration:

42

Experimental Studies – Results

•Tip acceleration displayed.

•Loop rate 1khz.

•Tracking filtered square wave.

Tip mass increased by 426%

Tip mass decreased by 70%

43

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.

44

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 apparent.

45

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.

46

Experimental Studies – Results

•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

Base Position:

47

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.

48

Agenda•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

•Project Roadmap

1.

2.

3.

4.

5.

49

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

50

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.

Scoring the Sliding Mode Observer

51

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.

Scoring the Sliding Mode Observer

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

ˆ ˆ ˆ ˆ( ) (sgn( ))L sx Ax Bu K y y K y y

1.01 4 0.056 4 3.25 6 0.066 4cK e e e e

1.2 4 0.343.2 8 4.5 5

2.8 4 0.481.1 7 1.9 4

s

ee e

Kee e

ˆcu Fr K x

0 0 1 01.195 3 0.391 1.195 3 0.391

Ce e

528 0.863713 0.184

1.08 3 0.9831.06 3 0.965

LKee

F = 2.24e4