Attitude Control in the Presence of Structural Disturbances

Attitude Control in the Presence of Flexible Structures

Steve Rogers

1


• Disturbance Rejection Techniques for Structural Modes

• PID/notch/lag control example using pidtune & sisotool.

• Fixed or adjustable Notch Filter Design for a known, invariable frequency disturbance. Design for width & depth from bode plots.

• Disturbance Accommodating Control Applications to systems with known & unknown disturbances.

• Active Noise applications (auxiliary signal available) – Narrowband (signal available with information) – Partially known

2

Example Structural Mode Poles & Zeros

10-1

100

101

102

0

45

90

135

180P.M.: -8.06 deg

Freq: 2.51 rad/s

Frequency (rad/s)

-60

-50

-40

-30

-20

-10

0

10

20

G.M.: inf

Freq: NaN

Stable loop

Open-Loop Bode Editor for Open Loop 1(OL1)

-0.08 -0.06 -0.04 -0.02 0-20

-15

-10

-5

0

5

10

15

200.000350.00070.001150.00170.00240.0034

0.0055

0.012

0.000350.00070.001150.00170.00240.0034

0.0055

0.012

2.5

5

7.5

10

12.5

15

17.5

20

2.5

5

7.5

10

12.5

15

17.5

20

Root Locus Editor for Open Loop 1(OL1)

Real Axis

1st 3 modes shown (6th order model) from the flexible spacecraft simulator at the US Naval Postgraduate School. (See ref) Sisotool data from model.

Step Response

Time (seconds)

Am

plit

ud

e0 200 400 600 800 1000 1200

-0.05

0

0.05

0.1

0.15

0.2

Ref: http://egr.uh.edu/smsl/Publication/Song_Attitude_Control_PWPFSpace_Technology_1997.pdf 3

Example Structural Mode Poles & Zeros 1st 3 modes used. Used pidtune to solve with pidf option. New PM = 90 deg. Response is good but slow (time constant ~ 334 sec).

10-1

100

101

102

0

45

90

135

180Nominal

P.M.: -8.06 deg

Freq: 2.51 rad/s

Frequency (rad/s)

-80

-60

-40

-20

0

20

Nominal

G.M.: inf

Freq: NaN

Stable loop


-0.08 -0.06 -0.04 -0.02 0-20

-15

-10

-5

0

5

10

15

20Root Locus Editor for Open Loop 1(OL1)

Real Axis

0 200 400 600 800 1000 1200-0.5

0

0.5

1Step Response

Time (seconds)

Am

plit

ud

e

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1Step Response

Time (seconds)

Am

plit

ud

e

4

Example Structural Mode PID with notch 1st 3 modes used. Compensators: 1) PIDF, 2) PIDF plus notches at 1st 2 modes & 3) PIDF plus notches at 1st 2 modes plus 2 poles @ -0.6, then used pidtune to solve with pidf option. Response is better (rise time 11.2, 21.5, vs 334 sec. Bandwidth much better).

5

10-1

100

101

102

0

90

180Nominal

P.M.: -8.06 deg

Freq: 2.51 rad/s

Frequency (rad/s)

-140

-120

-100

-80

-60

-40

-20

0

20

Nominal

G.M.: inf

Freq: NaN

Stable loop


-1.5 -1 -0.5 0-20

-15

-10

-5

0

5

10

15

200.0060.0130.0210.0320.0440.065

0.1

0.2

0.0060.0130.0210.0320.0440.065

0.1

0.2

2.5

5

7.5

10

12.5

15

17.5

20

2.5

5

7.5

10

12.5

15

17.5

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Root Locus Editor for Open Loop 1(OL1)

Real Axis

0 200 400 600 800 1000 1200-0.5

0

0.5

1Step Response

Time (seconds)

Am

plit

ud

e

Step Response

Time (seconds)

Am

plit

ud

e

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

System: untitled1(:,:,1)

Rise time (seconds): 334


Rise time (seconds): 21.5


Rise time (seconds): 11.2


• PID/notch/lag control example using pidtune & sisotool.

• Fixed or adjustable Notch Filter Design for a known, invariable frequency disturbance. Design for width & depth from bode plots.

• Disturbance Accommodating Control Applications to known & unknown disturbances.

• Active Noise applications (auxiliary signal available) – Narrowband (signal available with information) – Partially known plus narrowband

6

Disturbance Accommodating Control

• DAC relies on disturbance state estimation based control.

• Example of unknown & known formulation of disturbance states z.

• Most of the time there are no available or useful measurements.

• Frequencies wi are known/estimated from structural analyses.

• Attenuation of disturbance states by manipulating values of Q (states) & R (control) matrix elements in lqr method (as an example).

• Disturbance States augmented to plant.

• Pc is control input. • With this approach can use any

applicable state-space design method (pole placement, lqr, or H-inf).

7

unknown Known states

Typical DAC Schematic

Conventional Control

DAC Control signal

8

DAC controller is an add-on to conventional controller.

Plant/disturbance are not limited to linear

Adaptive DAC • The adaptive DAC method estimates changes in the control matrix of the

system under control. The changes are treated as disturbances & may occur due to model error, unmodeled dynamics, mfg variances, wear, etc.

• Adaptive DAC has none of the stability proof baggage associated with most adaptive methods, nor does it need occasional persistent excitation.

• This has the same approach as the above described DAC, but can estimate the disturbance associated with variations in the control matrix. Note: Da has the same eigenvalues as A & is usually set to A.

9

• Solution methods are the same as DAC. • Note that below shows that adaptive DAC

may be combined with DAC • The concept allows development of 3

separate controllers: 1) conventional, 2) normal DAC, & 3) adaptive DAC.


• Active Noise Control

• Adaptive Noise Cancellation (ANC)

• FIR/IIR Adaptive Architecture

• ANC with Partially Known Information

• Differencing Filters for ANC

• Results and Conclusions

10

Active Noise Cancellation Overview

• Active Noise Cancellation may be applied if auxiliary sensors exist. The design of an active noise cancellation system is achieved based on the principles of feedback control.

• Key Point - A system for adaptive noise cancellation has at least two inputs: 1) a noise-corrupted signal of interest and 2) an additional signal containing the offending noise, but without the signal of interest.

• This process is active/adaptive noise control and uses ideas from control theory.

• The active noise controller may be non adaptive or adaptive.

11

Active Noise Control Applied to Flexible Structures

• Active noise canceling relies on the creation of an ‘anti-noise’ or noise inversion, such as by using a bandpass or notch filter. Active Noise control may be a non adaptive or an adaptive component.

• The diagram shows 2 options for non-adaptive noise attenuation. The flexible structure sends disturbances that corrupt the pqr signals. The Anti-Noise filter receives a signal from a strain gage equivalent or an accelerometer in both cases. The filter output can be added either to ‘Sig’ or the control cmd.

• The Anti-Noise filter will have a normalized gain & will try to cancel the applicable structural modes of the flexible structure.

• Non adaptive approach is to first add a filter network (lag/lead/notch/bandpass) to the auxiliary signal & route it to the applicable input. The design objective is to move the poles away from the unit circle or improve stability. The new composite plant becomes the design plant for the traditional controller.

12

Option A: Used in Wind Turbines with accelerometers

Option B: Used in acoustic apps

Adaptive FIR Noise Cancelling • Adaptive noise canceling relies on an operation controlled in an adaptive manner for

the purpose of improving signal-to-noise ratio. The error term of this system is given by

• The following descriptions use option B. If the error e[n] signal is minimized by the adaptive component, e[n] converges to the signal s[n] as the filter output y[n] adaptively attenuates the common noise components of the input v1[n] also contained in v0[n].

• The signal e[n] becomes the signal used by the downstream systems in option B. Care must be taken in the location and selection of the additional sensor(s) that produces the signal v0[n]. They should not contain portions of s[n].

13

nynvnsne 1

1

0

0 )()(ˆ][M

k

k knvnwny

The reference signal v0(n) is processed by an adaptive FIR filter to produce the output signal.

The adaptive noise canceller should have no effect on the primary signal s(n), and the output signal-to-noise ratio is improved according to the amount of common (cancellable) frequency components.

][][][]1[0

nennwnw v

Adaptive IIR Filter • A typical adaptive IIR structure is given in the figure below. The residual

error is

14

][][][ nyndne

The output signal of the IIR filter y[n] is computed as

]1[][][][][ 0 nynbnvnany TT

•The objective of the adaptive system is to determine an optimum set of coefficients a[n] and b[n] to minimize a performance criterion that is based on the error signal e[n]. •As with the adaptive FIR filter, the steepest-descent algorithm can be used to update the filter coefficients. •A typical update equation for the general adaptive vector w[n] & then the specific a[n] & b[n] vectors is shown to the right. •More difficult than FIR to ensure stability.

][][][]1[ nenunwnw

][]1[][]1[

][][][]1[ 0

nenynbnb

nenvnana


• Motivation/Introduction






15

In the Case of Partially Known Frequency Modes

• When the principal disturbance frequency or frequency of interest is known this information may be incorporated into the filter structure.

• Some of the structures may be simple enough to enable prediction of a narrow range for the disturbance frequency.

• The end result is the possibility to improve accuracy and reduce the size of the adaptive filter when additional knowledge is available.

• In this case the adaptive (unknown) filter becomes an add-on component to a non-adaptive (known) filter.

16

Differencing Adaptive Filters - 1

• Differencing filters are those with exogenous inputs or ARMAX (auto regression moving average exogenous) models.

• An extra term based on a calculated deviation or differencing signal is included.

• There are two options for the differencing signal: input and output. • The typical ARMAX model equations for input differencing are:

17

]1[0][0][

]1[][]1[][][0][][

nvnvne

nenT

cnynT

bnvnT

any

• The ARMAX model equations for output differencing are:

]1[][][

]1[][]1[][][][][ 0

nynyne

nencnynbnvnany TTT

Differencing Adaptive Filters - 2 • The update equations for differencing adaptive filters are:

18

][][][

][][][]1[

][]1[][]1[

][][][]1[ 0

nyndne

nenencnc

nenynbnb

nenvnana

•The vector ‘c’ is of arbitrary but appropriate length. •The exogenous inputs incorporate a velocity vector into the filter so that it becomes slightly more responsive to transients and improves tracking. •The differencing filter may also be upgraded to include known information in the same way as the typical adaptive filter.







• Study Results and Conclusions

19

Adaptive Filters Considered

• There were seven different adaptive filter structures described above used in a Satellite flexible solar panel study. The model was a generic satellite model. It was modified to create the effects of a flexible solar panel. They are listed as follows:

• Cmode = 0 FIR filter • Cmode = 1 FIR filter with input differencing • Cmode = 2 IIR filter • Cmode = 3 IIR filter with input differencing • Cmode = 4 IIR filter with output differencing • Cmode = 5 IIR with partially known flexible model and input

differencing • Cmode = 6 FIR filter with partially known flexible model

20

Summary of Results • The error and fuel metrics were calculated by summing the absolute value of the

measurements at each time step. • These metrics are shown in the table below along with filter descriptions. • The data was generated from an 8000 second trajectory, which is approximately

one orbit. • The exogenous values were the differences between current and previous

adaptive filter input or output values. • The results show dramatic improvement regarding fuel consumption due to ANC

applications.

21

Active Noise Cancellation Type lags Euler error

(rad)

Fuel consumption

(N-m)

Turned off (baseline) 133 4963

FIR filter 970-num 120 1531

FIR filter with Input Differencing 815-num, 120-exog 133 2339

IIR filter 210-num, 120-den 117 1354

IIR filter with Input Differencing 155-num, 38-den, 40-exog 115 1064

IIR filter with Output Differencing 155-num, 38-den, 40-exog 115 1064

IIR filter with Partially Known Model

and Input Differencing

155-num, 80-den, 80-exog 115 1091

FIR filter with Partially Known Model 1230-num, 200-exog 130 2139

Backup Slides

22

Active/Adaptive Noise Cancellation Applied to Structural Vibration Attenuation







23

Motivation of Active Adaptive Noise Cancellation Systems

• Active Control Systems enable controllers to incorporate more information in order to improve performance.

• In many aerospace systems the weight budget dictates lighter structures which lead to more structurally flexible systems. – Long slender structural elements are characterized by

low frequency modes and low damping coefficients. – If the lower frequency vibration modes are low

enough they can interfere with control loops. Low damping exacerbates this condition and prolongs the vibration.

24

Active Noise Cancellation Overview • The objective of an active noise cancellation system is

to reduce the effects of or attenuate an undesired signal component from a given signal by use of an auxiliary signal.

• Similar adaptive filtering technology is used commercially in telecommunications for audio noise and echo cancellation.

• An FIR (numerator) filter is usually chosen as the adaptive filter structure due to its simplicity and ease of stabilization.

• The coefficients of the FIR filter are modified in real time according to a function of an error signal.

25

Adaptive filters with Additional Knowledge

The original IIR equation is:

26

]1[][*][][][][ 0 nynSxnbnvnanyTTThe new equation is:

]1[][][][][ 0 nynbnvnany TT

•Many flexible structures may be approximated as a simple transfer function derived from test data. •We would usually only know the frequency of the flexible structure and not the damping coefficient or the amplitude of an equivalent transfer function. •A transfer function Sx(z) may be composed of a known frequency plus an estimated damping coefficient.

Active/Adaptive Noise Cancellation Applied to Structure Vibration Attenuation






• Flexible Models for Simulation

• Simulink Model


28

Flexible Aerospace Structures

• Autonomous vehicles including UAV’s and satellites have missions that require light weight flexible appendages to reduce their weight and volume footprints.

• Large flexible appendages common to satellites include parabolic antennae, large synthetic-aperture radar, and solar arrays built from light materials in order to reduce their weight. Long duration UAV’s require light and high lift coefficient airframes.

• Spacecraft structures are now becoming extremely complicated because of these appendages, which also may induce structural oscillation under the excitation of external torques and forces.

• The vibration dynamics emanating from flexible appendages and liquid sloshing may interfere with the attitude control dynamics.

• In this study only the flexible solar array is modeled, which is constructed as an additive set of transfer functions

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Flexible Model Used in Simulation

• The development of the model is based on the Lagrange equations and one-mass modeling.

• The model is composed of the rigid satellite body with two symmetrical solar arrays, each modeled by a single mass.

• The input to the linear transfer functions is the three channel torque vector T(s) and the output is the flexible angular position vector θ(s) contribution from the solar array. Fi(s) is the optional contribution from other flexible structures.

• The output angles θ(s) are added to the rigid body Euler angles.

• The double integrator symbolically represents the rigid body system.

30

Active/Adaptive Noise Cancellation Applied to Structure Vibration Attenuation






• Flexible Models for Simulation

• Simulink Model


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Attitude Control System Simulink Model • The figure below shows the Attitude Control System and how the adaptive noise

control system integrates into it. • The adaptive noise control system is designated ANC toward the bottom center of

the figure. • It may be turned on/off by a switch. • We assume the strain gages may be oriented according to each pqr axis.

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Attitude Control

ANCout

3

torque cmd

2

atterr

1

zero cmd

(0 0 0)

alterrr smoothing

nav_mode

atterr

atterr_smooth

Sum of

Elements

Selector

Rate Transition 2

Copy

Rate Transition 1

NoOp

Quaternions to Rotation Angles

Rotation Order: ZYX

q [R1,R

2,R

3]

Quaternion

Multiplication

q

r

q*rQuaternion

Inverse

q Inv (q)

PD Controller

PD

Math

Function

uT

Abs

|u|

ANC

SG

pqr

pqrTruth

signals

yerr

pqrTruth

5

StrainGage

4

pointing mode

3

pointing command

2

q_ECI2Body

1[1x4]4 4

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4

4

3

3

3

3

4

4

3

3

3

3

3

3Roll

Pitch

Yaw

3{3}3

3

3 6

3 3

3

3

Simulation • The seven different strategies as explained above were tested on the satellite model flight software shown below. • An 8000 second trajectory was simulated for each of the seven strategies, which is approximately one orbit around the

earth. The mission was to allow a sensor to ‘stare’ straight down at the earth. • Two performance metrics were used for comparison: 1) Tracking error, difference between the ‘truth’ Euler angles and

the filtered output and 2) equivalent fuel consumption in N-m

33

Flight Software

Copyright 2006 -2007 The MathWorks , Inc .

ANCout

2

fs output

1

Signal

Conversion

Rate Transition 1

Copy

Rate Transition

Navigation

asbsatellite _Navigation

Position _ECEF

Velocity _ECEF

sensor failure

nav mode

pointing command

propulsion command

Goto

[nav _mode ]

From 2

[nav _mode ]

[nav _mode ]

From

[nav_mode ]

Failure Detection

asbsatellite _FailureDetection

sensor outputs

torque commands

sensor failure

actuator failure

Attitude Control

FlexSatellite _ACSq_ECI2Body

pointing command

pointing mode

StrainGage

pqrTruth

atterr

torque cmd

ANCout

pqrTruth

4SG

3

nav mode

2

sensor outputs

1

3{3}

atterr

9{11}

9{11}

4

4

4

<qecef 2b>

3

<xecef >

3

<vecef >

4{4}

actuator failure

9{11}

4{14}

4{14}

nav_mode

nav_mode

3 3

torque_cmds

3

3

nav_mode

3

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Sample Plots

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0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05

ANC turned off

total error = 133.3808 rad, total fuel = 4963.3052 N-m

roll

(rad)

0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05

pitch (

rad)

0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05yaw

(ra

d)

time in sec

0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05

FIR filter with partially known structure

total error = 129.8659 rad, total fuel = 2138.7937 N-m

roll

(rad)

0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05

pitch (

rad)

0 1000 2000 3000 4000 5000 6000 7000 8000

-0.05

0

0.05

yaw

(ra

d)

time in sec

Active Noise Cancellation Using an FIR filter with a Partially Known Model

Active Noise Cancellation Turned Off (Baseline)

Note that the standard controller has a dead band and only consumes fuel when outside the control band.

Attitude Control in the Presence of Structural Disturbances

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