14 th IFAC Symposium on System Identification, Newcastle, Australia, 2006

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1 14 th IFAC Symposium on System Identiication, Newcastle, Australia, 26 SUBSPACE IDENTIFICATION OF PERIODICALLY SWITCHING HAMMERSTEIN-WIENER MODELS FOR MAGNETOSPHERIC DYNAMICS 1 Harish J Palanthandalam-Madapusi Dennis S Bernstein Aaron J Ridley Department o Aerospace Engineering, University o Michigan, Ann Arbor, MI , USA {hpalanth,dsbaero}@umichedu Department o Atmospheric, Oceanic, and Space Science, University o Michigan, Ann Arbor, MI 4819, USA ridley@umichedu Abstract: Two existing Hammerstein-Wiener identiication algorithms and a third novel Hammerstein-Wiener identiication algorithm are considered or application to the magnetospheric system A modiied subspace algorithm that allows missing data points is described and used or identiying periodically switching Hammerstein-Wiener models, to capture the periodically time-varying nature o the system These models are used to predict ground-based magnetometer response using the ACE satellite measurements Keywords: System identiication, Identiication algorithms, Modelling, Models, Prediction, Space 1 INTRODUCTION The magnetosphere is the region o space dominated by the magnetic ield o the Earth By predicting magnetospheric conditions, damage to critical systems aboard satellites and large power transormers on the earth can be avoided Although large-scale, irst principles models o the magnetosphere exist (Powell et al, 1999), they are extremely expensive to run in real-time, and have to be run at a poor resolution or real-time perormance While empirical models are simplistic compared to irst principles models, they are useul or predicting speciic quantities at speciic locations Previous empirical models include neural network models (Weigel et al, 23) and Hammerstein- Wiener (H-W) models (Palanthandalam-Madapusi et al, 25), where the H-W model structure consists o linear dynamics with static input and output nonlinearities as shown in Figure 1 Since 1 This research was supported by the National Science Foundation Inormation Technology Research initiative, through Grant ATM measurements o magnetospheric conditions are made by ground-based magnetometers, which rotate with a one-day periodicity with respect to the magnetosphere, the system has a one-day periodicity In (Palanthandalam-Madapusi et al, 25), a sinusoid is used as an additional input to emulate the periodically time-varying nature o the system In the current work, periodically switching models are used to model the periodically time-varying nature o the system u NH z L NW y Fig 1 Hammerstein-Wiener Model In this paper, subspace-based identiication methods or periodically switching H-W models o the magnetospheric system are considered For this application, two existing H-W identiication methods (Goethals et al, 24; Palanthandalam- Madapusi et al, 25) are used, and a third H- W identiication algorithm is developed These three algorithms are then used as the basis or identiying periodically switching H-W models 535

2 Subspace identiication or periodic systems is developed in (Hench, 1995; Verhaegen and Yu, 1995) In particular, (Hench, 1995) uses liting to express the periodically time-varying identiication problem as a set o linear time-invariant identiication problems However, or a slowly timevarying periodic system with a large period p, both these methods are ineicient, since they require the calculation and storage o p state space matrices For example, in the magnetospheric system, which has a one-day periodicity with measurements at every minute, 144 sets o statespace matrices have to be calculated and stored Another issue with real-lie data such as the magnetospheric data is that there are oten missing data points To deal with these issues, irst a subspace algorithm that allows or missing data points is described Although this modiication is known, it is o great practical importance and is apparently unpublished Furthermore, this subspace algorithm is essential or identiying periodically switching models This technique is used with the three H-W identiication algorithms mentioned above to identiy periodically switching H- W models or the magnetospheric system With the ocus on prediction, all three methods are compared and the results reported 2 MAGNETOSPHERIC APPLICATION Much o the dynamics in the magnetosphere is controlled by the Sun s atmosphere, which lows supersonically away rom the Sun past the Earth and the other planets The solar wind carries embedded in it the Sun s magnetic ield, commonly reerred to as the interplanetary magnetic ield (IMF) The solar wind and IMF interact with the magnetosphere producing phenomena such as the aurora and ionospheric currents When the aurora and ionospheric currents become severe, the upper atmosphere can heat up dramatically and expand, causing increased drag on satellites In addition, large ionospheric currents can drive currents in power lines, which can overwhelm, and destroy transormers It is thereore important that we understand when and where these large currents may occur and be able to predict them The ACE satellite, which is positioned at the Lagrangian gravitational null point between the Sun and the Earth, measures the solar wind and IMF conditions approximately one hour beore they encounter the magnetosphere The exact delay is calculated based on the solar wind velocity This delay implies that measurements made by the ACE satellite at time t, reachtheearthat approximately time t + 1hour Thus, theace measurements are delayed by roughly an hour or use as the inputs to the identiied model Most importantly, because o this delay, the identiied model has a one-hour predictive capability since the inputs are available one hour into the uture The goal o the current study is to use the ACE measurements to predict the magnetosphere conditions The ionospheric currents cause magnetic perturbations that are measured by ground-based magnetometers Thus, the magnetometer data can be used as an indication o the magnetospheric conditions By predicting ground-based magnetometer response using the ACE data, one can determine possible disturbances and may be able to take steps to minimize damage 3 HAMMERSTEIN-WIENER IDENTIFICATION Three H-W identiication algorithms are considered here Based on these three algorithms, periodically switching H-W models are constructed in subsequent sections or the magnetospheric system The irst method (Palanthandalam-Madapusi et al, 25) consists o a two step procedure, in which the Hammerstein subsystem is identiied in the irst step and the Wiener nonlinearity is identiied as a second least-squares step To identiy the Hammerstein subsystem, a basis unction expansion o the known inputs is used The second method also consists o a two step procedure To identiy the Hammerstein subsystem in the irst step, the algorithm described in (Goethals et al, 24) is used, which uses least squares support vector machines The step or identiying the Wiener nonlinearity is the same Due to lack o space, the reader is reerred to (Palanthandalam-Madapusi et al, 25) and (Goethals et al, 24) or details o the irst two H-W identiication algorithms The third method, which is a novel one-step H-W identiication algorithm, is explained in detail in the next subsection 31 One Step Hammerstein-Wiener Identiication Consider a Wiener system o the orm x k+1 Ax k + Bu k, (31) y k Cx k + Gh(x k )+Du k, (32) where x R n,u R m,y R l,h: R n R q,a R n n,b R n m,c R l n, D R l m and G R l q The linear part o the output equation is Cx k, while Gh(x) is the nonlinear part o the output equation 311 Notation Deine the output block Hankel matrices 536

3 y y 1 y j 1 Y 2i 1 y i 1 y i y i+j 2 y i y i+1 y i+j 1 (33) y 2i 1 y 2i y 2i+j 2 Y i 1 Yp, (34) Y i 2i 1 where i and j are user-deined integers such that i n and j i Also, Y Y 2i 1 i Y + p Y i+1 2i 1 Y (35) The subscript p denotes the past and the subscript denotes uture The input block Hankel matrices U 2i 1,, U, + and U are deined as in (33)-(35) with y replaced by u The deinitions o the nonlinear block Hankel matrices H 2i 1, H p, H, H p + and H are analogous to (33)-(35) with y replaced by h(x)the extended observability matrix is C CA Γ i (36) CA i 1 Since i n, i(a, C) is observable, then Γ i has rank n Let M i R li mi be the lower block triangular Toeplitz matrix D CB D CAB CB D M i CA i 2 BCA i 3 B D, (37) and deine Ψ i R li qi as G Ψ i (38) G Finally, the state sequence X i R n j is X i xi x i+1 x i+j 2 x i+j 1 (39) 312 State Estimation Deinition 1 The nonlinear system (31)-(32) is observable i the triple (A, C, G) is such that Θ i Γi Ψ i R li (n+qi) has ull column rank or all i n Note that, or a system to be observable it is necessary that li n + qi or all i n, which implies l > q Also, a necessary condition or observability is that the pair (A, C) be observable in the linear sense Lemma 31 I the nonlinear system (31)-(32) is observable, then the intersection o the row space Up o the past block Hankel matrix W p and row space o the uture block Hankel matrix U W contains the state sequence X Y i Proo: From (31) and (32), Γ i X +Ψ i H p + M i, (31) Γ i X i +Ψ i H + M i U (311) Since the system is observable, (311) can be written as Xi Θ H i M i Θ U i, (312) where is the Moore-Penrose generalized inverse Furthermore, X i Θ i,n M i Θ U i,n, (313) where Θ i,n denotes the irst n rows o Θ i From (313), it is clear that the state sequence X i is contained in the row space o W or the uture input-output data Similarly rom (31), X Θ i,n M i Θ i,n (314) Next, X and X i are related as X i A i X +Δ i, (315) where Δ i A i 1 BA i 2 B B (316) Now, using (314) and (315), we obtain X i A i Θ i,n M i +Δ i A i Θ i,n (317) From (317), the state sequence X i is equally contained in the past input-output data W p Thus rom (313) and (317), the state sequence X i is contained in the intersection o W p and W Next, the notion o persistent excitation is deined Deinition 2 A sequence o inputs {u i } N i1 is persistently exciting or the system (31)-(32) i the ollowing conditions are satisied: H 2i 1 (1) The matrix has ull row rank U 2i 1 (2) The intersection between the row space o the state sequence X and row space o the H 2i 1 matrix is trivial U 2i 1 (3) The state sequence X has ull row rank n Theorem 31 I the nonlinear system (31) and (32) is observable and the input sequence {u i } N i1 is persistently exciting, then the intersection o the row spaces o the matrices W p and W yields the state sequence X i 537

4 Proo From Lemma 31, the intersection o W p and W contains the state sequence X i Next, using (31), we write Yp Γi Ψ i M i X H I p, (318) mi which implies that Yp rank rank X H n + qi + mi p Similarly, it can be shown that Y rank rank X i H U n + qi + mi U and rank U rank X H 2i 1 n +2qi +2mi U Y 2i 1 Now, using the Grassmann dimension theorem ((Bernstein, ( 25), Theorem 231) gives Up ) U dim row space row space Up U rank +rank rank U n (319) Thus, rom Lemma 31 and (319), the intersection o the row spaces o W p and W yields a valid state sequence There are several ways to compute the above intersection One method is to use a QR decomposition o the matrix, ollowed by a singular Wp W value decomposition Since the intersection can be calculated in any basis, the estimated state sequence ˆX i diers rom the actual states X i to within a similarity transormation Once the estimates ˆX i o the state sequence X i are computed, the system matrices can be computed by solving the least squares problems argmin A,B ˆX i+1 AB ˆXi, (32) U i i F argmin Θ CG,D Y i i Θ CG D CG ( ˆX i ), (321) U i i F where CG : R n Rŝ are basis unctions, and Θ CG CG ( ˆX i ) CX i + GH i i When additional noise terms are present in (31) and (32) the oblique projection method described in (Van Overschee and De Moor, 1996) is used to estimate the states The calculation o the state space matrices remain the same as (32)-(321) 313 Hammerstein-Wiener Extension Identiication or H-W systems is a straightorward extension o the Wiener identiication problem described in the previous section Consider the nonlinear discrete-time system x k+1 Ax k + F(u k )+w k, (322) y k Cx k + Gh(x k )+G(u k )+v k, (323) where F : R m R n and G: R m R l By writing F and G in terms o a set o basis unctions : R m R s (Palanthandalam-Madapusi et al, 24; Lacy and Bernstein, 21), the system can be rewritten as x k+1 Ax k + B(u k )+w k, (324) y k Cx k + Gh(x k )+D(u k )+v k, (325) where the matrices B and D contain the coeicients o the basis unction expansion Thus, by using a basis unction expansion or the input nonlinearity, the identiication problem is reduced to a Wiener identiication problem with generalized inputs For details o the above procedure see (Palanthandalam-Madapusi et al, 24) 4 SUBSPACE IDENTIFICATION WITH MISSING DATA This section considers the case in which some data points are unavailable Let q beatimestepor which a measurement o u q or y q is unavailable Then we deine the output block Hankel matrix Y 2i 1 y y q 2i+1 y q+1 y j 1 y i 1 y q i 1 y q+i 1 y i+j 2 y i y q i y q+i y i+j 1 y 2i 1 y q 1 y q+2i 1 y 2i+j 2 (41) Y i 1 Yp (42) Y i 2i 1 Note that Y 2i 1 has the same orm as the output block Hankel matrices deined in (Van Overschee and De Moor, 1996) and section 31, except that all the columns o the original Y 2i 1 containing y q are omitted When several data points are missing, the block Hankel matrix is constructed similarly by omitting all columns that contain the time steps corresponding to the missing data Furthermore, Y 2i 1 Y i Y i+1 2i 1 Y + p Y (43) The input block Hankel matrices U 2i 1,, U, U + p and U are deined analogous to (41)-(43) with y replaced by u Finally, the state sequences X and X i are deined as 538

5 X x x q 2i+1 x q+1 x i+j 1, X i xi x q i x q+i x i+j 1 (44) These state sequences have a gap o 2i 1time steps when one data point (u q or y q ) is missing X has states missing rom time steps q 2i +2 through q, while X i has states missing rom time step q i +1throughq + i 1 With these modiied deinitions o block Hankel matrices, it can be shown that, in the deterministic case, the state sequence estimated by the procedures described in (Van Overschee and De Moor, 1996) is indeed X i deined as (44) Furthermore, when the noise terms are present, the procedures in (Van Overschee and De Moor, 1996) yield optimal estimates o X i The above modiication is valid or the H-W identiication procedure described in section 31 as well 5 IDENTIFICATION OF PERIODICALLY SWITCHING MODELS To identiy periodically switching models, consider x k+1 A k x k + B k u k + w k, (51) y k C k x k + D k u k + v k, (52) where the period is p and switching occurs at time steps αp + r 1, αp + r 2,, αp + r β Hereα is any integer, β is the number o switching models and r 1 <r 2 < <r β are the oset switching times Hence, or all k, A k A k+p,and, or t 1,,β, A αp+rt 1+1 A αp+rt 1+2 A αp+rt, and similarly or B k,c k,d k To identiy the t th set o state-space matrices, the data points corresponding to all o the other sets o matrices are regarded as missing data points as in section 4 This procedure is applicable to the three H-W identiication algorithms described beore, but requires prior knowledge o the oset switching times r i, and requires that r t r t 1 2i, t 1,,β (53) 6 IDENTIFICATION FOR MAGNETOMETER DATA The ACE IMF and solar wind data are used as inputs to the model, while the ground-based magnetometer data are used as outputs o the model Knowledge o the physics and previous experience (Palanthandalam-Madapusi et al, 25) suggests that a combination o the B z and B y components is desirable or the inputs to the model, where B z and B y are the z and y components o the magnitude o the IMF, respectively On testing several combinations o these two components, it is ound that B t V x sin 4 (θ/2) works best, where B t B y 2 + Bz, 2 θ a cos B z /B t,andv x is the x-component o the velocity o the IMF Similar combinations o terms have previously been considered (Akasou, 2) Additional linearly entering inputs include the density o plasma (solar wind), B t, B z,andb y Since the low pattern o the inonospheric currents is in a Sun-ixed coordinate system with the magnetometer station rotating within it, the system has a one-day periodicity Periodically switching H-W models are used to capture this one-day periodicity and model the dierent dynamics governing the nightside and dayside o the magnetospheric system Switching between two, three, and our sets o state-space matrices within a one-day period are considered The oset switching times are chosen such that an equal amount o time is spent in each regime Thus, i two switching models are present in a day, switching occurs every 12 hours 7 IDENTIFICATION RESULTS For illustration, data or the month o January, 1999 measured by the ground-based magnetometer at Thule, Greenland, is used The irst 15 days data are used or identiication, while the next 16 days are used or validation The output o the identiied periodically switching H-W model using the irst method is shown in Figure 2 In this model, switching between three sets o statespace matrices occurs at regular time intervals o 8 hours each Data to the let o the black vertical line are used or identiication, while the identiied model is used to predict the data to the right o the vertical line When using real-time data, prediction o only one hour is possible, however prediction or 16 days is possible in the above example since the inputs or the entire period are available Figure 3 shows the output o the periodically switching H-W model identiied using the second method Again, switching occurs every 8 hours Finally, the output o the periodically switching H-W model identiied using the third method is shown in igure 4 The prediction eiciencies o the three periodically switching H-W models are calculated as PE 1 k (y k ŷ k ) 2 /σy 2 (Weigel et al, 23), where σy 2 is the variance o the measured output y From Table 1, the prediction eiciency or periodically switching H-W model based on the irst method is seen to be better than the eiciencies or the other two methods or both the amplitude B and the luctuation db/dt o the magnetic ield These prediction eiciencies compare avorably to the prediction eiciencies o the neural network model in (Weigel et al, 23) 539

6 Table 1 Prediction eiciencies Method 1 Method 2 Method 3 PE or B PE or db/dt CONCLUSION Two existing Hammerstein-Wiener identiication algorithms and a novel Hammerstein-Wiener identiication algorithm were considered or application to the magnetospheric system A modiied subspace algorithm that allows missing data points was described and used or identiying periodically switching models To capture the periodically time-varying nature o the system, periodically switching Hammerstein-Wiener models were then identiied using the three Hammerstein- Wiener identiication methods The inputs to the models were measurements rom the ACE satellite, while the outputs o the model were groundbased magnetometer readings The prediction eiciencies o the three methods were compared Future work will ocus on other lower-latitude magnetometers and identiication based on richer model structures 9 ACKNOWLEDGEMENT The authors are grateul to Ivan Goethals or helpul suggestions REFERENCES Akasou, S-I (2) Predicting geomagnetic storms as a space weather project In: Space Weather Study Using Multipoint Techniques (Ling-Hsiao Lyu, Ed) Vol 12 o Proceedings o COSPAR pp 3 2 Bernstein, D S (25) Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory Princeton University Press Goethals, I, K Pelckmans, JAK Suykens and B De Moor (24) Subspace identiication o hammerstein systems using least squares support vector machines preprint Hench, J J (1995) A technique or the identiication o linear periodic state-space models Int J Cont 62(2), Lacy, S L and D S Bernstein (21) Subspace Identiication or Nonlinear Systems That Are Linear in Unmeasured States In: Proc Con Dec Contr Orlando, Florida pp Palanthandalam-Madapusi, H, A J Ridley and D S Bernstein (25) Identiication and prediction o ionospheric dynamics using a hammerstein-wiener model In: Proc Amer Contr Con Portland, OR pp Palanthandalam-Madapusi, H, J B Hoagg and D S Bernstein (24) Basis-unction optimization or subspace-based nonlinear identiication o systems with measured-input nonlinearities In: Proc Amer Contr Con Boston, MA pp Powell, KG, PL Roe, TJ Linde, TI Gombosi and DL De Zeeuw (1999) A solutionadaptive upwind scheme or ideal magnetohydrodynamics J Comp Phys 154, 284 Van Overschee, P and B De Moor (1996) Subspace Identiication or Linear Systems: Theory, Implementation, Applications Kluwer Verhaegen, M and X Yu (1995) A class o subspace model identiication algorithms to identiy periodically and arbitrarily time-varying systems Automatica 31(2), Weigel, R S, A J Klimas and D Vassiliadis (23) Solar wind coupling to and predictability o ground magnetic ields and their time derivatives J o Geophysical Research 18(A7), SMP 16 1 SMP Magnetometer reading (Tesla) Actual data Model output Time (secs) Fig 2 Measured and predicted data o the Thule magnetometer using the irst identiied periodically switching H-W model Magnetometer reading (Tesla) Data Model output Time (secs) Fig 3 Measured and predicted data o the Thule magnetometer using the second identiied periodically switching H-W model Magnetometer reading (Tesla) Data Model output Time (secs) Fig 4 Measured and predicted data o the Thule magnetometer using the third identiied periodically switching H-W model x 1 6 x 1 6 x