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Attitude Control in the Presence of Flexible Structures Steve Rogers

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Attitude Control in the Presence of Flexible Structures • 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

Root Locus Editor for Open Loop 1(OL1) 20 0.0034

15

20

0.0024 0.0017 0.00115 0.0007 0.00035 17.5 15

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

Open-Loop Bode Editor for Open Loop 1(OL1) 20 G.M.: inf 10 Freq: NaN Stable loop 0

0.0055 12.5

-10 10

10

Step Response

-20

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7.5 -30

0.012

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0 2.5 -5

P.M.: -8.06 deg Freq: 2.51 rad/s

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0.012

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

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Example Structural Mode Poles & Zeros Root Locus Editor for Open Loop 1(OL1)

Open-Loop Bode Editor for Open Loop 1(OL1) 20

20

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

0

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-20 10 -40 5 Nominal -60 G.M.: inf Freq: NaN Stable loop -80 180 Nominal P.M.: -8.06 deg Freq: 2.51 rad/s 135

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Example Structural Mode PID with notch Root Locus Editor for Open Loop 1(OL1) 20 0.065

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0.044 0.032 0.0210.013 0.006 17.5

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).

Open-Loop Bode Editor for Open Loop 1(OL1) 20 0

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90 System: untitled1(:,:,3) 0.8 Rise time (seconds): 11.2 0.6

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System: untitled1(:,:,2) Rise time (seconds): 21.5

System: untitled1(:,:,1) Rise time (seconds): 334

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Attitude Control in the Presence of Flexible Structures • 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 unknown

Known states

• 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


Typical DAC Schematic Plant/disturbance are not limited to linear

Conventional Control

DAC Control signal

DAC controller is an add-on to conventional controller.

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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. • 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.

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Attitude Control in the Presence of Flexible Structures

• • • • • •

Active Noise Control Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Results and Conclusions

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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.

Option A: Used in Wind Turbines with accelerometers

Option B: Used in acoustic apps 12


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

en  sn  v1 n  yn

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]. The reference signal v0(n) is processed by an adaptive FIR filter to produce the output signal. M 1

y[n]   wˆ k (n)v0 (n  k ) k 0

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.

w[n  1]  w[n]  v0 [n]e[n] 13


Adaptive IIR Filter • A typical adaptive IIR structure is given in the figure below. The residual error is e[n]  d[n]  y[n] The output signal of the IIR filter y[n] is computed as

y[n]  aT [n]v0 [n]  bT [n] y[n  1] •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 steepestdescent 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.

w[n  1]  w[n]  u[n]e[n] a[n  1]  a[n]  v0 [n]e[n]

b[n  1]  b[n]   y[n  1]e[n] 14


Attitude Control in the Presence of Flexible Structures • • • • • •

Motivation/Introduction Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Results and Conclusions

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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:

T

T

T

y[ n ]  a [ n ]v 0 [ n ]  b [ n ] y [ n  1]  c [ n ]e [ n  1] e [ n ]  v 0 [ n ]  v 0 [ n  1]

• The ARMAX model equations for output differencing are:

y[n]  aT [n]v0 [n]  bT [n] y[n  1]  cT [n]e[n  1] e[n]  y[n]  y[n  1] 17


Differencing Adaptive Filters - 2

• The update equations for differencing adaptive filters are: a[n  1]  a[n]   v0 [n]e[n]

b[n  1]  b[n]   y[n  1]e[n] c[n  1]  c[n]   e [n]e[n] e[n]  d [n]  y[n]

•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. 18


Attitude Control in the Presence of Flexible Structures • • • • • •

Motivation/Introduction Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Study Results and Conclusions

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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

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• • • •

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. Active Noise Cancellation Type

lags

Turned off (baseline)

Euler error Fuel consumption (rad) (N-m) 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 155-num, 80-den, 80-exog and Input Differencing FIR filter with Partially Known Model 1230-num, 200-exog

115

1091

130

2139 21


Backup Slides

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Active/Adaptive Noise Cancellation Applied to Structural Vibration Attenuation

• • • • • •

Motivation/Introduction Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Results and Conclusions

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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.

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Adaptive filters with Additional Knowledge The original IIR equation is: y[n]  aT [n]v0 [n]  bT [n] y[n  1]

The new equation is:

y[n]  aT [n]v0 [n]  b[n] * Sx[n] y[n  1] T

•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. 26


Active/Adaptive Noise Cancellation Applied to Structure Vibration Attenuation

• • • • • • • •

Motivation/Introduction Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Flexible Models for Simulation Simulink Model Results and Conclusions 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

• • • • • • • •

Motivation/Introduction Adaptive Noise Cancellation (ANC) FIR/IIR Adaptive Architecture ANC with Partially Known Information Differencing Filters for ANC Flexible Models for Simulation Simulink Model Results and Conclusions 31


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.

• • •

Attitude Control

4

1 q_ECI2Body

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Inv (q)

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pqr pqrTruth yerr

ANC

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• • •

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 Flight Software

4

q_ECI2Body FlexSatellite _ACS

<qecef 2b>

atterr asbsatellite _Navigation Position _ECEF

3 <xecef >

4{14}

pointing command 3

4 4 [nav _mode ]

<vecef >

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propulsion command [nav _mode ]

sensor outputs Rate Transition

From 2

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pqrTruth

sensor failure

ANCout pqrTruth

atterr 3

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Attitude Control

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pointing command

nav_mode

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sensor failure 4{4}

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actuator failure

actuator failure

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Copy

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nav mode Rate Transition 1

[nav _mode ] Goto

Copyright 2006 -2007 The MathWorks , Inc .

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

Active Noise Cancellation Turned Off (Baseline) ANC turned off total error = 133.3808 rad, total fuel = 4963.3052 N-m

FIR filter with partially known structure total error = 129.8659 rad, total fuel = 2138.7937 N-m

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Note that the standard controller has a dead band and only consumes fuel when outside the control band. 34


Attitude Control in the Presence of Structural Disturbances  

Disturbance Rejection Techniques for Structural Modes. Disturbance Accommodating Control Applications to systems with known & unknown distur...

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