SYSTEM IDENTIFICATION FOR CONTROL SYSTEM DESIGN AND DEMONSTRATION BY FLEXLAB Jerrel R. Mitchell1, R. Dennis Irwin2 Cheng Professor of School Electrical Engineering and Computer Science, Ohio University, Athens, OH 45701 2 Moss Professor and Dean, Russ College of Engineering and Technology, Ohio University, Athens, OH 45701
[email protected] 1
INTRODUCTION In order to design high performance feedback control systems for complex systems such as large flexible, space-based structures or aero-elastic aircraft, accurate mathematical models are required. The applications of large flexible space structures include space-based telescopes, robotic systems, the space station, and any space-based platform using deployables, such as solar panels or antennae. A common problem with all of these applications is the structures are highly flexible because of their low mass and large size. As a consequence, vibration and accurate attitude control are challenging problems to solve. The National Aeronautics and Space Administration (NASA) in conjunction with the United States Air Force Research Laboratories (AFRL) and Boeing Phantom Works have recently been exploring ways to use flexible structures to their advantage at Dryden Flight Research Center in Edwards, California [1]. Active Aeroelastic Wing Flight Research was the topic of a research project that began in 1996 and was completed in 2005. The project goal was to use the aerodynamic forces acting on the control surfaces of an aircraft to bend a flexible wing to provide better roll maneuvering. Data was collected to combine control surface techniques and flexible wing structures to use in designing a more flexible, lighter weight wing for current aircraft [1]. In the Journal of Sound and Vibration, Yan-Ru Hu, and Charles Ng [2] of the Directorate of Spacecraft Engineering in the Canadian Space Agency, Saint Hubert, Quebec, Canada, wrote an article entitled “Active robust vibration control of flexible structures,” in which they developed an active vibration control technique with piezoelectric actuators. Their test bed was a flexible circular plate and their experiments showed that this control method suppressed structural vibrations adequately. The design of control system for complex systems as described above requires a model that accurately portrays the dynamics of the system. The most widely used control engineering analysis and design techniques require accurate linear, time-invariant (LTI) models; therefore, developing LTI models of physical systems is crucial. An accurate model provides the engineer with information regarding the response of the system in its practical operational frequency range [3]. Based on this model the engineer can make accurate predictions as to how the system will behave, as well as, produce good closed loop designs to achieve the specified dynamic performance requirements. It is essential to use models that accurately describe the relationship among system variables in terms of mathematical expressions, such as difference equations or differential equations [1]. Three approaches to developing mathematical models of dynamical systems are: (1) equations developed from the physics of the system (first principles); (2) equations developed from finite element methods; and (3) equations developed from system identification. For complex flexible structures, the first two methods often do not produce adequate models to design high performance control systems for such applications as pointing and vibration suppression. System identification is the process of extracting or inferring information about a mathematical model by numerical processing experimental data or data derived from experimentally collected data, e.g., frequency response data obtained by using time domain data that has been processed using signal processing techniques. One approach to System ID is to collect data in a laboratory environment for estimating key parameters to fine-tune models developed from dynamics and kinematics or finite element techniques. A typical scenario in this case would involve a test article that is instrumented with many sensors and then excited by tapping with a calibrated hammer at various locations while data is collected. Then the data is processed to get estimates of model frequencies, damping values, mode shapes, etc. The thrust of this paper system is system ID that is performed using the actuators and sensors that will be used to form feedback loops for proper operation after the system has been deployed. This is called System ID for Control System Design and is depicted as shown in the following figure. The implications are that the digital computer, that will be used to implement a feedback control law, is used to generate a random discrete signal as shown by the u’s in Figure 1. The
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Figure 1: Discrete Linear System digital sequence from the computer is converted to an analog signal that provides commands for the control system actuators (assumed to be integrated with the plant). Sensors (also assumed to be integrated with the plant) that will used to form feedback loops for control measure the effects of input excitations on the system outputs. The sensor outputs are converted to digital signals with A/D’s. The digital input and output data is processed through system ID algorithms to produce linear discrete models between the inputs to the D/A’s and outputs from the A/D’s. The models are difference equations, z-domain transfer functions, or discrete state space. The advantages of the system ID for control system design are as follows: (1) the dynamics of the actuators, and sensors are assumed to be part of the plant and, hence, are naturally included in the derived model; (2) by exciting the system with digital signals and collecting data at the same rate at which a digital controller will be used, the effects of D/A and A/D processes are inherently included in the model; (3) sampled-data models, needed to perform digital controller designs, are produced; (4) if need be, this type of system ID can be performed after the system has been deployed. The downside of system ID for control system design is that placement of actuators and sensors must be pre-determined. There basically are three approaches for performing system ID for control system design: (1) time domain least squares system ID; (2) frequency domain least squares system ID; and (3) time domain state space system ID. Time domain least squares system ID uses weighted least squares curve fitting to determine the coefficients of linear difference equation model(s) of a single-input, single-output (SISO) or single–input, multiple-output (SIMO) systems, given timesamples of input and output data [3]. Frequency domain least squares system ID (also known as the transfer function determination code or TFDC) determines linear models of SISO or SIMO systems, given frequency response data [4], [5]. The Eigensystem Realization Algorithm (ERA), an extension of the Ho-Kalman realization algorithm, determines a time domain state space realization given impulse response data between each input and each output [3], [6]. ERA is useful in obtaining multiple-input, multiple-output models (MIMO) as well as SISO models, even though the data must be obtained experimentally via multiple SIMO tests. TIME DOMAIN LEAST SQUARES SYSTEM ID A difference equation model of a SISO LTI system is as follows:
y(k) = a1 y(k "1) + a2 y(k " 2) +L+ an y(k " n) + b0 r(k) + b1r(k "1) +L+ bm r(k " n)
(1)
where y(k) and r(k) are, respectively, the output and input at the k-th point in time and the a’s and b’s are the coefficients of the difference equation model. From (1), it seen that the y(k) is computed as a weighted sum of the past n outputs, the current and!past n inputs. If p+1 measurements of the output are made, the above equations can be use to generate p+1 equations in the unknown coefficients, i.e., the a’s and b’s. These equations can be represented in matrix equation form as follows:
# y(k) & # y(k "1) % ( % % y(k "1) ( = % y(k " 2) % M ( % M % ( % $ y(k " p)' $ y(k " p "1) T
y(k " 2) y(k " 3) M y(k " p " 2)
L y(k " n) r(k) r(k "1) L y(k "1" n) r(k "1) r(k " 2) O M M M L y(k " p " n) r(k " p) r(k " p "1) T
L r(k " n) & ( L r(k "1" n) (#a& % ( ($b' O M ( L r(k " p " n)'
(2)
where a = [ a1 a2 L an ] , b = [b0 b1 L bn ] and [ ] T is transpose.. If p " 2n +1 an over determined set of ! equations result. Using ordinary or total least squares, the values for the a’s and b’s can be computed to form a difference equation model between an input and an output. (In the case of SIMO, the a’ for each difference equation are the same, while the b’s are different for each output. The y’s become vectors of length p where p is the number of ! ! outputs. The unknown!number of b’s increases to np. The part of the coefficient matrix containing the r’s is formed so that the b’s for each output are computed independently.) It should be noted that (2) implies that n+1 consecutive
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measurements of inputs and outputs equally spaced in time must be made [4], [8]. That is, individual data points cannot be omitted, even if it is known that some are highly corrupted with noise. FREQUENCY DOMAIN LEAST SQUARES SYSTEM IDENTIFICATION Frequency domain least squares system identification is a technique that uses frequency response data to estimate the coefficients of a transfer function of a linear system using experimental frequency response data or frequency response data derived from experimental time-domain data. A computer algorithm that implements this idea has been coined the Transfer Function Determination Code [5]. In this section the underlying theoretical basics of the TFDC algorithm are outlined. The development of TFDC starts with SISO frequency response data points given as N complex numbers, i.e.,
G(e j" iT ) = Xi + jYi ,
i = 1...n
(3)
where !i is the i-th frequency and T is the sampling period of the discrete system. A LTI z-transfer function model between the input and output can represented as: ! G(z) =
[z [z
n"1
n
] , #1& 1]% ( $ a'
z n"2 z n"3 L 1 [b] z
n"1
z
n"2
L
(4)
where b = [b1 b2 … bn]T and a = [a1 a2 … an]T are the real coefficients of the z-transfer function. Note that the leading coefficient of the denominator polynomial has been set to unity in order to assure uniqueness. ! The frequency response of a z-domain transfer function is obtained by letting z = z1, where z1 = ejk!iT = cos(k!iT) + j sin(k!iT ) from Euler’s equation and j = "##-#1. The ith frequency response point can be represented as G(zi) = Xi + j Yi . Using zi in (2) and multiplying both sides by the denominator gives
#1& zin"1 L 1 % ( = zin $a' By manipulating the equation the following results:
[
]
G(zi ) zin
[
[
n n"1 G(zi )zin"2 L G(zi ) !G(zi )zi = G(zi )zi
]
zin"1 L 1
zn
[ b]
(5)
# a& z n"1 L 1 % ( $ b'
]
(6)
where [a] and [b] are as defined above. This is a complex equation; hence the real part on the left must equal to the real part on the left and vice-vice for the complex parts, i.e., ! # a& Re G(zi )zin = Re G(zi )zin"1 G(zi )zin"2 L G(zi ) z n z n"1 L 1 % ( $ b' (7) and # a& Im G(zi )zin = Im G(zi )zin"1 G(zi )zin"2 L G(zi ) z n z n"1 L 1 % ( ! $b' (8)
{
}
{[
]}
{
}
{[
]}
As a consequence, each frequency point produces two linear equations in the unknown numerator and denominator coefficients ! of the transfer function. If there are N frequency response measurements, there will be 2N equations in 2n+1 unknowns. If these equations are concatenated, the general form is
" a% H$ ' = c # b&
!
3
(9)
where the 2N x (2n+1) matrix H and the 2n+1 column vector c are formed as per (7) and (8) for the N measurements as depicted by (6). If 2N is greater than (2n+1), the system of equations is an over determined set of linear equations that are suitable to solution by least squares or total least squares [8]. The above are equivalent to the basic equations of TFDC as presented in [4]. It should be noted that extension of TFDC to SIMO is similar as that described in time domain least squares above. It is interesting to note that the time domain least squares system ID and frequency domain least squares system ID techniques both estimate the same coefficients, because the coefficients of the z-transfer function model and the coefficients of the difference equation model are the same. Time domain least squares system ID needs, at a minimum, one or more sets of consecutive input/output data points of length n that are equally space in time. Frequency domain least squares system identification is different in that all frequency response points are not necessarily needed, consecutive data points are not required (only a minimum of n points), and the points do not have to be uniformly spaced in frequency. This means that if there are a significant number of frequency points present, not all need to be used in the system ID process. Obvious points to omit are points highly corrupted by noise. Knowing that data highly corrupted with noise can be omitted is a useful fact. Having a means of deciding which data can be discarded is a subject considered later. Eigensystem Realization Algorithm (ERA) The Eigensystem Realization Algorithm (ERA) uses impulse response data of a system to determine a state space realization. The algorithm extends upon the Ho-Kalman realization to estimate parameters from measurement data containing noise [6]. The algorithm can be used to develop models for SISO, SIMO, or MIMO linear time-invariant systems. However, as mentioned above, the algorithm requires impulse response data between each input and each output. The purpose of this discussion is to present the basics of the ERA algorithm. A more detailed discussion is given by [6]. Consider a discrete linear time-invariant system
x(k +1) = Ax(k) + Bu(k) y(k) = C(k)x(k)
(10a&b)
where k is the current time step, A is the discrete state transition matrix, B is the input weighting matrix, C is the output weighting matrix, x(k) is the system’s n-dimensional state vector, u(k) is the system’s m-dimensional input vector, and ! vector. The observability and controllability matrices are, y(k) is the system’s p-dimensional output
# C & % ( % CA ( % CA2 ( % ( O =% • ( % • ( % ( • % ( %$CA n"1('
(11)
and
! C = B AB A2 B ••• A n"1 B
[
! Multiplying these together gives
4
]
(12)
# CB CAB % 2 CAB CA B OC = % % M M % n"1 n $CA B CA
CA n"1 B & ( L CA n B ( = H (0) ( O M ( L CA2n"2 B' L
(13)
and is defined to be the block Hankel matrix H(0). The elements of this matrix are the impulse responses values between inputs and outputs. The elements of the matrix defined in (13) are called the Markov parameters [6]. ! In terms of impulse response outputs this becomes
" y(1) y(2) $ y(2) y(3) H (0) = $ $ M M $ # y(n) y(n +1)
L y(n) % ' L y(n +1)' O M ' ' L y(2n) &
(14)
If the system has m inputs and p outputs, each element of (14) is a p x m matrix. The elements of H(0) are formed from the first 2n impulse response matrices. Thus (13) and (14) are equivalent. The Hankel matrix H(0) can be decomposed ! using the singular value decomposition as follows:
H (0) = U1 "1 V1T where
(15)
"1 = diag(e1 ,e2 ,...en ) is a diagonal matrix containing the singular values and U1 and V1 are unitary matrices
respectively, that contain the right!and left eigenvectors of H(0)H(0)T and H(0)T H(0), respectively. The order of the system can be determined by observing the singular values. If en +1 ,K , eN are significantly smaller than e1 ,K , en ,
! then the rank of H(0) can be assumed to be n. [4] Equation (15) can be partitioned as follows: 1/ 2
1/ 2
H (0) = (U1 "1 )("1 V1T ).
(16)
Then O and C can be computed as
!
1/ 2
O = U1 "1
(17)
and 1/ 2
C = "1 V1T .
(18)
! Another Hankel matrix, H(1) is defined as follows: ! " CAB CA2 B L CA n B % $ 2 ' CA B CA 3 B L CA n+1 B' $ OAC = = H (1) $ M M O M ' $ n ' n+1 L CA2n B & #CA B CA
(19)
This matrix in terms of output values becomes
!
!
" y(2) y(3) $ y(3) y(4) H (1) = $ $ M M $ y(n +1) y(n + 3) #
L y(n +1) % ' L y(n + 3) ' ' O M ' L y(2n +1)&
5
(20)
From this, it is easily seen that an A matrix for the system can be computed as "1/ 2
A = O "1 H (1)C "1 = #1
"1/ 2
U1T H (1)V1 #1
(21)
The B and C matrices are easily obtained from the observability and controllability matrices by inspection. If the system has m inputs, the B matrix is simply the first m columns of the controllability matrix, and if the system has p outputs the C matrix is the first p rows!of the observability matrix [4], [5]. As mentioned above, ERA is useful for estimating a discrete state space realization of a linear time-invariant system using discrete impulse response data between each input and each output. From input and output data, a procedure for estimating the impulse response sequences between each input and each output can be devised if the system is stable. A pitfall in this approach is that the impulse response sequences are infinite in length. Thus, the best that could be hoped is to estimate approximations of the I/O impulse response sequences. The I/O data for a system provides little information as to the number of terms need in the approximation of an I/O impulse response sequence in order for the results to be useful in system ID. A better approach for obtaining the impulse response sequences is to estimate individual SISO systems from each input to each output by using least-squares time domain system ID or TFDC. Using either approach difference equations models can be obtained (directly with the first and indirectly with the latter). Using the difference equation models, impulse responses from each input to each output can be easily computed for use in ERA to generate a discrete state space model. If the I/O measurements are noisy, TFDC has the advantage since data that is highly corrupted with noise can be discarded using estimates of the coherence function. This is the next topic. TRANSFER FUNCTIONS AND COHERENCE FUNCTIONS VIA POWER SPECTRUMS Consider the LTI system shown in Figure 2, where the input and output are assumed to be stationary random processes [9], [10]. The autocorrelation functions of the input signal and output signal are
Rxx (" ) = E[ X(t)X(t + " )]
(22)
Ryy (" ) = E[Y (t)Y (t + " )]
(23)
and
!
respectively, where " represents time-difference and E[ ] is the expected value operation. Similarly the cross correlation function is !
Rxy (" ) = E[ X(t)Y (t + " )]
(24)
By the Wiener-Khintchine [20] relation for stationary processes, the power spectral density is the Fourier transform of the correlation function, thus for the signals X(t) and Y(t) these are ! +%
S xx ( j") = F { Rxx (# )} = &$% Rxx (# )e$ j"# d#
(25)
and
!
S yy ( j") = F { Ryy (# )} =
+%
&$% Ryy (# )e$ j"# d#
!
Figure 2: LTI Linear System with Input X(t) and Output y(t)
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(26)
respectively, where F is the Fourier transform operator. Similarly the cross power spectral density for X(t) and Y(t) is
S xy ( j") = F { Rxy (# )} =
+%
&$% Rxy (# )e$ j"# d#
(27)
If the cross power spectrum above is divided by the input power spectrum the result is the frequency response of the transfer function relating Y(s) to X(s), where s is the Laplace Transform variable, i.e., !
G( j") =
S xy ( j") S xx ( j")
(28)
Thus, if the cross-spectrum between input and output, and the auto-spectrum of the input are accurately known, an accurate frequency response of the system can be computed. However, the data for computing the spectrum above is measured or computed from data !that are likely to include measurement noise and system disturbances. Hence, the above ratio, in practice, only produces an estimate of the frequency response between the input and the output [9], [11]. The coherence function is related closely to the cross correlation of two processes but is a function of frequency, rather than time, and is normalized to only produce values from 0 to 1. The coherence function is defined as
" 2xy
=
S xy ( j#)
2
S xx ( j#)S yy ( j#)
(29)
by [4], [9]. The coherence function is sometimes called the coherency spectrum and can be viewed as a frequency dependent “correlation coefficient” between two random processes X and Y [4], [9]. It may also be viewed as a ! normalized signal to noise measurement in the frequency domain. Because this function is normalized to produce values from 0 to 1, where a 1 corresponds to frequencies in which there are perfect correlations between the random processes X and Y, and a 0 corresponds to frequencies where there are no correlations between the two random processes. This is a helpful function when examining input and output data of a linear system. When there is noise present, it is possible to determine on a frequencyby-frequency basis where the noise is significantly corrupting the output data by observing the coherence function. The frequency data that is most corrupted corresponds to a coherence value near zero. Likewise, the least corrupted frequency data correspond to a coherence values near one [9]. USE OF COHERENCE ESTIMATES TO DISCARD NOISY FREQUENCY POINTS To justify the importance of eliminating data with low signal to noise in least squares system ID requires an understanding of how the results of least squares are skewed by the data being processed. Least squares solutions basically fit curves to data so that the sum of the squares of the distances between the data points and the computed curve is a minimum. The outcome is weighted by the number of data points processed and by the accuracy of the data. If a large number of data points are highly corrupted by noise, the least squares solution will tend to compute a curve that fits the corrupted data, rather than the data that is most accurate. However, for frequency response estimates, the coherence estimates provide good evidence of which data points are highly corrupted with noise and which are not. Coherence Threshold is defined as the selected level of the coherence function to discarded data when using frequency response system ID. Frequency response data points with coherences below the threshold are considered to be too noisy to be useful in the frequency response system ID process. In fact, if the threshold is judiciously selected, the frequency response of the estimated transfer function and the estimated frequency response of the system will match closely at the retained data points. There is, however, an upper limit on the threshold. This derives from the fact that as the Coherence Threshold is increased and more data points are discarded, there may be too few data points to estimate the parameters correctly. Furthermore, the threshold might be so high that some of the salient features of the frequency response are lost, e.g., modal peaks. [12], [13]
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To illustrate the use of the Coherence Threshold approach consider the system shown in the Figure 1 where the plant is
G(s) =
5 "5 10 + 2 + 2 s + 0.1s +1 s + 0.2s + 8 s + 0.5s + 20 2
(30)
The model represents a continuous plant that is receiving digital commands through an analog to digital (A/D) device, and the output of the system is measured and converted to digital information with an A/D device. The sample period for ! this example was chosen to be T = 0.1 seconds. The input to the digital to analog (D/A) device is a sequence
u (0), u (T ),..., u (nT ) and the corresponding output sequence of the A/D device is y (0), y (T ),..., y (nT ) as shown in Figure 1. The system was simulated using MATLAB [14]. The input was chosen to be a zero mean, uniformly distributed random sequence over the range [-1,1]. Gaussian distributed discrete white noise with zero mean and a variance 0.1 was added to the output to simulate sensor noise. SPECTRUM of MATLAB was used to estimate the frequency response and the coherence function. The input to SPRCTRUM was 8192 I/O time points, producing 1025 frequency points [15], [16]. Figure 2 shows that the estimated and actual frequency responses (magnitude and phase), that are close matches if the MATLAB phase unwrapping is taken into account. Figure 3 shows the estimate of the coherence from SPECTRUM in MATLAB [2]. From this figure, it is obvious that many of the frequency data points are highly corrupted by noise, especially at high frequency. The problem with “bad” data is amplified even more by realizing that SPECTRUM uses linearly spaced frequencies, where the plot of the coherence estimate shown Figure 3 is on a log scale. With this in
Figure 2 Frequency Response Estimate
8
Figure 3 Coherence function and Threshold
Figure 4 Theoretical Response of Plant, TFDC Model with Noisy Data
Omitted, and TFDC Model Using All Data.
9
mind, it can easily be deduced that more than 80% of the frequency response data is highly corrupted by noise [12], [13]. Figure 3 shows a Coherence Threshold of 0.93. It was determined empirically that by setting the Coherence Threshold at this value, the data most corrupted would be omitted and the salient features of the frequency response would be retained. Figure 4 shows the frequency responses of a 8th order models obtained using a MATLAB TFDC code with data discarded using Coherence Threshold and without any data being discarded. With a Coherence Threshold of 0.93, an accurate model of the z-transfer function is obtained, whereas the model computed using all the frequency response datapoints is a poor representation of the true model. (Although not shown, it should be noted that a model generated by time domain least squares had a similar frequency response as the model generated using TFDC with no data discarded, i.e. also a poor representation of the system.) [12] MODEL DEVELOPMENT AND CONTROL SYSTEM DESIGN FOR FLEXLAB In this section it is shown how the frequency response system ID (TFDC) and the Coherence Threshold technique are used with data taken form the FLEXLAB facility at Ohio University in order to derive a model of the structure that is used to design a LQG controller for FLEXLAB to suppress residual vibrations induced by random excitation. First a brief overview of FLEXLAB is in order [17], [18]. FLEXLAB is a facility at Ohio University used as a test-bed for research in the area of modeling and control of flexible structures. The facility has been used to explore system identification methods, as well as closed loop control, and has even been used to investigate control systems for an active optical system [17]. Flexlab was built for the purpose of providing a significantly large flexible structure so students and faculty could perform these types of research activities. One configuration of FLEXLAB can be seen in Figure 5. The system consists of a 3/8 inch aluminum rod which is 12 feet in length. It is suspended from a motorized, two degree of freedom, gimbaled system that is mounted to the ceiling. The opposite end of the rod is hanging freely, resulting in an extremely flexible structure. There are two masses on the structure as shown in Figure 6. These masses can be relocated to change the dynamics of the structure. The two-degree of freedom, gimbaled system has two permanent magnet DC motors. The motors are positioned orthogonal to one another. The uppermost motor is oriented so that the shaft is parallel to the x-axis and referred to as the x-axis motor. When this motor is activated the rod moves in the y-z plane. The bottom-most motor shaft is oriented parallel to the yaxis and when activated, the rod moves in the x-z plane. These motors are used to disturb the system, as well as control the structure. When the motors are used as disturbances, they are used to randomly shake the structure. This simply simulates a structural vibration. The motors are also used as actuators to suppress the disturbances or vibrations within the system [17], [18]. There are two types of sensors used on the structure, Kistler K-Beam accelerometers and a Hamamatsu S1200 position sensing device (PSD). The accelerometers provide a voltage output proportional to the acceleration along the axes of measurement. The PSD senses the light image from an IR LED, positioned on the floor (not shown) at the bottom of the rod, and outputs a voltage proportional to the displacements in the x- and y- directions (Strahler, 2000). This configuration also includes an active optical mirror implemented by Strahler (2000). With the mirror included, the structure is asymmetric about the z-axis. Flexlab is interfaced to a Pentium II 333 MHz PC with National Instruments data acquisition cards. The software used for the task of collecting data and control implementation is National Instruments LabView[18], [19]. To demonstrate the use of TFDC and the Coherence Threshold in obtaining a model for a real system, the system was excited with the y-axis motor and data taken with the optical position sensor in the y-axis. The excitation signal was uniform white noise with amplitude on the range [-4, 4], mean 0, and variance of 5.38. Output data was obtained by using the y-axis position sensing device (PSDy). A 100 Hz sample rate was used for input and output data. The system was operated for 340 seconds to give 34000 samples of input and output time data. The first 32768 samples were used in the SPECTRUM of MATLAB to produce an estimated frequency response with 2049 frequency points. The frequency response data (magnitude and phase) obtained from MATLAB are shown in Figure 6. Figure 7 shows the estimated coherence function and the Coherence Threshold. For this case a Coherence Threshold of 0.45 was selected. This value was selected to avoid losing many of the salient features of the frequency response, e.g., the higher frequency modes that are much less in amplitude that the lower frequency modes. The frequency response data was input to a MATLAB TFDC code. Figure 8 shows the plot of the frequency response (magnitude and phase) of the data, plus the frequency
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response of a 10th order model obtained via TFDC using the data above the Coherence Threshold. It is evident from Figure 9 that the TFDC model obtained by using the data above the Coherence Threshold is a reasonable y-axis model. FLEXLAB CONTROLLER DESIGN USING THE TFDC DERIVED MODEL A significant rationale for system identification is to obtain a mathematical model with sufficient accuracy to design controllers using modern, optimal control design techniques such as a Linear Quadratic Gaussian Regulator (LQG). The z-transfer function model derived in the previous section was used to design a feedback controller using the LQG algorithm in MATLAB. The design goal is to add damping to the flexible modes of the system. The controller was then implemented on FLEXLAB as a feedback controller as shown in Figure 10, thereby demonstrating the usefulness of the model and the design. The LQG function in MATLAB was used to compute an optimal LQG regulator given the 10th order z-transfer model of the FLEXLAB. To accomplish the design, appropriate weighting matrices QXU and QWV had to be chosen. For the weighting matrices, two 11 x 11 dimension diagonal matrices were used to account for the 10 estimated states, as well as the single control entry. A diagonal matrix with all diagonal elements set to 0.01 was used for the measurement and input noise co-variances, QWV. For the state and control weighting matrix, QXU, a diagonal matrix was, also, selected with the first 10 diagonal elements set to 0.01, and the eleventh diagonal element set to 1000000. This provides significantly more weighting to control effort in order to avoid saturation of the y-axis motor. [19] To evaluate the effectiveness of the controller to suppress vibrations, a random input signal was applied for 30 seconds and then removed. Then the natural decay of the system vibration was recorded. First the decay was recorded with no feedback controller. Then the same input excitation was applied for the 30 seconds with no feedback, and then the LQG controller designed using the system ID model was used to close the loop. Figure 10 shows comparisons between the FLEXLAB open-loop system and the closed-loop compensated system behavior. When the random input was removed at the 30-second point and the controller was used to close the loop, the transient response due to excitation by the input disturbance decayed to nearly zero in less then 10 seconds. However, with no feedback controller, significant excitation still existed at the 60-second point. Thus, the controller designed using the model obtained by TFDC is demonstrated to have significantly dampened the modes of the system. [12], [19] CONCLUSION Frequency domain least squares system ID uses frequency response data, obtained either directly from experimentation or computed using signal processing techniques and experimental time data, in order to obtain estimates of the coefficients of z-transfer functions. Advantages of frequency domain least squares approach are: (1) not every frequency domain data point is needed, (2) the frequency data does not have to be uniformly spaced with respect to frequency, (3) the frequency response of a computed model can be directly compared to the experimental derived data, (4) if the frequency response data is derived from sampled time data, the model can be computed at a different sample rate than the time data, and (5) if frequency response data is derived from experimental time data using power spectral density functions, ensemble averaging and long data sets can be used to reduce the noise level on the frequency response data. The coherence function is a means of determining which frequency data points are most corrupted by noise. The concept of the Coherence Threshold has been shown as a means for deciding which frequency response data points could be deleted in order to improve the results in computing models using frequency domain least squares. An illustrative example was used to demonstrate the application of the Coherence Threshold technique. This was followed by the application of the technique to obtain a model for the FLEXLAB facility at Ohio University. The derived model was then used to design an LQG controller to add damping to the flexible modes of the system. The design was implemented in the FLEXLAB system and used to physically demonstrate that with controller designed using the model from system ID suppress vibrations in the system. [12], [13]
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DC motors
3/8” Aluminum Rod
Masses
Accelerometers
Steering Mirror Assembly
PSD z
Diode Laser y
Figure 5 Current Flexlab Configuration [17]
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Figure 6 Estimate of Frequency Response (Flexlab y-axis)
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Figure 7 Coherence Function and Threshold
Figure 8 TFDC Estimate vs. SPECTRUM estimate
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Figure 9 Controller Design Configuration
Figure 10 Open vs. Closed Loop using an LQG regulator REFERENCES [1] National Aeronautics and Space Administration (24 July, 2006). [Online]. Available: http://www.nasa.gov/lb/centers/dryden/news/FactSheets/FS-061-DFRC.html [2006, September 20] [2] Hu, Y. (2005) Active robust vibration control of flexible structures. J. Sound Vib. 288, 43-56. [3] Medina, E.A. (1991) Multi-Input, Multi-Output System Identification from Frequency Response Samples with Applications to the Modeling of Large Space Structures. MS Thesis, Ohio University, Athens, OH. [4] Ljung, L. (1999) System Identification. Theory for the User. Second edition. Prentice Hall, Saddle River, New Jersey. [5] Mitchell, J., Gresham, L., Irwin, D., Tollison, D. (1982) Using parameter estimation techniques for transfer function modeling. The Fourteenth Southeastern Symposium on System Theory (Blacksburg, Virginia), April 15-16. pp. 243-247.
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[6] Juang, J.N. (1994) Applied System Identification. Prentice Hall, Englewood Cliffs, New Jersey. [7] Phillips, C. L. and Nagle, H. T. (1995) Digital Control System Analysis and Design. Third edition. Prentice Hall, Englewood Cliffs, New Jersey. [8] Huffel, S. A. and Vandewalle, J. (1991) The Total Least Squares Problem – Computational Aspects and Analysis, SIAM, Philadelphia, PA. [9] Brown, R.G. and Hwang, P.Y.C. (1992) Introduction to Random Signals and Applied Kalman Filtering. Second edition. John Wiley and Sons, Inc., Canada. [10] Peebles, P.Z. (2001) Probability, Random Variables, and Random Signal Principles. Fourth edition. IrwinMcGraw-Hill, New York, New York. [11] Couch II, L.W. (2001) Digital and Analog Communication Systems. Sixth edition. Prentice Hall, Upper Saddle River, New Jersey. [12] Thomas, J. (2007) Using The Coherence Function as a Means To Improve Frequency Domain Least Squares System Identification. MS Thesis, Ohio University, Athens, OH. [13] Mitchell , J. R., Thomas, J., Bukley, (2007) A., Improvement of Frequency Response System ID Using Estimates of the Coherence Function, AIAA Navigation and Control Conference, Hilton Head, SC. [14] Control System Toolbox For Use with MATLAB, User's Guide (1999) The Mathworks, Inc. [15] Little, J., Shure, L. (1988) Signal Processing Toolbox For Use with MATLAB, User's Guide. The Mathworks, Inc. [16] Signal Processing Toolbox For Use with MATLAB, User's Guide (2001) The Mathworks, Inc. [17] Strahler, J. A. (2000) Integration of an Active Optical System for Flexlab. MS Thesis, Ohio University, Athens, OH. [18] Saunders, C. (2006) Labview Software Development for Input and Output Measurement and Control of Flexlab. MS Thesis, Ohio University, Athens, OH. [19] Simulink, Dynamic System Simulation for MATLAB, User's Guide (1999) The Mathworks, Inc.
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