6.2011-6395

AIAA 2011-6395
AIAA Guidance, Navigation, and Control Conference
08 - 11 August 2011, Portland, Oregon
Identification of Gimbaled Gyroscopic Systems Using
Higher Order Sliding Mode Observers
Nathan G. Brown1
U.S. Army Research and Development Command, Huntsville, AL, 35808
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Yuri B. Shtessel2
Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL,
35899, USA
In industry it is often necessary to create a representative mathematical model of a piece
of unknown or partially known hardware. Solutions to this problem exist for linear systems,
but depend heavily on the availability of the system state vector and its derivative for
calculation. This paper proposes the use of recent advances in sliding mode control and
observation theory to overcome this data requirement. The higher order sliding mode
(HOSM) control based identification algorithm that incorporates higher order sliding mode
differentiators has been used for identification of a linearized 2-axis gimbaled gyroscope
system. A full non-linear model of the system is created via system identification using the
results of the identification. The identification process is performed for both exact and noisy
measurements showing exceptional performance of the HOSM-based identification
algorithm.
Nomenclature
=
=
=
=
=
=
=
=
=
=
=
=
=
1
2
Pitch gimbal angular deflection from zero
Yaw gimbal angular deflection from zero
Spring Torque Constant, Pitch Channel
Viscous Friction Constant, Pitch Channel
Spring Torque Constant, Yaw Channel
Viscous Friction Constant, Yaw Channel
Rotor Spin Speed, rad/sec
Spin angular momentum of the rotor
Moment of Inertia of the r object about the x axis
Denotes the rotor
Denotes the inner gimbal
Denotes the outer gimbal
Lie Derivative
Civilian Engineer, AMRDEC, Nathan.G.Brown@us.army.mil
Professor, Associate Fellow of AIAA and IEEE Senior Member, Shtessel@ece.uah.edu.
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Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
I. Introduction
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A
s time passes and technology improves the intended environment for a mechanical system will often change
before the hardware ceases to be of use or its inventory is depleted. When this type of requirement change
occurs it is usually cost effective to update the given hardware stores to the new environment or in some way recycle
it for other applications. In either case, a complete understanding of the hardware is needed to perform the required
modifications. When attempting to thoroughly understand such a device, attention inevitably turns to test data and
documentation saved from the original design process of the mechanism. Unfortunately, design documentation is
frequently insufficient, lost or was never created. This occurs with both private sector and government technologies.
Considering that documentation costs money, data storage mediums degrade and people retire this is not a surprise.
It does, however, create a distinct problem when working with older systems.
To fill in the holes of inadequate data about a device, system identification methods are regularly used. In
literature system identification deals primarily with linear systems and in most instances requires full knowledge of
all of the system states and their derivatives. However, in the majority of realistic environments all of the system
states and state derivatives are rarely, if ever, available. This paper proposes the use of recent developments in
higher order sliding mode theory to help provide the required data. Acknowledging the fact that many mechanical
systems exist in which the states are either measureable or are direct derivatives of a measurable state, the problem
of obtaining the required state information often reduces to the problem of taking multiple derivatives accurately in a
non-ideal environment. This is not easily done through conventional, linear means but is readily accomplished
through higher orders sliding mode techniques.
To demonstrate the effectiveness of this technique in practical application a study is performed in which an
unknown 2-axis gimbaled gyroscope system is identified. The gyroscope considered is typical of many commercial
and military applications and differs from academic models in that the rotor is outside of the gimbal structure. When
this is true, the moments of inertia of the gimbals become negligible and the full non-linear model and its linear
approximation may be directly related. A simple illustration of this type of gyroscope is provided in Fig. 1.
Figure 1. Illustration of Gyroscope Rotor Configuration
II. Mathematical Model of Gyroscope System and Problem Formulation
Specifically, this paper considers a yaw-pitch gimbaled gyroscope which experiences both spring torque,
and
, and viscous friction,
and , on each gimbal respectively. Mathematically, the system can be fully described
by the differential equations,
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,
(1)
and
(2)
where
and
are known user generated control torques and the total moments of inertia are defined as,
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,
(3)
,
(4)
,
(5)
.
(6)
and
Since a configuration was chosen such that the gimbal inertias are much less than those of the rotor, the gimbal
moments of inertia in Eqs. (1) through (4) may be safely ignored leaving the total moments of inertia approximately
equal to those of the rotor. Using this and typical small angle approximation, Eqs. (1) and (2) may be modeled by
linear state equations as,
(7)
where
and
.
Therefore, it can be stated that the ultimate goal of this demonstration is to obtain accurate estimates of all of the
parameters composing the A and B matrix of Eq. (7) using a higher order sliding mode supported system
identification effort. From that result, the parameters are reorganized into the form of Eqs. (1) and (2) producing a
viable model of the gyroscope which is valid for large deflections. Furthermore, the available data is assumed to be
only the potentiometer measured states and and the control torques
and
.
III. Method of System Identification
Many methods of system identification exist. For example, there are methods that rely on curve fitting phase and
gain data to a desired order of transfer function and others that even use genetic algorithms to estimate system
parameters based on scoring model performance. The method considered for this study is based on the modified
HOSM-based identification algorithm2. Consider the general state space representation of a system given as,
.
(8)
Secondly, assume that the control inputs,
, are known and that all of the states and their derivatives are
measurable and without noise. If these conditions are met, then it is possible to determine values for exactly via2,
.
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(9)
The matrices
are formed by appending the vectors
, respectively, as samples are taken versus time, as
shown in Eq. (10). The order and spacing of the samples is not important so long as the entries are consistent for
each matrix.
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(10)
By constructing the matrices using this process, the matrix becomes defined exactly. This means that there is
precisely enough information, measurements, to define . The matrices constructed,
, are square and will be
invertible based on the linear independence of the states being sampled.
This method works well for perfect samples. However, if the samples are noise corrupted even slightly the
resulting estimation will degrade rapidly. To make the method more robust, more samples can be incorporated into
the calculation. The rationale is that the estimation of A will be over specified and white noise will effectively
average out. This will however produce a dimension problem for the inverse which is dealt with via the MoorePenrose pseudo inverse as6,
(11)
where
and
. This modification produces significantly improved results over using n
values. In order to implement the identification algorithm given by Eqs. (9) and (11), the state vector
measured and differentiated. For this purpose, the higher order exact sliding mode differentiator is used.
must be
IV. The Higher Order Sliding Mode Exact Differentiator
Equation (11) has stringent data availability requirements. In order to cope with that requirement, this paper
proposes the use of sliding mode control theory (SMC). SMC has become a major field of study within control
system engineering since it was introduced in the 1960’s. SMC differs from the widely accepted linear control
methods in that it is by definition a discontinuous, high speed switching control based on a sliding variable, , being
driven to zero. There are many advantages to SMC with the foremost being that it does not require a linear plant and
is quite robust to disturbances. In fact, under certain conditions, disturbances can be completely rejected by the
controller. Furthermore, it is a relatively straight forward process to use sliding mode theory as an observer or signal
differentiator.
Unfortunately, classical sliding mode as described suffers from so-called chattering in instances when the rate of
switching is insufficient. This phenomenon makes classical SMC inappropriate for digital processing applications
requiring an extreme degree of accuracy such signal differentiation for system identification. Luckily, recent work
by Levant in the field of higher order sliding mode has produced a viable, high accuracy alternative.
A. Basics of Higher Order Sliding Mode (HOSM)
HOSM techniques differ from traditional techniques in that a number of derivatives of the sliding variable, , are
driven to zero as well as itself. Accordingly, the number of derivatives of that a controller may exactly drive to
zero is called the order of the sliding controller. Consider a dynamic system
that is closed by
discontinuous feedback and provides a continuous output
in the state space of the system. If the
time derivatives
are also continuous in the state space and Eq. (12) defines a non-empty set
containing Fillipov trajectories the resulting system motion for this condition is termed r-sliding motion10.
As with first order SMC, this sliding motion will be constrained by the dynamics described by,
(12)
according to the definition of in terms of system states. Clearly, since continuity was assumed for the first
derivatives, the discontinuous control term will not be present in those derivatives of . However, if
discontinuous, the set given by Eq. (12) is called strict10.
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is
Secondly, Levant proposes that if Eq. (12) is true and the rank test,
where
(13)
is also shown to be true, the set given by Eq. (12) is said to have met the r-sliding regulatory condition and
represents a differentiable manifold for
order SMC. It is noteworthy that if the relative degree of the system is
equal to r, then it is implied that
contains
explicitly and Eq. (13) will always be met10.
Furthermore, Levant proposes that r-sliding modes will exist if the controller is allowed sufficient gain.
Consider a system of the form,
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(14)
with relative degree r near some r-sliding point
. Next, it is assumed that the magnitude of the discontinuous
control
is allowed to take values according to
. If the value of K is sufficiently large, it can be
shown that r-sliding modes will exist in the vicinity of
and r-sliding control may be possible.
The existence of r-sliding modes is less important than the existence of stable, finite time convergent r-sliding
modes. Since the regulatory condition was met, it is known that all of the partial derivatives of
with
respect to
are zero as,
(15)
where
represents a Lie Derivative. It is also apparent that
and may be expressed by,
(16)
where
. Simply stated,
is the linear combination of the unforced dynamics of
and the forced
dynamics of . Clearly, is a continuous function since it is composed of
. If it is globally bounded, it is
reasonable that a Fillipov construct could always be found to satisfy Eq. (16). Levant has provided a condition for
stability of a sliding mode10. It states that for some
and
the relationship,
(17)
must hold. If this condition can be met through design choices, stability and accuracy of the controller are
guaranteed.
B. Form of the Differentiator
Consider that it is desired to, in the absence of any noise, form exact derivatives of some input signal ,
where the
derivative has a known Lipschitz constant . This is readily accomplished using HOSM techniques in
a recursive fashion. This is a powerful result since the mechanism can be extended to obtain many derivatives.
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In an earlier work10, Levant shows that a first order exact differentiation using super-twisting control may be
performed as,
(18)
.
In Eq. (18), it is clear that
represents an error signal. Thus, the variable
. Subsequently, the desired derivative
may be found according to,
is being forced to track
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.
(19)
The number of derivatives tracked may be increased by joining subsequent systems of form similar to Eq. (18).
In general, a differentiator of arbitrary order may be created according to 10,
(20)
.
where
in finite time.
C. Differentiator Tuning
An important activity in the design of the HOSM differentiator is the selection of appropriate gains. These gains
must be selected based on the Lipschitz constant of the derivative
of the input signal . The best method for
choosing these gains is to first select gains that stabilize an equivalent differentiator for Lipschitz constant
.
This is usually done through the use of computer simulation. Once complete, the gains designed for
may be
extended to accommodate any signal of interest with
having a constant
. This extension is performed for
the
gain according to the relationship,
(21)
where
represents the gains that provide a stable differentiator for
.This will provide an accuracy of,
(22)
for some
, which depends on the choice of
, and noise bound 10. In order to mitigate the effect of the
measurement noise it is recommended10 that the HOSM differentiator of Eq. (20) be implemented with n greater
than the number of derivatives actually required. This will improve the noise filtration of the lower order derivatives
and improve accuracy.
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V. Gyroscope Specific Solutions
According to the system HOSM-based identification method given by Eqs. (9)-(11), the control matrix B needs
to be available to effectively estimate the state matrix A. For the gyroscope system studied this is not assumed to be
the case. To proceed, some system specific methods are developed to obtain an estimate of B.
A. Estimating the Spin Angular Momentum, h
The first step in estimating the matrix B is to generate a good estimate of the spin angular momentum, h. This
can be done through the so-called law of the gyroscope which is given as,
(26)
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and
(27)
where the superscript I implies motion with respect to inertial space as opposed to another gimbal. Noting that the
control torque is much larger than the disturbances, it is reasonable to assume that
and
for the
moment. In this case, it is easy apply a known torque to each channel independently and solve the corresponding
equation for h. Doing so for each sample as it’s taken and averaging them together produces a good estimation of h.
Mathematically, this appears as,
(28)
where
and
. Note that L is the number of samples taken.
B. Estimating the Total Moments of Inertia and
Next, it is necessary to solve for each of the moments of inertia composing B. From the state space description in
Eq.(7), the second equation can be written as,
.
(29)
Applying the system identification process as given will yield the row vector,
:
(30)
where the fourth element is the spin angular momentum h. Since this value was estimated in Section A, it can serve
as a tool to obtain . Selecting only the fourth element as indicated by the subscript and subtracting from it yields,
=0.
(31)
The Newton-Raphson technique for non-linear equations solves this quite easily. Convergence usually occurs
within four iterations limiting the workload of applying the system identification within the loop. This technique
works equally well with the fourth equation of the system. Unfortunately, as noise corruption is applied to the signal,
the effectiveness of this method degrades.
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C. Separating the Identification
It is well known in literature that least squares type identification methods struggle when there is a high level of
correlation between any of the input signals. Choosing some realistic values for the parameters yields a state matrix
as,
.
(32)
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Notice that the parameter
is much larger than any other coefficient in the second equation. When this occurs, the
second and third equations become highly correlated. This is also the case for the first and fourth equations.
Performing identification on this in simulation initially failed as a result. The solution is to separate the identification
process and isolate the equations of interest. In doing so, the equation,
(33)
can be re-written equivalently as,
.
(34)
Although this is mathematically equivalent, it makes a large difference when identifying the system. From that point
of view, the second equation contains more information since the parameter
has been dictated to be equal to zero
according to the mathematical derivation of the system. This arrangement simply enforces that criterion.
VI. Simulation
To confirm the proposed application of this differentiator, truth data was generated from the non-linear
model using a high speed computer program with parameters corresponding to Eq. (32) given as,
(35)
Care was taken to ensure that the control inputs would never cause the model to operate outside of the region in
which it behaves linearly. In a second program, the described methods were used to identify the system and
reconstruct a valid linear and subsequently a non-linear model. To evaluate the developed model, a comparison plot
between model and true system response is provided which operates well outside of the linear region.
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A. Results for Noiseless Measurements
The algorithm is first performed with noiseless data samples taken at a data rate of 3 KHz. The results shown in
Table 1 were obtained for each of the parameters. Notice that one of the parameters is in error by 35%. This may
appear large, but is actually quite good considering the size of the moment disturbance being measured. It does
however show up in the model comparison plot Fig.5.
Truth
Estimate
Error
h
0.44
0.4401
0.023%
Jy
0.0007
7.00E-04
0.000%
Jz
0.0009
9.00E-04
0.013%
k1
0.004
0.0045
12.500%
k2
0.0031
0.0032
3.226%
k3
0.004
0.0032
20.000%
k4
0.0031
0.002
35.484%
In addition to the upcoming comparison plots, Figs.2 and 3 show how the differentiator performed during the
execution of the algorithm.
Differentiator Performance for Pitch 1st Derivative
1
Truth
HOSM
0.5
Degrees/s
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Table 1: Algorithm Results for Noiseless Samples
0
-0.5
-1
-1.5
-2
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
Time, s
Figure 2: Pitch Differentiator Performance for First Derivative
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4.8
5
Differentiator Performance for Pitch 2nd Derivative
4
Truth
HOSM
Degrees/s 2
2
0
-2
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
Time, s
Figure 3: Pitch Differentiator Performance for Second Derivative
4.8
5
Comparison of Reconstructed Model and Truth for Large Deflection Angles: Pitch
Truth Model
Reconstructed Model
20
Pitch, Degrees
15
10
5
0
3.8
4
4.2
4.4
Time, s
4.6
4.8
5
Figure 4: Comparison of Truth and Model Response in Non linear Region - Pitch Channel
Comparison of Reconstructed Model and Truth for Large Deflection Angles: Yaw
5
Truth Model
Reconstructed Model
0
Yaw, Degrees
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-4
-5
-10
-15
-20
3.8
4
4.2
4.4
4.6
4.8
Time, s
Figure 5: Comparison of Truth and Model Response in Non linear Region - Yaw Channel
B. Results for Noisy Measurements
This section repeats the entire process except that 1E-6 1-sigma Gaussian noise has been additively applied to
the samples taken. An example of the noise is shown in Fig.6.
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Input Signal Compared to NoiseCorrupted Input Signal
Exact
Noisy
1.1154
1.1154
1.1154
1.1154
1.1154
1.1154
1.1154
1.1154
0.873
0.8735
0.874
Time, s
0.8745
0.875
0.8755
0.876
Figure 6: Example of Noise Corrupted Signal
Using this input signal leads to the results of Table 2. In this case, performance has deteriorated quite a bit. Most
importantly, the moments of inertia are now in error by up to 77% and one of the other errors has risen by orders of
magnitude. Figures 7-10 show the differentiator performance with the addition of zoomed sections. Notice the small
deviations that have developed.
Table 2: Algorithm Results for Noisy Samples
Truth
Estimate
Error
h
0.44
0.44011
0.025%
Jy
0.0007
9.60E-04
37.143%
Jz
0.0009
1.60E-03
77.778%
k1
0.004
0.0051
27.500%
k2
0.0031
0.0027
12.903%
k3
0.004
0.042
950.000%
k4
0.0031
0.0013
58.065%
Differentiator Performance for Pitch 1st Derivative
1
Truth
HOSM
0.5
Degrees/s
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0.8725
0
-0.5
-1
-1.5
-2
3
3.2
3.4
3.6
3.8
4
Time, s
4.2
4.4
4.6
Figure 7: Pitch Differentiator Performance for First Derivative
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4.8
5
Differentiator Performance for Pitch 1st Derivative
-0.6
Truth
HOSM
Degrees/s
-0.65
-0.7
-0.75
-0.8
3.53
3.54
3.55
3.56
3.57
3.58
3.59
3.6
Figure 8: Zoomed Section of Pitch Differentiator Performance for First Derivative
Differentiator Performance for Pitch 2nd Derivative
4
Truth
HOSM
Degrees/s 2
2
0
-2
-4
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
Time, s
Figure 9: Pitch Differentiator Performance for Second Derivative
4.8
5
Differentiator Performance for Pitch 2nd Derivative
-1.08
Degrees/s 2
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Time, s
Truth
HOSM
-1.1
-1.12
-1.14
3.968
3.9685 3.969
3.9695
3.97
3.9705 3.971 3.9715 3.972 3.9725 3.973
Time, s
Figure 10: Zoomed Section of Pitch Differentiator Performance for Second Derivative
Again, comparative plots are shown of the truth and model responses. Figure 12 shows noticeable deviation from the
truth response. This response, though not terrible, represents about 2 degrees error and will grow as the simulation
runs.
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Comparison of Reconstructed Model and Truth for Large Deflection Angles: Pitch
20
Truth Model
Reconstructed Model
Pitch, Degrees
15
10
5
0
4.65
4.7
4.75
4.8
Time, s
4.85
4.9
4.95
5
Figure 11: Comparison of Truth and Model Response in Non linear Region - Pitch Channel
Comparison of Reconstructed Model and Truth for Large Deflection Angles: Yaw
5
Truth Model
Reconstructed Model
0
Yaw, Degrees
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4.6
-5
-10
-15
-20
4.4
4.5
4.6
4.7
4.8
Time, s
4.9
5
5.1
5.2
5.3
Figure 12: Comparison of Truth and Model Response in Non linear Region - Yaw Channel
VII. Conclusion
In this paper a HOSM-based system identification algorithm has been applied for the parameter identification
of gyroscope system. The identified parameters then have been used in the nonlinear model of the studied device.
Simulation study showed that the HOSM-based identification algorithm method works acceptably for ideal and low
noise condition. It is, however, conceded that the addition of noise degraded overall performance significantly and is
in part due to the order of calculation with the disturbance parameters being calculated using estimated values for
the moments of inertia. Clearly the accuracy of subsequent calculation will be degraded. If more information about
the moments of inertia were available, the disturbance term estimations can be noticeably improved.
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Cannon, Robert H., Dynamics of Physical Systems, McGraw-Hill, 1967
2
A. Poznyak, Y. Shtessel, L. Fridman, J. Davila, and J. Escobar, “Identification of Dynamic Systems Parameters via
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Information Sciences, E. Fossas, C. Edwards, and L. Fridman (eds.), Springer-Verlag, Berlin, June 2006, pp. 313
351.
3
Brogan, William L., Modern Control Theory, 3rd edition, Prentice Hall, 1990.
4
Levine, William S., The Control Handbook, CRC Press, 1996
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Edwards, C., and Spurgeon, S Sliding Mode Control, Taylor & Friends, 1998.
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Gibson, John E., Nonlinear Automatic Control, McGraw-Hill, 1963
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Spong, Mark W., Hutchinson, Seth, Vidyasagar, M., Robot Modeling and Control, John Wiley & Sons, Inc. 2006
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Levant, Arie, Higher-order sliding modes, differentiation and output-feedback control, International Journal of
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11
Hartfield, D.J., Matrix Theory and Applications with MATLAB, CRC Press, 2001
12
Dorf, Richard C., Bishop, Robert H., Modern Control Systems, 9th edition, Prentice Hall, 2001
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