A NONLINEARITY MEASURE FOR ESTIMATION SYSTEMS

Transcription

1 AAS A NONLINEARITY MEASURE FOR ESTIMATION SYSTEMS Andrew J. Sinclair,JohnE.Hurtado, and John L. Junkins The concept of nonlinearity measures for dynamical systems is extended to estimation systems, which include both a dynamical model and a measurement model. For linear systems, propagation through linear dynamics and update by linear measurements both preserve the Gaussian property of a distribution of estimates. For nonlinear systems, how well a Gaussian distribution of estimates stays Gaussian is identified as a measure of the nonlinearity in the total estimation system. This measure can be used to compare the nonlinearity introduced by both the dynamics and the measurements for various representations of a system. INTRODUCTION In the estimation of dynamical systems, a variety of representations are typically available for any given physical system. When dealing with linear representations, estimation processes such as least squares or the Kalman filter are optimal in many senses. Estimating nonlinear systems, or even analyzing their observability, though, can be difficult. 1, 2 If no linear realization of the system is available, however, intuitively it is desirable to select the nonlinear realization that is the least nonlinear. This should allow application of linear techniques, such as the extended Kalman filter (EKF), with near-optimal performance. This motivates the desire for a test to measure the nonlinearity of an estimation system. Nonlinearity measures for dynamical systems have been developed previously. An example of this is the nonlinearity index introduced by Junkins 3 and Junkins and Singla. 4 This index uses the fact that for linear systems the state-transition matrix is independent of the initial conditions of the states. Given a nonlinear system, the Assistant Professor, Department of Aerospace Engineering, Auburn University, 211 Aerospace Engineering Building, Auburn, AL , sinclair@auburn.edu. Assistant Professor, Department of Aerospace Engineering, Texas A&M University, TAMU-3141, College Station, TX , jehurtado@tamu.edu. Distinguished Professor, Department of Aerospace Engineering, Texas A&M University, TAMU-3141, College Station, TX , junkins@tamu.edu 1

2 state-transition matrix can be integrated for a distribution of initial conditions, and the variability in the sample matrices is measured as the nonlinearity index. An estimation system, however, consists of both a dynamical model, ẋ = f(x, u), and a measurement model, y = h(x). Measuring the nonlinearity of the dynamics is only half of the story. To describe the nonlinearity of an estimation system a measure is needed that accounts for both the dynamical and measurement models. In this paper, a nonlinearity measure is defined for an estimation system by looking at a distribution of candidate estimates. The measure is then applied to an example orbitdetermination problem to investigate its usefulness in describing the performance and behavior of the estimation process. KOLMOGOROV-SMIRNOV TEST For a completely linear estimation system, in a typical Kalman filter both propagation through the continuous, linear dynamics and update by the discrete, linear measurements will preserve the Gaussian property of a distribution of estimates. In other words, given a Gaussian distribution of initial estimates, Kalman filtering of each estimate will result in a Gaussian distribution of final estimates. This suggests a measure of nonlinearity for an estimation system based on the distribution of a set of estimates. The following procedure will be used to investigate the nonlinearity of a given estimation system. First, take a Gaussian sample of initial conditions. Then, apply EKF filtering to each estimate. Finally, measure how well the set of final estimates fit a Gaussian distribution. A common statistical test to measure the fit of data samples, which will be used in this paper, is the Kolmogorov-Smirnov test (KS test). 5 7 The KS test is a comparison of two probability distributions. Most commonly it is used to measure how well the probability distribution of a data sample matches a Gaussian distribution function. Possible alternatives to the KS test include the chi-squared test or looking at the skewness and kurtosis (related to the peakiness of the data). The advantage of the KS test is that it measures the departure from Gaussian behavior in a single quantity and does not require binning of the data. To evaluate the KS test, the sample distribution, x i, is first normalized to a mean of zero and a standard deviation of one. The resulting probability distribution is then plotted on top of the normal probability distribution function. The KS test value is the largest probability difference between the two distributions over the range of the distribution, i.e., the largest vertical difference between the plots. Defining y i to be a normalized data set, the KS test is shown below. y i = x i x (1) σ x D =max( F data (y) F N (y) ) (2) y 2

3 1 Distribution Functions F N (y) F data (y) y D Figure 1 KS Test of Example Non-Gaussian Data Sample Here, F data (y) is the distribution function of the normalized data set, and F N (y) is the normal distribution function. Figure 1 illustrates the KS test for an example non-gaussian data sample. For this sample the KS test value is D =.2169, and this maximum separation occurs at a value slightly greater than y =. Figure 1 also illustrates the sensitivity of the KS test to the size of the data sample. Small data sets produce large stair steps in the probability distribution function. Therefore a small data sample will give large values in the KS test even if the data is taken from a truly Gaussian distribution. The KS test value will be on the order of D 1/N where N is the number of samples, because this is the size of the steps in the distribution function. This stems from the fact that small samples are necessarily a poor approximation of the Gaussian distribution and represents a limit in the precision of the KS test. Significantly, the KS test is defined for a single data set. Typical EKF filters, however, estimate several states. In this sense the KS test can be used to measure the nonlinearity along each state, by applying the test to the distribution of final estimates for each coordinate. This is beneficial because it allows the measurement of differing degrees of nonlinearity in each state. Linear combinations of the states could also be taken to measure nonlinearity along any desired direction in state space. It must also be noted that the behavior of the EKF itself has some bearing on the design of the testing procedure. The process noise in any Kalman filter makes the filter forget the past, because previous estimates can no longer be trusted due to the uncertainty in the dynamical model. Therefore, for a stable filter, any initial distribution of estimates will eventually converge to a single point regardless of nonlinearity. This would make testing the distribution of the points impossible. In order to avoid 3

4 this, in this paper each initial estimate has been filtered using a separate set of noisy measurements. This prevents convergence of the distribution to a single point. Using separate measurement sets does limit this test in some ways to an analysis tool, rather than a real-time application where only one set of true measurements is available. Having defined the nonlinearity measure, the question remains to investigate its usefulness. What does it reveal about the effect of nonlinearity on EKF performance? This issue will be investigated in the following orbit determination example. The nonlinearity measure will be applied to an orbit determination filter implemented with both rectangular-coordinate and orbital-element dynamics and spherical-coordinate and rectangular-coordinate measurements. ORBIT DETERMINATION As an illustrative example, consider the ground-based radar tracking of a satellite. The nonlinearity of the orbit-determination EKF will be considered for various dynamics and measurement models. The dynamics models to be considered will use rectangular-coordinate and orbital-element coordinatizations, and the measurement models will use either spherical or rectangular coordinates. Model Development In rectangular coordinates the spacecraft state is given by [x RC ]=[r T v T ] T = [ xyzẋ ẏ ż ] T. The dynamical model for a Keplerian orbit is given by the following state equations. [ ] v [ẋ RC ]=[f RC (x RC )] = μ (3) r r 3 In implementing linear estimation techniques, the Jacobian of the state equations is also needed. [ ] [ ] frc 3 3 I = 3 3 (4) x RC G Here, G v/ r is the gravity-gradient matrix. An alternative state representation is the classic orbital elements [x OE ] = [aeiω ωf] T. For a Keplerian orbit, a, e, i, Ω, and ω are constants with respect to time. Thus the only nonzero component of the state equation is the derivative of the true anomaly. [ẋ OE ]=[f OE (x OE )] = [ f ] T (5) The state equation for the true anomaly is found from Kepler s second law. f = μ(1 + e cos f) a(1 e2 ) (6) 4

5 ~ y ~ SC Geometric y ^ RC x Nonlinear EKF R SC Transformation R RC P Pre-transformation of Measurements to Reduce System Non- Figure 2 linearity Again, the Jacobian of this state equation with respect to the orbital elements is also required but is not reproduced here. The measurement model describes the position of the satellite relative to a radar site. The measurements provided by the radar are the range, azimuth, and elevation. r = ( x 2 + y 2 + z 2) ( 1 2 y ) ( z ) ; φ =tan 1 ; θ =sin 1 (7) x r Note that these models were used even though they correspond to the dubious prospect of a radar site located at the center of the Earth. These measurements are nonlinear functions of the states, [ỹ SC ]=[ r φ θ] T. However, the measurements can be geometrically solved for a single-point estimate of the rectangular-coordinate position. x = r cos θ cos φ ; ỹ = r cos θ sin φ ; z = r sin θ (8) Clearly, these equations represent an alternative measurement model [ỹ RC ]=[ x ỹ z] T that is linearly related to the rectangular-coordinate states. Of course, most estimation techniques will also require that the uncertainty in ỹ SC be transformed to give the uncertainty in ỹ RC. If the errors in the original measurements were uncorrelated, Gaussian, white noise, the errors in the new measurements will not be so, due to the nonlinear mapping above. However, a linear approximation can be used to map the covariances. [ ] [ ] T yrc yrc R RC R SC (9) y SC y SC The implementation of the rectangular-coordinate measurements is illustrated in Figure 2. Nonlinearity Measure Using x RC Based on the above models several EKF filters have been implemented. The results from the filters using rectangular-coordinate state models are described in this section, and the results of the orbital-element state model are covered in the following section. The trajectory investigated was associated with initial conditions [r ] = [1, 5, ] T km and [v ]=[6.5 ] T km/s and a non-keplerian disturbance acceleration of [a d ]=1 6 sin(2πt)[111] T km/s with the time t measured in hours. 5

6 The presence of this deterministic perturbation does violate the assumptions of the Kalman filter but is not believed to significantly affect the results presented here. The measurements used in the filters had update rates of one minute and were corrupted by Gaussian random noise. The standard deviations of the noise were 1 km in the range measurement and.1 degrees in the two angle measurements. The systems were filtered over a period of 1 seconds. An EKF filter was implemented in rectangular-coordinate dynamics using both spherical-coordinate and rectangular-coordinate measurements. For both filters a variety of filter tunings were tested. Three different process noise settings were used. [ ] 2 [ ] [Q A ]= (1 3 km/s 2 ; [Q )I B ]= (1 4 km/s 2 )I 3 3 [ ] [Q C ]= (1 5 km/s 2 (1) )I 3 3 Additionally, three different measurement noise settings for the spherical-coordinate measurements were tested km 1 km [R A ]= π rad 18 ; [R B ]=.1π rad π rad 18.1π rad km [R C ]=.1π rad (11) 18.1π rad 18 For both the process and measurement noise, progression from A to B to C settings corresponds to progression from conservative tuning to aggressive tuning, with Q C and R C corresponding to the true amount of unmodeled disturbance in the system. The nonlinearity of each filter was tested using each of the tuning settings. To do this a Gaussian distribution of 5 initial estimates was taken with mean equal to the true initial states and standard deviations of 1 km in the position coordinates and.1 km/s in the velocity components. The EKF implementations were then used to filter each of these initial estimates. As mentioned, a new set of noisy measurements was used for each initial estimate, however, these measurements were all based on the same true motion. The same initial covariance was used for all trials. [ ] 2 (1 km)i3 3 [P ]= 3 3 (12) 3 3 (.1 km/s)i 3 3 After filtering each of the initial estimates, the statistics of the final distribution of estimates can be analyzed. In addition to the KS test, the standard deviation of 6

7 the distribution can be computed. In a real application, both of these statistics could be computed based on a distribution of estimates. In simulation, the distribution of estimates can also be compared to the true states. The performance of the filter can be judged by computing the mean error in the estimates relative to the true states. Tables 1 through 6 summarize these statistics of the final estimates for the x position coordinate for the various filter tunings. Tables 1 to 3 correspond to the filter implementation using spherical-coordinate measurements. Tables 4 to 6 correspond to the filter using rectangular-coordinate measurements. These results can be interpreted in terms of the expected behavior of the EKF. The Kalman filter of course is optimal and has guaranteed stability for linear systems. Applying the EKF to nonlinear systems, however, risks poor performance. One method to compensate for the nonlinearity is to set conservative values of the process and measurement noise. This corresponds to values of Q A and R A in the tables. This approach attempts to cover up the nonlinearity by treating it as noise and discounting the dynamics and measurement models. This can help stabilize the filter but at the cost of poor performance. Alternatively, for very accurate estimates the linear approximation becomes more valid. Therefore, if the EKF can successfully converge to an accurate estimate, the nonlinearity in the system becomes less significant. This behavior can result from setting aggressive tuning in the EKF. This approach corresponds to values of Q C and R C in the tables. Although this aggressive tuning does not reflect the nonlinearity, if the filter converges, then the nonlinearity becomes insignificant. Considering Tables 2 and 3, these two trends can be seen in the performance of the EKF using spherical-coordinate measurements. For the conservative tuning of R A the filter has moderate performance with large standard deviations around 26 km but small estimate biases around 3.2 km. For the more aggressive tuning of R C and Q A, the filter is able to accurately converge to the linear region. This results in a small standard deviation of 1.44 km and a bias of only.4 km. Interestingly, the intermediate cases of filter tuning produce very poor results. In the intermediate case the estimates remain in the nonlinear region, but the filter essentially trusts the models too much. The nonlinearity manifests itself in biased estimates. This is an example of a little confidence being a dangerous thing. Somewhat surprisingly, the filter also performs poorly with aggressive setting of the measurement noise R C and the more aggressive settings of the process noise Q B and Q C. These settings lead the filter to rely on the dynamics model more than when using Q A. In this case, the result is that the filter takes longer to converge to the linear region. Simulations over longer time periods, though, showed that these tunings eventually reached performance similar to Q A. 7

8 Table 1 KS TEST OF x COORDINATE USING ỹ SC (initial KS statistic:.314) Q A Q B Q C Table 2 STANDARD DEVIATION OF x COORD. ESTIMATES USING ỹ SC Q A km 8.47 km 1.44 km Q B 26.6 km km 8.69 km Q C 26.7 km km 12.9 km Table 3 MEAN OF x COORDINATE ERROR USING ỹ SC Q A 3.26 km 7.6 km.4 km Q B 3.23 km 1.6 km 7.27 km Q C 3.23 km 1.65 km 1.82 km 8

9 Table 4 KS TEST OF x COORDINATE USING ỹ RC (initial KS statistic:.314) Q A Q B Q C Table 5 STANDARD DEVIATION OF x COORD. ESTIMATES USING ỹ RC Q A km 1.5 km 1.43 km Q B km 1.58 km 1.45 km Q C 25.9 km 1.58 km 1.46 km Table 6 MEAN OF x COORDINATE ERROR USING ỹ RC Q A 1.48 km.42 km.3 km Q B 1.51 km.58 km.43 km Q C 1.52 km.59 km.58 km 9

10 The usefulness of the KS test as a nonlinearity measure is demonstrated by comparing Table 1 with Tables 2 and 3. This shows that the KS statistic reflects the trends described above. First, note that the KS statistic value of the initial randomly selected distribution was.314. This therefore represents a numerical zero in the KS statistic. For conservative tuning of the filter, the KS value increases slightly but only to a value of.427, indicating the filter is largely unaffected by the nonlinearity. For the aggressive tuning R C and Q A, the KS value of.274 is very small. As mentioned, with this tuning the estimate converges to the linear region and the nonlinearity is insignificant. Importantly, for the intermediate tunings of the filter, the KS test results in large values around.13 or.14. The KS test of the final estimates for the x coordinate for filter tunings Q A and R B is demonstrated in Figure 3. The distribution of final-position estimate errors for this tuning is shown in Figure 4, clearly showing the non-gaussian nature of the final distribution. Unlike the standard deviation, the KS test indicates the impact of the nonlinearity for these tunings. The KS test gives better indication of the estimate bias than the standard deviation alone. Additional insight into the orbit-determination EKF and the KS test can be gained by comparing Tables 1 through 3 to Tables 4 through 6. Here, the impact of switching from spherical-coordinate measurements to rectangular-coordinate measurements is revealed. Using ỹ RC, acceptable performance is obtained for all filter tunings, though more conservative settings of R A still result in greater uncertainty as expected. Significantly, Table 4 shows that the nonlinearity has been greatly reduced. Figures 5 and 6 show the distribution of final estimates in the x coordinate for the same filter tuning as in Figures 3 and 4. These figures demonstrate that the final distributions are much more Gaussian than the distributions obtained using ỹ SC. This reduction in nonlinearity is as expected because switching to ỹ RC results in a linear measurement model. In fact, the KS values show that nearly all of the nonlinearity has been removed, indicating the nonlinearity in the system was primarily due to the nonlinear measurements ỹ SC. It is intuitive that, using 6-second measurement update rates, the nonlinearity in the dynamics is not very significant. The orbital equations are sufficiently linear over a span of just one minute. These results demonstrate that the measurement of nonlinearity in an estimation system must account for both the measurement and dynamics models; a measurement of dynamical nonlinearity is insufficient. Finally, it is interesting to note that the results for the two filters were nearly identical for the aggressive tuning of R C and Q A. As mentioned, once the filter converges to the linear region, the nonlinearities in the dynamic and measurement models become insignificant. During the transient phase, however, the nonlinearities are important. Using the linear measurements ỹ RC resulted in quicker convergence than the filter using ỹ SC. Although not shown, during the convergence phase the filter using ỹ SC did have significantly higher KS values. Once the filter converged, however, the distribution returned to Gaussian statistics. Because of the the effect of the 1

11 1 Distribution Functions F (y) F F (y) N y Figure 3 KS Test of x Coordinate for Tuning Q A and R B Using ỹ SC z y 2 x Figure 4 Final Position Errors (km) for Tuning Q A and R B Using ỹ SC 11

12 1 Distribution Functions F (y) F F (y) N y Figure 5 KS Test of x Coordinate for Tuning Q A and R B Using ỹ RC 5 z y 5 x 5 Figure 6 Final Position Errors (km) for Tuning Q A and R B Using ỹ RC 12

13 process noise, which causes the filter to forget old information, the KS test represents a test of nonlinearity experienced over a recent period with a sliding horizon. Nonlinearity Measure Using x OE The nonlinearity measure was also performed on filter implementations with the orbital-element model, using the same baseline trajectory. As noted, the nonlinearity measure is sensitive to the distribution of initial conditions used in the computation. To compare the results from the orbital-element model to the rectangular-coordinate model, therefore, it is desirable to pick a Gaussian distribution of initial estimates for the orbital elements that is in some sense equivalent to the Gaussian distribution used for the rectangular coordinates. Based on the initial distribution for x RC, the following standard deviations were chosen for the orbital elements. σ a = 369 km ; σ e =.18 ; σ i =.832 rad σ Ω =.379 rad ; σ ω =.841 rad ; σ f =.729 rad (13) The distribution of initial conditions was further modified, however, to prevent initial estimates with negative semimajor axis or eccentricity. Of course, this introduced a small non-gaussian nature into the initial distribution. The Kalman filter was implemented in the same manner as described for x RC ; however, the following process noise settings were used for x OE. [ ] [ ] [Q A ]= (1 5 ; [Q )I B ]= (1 6 )I 5 5 [ ] [Q C ]= (1 7 (14) )I 5 5 Tables 7 through 12 summarize the statistics of the final estimates for the semimajor axis using the various filter tunings. Tables 7 through 9 show the results for the filter using y SC, and Tables 1 through 12 show the results for y RC. The results for the orbital-element filter display some of the same trends as the rectangular-coordinate filter with respect to filter tuning. Using the orbital-element implementation, however, there is no longer a dramatic reduction of nonlinearity in switching from y SC to y RC. Whereas the pairing of x RC and y RC produced very low nonlinearity, the same benefit is not achieved by pairing x OE and y RC. Unlike some of the results from the rectangular-coordinate filter, here both the KS test and the distribution standard deviation show good correlation with the performance of the orbital-element filter (indicated by the estimation error mean). This 13

14 Table 7 KS TEST OF a USING ỹ SC (initial KS statistic:.346) Q A Q B Q C Table 8 STANDARD DEVIATION OF a ESTIMATES USING ỹ SC Q A 1236 km 22 km 1974 km Q B 1236 km 2183 km 2348 km Q C 1236 km 221 km 255 km Table 9 MEAN OF a ERROR USING ỹ SC Q A 214 km 283 km 289 km Q B 214 km 27 km 358 km Q C 214 km 271 km 374 km 14

15 Table 1 KS TEST OF a USING ỹ RC (initial KS statistic:.346) Q A Q B Q C Table 11 STANDARD DEVIATION OF a ESTIMATES USING ỹ RC Q A 1129 km 2294 km 8377 km Q B 1125 km 2326 km 13,3 km Q C 1125 km 2356 km 14,16 km Table 12 MEAN OF a ERROR USING ỹ RC Q A 166 km 49 km 1175 km Q B 172 km 47 km 1773 km Q C 172 km 436 km 1788 km 15

16 1 Distribution Functions F (y) N F (y) F y Figure 7 KS Test of Semimajor Axis for Tuning Q A and R B Using ỹ SC indicates that the main manifestation of nonlinearity in the orbital-element filter is a convergence domain behavior. Although estimates near the truth converge well, poor estimates do not converge as well as would be expected in a linear system. This results in farther lying outliers than would be expected for a Gaussian distribution. The presence of these outliers is reflected in both the KS statistics and the standard deviations. The impact of the convergence domain and outliers is also illustrated by looking at the distribution functions of the final estimates. The KS test of the final estimates for the semimajor axis for filter tunings Q A and R B is demonstrated in Figure 7. Some asymmetry is apparent in the distribution function, but significantly the distribution exhibits much steeper transition near the mean than would be expected from a Gaussian distribution. DISCUSSION The results presented demonstrate that the KS test is a useful measure of the nonlinearity in an EKF system, which depends on both dynamical and measurement model nonlinearity and the filter tuning. The results from the rectangular-coordinate state model demonstrated that the KS test can give an indication of nonlinearity, which might warn of biased estimates even though the distribution of estimates is tightly converged. The results from the orbital-element model demonstrated that correlation between the KS statistic and sample standard deviation can indicate a domain of convergence. Beyond these sample results for orbit determination, the nonlinearity measure could be used to aid in the development of estimation implementations for a variety of other physical systems. Because the KS test provides important information about the nonlinearity of the 16

17 system and the performance of the EKF, it seems that this measure could have applications in adaptive filtering. The test indicates the amount of nonlinearity recently experienced by the filter and could be potentially used as a health monitor for the filter. Previous methods of adaptive filtering have focused on the measurement residuals. These methods have attempted to tune the filter based on observed values of the residual covariance or autocorrelation (testing whiteness). 8 1 Methods based on the departure from Gaussian behavior, however, have not been developed to the authors knowledge. Additionally, the methods used in the test are akin to those used in unscented Kalman filtering and particle filtering. The information provided by the KS test could be applied in studying the distribution of points used in these filtering techniques. REFERENCES 1. Lefferts, E. J., Markley, F. L., and Shuster, M. D., Kalman Filtering for Spacecraft Attitude Estimation, Journal of Guidance, Control, and Dynamics, Vol. 5, No. 5, 1982, pp Hermann, R. and Krener, A. J., Nonlinear Controllability and Observability, IEEE Transactions on Automatic Control, Vol. AC-22, No. 5, 1977, pp Junkins, J. L., Adventures on the Interface of Dynamics and Control, Journal of Guidance, Control, and Dynamics, Vol. 2, No. 6, 1997, pp Junkins, J. L. and Singla, P., How Nonlinear Is It? A Tutorial on the Nonlinearity of Orbit and Attitude Dynamics, The Journal of the Astronautical Sciences, Vol. 52, No. 1 & 2, 24, pp Kirkpatrick, E. G., Introductory Statistics and Probability for Engineering, Science, and Technology, Prentice-Hall, Englewood Cliffs, New Jersey, 1974, Section Mood, A. M., Graybill, F. A., and Boes, D. C., Introduction to the Theory of Statistics, McGraw Hill, New York, 3rd ed., 1974, Chapter 11, Section DeGroot, M. H., Probability and Statistics, Addison-Wesley, Reading, Massachusetts, 2nd ed., 1986, Section Maybeck, P. S., Stochastic Models, Estimation, and Control, Vol. 2, Academic Press, New York, 1982, Chapter Stengel, R. F., Optimal Control and Estimation, Dover, New York, 1994, Section Crassidis, J. L. and Junkins, J. L., Optimal Estimation of Dynamical Systems, Chapman & Hall/CRC, Boca Raton, Florida, 24, Sections and