control team welcome dr. spanos

Control TeamControl TeamWelcome Dr. SpanosWelcome Dr. Spanos

Control TeamControl TeamWelcome Dr. SpanosWelcome Dr. Spanos

Faculty Advisors

Dr. Helen Boussalis

Dr. Charles Liu

Student Assistants

Jessica AlvarengaAllison Bretaña

04/21/23 NASA Grant URC NCC NNX08BA44A 1

State Estimation Methods:Observer and Kalman Filter

04/21/23 2NASA Grant URC NCC NNX08BA44A

Outline• Objective

• Project Background and Luenberger Observer

• Kalman Filter

• Single Panel Simulations

• Noise Modeling

• Future goals

• Timeline

• References


Fault Detection

• Component Failures cannot be allowed to cause a total malfunction

• Used to achieve a fault tolerant reconfigurable controller



• Project Background and Luenberger Observer

• Kalman Filter

• Single Panel Simulation

• Noise Modeling

• Future goals

• Timeline

• References


Fault Detection and Isolation


State Observer

Discrete System Model

Observer Design


sfkCxky

kBukAxkx

)()(

)()()1(

)(ˆ)(ˆ

))(ˆ)(()()(ˆ)1(ˆ

kxCky

kykyLkBukxAkx

Residual ErrorErrorOutputkykykey :)(ˆ)()(

ErrorStatekxkxke :)(ˆ)()(

Dynamic State Error

State Observer

sLfkeLCAke )()()1(

PD KsKL ]1[

Dynamic Error Equation

PD Gains State Feedback (L)


State Observer

)(ˆ)(ˆ

))(ˆ)(()()(ˆ)1(ˆ

kxCky

kykyLkBukxAkx


Simulink Observer Realization

State Observer

ErrorOutputkykykey :)(ˆ)()(


Simulink Simulation Results

State Observer


Residual Error

Observer Simulated Output

Real System Output

Initatied Actuator Fault

Observer Discrepencies


• Project Background and Luenberg Observer

• Kalman Filter


• Noise Modeling

• Future goals

• Timeline

• References


Kalman Filter Methodology

– Two Phases:– Predictions

• Previous Estimate Current Estimate

– Update

• Current Measurement Refines Current State estimate

PredictionPredictionPredictionPrediction UpdateUpdateUpdateUpdate


[1] http://www.nps.gov/gis/gps/glossary.htm

– “A numerical method used to track a

time-varying signal in the presence of noise.”[1]

– A method of estimating the internal states of a system

Kalman Equations

1111

kkk

kkkkk

vCxy

wBuxx

1111

111 ˆˆ

kkkkk

kkkk

QPP

Buxx


1][

]ˆ[ˆˆ

kkk

kkkkk

PCKIP

xCyKxx

RvvE

QwwE

kk

kk

),0(~

),0(~

Rv

Qw

k

k

System State Equations

Noise Distributions

Noise Variances

A Priori Equations

A Posteriori Equations

][ kkkk

kkk RCPC

CPK

Kalman Gain Equation

GainKalman :

CovarianceError

PosterioriA :

Covariance

Error PrioriA :

Estimate PosterioriA :ˆ

Estimate PrioriA :ˆ

Noiset Measuremen:

Noise Process:

MatricesState:,

MatrixTransition State:

Output:

Control:

States:

k

k

k

k

k

k

k

k

k

k

k

K

P

P

x

x

v

w

CB

y

u

x

∑∑ ∑∑

DelayDelay

+

-

+

+

∑∑

DelayDelay

∑∑

+

++

+

+

Kalman Filter Realization




• Kalman Filter

• Single Panel Simulations

• Noise Modeling

• Future goals

• Timeline

• References


Single Panel Model


trans1* u

Transformation

Input Z

System Model with Noise

Output Scopes

Sensors w/ PID & Faults1

Output Scopes

Sensors w/ PID & Faults

Scope

Primary ReferenceInputs

PIDController

u

y

Kalman Output: Residual

Kalman Output Estimate

Kalman Filter Model

Addetive Faults

Output Scopes

Actuators w/ PID & Faults

Single Panel Model


1

Z

System Output Z

c1* u

C1

Band-LimitedWhite Noise, w Band-Limited

White Noise, v

b1* u

B1

a1* u

A1

1

Input

Single Panel Kalman Filter


y - y ^ = y - H x-^

xk-^

xk^x

k-1^

Residual

y ^

y ^

2

Kalman Output Estimate

1

Kalman Output: Residual

z

1

Unit Delay

System Output Z

y - y ^K ( z - H x _̂ )

Kalman Gain

c1* u

C1

b1* u

B1

a1* u

A1

2

y

1

u

Single Panel Kalman Gain


P

PC'

CPC'

K

CP

Kk = ( P

k * CT ) / (C * P

k * CT + R)

Pk- = ( A * P

k-1 * AT ) + Q

Pk = ( I - K

k * C ) * P

k-

1

K ( z - H x^_ )

z

1

MatrixMultiply

MatrixMultiply

Inv

Divide

R

Constant1

Q

Constant

c1* u

C1

u*K

C'

c1* u

C

a1* u

A1

u*K

A'

1

y - y ^

No Noise, No Fault


System Simulation Edge Sensor Estimates

KF Edge Sensor Estimates

KF Edge Sensor Residuals

Magnified View of KF Edge Sensor Residuals

No Noise, Additive Sensor Fault





Simulated Noise and Additive Sensor Fault





Issues with Simulation

• Long run times (10 sec took ~10 minutes)

• Faulty residuals

• Difficult to tune noise


Solution

• Develop a new and efficient simulation code

• Create accurate process and measurement noise models

• Simulation of an open-loop system




• Kalman Filter


• Noise Modeling

• Future goals

• Timeline

• References


• Case 1: Assume no process noise– All noise attributed to sensors

• Case 2: Assume no sensors noise– All noise attributed to process

• Case 3: Combination of process and sensor noise (Real Scenario)


Noise Scenarios

Case 1: No process noise


w=0, v~N(0,R)

Sensor noise is attributed to the measurements.

111

kkk

kkkk

vCxy

Buxx

DIRECTDIRECTMeasurement NoiseMeasurement Noise

04/21/23 30NASA Grant URC NCC

NNX08BA44A

Edge Sensor Data Edge Sensor Data (System at rest)(System at rest)

04/21/23NASA Grant URC NCC NNX08BA44A

31

Single Panel Edge Sensor Single Panel Edge Sensor Data (System at rest)Data (System at rest)


Case 2: No Sensor Noise

w~N(0,Q), v=0

Sensor noise is attributed to noise in the process.

Are not directly observing states.


kk

kkkkk

Cxy

wBuxx

111

Inversion of State Space

A: n x n B: n x m C: p x n

However, C may not be square, as in our case, and is not invertible.


111

11

111

1

111

111

)(

kkkkk

kk

kk

kk

kkkkk

kkkkk

BuyCyCw

xyC

similarlyxyC

Cxy

Buxxw

wBuxx

Moore-Penrose Pseudo Inverse

• Use the Moore-Penrose Pseudo Inverse to invert the state space model and allow us to make process noise calculations using sensor measurements.


. transposeconjugateor TransposeHermetian theIs and

,)(

satisfies which Inverse Pseudo Penrose-Moore theis

, where

,

:gives This

*

*

1

111

C

CCCC

C

CCC

Buyyw

C

kkCkkCk

• Use mathematical equation to determine process noise

where

• Calculate mean, standard deviation and variance of process noise using MATLAB


Noise Modeling


PANEL 1 PANEL 2 PANEL 3 PANEL 4 PANEL 5 PANEL 6STATE mean std. dev. variance mean std. dev. variance mean std. dev. variance mean std. dev. variance mean std. dev. variance mean std. dev. variance

1 -6.16338 0.151601 0.022983 7.549682 0.094799 0.008987 -0.50693 0.541283 0.292988 3.442127 0.763661 0.583179 1.613433 0.845681 0.715177 -1.66758 0.306442 0.0939072 -0.02756 0.131574 0.017312 -1.23316 0.060106 0.003613 -1.29316 0.789476 0.623273 10.80916 0.639489 0.408947 6.087263 0.650141 0.422684 2.313367 1.290944 1.6665363 -6.56202 0.345914 0.119657 4.736639 0.121153 0.014678 3.138532 0.540469 0.292107 1.797797 0.353535 0.124987 2.394228 1.01984 1.040073 -1.25442 0.310149 0.0961924 -4.60967 0.087644 0.007682 3.2591 0.080688 0.006511 1.316403 0.065577 0.0043 3.96695 0.106595 0.011362 2.08668 0.215191 0.046307 1.43054 0.438764 0.1925145 14.10429 3.075365 9.457871 13.38073 0.824995 0.680617 -32.906 0.683676 0.467413 14.57379 2.741044 7.513324 -3.37381 2.541154 6.457463 -7.31638 1.792657 3.213626 -1.01944 0.947898 0.898511 9.360859 0.250873 0.062937 -7.39944 0.806468 0.650391 -1.14779 1.516844 2.300817 -3.83095 0.662759 0.439249 -4.07433 1.240767 1.5395037 6.732807 0.682667 0.466034 -0.90668 0.187941 0.035322 -8.55227 0.610493 0.372702 6.071319 0.306403 0.093883 0.866875 1.14848 1.319007 -0.56379 0.5888 0.3466868 -31.4423 4.464805 19.93448 1.021753 1.226868 1.505206 33.65548 1.319123 1.740087 20.05731 2.236104 5.000161 28.00057 4.670018 21.80907 1.590931 3.430741 11.769989 5.057215 1.492933 2.22885 6.763659 0.397679 0.158149 -14.2327 0.376554 0.141793 6.644327 1.218853 1.485602 -1.99029 1.239344 1.535973 -1.91605 0.769243 0.59173410 -4.88285 0.942825 0.888919 -6.33453 0.267585 0.071602 10.81644 0.829386 0.68788 2.651626 1.521532 2.315058 4.531653 0.529614 0.280491 6.856123 1.650637 2.72460311 -9.0353 0.658259 0.433305 7.20533 0.262258 0.068779 9.54112 1.843249 3.397568 -29.2856 1.746427 3.050008 -17.9683 1.686115 2.842985 1.018385 2.692463 7.24935512 -3.93729 0.955842 0.913635 -2.01874 0.263814 0.069598 3.48422 0.779793 0.608077 15.70021 1.06811 1.140858 12.03615 0.386468 0.149357 1.737163 1.813873 3.29013413 -5.04278 1.251015 1.565039 11.92345 0.361935 0.130997 -8.27734 0.296692 0.088026 10.17356 1.220009 1.488422 0.679103 0.724412 0.524773 1.262926 0.684781 0.46892514 13.17072 0.750368 0.563052 -5.27878 0.323097 0.104391 -2.8995 1.568924 2.461523 -33.4333 1.368429 1.872599 -20.2629 0.555023 0.308051 -7.14973 3.737233 13.9669115 -32.1695 2.646428 7.003581 -0.13942 0.91032 0.828682 25.35087 3.526859 12.43873 51.91797 4.149316 17.21682 36.17415 1.256982 1.580005 23.82574 8.265156 68.312816 -0.49436 0.710994 0.505513 -6.87684 0.183327 0.033609 8.663878 0.211037 0.044536 -7.68765 1.014574 1.029359 -1.80446 0.446535 0.199393 3.207253 0.42487 0.18051517 -1.36616 0.820784 0.673687 5.465606 0.230778 0.053258 -3.24015 0.362225 0.131207 -3.67876 0.831127 0.690772 -5.14577 0.34753 0.120777 0.334765 0.674779 0.45532718 1.259203 0.493048 0.243096 5.784451 0.143753 0.020665 -10.5391 0.417255 0.174102 17.60379 0.852202 0.726249 7.960239 0.62766 0.393957 -2.94512 0.83618 0.69919719 -41.9464 2.137926 4.570727 24.3815 0.777469 0.604458 23.29307 1.969631 3.879448 17.17468 0.947347 0.897466 17.27459 4.913287 24.14039 1.419739 1.993257 3.97307220 20.56158 1.339986 1.795562 -6.74163 0.453423 0.205592 -28.9581 3.748309 14.04982 49.38333 2.301965 5.299044 21.75745 4.7632 22.68807 2.183285 4.657003 21.6876721 -43.843 2.299461 5.287521 61.1673 0.871758 0.759962 -6.3887 7.941124 63.06145 29.76193 7.608398 57.88773 28.44375 11.83002 139.9495 -45.226 6.180394 38.1972722 -13.9888 0.883045 0.779769 3.86685 0.294509 0.086736 17.33633 1.611491 2.596903 -20.7948 1.023012 1.046554 -8.17725 2.399821 5.75914 1.426667 1.756177 3.08415623 11.83478 1.848239 3.415989 7.652212 0.50636 0.256401 -23.9567 0.426612 0.181998 13.36027 1.825611 3.332854 0.774112 1.647124 2.713019 -7.29733 1.191057 1.41861624 4.693166 1.262302 1.593405 1.144195 0.351107 0.123276 -3.04765 0.713218 0.50868 -20.0107 1.134723 1.287596 -15.7416 0.672405 0.452129 0.470809 1.909501 3.64619325 13.56646 0.923029 0.851983 -12.3327 0.410051 0.168142 -18.6624 4.630475 21.4413 67.34064 3.800936 14.44711 38.4976 4.280382 18.32167 5.051207 6.912096 47.7770726 56.90523 2.252087 5.071897 -21.8818 1.719363 2.956211 -12.3691 9.896664 97.94397 -118.894 5.991901 35.90288 -53.115 7.571164 57.32252 -66.0185 18.37947 337.80527 20.35218 1.917848 3.678142 -7.47378 0.973209 0.947135 3.304651 6.515976 42.45794 -70.1821 3.611628 13.04386 -28.2458 6.100969 37.22182 -37.6526 10.72388 115.001628 -7.45211 0.556038 0.309178 7.112312 0.179828 0.032338 1.928492 0.97547 0.951542 4.352816 0.711235 0.505856 4.924443 1.666709 2.777919 -4.8556 0.604823 0.3658129 12.43557 0.600764 0.360917 -11.6193 0.309096 0.095541 2.283272 0.925069 0.855753 -22.9529 0.574817 0.330415 -9.86732 0.60787 0.369506 -6.40242 2.341702 5.4835730 -2.48396 2.52386 6.369871 14.54849 0.718792 0.516662 -16.2446 1.207237 1.457422 16.30463 1.463358 2.141416 -1.3273 2.282925 5.211746 6.044186 1.718371 2.95279831 34.71309 1.118392 1.250801 -23.4768 0.841522 0.70816 -1.82415 3.439076 11.82725 -63.8923 1.670249 2.78973 -29.8742 1.959569 3.83991 -23.7484 7.785847 60.6194132 -8.95341 0.452141 0.204432 18.76359 0.366565 0.13437 5.434745 5.872934 34.49135 -46.0202 4.616042 21.30784 -21.5535 6.217607 38.65864 -24.1951 7.939902 63.0420433 -11.1308 1.916075 3.671345 -6.88849 0.536843 0.2882 15.59439 1.519765 2.309687 18.10864 2.478354 6.142238 16.39469 0.830282 0.689369 9.256328 3.499765 12.2483534 -18.0303 0.395919 0.156752 9.521829 0.338234 0.114402 8.673757 0.450529 0.202976 11.92681 0.304003 0.092418 6.427131 0.546304 0.298448 8.48887 1.977068 3.90879835 -20.3066 2.301924 5.298854 -12.6436 0.689615 0.475569 44.35304 1.040692 1.083039 -44.9621 3.07783 9.473035 -17.216 2.630995 6.922137 16.45427 1.835032 3.36734236 26.17034 3.823725 14.62088 -10.7961 1.069526 1.143886 -13.8137 1.441627 2.078288 -45.3406 1.583323 2.506911 -40.0176 4.525706 20.48201 8.817662 3.280777 10.763537 -3.3361 2.740591 7.510838 4.100992 0.868899 0.754986 10.02466 4.836579 23.39249 -25.9582 2.27757 5.187325 -1.44901 6.264672 39.24611 -27.2205 5.961346 35.5376538 -48.1625 4.848361 23.50661 32.89802 1.431249 2.048475 42.33865 9.91423 98.29195 -50.0319 5.238495 27.44183 -5.32283 14.17606 200.9606 -35.0797 10.08864 101.780739 0.469042 7.722104 59.63089 -36.5977 2.136976 4.566667 46.82179 3.508144 12.30707 -32.1143 4.081489 16.65855 13.7346 7.148948 51.10745 -18.4437 4.445823 19.7653440 150.4421 3.420579 11.70036 -140.167 2.922579 8.541469 20.50118 6.519705 42.50655 -280.586 7.923674 62.78462 -144.059 5.822045 33.89621 -36.5953 23.46778 550.736641 2.619508 1.54821 2.396954 7.612411 0.559561 0.313109 -12.8849 1.583765 2.508312 27.65688 1.808188 3.269543 22.26516 2.648602 7.015094 -21.1654 1.218967 1.48588142 -18.603 3.576372 12.79044 -11.5922 0.965438 0.932071 30.461 1.316825 1.734027 13.55685 3.494879 12.21418 20.22511 2.118491 4.488005 10.18664 3.980243 15.8423343 34.4221 15.02156 225.6471 224.1288 4.462938 19.91781 -269.087 20.4383 417.7242 157.9006 38.98844 1520.098 46.19118 20.35599 414.3663 -201.513 27.7892 772.239644 47.48899 3.650561 13.3266 5.847482 1.276171 1.628612 -55.1593 2.920337 8.528368 -15.3051 5.334263 28.45436 -18.3124 1.940334 3.764896 -37.6812 7.866977 61.8893245 -59.7554 2.674121 7.150921 33.57174 1.531525 2.345567 -9.02921 9.906136 98.13152 209.76 7.778674 60.50777 119.3493 4.942831 24.43158 35.61774 21.68449 470.217246 4.286548 8.015477 64.24787 17.51157 2.448981 5.997507 -64.6048 13.96711 195.0801 153.3645 5.453322 29.73872 47.67032 17.01947 289.6623 57.78278 19.53231 381.511347 33.25264 8.769659 76.90692 -100.74 2.500666 6.253331 69.40679 4.161419 17.31741 -52.0901 12.48055 155.7641 3.530202 5.279189 27.86984 19.03135 5.07586 25.7643648 -83.7287 2.317389 5.37029 19.49981 2.345878 5.503145 28.33433 15.5565 242.0048 187.9286 11.05429 122.1974 93.20885 11.03403 121.7498 98.7924 29.1192 847.9278

PANEL 1STATE mean std. dev. variance

1 -6.16338 0.151601 0.0229832 -0.02756 0.131574 0.0173123 -6.56202 0.345914 0.1196574 -4.60967 0.087644 0.0076825 14.10429 3.075365 9.4578716 -1.01944 0.947898 0.8985117 6.732807 0.682667 0.4660348 -31.4423 4.464805 19.934489 5.057215 1.492933 2.2288510 -4.88285 0.942825 0.88891911 -9.0353 0.658259 0.43330512 -3.93729 0.955842 0.91363513 -5.04278 1.251015 1.56503914 13.17072 0.750368 0.56305215 -32.1695 2.646428 7.00358116 -0.49436 0.710994 0.50551317 -1.36616 0.820784 0.67368718 1.259203 0.493048 0.24309619 -41.9464 2.137926 4.57072720 20.56158 1.339986 1.79556221 -43.843 2.299461 5.28752122 -13.9888 0.883045 0.77976923 11.83478 1.848239 3.41598924 4.693166 1.262302 1.59340525 13.56646 0.923029 0.85198326 56.90523 2.252087 5.07189727 20.35218 1.917848 3.67814228 -7.45211 0.556038 0.30917829 12.43557 0.600764 0.36091730 -2.48396 2.52386 6.36987131 34.71309 1.118392 1.25080132 -8.95341 0.452141 0.20443233 -11.1308 1.916075 3.67134534 -18.0303 0.395919 0.15675235 -20.3066 2.301924 5.29885436 26.17034 3.823725 14.6208837 -3.3361 2.740591 7.51083838 -48.1625 4.848361 23.5066139 0.469042 7.722104 59.6308940 150.4421 3.420579 11.7003641 2.619508 1.54821 2.39695442 -18.603 3.576372 12.7904443 34.4221 15.02156 225.647144 47.48899 3.650561 13.326645 -59.7554 2.674121 7.15092146 4.286548 8.015477 64.2478747 33.25264 8.769659 76.9069248 -83.7287 2.317389 5.37029

•Simulation focuses on Panel 1

•Apply the calculated variance to Gaussian White noise in simulation

2/18/2010NASA Grant URC NCC NNX08BA44A

38

Simulated Noise



• Kalman Filter

• Implementation into a SISO System

• Initial simulations

• Noise Modeling

• Future goals

• Timeline

• References


Future Goals

• Improve the noise model for the homogenous case

• Noise analysis for non-homogenous cases– Step input– Impulse– Chirp– Sinusoid

• Develop algorithm for Testbed implementation




• Kalman Filter



• Noise Modeling

• Future goals

• Timeline

• References


Timeline


2009 MAR APR MAY JUN JUL

Jessica

Alvarenga

Introduction to SPACE Laboratory and Testbed Kalman Filter Familiarization and Paper Surveying.D

O

C

U

M

E

N

T

A

T

I

O

N

Learn Matlab, LabVIEW and C

Chris

TorresObserver

Timeline


2009 AUG SEP OCT NOV DEC

Jessica

Alvarenga

Kalman Filter Simulation in Matlab.

Initial Simulation of Testbed NoiseFinalize Matlab Simulation.

D

O

C

U

M

E

N

T

A

T

I

O

N

NSF GK-12 IMPACT LA

Allison

Bretaña

Introduction to Testbed

Initial Training

Chris

Torres

Kalman Filter Design

Testbed Noise Analysis Kalman Filter Simulation

Timeline


DEC JAN FEB MAR APR MAY JUN JUL

Jessica

Alvarenga

Noise Modeling and Investigation of Plant Model Coding Implementation of KF in C codeD

O

C

U

M

E

N

T

A

T

I

O

N

NSF GK-12 IMPACT LA

Allison

Bretaña

Familiarization with Testbed Simulink Modeling of KF FDI SchemaIntegration of sensor noise statistics

into KF FDI Schema

Initial training period Sensor Noise Modeling Integration of Noise Model into C code


• Project Background

• Lyapunov Observer

• Kalman Filter



• Noise Modeling

• Future goals

• Timeline

• References


ReferencesAndrews, A. and Grewal, M. (2001). Kalman Filtering: theory and practice using MATLAB. New

York, NY: John Wiley and Sons Inc.

Boussalis, H., “Stability of Large Scale Systems”, New Mexico, USA, November, 1979.

Boussalis, H., Guillaume, D., Wu, C., Liu, C. (2009). Space URC Annual Report. NASA, 139.

Boussalis, H., Mirmirani, M., Chassiako, A., Rad, K., “The Use of Decentralized Control in Design of a Large Segmented Space Reflector”, Control and Structures Research Laboratory, California.

Cao, Yi (February 5, 2010 information retrieved). MATLAB Central. http://www.mathworks.com/matlabcentral/fileexchange/18465

Clark, B., Larson, E., Parker,E. Model-Based Sensor and Actuator Fault Detection and Isolation. NASA Langley Research Center,5.

Greg, W. & Bishop, G. (2006). An Introduction to the Kalman Filter. University of North Carolina at Chapel Hill, NC 27599-3175.

NASA. (November 30, 2009 revision). James Webb Space Telescope. Retrieved from www.jswt.nasa.gov/

Simon, D. (2001). Kalman Filtering. Embedded Systems Programming, 73-79.

Simon, D. (2006). Optimal State Estimation: Kalman, H Infinity and Nonlinear Approaches. Hoboken, NJ. John Wiley and Sons Inc.


Questions?

Thank You


control team welcome dr. spanos

Documents

current state estimate

presence of noise

numerical method

internal states

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