Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00082439/00001
## Material Information- Title:
- Autoregressive moving-average (ARMA) model identification for degenerate time series with application to maneuvering target tracking
- Creator:
- Speakman, Norman Owen, 1949- (
*Dissertant*) Bullock, Thomas E. (*Thesis advisor*) Shaffer, Charles V. (*Reviewer*) Basile, Giuseppe (*Reviewer*) Rao, P. V. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1985
- Copyright Date:
- 1985
- Language:
- English
- Physical Description:
- vi, 90 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Acceleration ( jstor )
Aircraft maneuvers ( jstor ) Autocorrelation ( jstor ) Autoregressive moving average ( jstor ) Eigenvalues ( jstor ) Parametric models ( jstor ) Stochastic models ( jstor ) Time series ( jstor ) Time series models ( jstor ) Trajectories ( jstor ) Autoregression (Statistics) ( lcsh ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Stochastic processes ( lcsh ) Time-series analysis ( lcsh ) Tracking radar ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- Research was conducted in the general areas of time series analysis and stochastic realization. Results were then applied to the specific problem of tracking a highly maneuverable aircraft target. An algorithm was developed to identify the order and parameters of the minimum autoregressive moving-average (ARMA) model of a multi-variable system given the output autocorrelation sequence. Studies were also conducted in the area of degenerate time series modeling. It was found that degeneracy in vector-valued time series is caused by the presence of one or more deterministic relationships in the time series. ARMA models for degenerate time series can be identified by finding and extracting the deterministic relationships from the time series. The result is a reduced dimension atochastic model of the system, The model found will have fewer white noise inputs than outputs. An AR<A identification algorithm for use with degenerate time series was developed. The equivalence between the ARMA model found and the Kalman filter innovation representation was discussed. A thorough numberical example demonstrating this equivalence as well as degenerate time series modeling was presented. Application of the ARMA model identification procedure to the target racking problem was investigated. It was discovered that the time series formed from target inertial velocity vectors is degenerate. This fact has physical significance and allows one to determine the orientation of the maneuver plane (the plane containing all the target motion) in inertial space. The ARMA modeling technique was used to develop an adaptive modeling and tracking algorithm. Operation of the algorithm was demonstrated using a target trajectory simulation.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1985.
- Bibliography:
- Bibliography: leaves 87-89.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Norman Owen Speakman.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 029355001 ( ALEPH )
14281429 ( OCLC ) AEG0115 ( NOTIS )
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AUTOREGRESSIVE MOVING-AVERAGE (ARMA) MODEL IDENTIFICATION FOR DEGENERATE TIME SERIES WITH APPLICATION TO MANEUVERING TARGET TRACKING By NORMAN OWEN SPEAKI'AN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 This dissertation is dedicated to my wife, Donna, and our children, Erik, Timothy, and Amy. ACKNOWLEDGMENTS The author wishes to express sincere appreciation to the chairman of his graduate committee, Dr. Thomas E. Bullock, for the guidance, patience and, above all, the dedication shown during the course of his doctoral program. Thanks are also extended to Dr. Giuseppi Basile, Dr. Charles V. Shaffer and Dr. Pramod P. Khargonekar for the help they willingly and professionally provided. The author is also indebted to Dr. William P. Albritton and Dr. Donald C. Daniel at the Air Force Armament Laboratory for the tremendous encouragement provided during the writing of this dissertation. The author wishes to offer a special thanks to his wife and children for the seemingly endless patience and understanding shown during the more trying times of this endeavor. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . ABSTRACT . . INTRODUCTION . . General Description of the Problem . Survey of Past Approaches . THEORETICAL DEVELOPMENT . . Background . . ARMA Model Parameter Identification . ARMA Model Identification Algorithm . ARMA Model Parameter Identification (Degenerate Case) . ARMA Model Identification Algorithm (Degenerate Case) .. .. Relationship to the Kalman Filter . The Multichannel Burg Algorithm . The Multichannel Burg Algorithm (Degenerate Case) . Numerical Example . . APPLICATION TO TARGET TRACKING . . The Problem . Engagement Geometry . Trajectory Generation . Target Dynamic Model Synthesis (Off-line) . Target Dynamic Model Synthesis (On-line). Maneuvering Target Tracking Algorithm . Numerical Results and Predictor Performance SUMMARY AND RECOMMENDATIONS. . . Summary . . Recommendations for Further Research. . BIOGRAPHICAL SKETC . . * . . . Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AUTOREGRESSIVE MOVING-AVERAGE (ARMA) MODEL IDENTIFICATION FOR DEGENERATE TIME SERIES WITH APPLICATION TO MANEUVERING TARGET TRACKING By Norman Owen Speakman December 1985 Chairman: Dr. Thomas E. Bullock Major Department: Electrical Engineering Research was conducted in the general areas of time series analysis and stochastic realization. Results were then applied to the specific problem of tracking a highly maneuverable aircraft target. An algorithm was developed to identify the order and parameters of the minimum order autoregressive moving-average (ARMA) model of a multi- variable system given the output autocorrelation sequence. Studies were also conducted in the area of degenerate time series rmdeling. It was found that degeneracy in vector-valued time series is caused by the pre- sence of one or more deterministic relationships in the time series. ARMA models for degenerate time series can be identified by finding and extracting the deterministic relationships from the time series. The v result is a reduced dimension stochastic model of the system. The model found will have fewer white noise inputs than outputs. An ARMA iden- tification algorithm for use with degenerate times series was developed. The equivalence between the ARMA model found and the Kalman filter innovations representation was discussed. A thorough numerical example demonstrating this equivalence as well as degenerate time series modeling was presented. Application of the ARMA model identification procedure to the target tracking problem was investigated. It was discovered that the time series formed from target inertial velocity vectors is degenerate. This fact has physical significance and allows one to determine the orien- tation of the maneuver plane (the plane containing all the target motion) in inertial space. The ARMA modeling technique was used to deve- lop an adaptive modeling and tracking algorithm. Operation of the algorithm was demonstrated using a target trajectory simulation. INTRODUCTION General Description of the Problem The ability to accurately estimate aircraft target motion is a cru- cial factor in the accuracy of modern weapon systems. A great deal of attention has recently been devoted to the problem of tracking aircraft targets which are highly maneuverable (i.e., capable of achieving and maintaining relatively large accelerations). The level of attention has risen primarily because rapidly advancing microcomputer technology per- mits missile guidance systems to utilize more information about pre- dicted target motion and secondarily because modern fighter aircraft are more maneuverable. Sustained acceleration on the order of 9 g's is com- mon. Effective tracking basically requires that the orientation of an observer-based coordinate system (usually associated with the centerline of a seeker) is maintained relative to the line-of-sight vector between the observer and target as the target moves relative to the observer. Improved tracking results in enhanced estimates of target kinematic variables which are critical parameters in the computation of guidance commands for missiles using advanced guidance laws. The mathematical science of estimation theory lead to the develop- ment of optimal (in the minimum mean-square-error sense) estimators which accept noise corrupted measurements, relates them to the system states and accounts for the propagation of the states between 1 2 measurements. The celebrated Kalman filter is probably the most widely used product of estimation theory and has been applied to the target tracking problem for several years. A key element in the design of a Kalman filter is the construction of a statistical model which accounts for the behavior of the system between measurements, i.e., a system dynamic model. Several major technical difficulties are encountered when for- mulating a dynamic model for the tracking problem when it is known that the target is capable of executing abrupt maneuvers. First, there is no direct measure of target acceleration although acceleration, a state variable, is observable from measurable parameters. Second, there are uncertainties in the measurements and in the system dynamic model. Uncertainties exist in all estimation problems but the uncertainty asso- ciated with the target dynamic model is somewhat unique. A very agile and highly maneuverable aircraft can change its direction of flight very quickly and in an erratic fashion. As a result, tracking filters require high bandwidths to be effective during maneuvering flight. Lower band- widths can of course be used when it is known that the target is not maneuvering. Since the transition from non-maneuvering flight to maneuvering flight is sudden and, as far as the tracking filter is con- cerned, occurs at random times, tracking is usually done using high filter bandwidths. Tracking performance is therefore degraded when the target is not maneuvering. Finally, measurements (angles and ranges) are usually best expressed in spherical coordinates whereas the dynamics (kinematic variables such as position, velocity and acceleration) are best written in cartesian coordinates. Linearization of the measurement equations is required which leads to errors. 3 A class of theoretical problems which has application to statisti- cal modeling is contained in the area known as stochastic realization theory. The stochastic realization problem is to determine a Markovian representation for a rational, stationary time series given the covariance sequence of the time series. A common technique used to model a system which has unknown dynamics is to examine the time correlation of cinematic variables such as velocity and acceleration. Once the covariance sequence is defined, results from the theory of stochastic realization can be used to formulate a statistical model of the system under consideration. Problems to be solved in this research effort are two-fold. The first objective is to formulate an innovative dynamic model of an aircraft target suitable for use in a tracking filter. Second, to accomplish the first goal, a new development in stochastic modeling will be explored which extends the theory of stochastic realization. The latter objective has applications far beyond that of target dynamic modeling and some interesting theoretical implications will be pursued. Survey of Past Approaches Research in the maneuvering target tracking problem can be divided into three major categories: (1) modeling, (2) tracking schemes and (3) maneuver detection. Although research efforts can be accomplished in any one of these discrete areas, they are actually quite closely related. Some researchers have dealt with more than one of the above categories s imultaneous ly. 4 The most commonly used discrete target dynamic model in use today is credited to Singer [i1. The model is based on the assumption that the target undergoes random accelerations in each of the three inertial axes and that the acceleration is exponentially correlated. This results in the well-known first-order Gauss-Markov model, a zero mean, time- correlated Gaussian noise process. Fitts [2] and McAuley and Denlinger [31 adopted the Singer approach to target modeling. There are certain heuristics associated with the Singer model. The model requires a priori knowledge of the acceleration time constant and the acceleration probability density. Singer claims that the simple model proposed is a good one provided that these parameters are correctly chosen. Another method of modeling target motion was proposed by Gholson and Moose [41 and later extended by Moose, Van Landingham and McCabe [51. It basically involves the use of semi-Markov processes to fodel major changes in trajectories. Target motion is modeled as being driven by a time-correlated process which has a randomly varying mean. This is actually a combination of Singer's time-correlated input and semi-Markov modeling. The input, chosen randomly from a finite set of possible discrete levels, is applied for a random time interval after which a new input level is chosen. The motivation for this model is that it more closely approximates real-world maneuvers. Kendrick et al. [6 extended the state-of-the-art in target modeling by developing a new statistical model for aircraft normal acceleration. The basic premise is that the dominant acceleration is normal to the wings. The acceleration magnitude is modeled as an assymetrically 5 distributed time-correlated random process. This is done by modeling acceleration magnitude as a deterministic nonlinear function of a zero- mean time-correlated random variable and has the advantage of allowing limits on the acceleration level which cannot be done using the simpler Singer model. Non-normal acceleration, resolved into components along the three inertial axes, are modeled by the familiar first-order Gauss-Markov process with a symmetrically distributed white noise input. Obviously this method, while providing more latitude in modeling, requires parameter values to define the nonlinear acceleration function. Again, as in the Singer model, certain parameters must be specified a prior even though the parameters are chosen to be constants for a par- ticular class of aircraft. An innovative approach to modeling target motion was developed by Bullock and Sangsuk-Iam [7]. In this nonlinear model it is assumed that the aircraft trajectory is composed of circular arcs in a maneuver plane and the velocity magnitude and turning rate are constant. A model writ- ten in polar coordinates follows from this formulation. An interesting point to note here is that acceleration has been removed from the state vector and is therefore not modeled explicitly. Using the usual model in linear cartesian coordinates requires piecewise constant approximations to the natural sinusoidal variation in inertial acceleration components. Difficulties encountered in estimating sinusoidal variations have been removed using the nonlinear model. Marked improvement over the Singer model has been shown. Hull, Kite and Speyer [8] derived a new stochastic target model by assuming that the angle defining the orientation of the acceleration 6 vector in inertial space is a random process. The model is then written in spherical rather than cartesian coordinates. They studied two cases: (1) acceleration magnitude is constant and (2) acceleration magnitude is a random process. Only the two-dimensional case was considered. Results indicate that the variable magnitude model exhibits better tracking per- formance than the constant magnitude model. Under certain conditions it was observed that the constant model was comparable to the Singer imdel. One advantage that the circular model has over the conventional car- tesian model is that the acceleration magnitude can be limited to a maximum value. The Singer model, which assumes the inertial components of acceleration are independent, can result in magnitudes which exceed maximum achievable acceleration levels. Several studies have been done to compare various proposed tracking methods. Vergez and Liefer [91 compared four commonly used models: (1) first-order Gauss-Markov, (2) second-order Gauss-Markov, (3) zero acce- leration and (4) constant acceleration. The first-order Gauss-Markov model was found to perform the best for the conditions considered. Lin and Shafroth [101 also performed a similar study. They, however, con- sidered different tracking algorithms: (1) Singer's approach, (2) Input Estimation, (3) Bayesian and (4) Re-initialization, which employs maneuver detection. Each tracking method implied a different accelera- tion model and, as expected, performed well for certain given trajec- tories. Lin and Shafroth concluded that the tracking algorithm which performed best during maneuvers was the Bayes tracker. Many researchers have sought solutions to the tracking problem by devising new tracking schemes. Most of the algorithms proposed use the 7 Singer model for target dynamics. Pearson and Stear [11i developed a radar tracker which models target motion in the rotating coordinate frame of the tracking system, i.e., the line-of-sight coordinate frame. Two acceleration components normal to the line-of-sight vector are modeled as independent first-order Gauss-Markov processes. Two approaches were derived: angle and range tracking systems. The angle tracker estimates angular variables such as line-of-sight rate and the range system provides estimates of linear scalar variables such as range and range rate. It should be noted that any model written in a rotating coordinate system is inherently complex due to the necessity of including non-neglible Coriolis accelerations. Spingarn and Weidemann [12] applied linear regression techniques to both the non-maneuvering and maneuvering tracking problems. The non- maneuvering problem results in an expanding memory filter. When the same technique is applied to maneuvering targets it becomes necessary to truncate older data and form what is known as a fading memory filter. Filtering in both line-of-sight and inertial cartesian coordinates revealed that filtering should be done in the inertial frame to obtain smaller prediction errors. The use of a combination of electro-optical sensor measurements and pattern recognition algorithms to deduce target aspect angle was pro- posed by Kendrick et al. [6]. The aspect angle is the angle between the missile's velocity vector and the target velocity vector. Estimates of target orientation can be obtained from the aspect angle measurement. Target motion is then estimated using a well-known relationship between attitude and kinematic states. 8 Many tracking schemes involve the use of maneuver detectors. Maneuvers are usually detected by monitoring the innovations sequence of a tracking filter which has been designed for a non-maneuvering target. When a bias is encountered in the residuals which exceeds a predeter- mined threshold, a maneuver is assumed to have occurred. The strategy used after detection varies. Thorp [13] used a likelihood ratio test for maneuver detection and a binary random variable in the target state equations. The tracking filter combines estimates from two Kalman filters using coefficients obtained from the likelihood ratio. Bullock et al. [71 and Chan et al. [14] use the statistic X2 to detect maneuver occurrence. When a maneuver is detected by performing a statistical hypothesis test, the tracking filter is reinitialized and the filter gain is adjusted accordingly. The process continues each time a measurement is made. Chan et al. [14, 151 use a least squares technique to estimate the target's input acceleration. At each measurement the magnitude of the estimate is compared with a threshold. When a detec- tion is made, a Kalman filter which assumes constant velocity target motion is updated. The purpose of the update is to remove the bias caused by the target deviating from straight-line motion. The filter is thus able to maintain track even in the presence of maneuvers. Although numerous methods using combinations of modeling, tracking and maneuver detection techniques have been devised and implemented, the problem of tracking a highly maneuverable aircraft still remains a key technological challenge. Maybeck [16, p. xiii] states that "system modeling is a critical issue and typically the 'weak link' in applying theory to practice." It is with this same conviction that this research effort is undertaken. THEORETICAL DEVELOPMENT Background Time series analysis is a very useful and powerful method used in applications such as prediction and control. In many of these applica- tions, a continuous signal is sampled at some regular time interval to form a discrete-time signal known as a time series. Having obtained a discrete time history of the signal, one may use the time series to determine a parametric model of the system under consideration and use the model to predict future behavior of the system. A general model which is commonly used in time series analysis is based on the assump- tion that a signal can be modeled as the output of a system which is driven by a Gaussian white noise input. In such a model the present out- put is a linear combination of past outputs and present and past inputs. This model is usually written as a discrete-time difference equation: AOYk + Al1k-1 + ... + AnYk-n = Boek + Blek- + ... + Bmek-m (1) where yk(k>O) is a p-dimensional vector sequence of outputs (time series) and ek (k>O) is a p-dimensional zero-mean vector sequence of Gaussian white noise. The matrices Ai (Oi square. Equation (1) can also be written using z-transform notation, A(z)Y(z)=B(z)E(z) (2) 9 where n A(z)= I Aiz-i (3) i=0 m B(z)= X B z-J (4) j=0 and z-1 is the unit delay operator. Equation (2) is termed a pole-zero model. There are also two special cases of equation (2) which are useful: the all-pole model where Bj=0 (14j known as an autoregressive (AR) model and the all-zero model is called a moving-average (MA) model. The more general pole-zero nmdel is termed an autoregressive moving-average (ARMA) model. In the remainder of this dissertation we will be primarily concerned with ARMA models but may occasionally have need to use AR models. ARMA Model Parameter Identification Much effort has been spent devising suitable methods for identifying ARMA models [17, 18, 19 to name a few]. Most of these methods are restricted to the time domain. We shall also concentrate our attention on time domain model identification. Suppose we have a stationary output time series, yk, and we wish to construct an ARMA model of the time series. It is well-known that an ARMA model of finite order (n,m) can be written as an infinite order AR model [17, 20]. This equivalence exists if we assume the AIMA model is invertible. The condition for invertibility is [20] det[B(z)]#0 IzI>l. 11 If the invertibility condition is met we can write equation (2) as B-l(z)A(z)Y(z)=E(z), (6) which can also be written as an AR model if we let y(z)=B-l(z)A(z) (7) so that y(z)Y(z)=E(z). (8) A major problem associated with multivariable ARMA model iden- tification is that of uniqueness of the realization termed iden- tifiability. That is, given that Yk can be modeled as an ARMA process, is it possible to determine unique values for n and m and the coef- ficient matrices Ai (O
sequence of yk? To see the difficulty with identifying a unique ARMA model from the output autocorrelation sequence, one need only multiply both sides of the model by the same matrix polynomial in z and observe that the autocorrelation structure is unaltered. This gives rise to the notion of "classes" of models for a given autocorrelation sequence. The problem now becomes one of determining conditions to apply to A(z) and B(z) so that one APRMA model from the class of models will be identified. This will insure that, given an autocorrelation sequence, only one model will result which fits the data and meets the constraints. Hannan [211 discusses this problem and presents three sufficient conditions for identify ability: (1) A0 = BO = I 12 (2) A(z) and B(z) have no common left divisors except unimodular ones (3) det[A(z)]0 I|z >1 det[B(z)] 0 |z| >1. (10) (11) We note that condition (1) can be met without loss of generality as long as the covariance matrix of ek, I, is allowed to be general sym- metric non-negative. To show this, consider an ARMA model which has AO0I and B0:I, AOYk + Alyk-1 + ... + AnYk-n = Boek + Blek.- + ... + Bmek-m. (12) We can pre-multiply equation (12) form, (Ai)new by AO-1 to put the equation in reduced A0-1 Ai i=0,l,2,...,n. (13) The lead coefficient on the right hand side can be made identity by a proper change of input basis. Let (Bi)new = A0-1 Bi BO-1 AO (ek)new = AO-1 BO ek i=0,1,2,...,m k>0. We now have an equivalent ARMA model in which (AO)new = (BO)new = 1. One may ask, what happens if det[AO]=0 or det[BO]=0? We can that these conditions cannot occur for a stationary, invertible series. The determinant of A(z) is a polynomial in z-1, det[A(z)] = fo + flz-l + f2z-2 + ... + fNz-N . and (14) (15) show time (16) Equation (16) can also be written det[A(z)] = det[AO + Alz-1 + ... + Anz-n] z (17) By setting z-1 = 0 and combining equations (16) and (17) we can solve for fo: fo = det[Ao]. (18) If we assume that det[AO]=0 and det[A(z)]=0 we find from equation (16) that z-1 (fl + f2z-1 + ... + fNz-N+ ) = 0. (19) One solution to equation (19) is z= We know that det[A(z)]+0 for zl >1 if the time series is stationary. Therefore, a necessary condition for stationarity is det[Ao010. An analogous argument exists which says that a necessary condition for invertibility is det[Bolg0. We can modify the constraint on B0 and specify conditions which must be met by II. It has been shown [20] that requiring BO to be lower triangular and ==I also results in the elimination of redundant para- meters and thus allows the unique determination of an ARMA model. We shall choose the former condition, i.e., that BO=I since this gives us latitude in selecting II. This becomes important later in this develop- ment. Condition (2) is imposed to eliminate the possibility that a can- cellation may occur when the product B-l(z)A(z) is formed. The require- ment imposed by condition (3) amounts to no more than requiring that the model be both stationary and invertible. Having stated conditions which insure the unique identification of ARMA model parameters, we can proceed by deriving an expression for B-l(z). in B(z) = Bjz- BO=I (20) j=0 and B-1(z) = kz-k (21) k=0 Performing the multiplication B-l(z)B(z) we find B-l(z)B(z)= 80 + fk(i.,Bj)z- i,j Also we require that B-l(z)B(z)=I (23) Equating like powers of z in equations (22) and (23) it is obvious that O = I (24) and fk(i,Bj)=O k>0 (25) Each Ck(Bi,Bj) is linear in Bk. We can therefore solve each equation for Sk in tens of Bi(i are 0O = I (26) 01 = -B1 (27) 82 = -1B B2 = B12 B2 (28) 03 = -2B1 -B1B2 -B3 = B2B1+BIB2-B13-B3 (29) Referring to equation (7), we see that by post-multiplying B-l(z) by 15 A(z) we obtain a matrix polynomial which is the coefficient matrix poly- nomial for an equivalent AR model. That is, B-(z)A(z) = I (B1 Al)z-1 + (B12 B2 + A2 BAl)z-2 +... (30) and y(z) = B-l(z)A(z) = I + ylz-1 + Y2Z-2 + ... (31) Equating coefficients of like powers of z in equations (30) and (31) yields equations which relate the AR model coefficients to the ARMA model coefficients. We can also use these equations to write the MA part of the ARMA model in terms of the AR part and the coefficients of the pure AR model. The result is B0 = I (32) B1 = A1 Y1 (33) B2 = A2 Y2 B1 Y1 (34) B3 = A3 Y3 B1 Y2 B2 Y1 (35) etc. In general, B0 = I (36) and N-1 BN = AN I Bk YN-k N>0. (37) k=0 The coefficients in the pure AR model can be found by solving the normal, or Yule-Walker, equations [22, 23, 24, 25, 26]. An efficient recursive method used to solve the Yule-Walker equations is the Levinson algorithm [27, 28, 291. The Levinson problem is one of determining 16 the coefficients in an AR model such that the one-step-ahead prediction is optimal in the minimum mean-square-error sense. That is, N-1 YNIN-1 =- YN-i Yi i=O (38) where YNI1N- is the one-step-ahead prediction of YN and N is the number of data points in the sequence being modeled. Application of the projec- tion theory leads to EI(YN YNIN-1) Yj'] = 0 j=0,1,2, ..., N-I (39) which, when combined with equation (38), results in N-1 E[yNYj'] = YN-i E[yiYj'] i=0 j=0,1,2, ..., N-1. Equation (40) can be written as N-1 CN-j = I YN-iCi-j i=0 (40) (41) by letting Ci-j = E[yiYj'] . (42) The minimized mean-square-error of the prediction can be shown to be N-1 N = CO + YN-i Ci-N (43) i=0 17 To find A(z) we post-multiply equation (1) by Y'k-r and take expec- tations to give n m Cr = AiCr-i + i BjDrj , i=l j=l (44) (45) Cr=E[yk+ryk'l Dr=E[ek+ryk']. (46) Since a future input cannot affect the present output of a causal system we can write (47) Dr-_ = 0 which further leads to n m Cr = I AiCr-i + Z BjDr-j i=l j=l O (48) n Cr = C AiCr-i r>m. i=l We can determine the Ai (l
solving the set of linear equations, (49) I A1 A2 *An ]I Cm Cml1 Cm+1 ... Cm ... Cm-n-l Cm-n ** Cm+n-l Cm+n-2 C Cm S= [Cm+l *** m+nI- (50) Equations (50) are commonly referred to as the modified, extended [23], or shifted Yule-Walker equations. where and 18 ARMA Model Identification Algorithm The purpose of the algorithm is to find the autocorrelation matrix at the smallest lag which is a linear combination of other autocorrela- tion matrices. The model order is related to the position of this matrix in a larger block Toeplitz matrix formed from the autocorrelation sequence. The matrix is found by searching the block Toeplitz for a reduction in rank of submatrices. Rank reduction is determined by com- puting eigenvalues. When the rank of a submatrix is reduced by the dimension of the output vector, p, we know that p columns (or rows, since the matrix is Toeplitz) can be written as a linear combination of the other columns (or rows). We can write Ci = -AICiI A2Ci-2 ... AnCin. (51) We determine i by keeping track of our position within the large block Toeplitz. Comparing equation (51) with equation (49) we see that m is related to the subscript i in equation (51). More precisely, m = i-1. (52) Furthermore, we see from equation (50) that n is determined from the size of the submatrix in which the rank reduction occurred. By per- forming the search in a systematic fashion, the values computed for m and n are minimum. Determination of AWMA model coefficients is rather straightforward using results presented in the previous section once the order of the model is known. The algorithm: 1. Construct the Toeplitz matrix of autocorrelations, CO C1 C2 Ck C_1 CO Cl Ck-i C_2 C_1 Co Ck-2 T = (53) Ck Cl-k C2-k. CO Note that this is the same Toeplitz matrix used in the Levinson algorithm to solve the Yule-Walker equations. We have therefore not imposed any additional computational burden (e.g., computing autocorrelations) beyond that necessary to generate an AR model of the data. 2. Set q=2. 3. Systematically examine submatrices within T. Begin with the northwest corner qpxqp submatrix and proceed to the right. For example, when q=2 we will have CO C1 Cl C2 Cr-1 Cr C-1 CO CO Cl Cr-2 Cr-1 Causality requires that n>m. It is therefore not necessary to examine every submatrix within T, but only those which admit. causal realizations. The condition r<2q-l (55) will insure causality. 20 4. Compute the eigenvalues of each submatrix. If p eigenvalues are found which equal zero, stop the search and go to step 6. 5. If p zero eigenvalues are not found, increment q and go to step 2. 6. The submatrix with p zero eigenvalues will be Ci Ci+l C2i-j Ci-l Ci 2i-j-1 .. (56) Cj Cj+I Ci Then, n=i-j (57) and m=i-1. (58) Having determined the values for m and n, we can find Ai (I
and Bj (1 ARMA Model Parameter Identification (Degenerate Case) The vast majority of literature devoted to the modeling of time series is concerned with the nondegenerate case. This case requires that the Toeplitz matrix of autocorrelations is full rank. Some of the literature dealing with degeneracy of time series is applicable only to the scalar case but this is a rather trivial case since scalar dege- neracy implies that the system generating the time series is deter- ministic. The AR model identified when the singularity occurs is the deterministic model of the process. The fitting procedure stops because 21 the next value in the sequence being modeled can be predicted with no error using the model found. A more difficult problem is encountered when an attempt is made to model a degenerate vector time series. Techniques used to make AR models for vector processes also break down but the reason for failure is not as obvious. Degeneracy results when the rank of the covariance matrix of the one-step-ahead prediction error is less than the dimension of the vector sequence being modeled. The vector Levinson algorithm fails when this condition is encountered because the inverse of the error covariance matrix must be computed. A singular error covariance indica- tes that a deterministic relationship exists between two or more com- ponents of the vector. Inouye [30] has extended the vector Levinson algorithm to handle the degenerate case by using the pseudoinverse of the error covariance when required. With the exception of this modifica- tion, the algorithm is unaltered. This method does not treat the real cause of the degeneracy. In this research we have also developed a procedure which can handle degenerate time series using the Levinson algorithm. This approach is more pragmatic in that it takes into account the actual cause of the degeneracy. This basically involves the identification of the deter- ministic part of the system, reducing the dimension of the original vec- tor and modeling the reduced dimensional vector sequence in the usual way. We begin by writing the AR equation which results after completing N steps of the Levinson algorithm, Yk + Yk-1 + + YNYk-N = ek. (59) From the algorithm we also compute HN = E[ekek'I. (60) Suppose that det[InNI = 0. The time series is therefore degenerate. Since HN is a real symmetric matrix, it can be diagonalized by an ortho- gonal transformation, M'IHN M = A (61) where A = diag (XO,X 1,. ..,pI) and Xi (0
I[N. The columns of M are the normalized eigenvectors of HN. Let the eigenvector associated with the zero eigenvalue be denoted by V0. From equations (60) and (61) we have VO'E[ekek']VO = 0 (62) E[(VO'ek)(VO'ek)'] = 0 => VO'ek = 0. vk (63) Using this result in equation (59) yields N V'( I Yi Yk-i ) = 0 i=0 Y = I. (64) Equation (64) is a deterministic difference equation relating the p com- ponents of the output. The problem now becomes one of modeling the remaining stochastic part of the output. This is done by transforming or 23 the original data sequence (or equivently the original autocorrelation sequence) using M. Let M = I v 2 v -1 V1 (65) where Vi (l eigenvalues of uiN. The transformed autocorrelation sequence will be Cj = M'Cj M[4 jo0. (66) The Cj sequence can now be used in the ARMA model identification algorithm described in the previous section. If a singular I[ is again encountered, the process is repeated with a further reduction in dimen- sion. The important thing to note here is that the dimension of the stochastic model has been reduced by the identification of a deter- ministic relationship in the output. After the stochastic model is iden- tified it is combined with the deterministic model to form a model with p outputs and less than p inputs. ARMA Model Identification Algorithm (Degenerate Case) Before an ARMA model can be constructed using the algorithm pre- sented earlier, an AR model must be found by solving the normal equations. If the time series is degenerate, a singular error covariance matrix will result as discussed in the previous section. The following algorithm allows one to determine the ARMA model parameters for a dege- nerate stationary time series. The algorithm: 1. Set v=p. 2. Use the Levinson algorithm to determine the AR model coef- ficients and II. Also, at each step compute the eigenvalues and eigenvectors of ]I. 3. If rank (II) = v, use the nondegenerate ARMA identification algorithm with the values computed for yi (l
step 6. 4. If H is singular, use the eigenvector, V0, associated with the zero eigenvalue to transform the AR model to the null space of II. Decrement v. 5. Form M_ from the eigenvectors normal to VO. Transform the original autocorrelation sequence using M*. Go to step 2. 6. The stochastic and deterministic models found are combined to form an ARMA model with v inputs and p outputs. Relationship to the Kalman Filter In this section we discuss the equivalence between the ARMA model identified and the innovations representation of the Kalman filter. Given an output autocorrelation sequence, the ARMA model identified has the same transfer function and driving noise covariance as the innova- tions model (Kalman filter) of the system which generated the watocorre- lation sequence. The innovations model is a Kalman filter which has the zero-mean, white innovations sequence as input and the measurement sequence as out- put. The innovations model is sometimes termed an inverted Kalman 25 filter. The relevant equations and a block diagram are given below: A A Xk+lIk = F xk k-l + K yk Yk = k + Yk = H xklk-1 + Yk (67) (68) Figure 1. Innovations model where K is the Kalman gain matrix, F and H have the usual meaning and Yk is the innovations sequence. The transfer function of the innovations model is HK(z) = I + H(zI-F)-1K. (69) The ARMA model given by equation (2) has the transfer function HA(z) = A-l(z)B(z). (70 We observe that the AHMA model with transfer function given by equation (70) is equivalent to the inverted Kalman filter with transfer function given by equation (69). Furthermore, E[eeke']=E[1kYk']. These observations are based on the following facts: 26 1. The innovations model and the AR model both solve the one- step-ahead prediction problem with minimum error variance. 2. The minimum variance estimate is unique. 3. The ARMA model is equivalent to the AR model. By using the orthogonality principle, the Levinson solution yields A an estimate, yklk-l which is the minimum variance estimate of Yk given the sequence Yk-l, Yk-2,**' YO' That is, Yklk-l = E[yk Yk_-1 (71) where k-1 ElyklYk-l = k Yk-iYi (72) i=0 and Yk-l denotes the sequence Yk-l, Yk-2, *** Y0' The minimized mean-square-error of the ARMA model output is II = E[(yk Yklk-1)(k YkIk-1)'I = E[kek'l (73) The Kalman filter innovations sequence is given by A (74) Yk = Yk Yklk-l Since the minimum variance estimate is unique we conclude E[ekek'l = E[ykYk'l. (75) Equivalently, E[ekek']=HEH'+R (76) where Z is the steady-state solution to the discrete-time Riccati equation. 27 The ARMA model found using the identification method presented earlier is equivalent to the AR model found using the Levinson algorithm. The systems described by equations (59) and (60) are linear systems driven by whibe noise processes. The output sequences of the two systems with equivalent white noise inputs are identical. We conclude that HA(z) = HK(z). This result is significant because, unlike the inverted Kalman filter, the ARMA model does not require the solution to a Riccati type equation. All equations necessary to find HA(z) are linear and have ana- lytic solutions. The Multichannel Burg Algorithm In practice, when the Levinson algorithm is used in conjunction with a finite length time series, estimates of the lagged autocorrelations must be used. These estimates are frequently obtained from N-k Ck = 1/N C Yk+iYi' k=0,1,2,...,N-1 (77) i=l which yields a biased estimate of the autocorrelation but is attractive in that T>0, where T is the block Toeplitz of autocorrelations given by equation (53). The unbiased autocorrelation estimate, obtained by replacing N with N-k in the denominator of equation (77), sometimes results in T<0. This result is, of course, undesirable. Methods which rely on these types of lagged autocorrelation estimation are referred to as Yule-Walker estimation [31]. 28 An autoregressive modeling method which has received considerable attention in recent years is the Burg reflection coefficient technique [31, 32, 33, 34]. The original work, which covered the single-channel complex case, was done by Burg [35, 36]. Strand [31] generalized the method to include multichannel complex time series. The Burg process avoids the problems of autocorrelation estimation by determining the AR coefficients directly from the data. This is a major advantage of the reflection coefficient method over Yule-Walker estimation. An inter- mediate step in the procedure involves the calculation of the reflection coefficients, from which the AR coefficients can be found recursively. A brief description of the multichannel Burg algorithm is presented here for convenience. We begin by writing the N-element forward filter, Yk + 1,N Yk-l + + YN,N Yk-N = ek (78) and likewise the N-element backward filter, Yk + 01,N Yk+l + ... + ON,N Yk+N = bk (79) where yk (k>O) is a p-dimensional vector-valued time series, Yi,N (1
backward pxp matrix coefficients and ek and bk are forward and backward, respectively, zero-mean white noise prediction errors. 29 Burg derived the following recursion for the forward and backward filter coefficients: FN = [FN-1 01 + RN [0 B*N-l] (80) and BN = R [FN-[ 1 0] + [10 B*n1 (81) where FN-1 = II Y1,N-1 ** YN-1,N-1i (82) B N-1 = [N-1,N-1 *'* 1,N-1_ I] (3) and FO = B* = I. (84) In this notation a denotes time reversal. The matrices RN and R N are the forward and backward reflection coefficients, respectively. It should be noted here that yN,N = RN and 0N,N = R*N. It is generally the case that the reflection coefficients are chosen which yield optimal (in the minimum inean-square-error sense) forward filters. A derivation of the RN which minimizes the expected mean-square-error of the forward filter is presented later in this section. Using equations (78) through (81) it can be shown that N = "N-l RN 1N-1 (RN)' (85) and (86) FN = NI R*N HN-1 (RN)' 30 where HN and TN are the forward and backward prediction error covariance matrices, respectively. If we post-multiply equations (78) and (79) by Y'k-N and y'k, respectively, and note that for an N-element backward filter, E[bk'k] = rN (87) then we can show, using equations (80) and (81), CN + YI,N-1CN-1 + ... + YN-1,NI-1C + RNFN-1 = 0, (88) where CN is defined by equation (45). Likewise we can obtain CN + 1,N-1C'N-1 + ..* + ON-1,N-lC'l + R*NIIN-l = 0 (89) by using E[ekY'k] = [N (90) for an N-element forward filter. By defining N-1 AN = CN + Yi,N-lCN-i (91) i=l and N-1 A*N = C'N + 6 i,N-1C' -i (92) i=l and using a result attributed to Burg [351, namely that, A' = A*N (93) we obtain the expression for the backward reflection coefficient, R*N= rN-1 R'N (~"N-1)-1 (94) Define the (N+l)pxl vector ym(N) = I y'm+ y'm+ N-l1 *** I Y'nN- .. Yr I ' iN=0,l,...N- m=1,2, . (95) where Nd is the number of data points in the time series and M=Nd-N. Using equations (78) through (81) we can write the forward filter and backward filter mth errors as um = [(FN-1 0) + RN (0 B*I~-)] ym(N) Vm = [ RN (FN-1I 0) + (0 B*N-1)] yIm(N). (96) (97) Equations (96) and (97) can be written um = el(N) + RN bm(N) (98) vm = bm(N) + (R*N) em(N) em(N) = (FN_1 0) ym(N) (99) (100) bm(N) = (0 JB-1) ym(N) (101) are the forward and backward residual sequences, respectively. The objective now is to find the forward reflection coefficient which minimizes the trace of the forward filter mean-square-error. and where The quantity to minimize is M J(HN) = (1/M) I (u'mu) (102) which, using equation (96), can be written M J(RN) = (1/M) ) [e,n(N)+ RN bl(N)]' le(N)+ RN bm(N)]. (103) m=1 Using the identities x'y=tr(x'y)=tr(yx'), tr(AB)=tr(BA) and tr(A')=tr(A) for vectors x and y, square matrices A and B and where tr is the trace operator, we can write J(RN) = tr(E) + 2tr(RNG) +tr(RNBRN), M E = (1/M) I em(N)e'm(N), m=l M G = (1/M) m bm(N)e'm(N) m=1 (104) (105) (106) (107) M B = (1/M) 7 bm(N)b'm(N). m=l To find RN which minimizes J(RN) consider any two general real square matrices X and Y. Using equation (104) we can write J(Y)-J(X) = 2tr[(Y-X)G] + tr(YBY'-XBX'). (108) Now note that tr[(Y-X)B(Y-X)'] = tr(YBY'+XBX'-XBY'-YBX') where (109) tr(XBY')=tr(YBX'). Using equations (109) and (110) we can write J(Y)-J(X) = 2tr[S(G+BX')] + tr(SBS') (111) where S = Y-X. (112) A necessary and sufficient condition for X to minimize J(X) is G + BX' = 0. (113) For sufficiency note that if G+BX'=0 then, since tr(SBS')>0, J(Y)>J(X) for any choice of X and Y. Now consider the case G+BX'*O. We can choose Y = X-k(G+BX') (114) where k is a real positive constant. Substituting this expression for Y into equation (111) results in J(Y)-J(X) = k2 tr[(G+BX')B(G+BX')']- 2k trl(G+BX')2] J(Y)-J(X) = k2ql kq2 :q = tr[(G+BX')B(G+BX')'] (115) (116) (117) and q2 = 2tr[(G+BX')2]. where (110) (118) 34 Since G+BX'1O we have ql>0 and q2>0. Therefore, if 0 then J(Y) Thus, X minimizes J(X) iff G+BX'=0. If we substitute RN for X in equation (113) we obtain RN = -G'B-1, which is the RN which minimizes J(HN). The residuals are updated by em(N) = em+l(N-1) + (RN-)bm+1(N-1) m1=,2, bm(N) = bm(N-1) + (R*N-_)em(N-l). N>1 m=l,2,...,M N>1 Algorithm initialization is accomplished by setting em(1) = Ym+1, m=1,2,...,Nd-1 bm(l) = yi m=1,2,...,Nd-1 and Nd HO = 0 =(1/Nd) I YmYm. m=1 (119) (120) (121) (122) (123) (124) (125) (126) . .,M 35 It is interesting to observe that the Levinson algorithm and the Burg algorithm produce the same model. When equations (124) and (125) are combined with equations (106) and (107), we note that B and G' are the biased estimates of the zeroth lag and first lag autocorrelations, respectively. For N=l, equation (121) becomes RI = -CCO-1. (127) This is also the expression for the first AR coefficient using the Levinson algorithm. The reflection coefficient found using the Levinson algorithm is Yk+l = k(k)-l, k>l (128) where ak = E[(yk+1-yk+llk) (o-Yo k)' (129) and Fk = E[(yo0-oIk)(yo-ok)']- (130) Equation (129) is the cross-covariance of the forward and backward residuals. This is the same quantity as G' (for N>1), where G is given by equation (106). Likewise, rk corresponds to B (for N>1) from equation (107). We conclude that the Burg algorithm produces the same realization as the Levinson algorithm. The Multichannel Burg Algorithm (Degenerate Case) The multichannel Burg algorithm is subject to the same problems exhibited by other identification techniques when degenerate time series 36 modeling is attempted. As before, we find that det[lN]=0 and the algorithm fails. The solution for the Burg algorithm is the same as discussed previously; identify the deterministic relationship in the data and reduce the dimension of the output vector. This requires a modification to the algorithm to handle the possibility that a dege- nerate time series is encountered. The modified Burg algorithm is pre- sented below: 1. Set N=l. 2. Initialize using equations (124),(125) and (126). 3. Compute E, B and G. Use equations (105),[(106) and (107). 4. Solve equation (121) for RN and compute R* from equation (94). 5. Compute HN and FN Use equations (85) and (86). 6. Compute the eigenvalues and associated normalized eigenvec- tors of HN. If det[HNI=0, go to step 11. 7. Update FN and BN using equations (80) and (81). 8. Increment N. 9. Update em(N) and b,(N) using equations (122) and (123). 10. Go to step 3. 11. Transform the Nth order AR model using VO, the eigenvector associated with the zero eigenvalue of 11N. The result is V'0 ( Yk + Y1,N Yk-l + *** + YN,N Yk-N ) = 0. (131) This is the deterministic relationship in the output. 12. Transform the output vector to a new basis using M, As before, the columns of M are the eigenvectors of HN normal to V0. The reduced dimension output vector is m=l,2,...,Nd (132) ym(new) = M' Y1m. 37 Of course, all subsequent output data obtained are also transformed to the new basis. 13. Using ym(new), go to step 1. When the stochastic portion of the time series has been identified, it is combined with the deterministic part to form a complete model of the system. Numerical Example In this section a numerical example is presented which illustrates the major results discussed in previous sections. Consider a dynamical system described by the vector difference equation, Xk+l = Fxk + Wk (133) with measurements given by Yk = Hxk + Vk (134) where xk is a qxl state vector, yk is a pxl output vector, wk and vk are vector sequences of Gaussian white noise and F and H are ratrices of appropriate dimension. The output autocorrelation sequence is given by Ci = HFipH' + R60i i>O (135) where P is the steady-state solution to the discrete-time Lyapunov equation, Pk+l = FPkF' + Q , (136) E[wjWk'] Q&jk E[vjvk'] = Rjk. For this example let H= 1.0 4.28 1.0 0.0 1.0 0.0 1.0 -5.77 -5.82 0.0 0.0 0.0 0.0 -1.0 Q= diag (10.0 5.0 0.0 0.0) , R= null matrix. The steady-state solution to equation (136) is 167.702 448.902 132.140 277.985 448.902 1408.73 451.145 1023.46 132.140 451.145 167.702 448.902 277.985 1023.46 448.902 1408.73 (137) (138) 1.0 1.0 0.0 0.0 (139) 0.0 0.0 0.0 (140) (141) (142) (143) 39 The resulting output autocorrelation matrices at the zeroth and first lags are 167.702 Co = 448.902 35.562 132.140 CI = 451.145 -35.562 448.902 14o8.73 -2.2428 277.985 1023.46 -170.917 35.562 -2.2428 71.1237 67.594 187.136 32.032 (144) (145) Initial estimates by of YI and I, the error covariance matrix, are given Y1 = -CICo-1 (146) III = CO CIC0-1C' . (147) These values are used to initialize the Levinson recursion. Substituting equations (144) and (145) into equations (146) and (147) yields -1.4393 Y1 = -4.6693 -0.4393 and 35.1067 25.1067 35.1067 0.2609 0.7609 0.2609 25.1067 30.1067 25.1067 -0.2225 -0.2725 -0.2225 35.1067 25.1067 35.1067_ (148) (149) n 1 which has eigenvalues, e0 = 0.0 (150) el = 9.3802 (151) eg = 90.670 (152) Since H1 is singular we have a degenerate time series. The normalized eigenvector of 1l associated with e0 is 0.70711 V = 0.0 (153) -0.7071 and the matrix Mi is -0.3563 -0.6108 ML = 0.8638 -0.5038 (154) -0.3563 -0.6108 We are now ready to identify the deterministic relationship in the out- put and transform the output to a new basis. Referring to equation (64) we can write VO'(Yk + Y1 Yk-1) = 0 (155) from which we obtain (3) = k(1) Yk(1) (156) Equation (156) describes the deterministic relationship which resulted in the singularity of I[. We obtain the reduced dimension autocorrela- tion sequence by the following transformation: j>0 (157) Cj = 1,4_LCjm-L 41 We find that the Cj(j>0) sequence is generated by a non-degenerate time series. Using the algorithm for non-degenerate AIMA models, we find that the stochastic portion of the system can be described by an ARMA(2,1) model. The coefficients found are 1.1825 A = -2.5635 A2 -0.2265 A2 = 0.3982 -0.2502 l -2.8504 1.8772 1 -2.6825 0.1153 1 -0.4235 -0.0219 -0.2498 (158) (159) (160) The error covariance matrix found is 6.5283 16.1920o The ARMA model transfer function is HA(z) = A-l(z)B(z) where A(z) = I + Alz-1 + A2z-2 (161) (162) (163) (164) B(z) = I + Blz-1. Carrying out the operations called for in equation (162) yields 5 8.8080 6.5283 h11(z) h21(z) hl2(z) h22(z) HA(z) = 1.0 1.5z-1 + 0.99z-2 0.345z-3 + 0.05z-4 where h11(z) = 1.0 2.9327z-1 + 5.5983z-2 + 0.4347z-3 hl2(z) = -1.899z-1 + 0.4123z-2 + 0.0381z-3 h21(z) = -0.2869z-1 4.4101z-2 + 0.7452z-3 h22(z) = 1.0 + 0.9327z-1 0.5780z-2 + 0.0653z-3 The poles of the system are found to be zl = 0.3 + j0.4 z2 = 0.3 j0.4 z3 = 0.4 and z4 = 0.5 - The ARMA model identified has two inputs and three outputs. The intermediate vector u(z)=M 'y(z). A complete description of the system is displayed in figure 2. -.3563 1-.5z-1 .8638 .8638 -.6108 1-.5z-1 -.5038 Figure 2. Block diagram of system in numerical example. (165) (166) (167) (168) (169) (170) (171) (172) (173) e(z) 43 The equivalent innovations model for the system can be found which has the transfer function, HK(Z) = I + H(zI-F)-1K. (174) To find HK(z) we must compute K, the steady-state Kalman gain matrix. This matrix is found from Kk = FEkk-lH'(HEklk-lH'+R)-1 (175) where Eklk-_1 is found by solving the discrete-time Riccati equation, Ek+llk = F[EkJk-l- Eklk-1H'(HEklk-lH'+R)-HEZklk-l]F'+Q (176) Initializing equation (176) with EO|-1 = P (177) results in det[HE110H' + R] = 0. (178) This result is expected since I[1, from equation (147), is equivalent to HEIOIH'+ R. To see this set k=0 and 201-1 = P in equation (176). Rewriting equation (176) we obtain Eli0 = FPF' + Q FPH'(HPH' + R)-1HPF'. (179) Recognizing that in steady-state P = FPF' + Q (180) we have ElIO = P FPH'(HPH' + R)-HPF', (181) HEl1OH'+ R = HPH' + R HFPH'(HPH' + R)-1HPF'f'. Using equation (135) in conjunction with equation (182) we find HE1|OH'+ R = CO CICo-1C1' (183) and from equation (147), HEI IOH' + R = 1[1. (184) Since we have defined R to be the null matrix we have HE1IOH' = I[I. (185) Of course the eigenvalues and eigenvectors of HE|IOH' are the same as those of i[1. If we make the same transformation that we did in the ARMA modeling procedure, i.e., transform to the null space of HEIlO', we get Vo' [ I O' I Vo = 0 (186) which is equivalent to E[(V01y-J)(V01-yl)1J= 0. (187) Equation (187) implies Vo' 1 = 0. (188) The Kalman filter equations can be manipulated in a way such that yk+l = Yk+l HKkYk H(F-KkH)xkIk-1. (182) (189) 45 If we set k=0 and xO_-1 = x0 = 0 in equation (189) we obtain Yl = Yl HKO YO (190) Combining equations (188) and (190) we can write VO'( yi HKO YO ) = 0. (191) This result is identical to equation (155) if Yl = -HKO. (192) This equivalence is easily obtained from equations (135),(146),(175) and (177). For this specific example we find -1.4393 0.2609 -0.2225 HK0 = -4.6693 0.7609 -0.2725 (193) -0.4393 0.2609 -0.2225 which checks with equation (148). Substituting equation (193) into equation (191) and using V0 as defined in equation (153) yields (3) = y(1) yo(1) (194) which agrees with equation (156) for k=l. To complete the transformation of the output to a new basis we write Ynew = M' y (195) and Ynew = ML'Hx. (196) By defining Hnew = MH'H and using Hnew to obtain the steady-state solution to equation (176), we find -0.7126 Hnew = -1.2216 0.8638 -0.5038 0.3563 0.6108 0.0 0.0 (197) (198) E = diag ( 10.0 5.0 0.0 0.0 ) . From equations (193) and (194) we compute HnewEH'new = 8.8080 6.5283 6.5283 16.1920 (199) and note that, using equation (161), H = HnewvH'new" (200) We can now use E to compute the steady-state Kalman gain matrix and substitute fHnew for H in equation (174) to find the transfer function for the innovations model. Doing so we find that HK(z) = HA(z) (201) where HA(z) is given by equations (165)-(169). 47 Through this numerical example the following major results have been demonstrated: (1) The modeling of degenerate time series, (2) ARMA model identification for vector-valued time series and (3) The equivalence between the ARMA model found and the innovations model of the system. APPLICATION TO TARGET I'. lji', The Problem In most missile-target engagements involving very agile targets, the missile is by necessity steered or guided to the target wit, the aid of a tracking system. A target tracker located on the missile is called a seeker. In many surface-to-air defense systems, the tracker is located on the ground and missile and target inertial position are maintained by tne tracker. In either case, the function of the tracker is to piurov.de relative position information (angles, and in some cases, range and range rate) to the missile guidance system so steering commands can be issued to the control system. The control system then commands aerodyna- mic control surface deflections to maneuver the missile onto a collision course with the target. Many guidance schemes developed using modern optimization techniques rely on estimates of future target states to be effective [37]. This naturally leads to the utilization of a tracking filter. At present, tracking filters use target dynamic models which have been devised assuming a priori statistical knowledge of the target's behavior. These models work well on the average but show severe deficiencies in certain specific cases. The research presented here involves the use of sta- tistical modeling techniques developed in the preceding chapter to synthesize a target dynamic model. This idea is novel in that the model 48 49 can be made to adapt to changes in the target's trajectory by utilizing a time history of past measured kinematic variables. The measurements come, quite naturally, from the tracker. Engagement Geometry The definition of the coordinate frame in which the missile-target engagement occurs is a rather important aspect of the problem formulation. The coordinate frame chosen dictates the cinematic variables which describe the target's motion and also defines the frame of reference for the measurements made by the tracker. The target trajectory is described by a time-varying range vector in three-space. The range magnitude is defined relative to the origin of an inertial reference frame. Inertial, in this case, means neither translating nor rotating with respect to a point fixed on the earth's surface. Three mutually perpendicular unit vectors, i,j and k, which are colinear with the inertial x-axis, y-axis and z-axis, respectively, are used to establish the coordinate system. Henceforth, underlined quan- tities are vectors. The x-y plane is tangent to the earth's surface at the origin of the inertial frame. The z-axis is directed away from the earth's surface and completes the right-handed triad. Orientation of the range vector is determined by two angles: azimuth and elevation. Azimuth is the angle between the inertial x-axis and the projection of the range vector onto the x-y plane. Elevation is the angle between the range vector and the range vector projected onto the x-y plane. Define It = xti + Yti + ztk (202) as the target range vector in inertial space. Then we can write Rtxy = xt2 + Yt2 (203) at = tan-1 (Yt / Xt) (204) and Et = tan-1 (zt / Rtxy) (205) where Rtxy is the magnitude of the projection of It onto the inertial x-y plane, at is the azimuth angle and et is the elevation angle. These quantities are shown in figure 3. Target Trajectory Origin x \ i \-- ,xy _ Figure 3. Target trajectory 51 Trajectory Generation To illustrate the use of the modeling technique in the tracking problem, a maneuvering target trajectory was generated using elementary kinematic equations. For simplicity we assume a point-mass target and therefore model only the motion of the center-of-gravity. Aircraft flight can be categorized as (1) non-accelerating or (2) accelerating. The resulting trajectory in each case is quite different. In non-accelerating flight we obtain a trajectory which is a straight line in inertial space. If the target is accelerating, as in the case of a target performing evasive maneuvers, two effects are observed. The component of acceleration along the velocity vector increases the target speed (magnitude of velocity) and the component normal to the velocity vector rotates the velocity vector relative to the inertial frame, thus changing the flight path. The result is a circular arc in inertial space for constant speed targets. The turning rate is a function of speed and acceleration magnitude and can be calculated from a = at / Vt (206) where at is the magnitude of the acceleration component normal to the velocity vector, Vt is the target's speed and W is the turning rate of the velocity vector relative to the inertial frame of reference. At this point we assume that the target acceleration vector is normal to the velocity vector. This is a good assumption for aircraft targets and also simplifies the kinematics. 52 For this illustration, consider a trajectory composed of segments of non-accelerating and accelerating flight. Furthermore, assume that the non-accelerating portions are at constant altitude (z component of posi- tion in the inertial frame). The accelerating portion is in a maneuver plane defined by the velocity and acceleration vectors. In general the maneuver plane is skewed to the principal inertial planes. The flight segments are defined as non-accelerating 0.0 accelerating 10.04 non-accelerating 31.5 where t is time in seconds. Transition from one flight segment to the next occurs instantaneously. Target initial position and velocity are (Rt)o = ( 1299 i 750 + 2598 k ) m. (207) and (Yt)o = ( 150 i + 259.8 j ) m./sec. (208) At t= 10.0 seconds the aircraft executes a turn in a maneuver plane inclined 30 to the inertial x-y plane. The aircraft completes the 3600 turn in 21.5 seconds. This corresponds to an acceleration of 87.672 m./sec.2 (8.9 g's). The trajectory is shown in figures 4 through 7. Target Dynamic Model Synthesis (Off-line) A statistical model of the target trajectory presented in the pre- ceding section will be developed. We begin by noting that the measure- ments made by the tracker can be used to create a time history of target 53 inertial position. The transformation is relatively simple for trackers with fixed inertial position. However, for missile-borne seekers the transformation is more complicated since missile attitude and position relative to the inertial frame must be taken into account. These quan- tities are measured by an inertial navigation systems (INS) onboard the missile. For the present example let us assume that we have available a time series of target inertial position. The position vector is three- dimensional with components xt, Yt and zt. Let us further assume that we have the entire time series available at the outset. This will allow us S-- r-t () 0 < -1 -1 0 1 2 3 5 Inertial x-axis (km.) Figure 4. Target trajectory in the inertial x-y plane 54 to develop a dynamic model off-line. Of course, off-line modeling is not practical for real-time tracking but is useful when investigating model characteristics such as model order, error covariances and coefficient values. Later in this chapter an on-line modeling technique will be demonstrated. Finally, for this example we assume that the measurements are deterministic. This means that the errors in prediction are due to modeling errors and not measurement errors. The error covariance iden- tified will therefore approximate the process noise covariance. 5 F 4 C H 4-1 c 2 0 0 O ^ ----------- -- -- -- 0 10 20 30 40 50 time (seconds) Figure 5. Target position versus time (x-axis) 55 Since the target motion is planar we can describe the trajectory in a new basis in which one component of the position vector is constant. That is, one axis of the new coordinate frame will be normal to the maneuver plane. If we denote the position vector components in the new basis with a *, for example xt and let the z-axis of the new basis be normal to the maneuver plane,then, since (zt*)k is constant for all k>-, we can write (209) 2 1 n -11 0 20 30 time (seconds) Figure 6. Target position versus time (y-axis) I I I _I (1-z-l)zt*(z)=o. I 56 But (zt*)k can also be written as a linear combination of the position vector components in the original basis, zt*(z) = Vo'_.t(z). (210) Combining equations (209) and (210) yields (l-z-1)yVO'R(z)=0. (211) Therefore, we anticipate finding that the position data forms a dege- nerate time series and furthermore the degeneracy occurs in the first difference of the transformed data. Interpreted physically this means 5 .0- -P 0 P, b.l -P -P a) 0O 0 3 ----__ -------_ -------------- cd 0 0 10 20 30 4o 50 time (seconds) (+ a 0 0 10 20 30 40 50 time (seconds) Figure 7. Target position versus time (z-axis) 57 that the velocity component normal to the maneuver plane is zero. Equation (211) can be found using the ARMA modeling algorithm for dege- nerate time series if we assume we can precisely, or at least with insignificant error, identify the model for a constant. Let us consider the problem of modeling a constant scalar using the autocorrelation method with biased estimates of the autocorrelation sequence. We find that Co = K2 (212) CI = (N-1)K2/N (213) and l = -(N-1)/N (214) where K is the constant we wish to model. Furthermore, the error covariance matrix is given by H = (2N-1)K2/N2. (215) For the trajectory under consideration here K = 3 km (z component of position in the new basis) and N = 161 for a sample rate of 4 Hz. Using these values in equations (214) and (215) yields YT = -0.99379 (216) and H = 1.11454 x 105 m2. (217) Ideally we should get Y- = -1.0 and H = 0. Note that the expressions for yl and I given by equations (214) and (215) converge to the true values 58 as N oo However, the error associated with using a finite length data sequence is excessive. Box and Jenkins [261 suggest a technique to handle nonstationary situations such as this. We place a pole at z=l by differencing the ori- ginal data sequence and then model the difference data in the usual fashion. Thus, we obtain the new sequence, A(Rt)k = (t)k (Rtk-1 k>0. (218) The model of the original data can easily be constructed from the model of the difference data. The time series now consists of 160 vectors sampled at 4 Hz. The biased sample autocorrelation matrices for the zeroth through nine- teenth lag were computed. We find that CO is singular. This means that a linear combination of the components of A(Rt)k is zero. Since the time series used to compute CO is the first difference of the original time series, we can write (l-z-l)V'Rit(z)=0 (219) where V0 is the eigenvector associated with the zero eigenvalue of CO. Comparing equations (211) and (219) we see that we have found the deter- ministic relationship that we expected to discover earlier. The components of V_ are the direction cosines of the unit vector normal to the maneuver plane. This normal unit vector uniquely defines the plane of the maneuver in the original coordinate frame. The fact that we can determine the maneuver plane by processing position measure- ments in the fashion described here is significant. 59 Having found the deterministic relationship in the data, we can proceed with the modeling process by transforming the autocorrelation sequence to the new basis using Mi. For this example we find Vo'= (0.4330 -0.2500 0.8660) =-0.5000 -0.8660 o.ooo0 M .75' =-0.5 0.7500 -0.4330 -0.5000 (220) (221) When the transformed autocorrelation sequence is used in the Levinson algorithm, we find that the reduced dimensional time series is also degenerate. The eigenvalues of HI, corresponding to an AR(1) model, are eo = 4.9907 el = 69.9572 . (222) (223) However, the eigenvalues of 112 are e0 = 0.0186 el = 69.8288 . The eigenvector associated with e0 is Vo'= (0.0000 1.0000) and the deterministic relationship is found from (224) (225) (226) = 0 1O' ( I + ylz- + y2z-2 ) and (227) -1.97598 1 = _-.0806 Y 0.9812 Y2 = 0.1535 1 Axt*(z) -0.0003 ] -0.9931 -0.00021 0.0068 (228) (229) Ayt*(z) P' = M I' A.(z) . (230) Substituting equations (226), (228) and (229) into equation (227) yields the second deterministic relationship, ( 1 1.97598 z-1 + 0.9812 z-2 ) Ayt*(z) = 0. (231) The problem has now been reduced to one of modeling a scalar. The new transformation matrix is M = ( 1.0000 0.0000 )' (232) The remaining variable to model is (Axt*)k. Performing the transfor- mation on the autocorrelation sequence -- which in this case amounts to no rxre than picking out the (1,1) element -- we find ( 1 0.99407 z-1) Axt*(z) = w(z) (233) n = E[wk21 = 78.0 m.2 where (234) 61 The AR(1) coefficient in equation (233) is approximately -1.0 indi- cating that the time series is possibly nonstationary. Therefore, it may be prudent to again difference the data and model the resulting time series. In this case the data sequence to difference is (Axt*)k. The new sequence is A2(xt*)k = (Axt*)k (Axt*)k-l k>l. (235) We now find that a deterministic AR(2) model fits the A2(xt*)k sequence. The model found is ( 1 1.97595z-1 + 0.9812z-2) A2xt*(z) = 0 (236) This is the final equation we need to complete the model. Using equations (221) and (230) we can write the following discrete- time difference equations: (Axt )k = -0.5 (Axt)k -0.866 (Ayt)k (237) and (Ayt*)k = 0.75 (Axt)k -0.433 (Ayt)k -0.5 (Azt)k (238) From equation (209) we have (Azt*)k = 0. (239) We can solve equations (237) through (239) simultaneously to obtain (Axt)k = 0.75 (Ayt )k -0.500 (Axt*)k, (240) (Ayt)k = -0.433 (Ay*t)k -0.866 (Axt*)k (241) and (Azt)k = -0.500 (Ayt*)k. (242) 62 From equations (231) and (236) we have (Yt*)k = 1.97598 (Ayt*)k-1 -0.9812 (Ayt*)k2 (243) and (Ab*)k = 2.97595 (Axt*)k-1 2.95715 (Axt*)k2 + .9812 (Axt*)k-3.(244) Estimates of (Axt)k, (Ayt)k and (Azt)k are obtained from equations (240) through (242) by placing hats on the variables on the right hand side of these equations. Our objective is to model xt, Yt and zt but the above equations involve Axt, Ayt and Azt. Using equation (218) we can write A A (.t)k t)k-1 + (At)k k>O (245) A since (ARt)k is conditioned on (Rt)k-l' Combining equations (237) through (245) we obtain 4 (_t)k= I Fi i=l 2.8555 FI = 0.6469 L-0.7410 -3.1467 F2 = -1.6087 1.1089 0.6469 3.6023 0.4278 -1.6087 -5.0040 -0.6402 (Rtt)k-i k>3 (246) -0.7410 0.4278 1.4940 1.1089 -0.6402 -0.7393 (247) (248) where 1.5365 1.3867 -0.3680 F3 = 1.3867 3.1375 0.2124 (249) -0.3680 0.2124 0.2453 and -0.2453 -0.4249 0.0000 F4 = -0.4249 -0.7358 0.0000 (250) 0.0000 0.0000 0.0000 It should be emphasized that equation (246) is a deterministic equation. Therefore, for this particular example, we can predict (At)k from measurements of (Qt)k-l, ... (Rt)o with no error. Several points concerning this model should be mentioned. First, the trajectory modeled is very idealistic. We have assumed that the aircraft can accelerate instantaneously and that constant speed and acceleration are maintained in the turn. These assumptions lead to a perfectly cir- cular trajectory when the aircraft is accelerating. Second, we have assumed that the maneuver occurs in a plane. This is actually a rather good assumption for piloted aircraft executing a sustained high acce- leration maneuver. Also, deterministic measurements were used to build the model. This, of course, helps significantly in determining the pre- sence of degeneracy in the time series and the correct model order. Finally, the computations were done on a Cyber 176 a 60 bit wordlength digital computer. Computational precision is extremely high. With these facts clearly in mind, we should not be surprised that the model found is deterministic. 64 Target Dynamic Model Synthesis (On-line) In actual practice the measurements made by a radar tracking system are, of course, noisy. We assume that the measurements are corrupted by additive, white, zero-mean Gaussian noise. The quality of the measure- ments is a function of many factors such as transmitted/received power, pulse repetition frequency and the sizes of the range and angle cells, to mention only a few. Our goal here is not to model or analyze a par- ticular radar system but rather to illustrate a tracking concept. We shall therefore choose noise characteristics which approximately repre- sent state-of-the-art radar performance. The standard deviation of the noise in the range measurement is typically on the order of 10 meters. Angular measurement noise has a standard deviation on the order of a few tenths of a degree. For this example we choose or = 10 m. (251) and a, = aF= 5 mrad. 0.3* (252) where or is the range noise standard deviation and oa and ap are the azimuth and elevation noise standard deviations, respectively. These statistics are assumed to remain constant during the entire tracking process. The noisy measurements are presented in figures 8 through 10. In order to make the proposed tracking scheme practical for real- time use, we must first convert the measurements of range, azimuth and elevation to inertial position. To do this let us assume that the tracker is located at the origin of the inertial frame. This is by no means restrictive since a simple coordinate frame translation can be made to put the origin at the tracker. 65 Once we have the range and angle measurements we can obtain inertial position using the inverse transformation, xt = Rt cos(et)cos(at) (253) Yt = Rt cos(et)sin(at) (254) and zt = Rt sin(st). (255) These equations can easily be obtained from figure 3. Now that we have a time series of position vector components we can create a dynamic model of the trajectory in the same manner that we did 7 6 5 d3 2 1 0 0 10 20 30 0O 50 time (seconds) Figure 8. Range measurement versus time. 66 in the off-line case. The difference now is that the output sequence is corrupted with measurement noise -- not deterministic. Since the measurement noise is relatively small we might expect to observe the same degenerate time series phenomenon that we did in the deterministic case. Before proceeding any further with real-time model identification, let us model the entire noisy output time series to see if we can identify the same deterministic relationship that we found in the deterministic model. As before, we can check for linear dependence in the difference data by computing the eigenvalues of CO. Using noisy 1. 1.' 5 0 ,C _P s N < U.3 / 0.0 -0.5 -i.0 0 10 20 30 40 50 time (seconds) Figure 9. Azimuth angle measurement versus time. 67 measurements we know CO>0 and, therefore, no eigenvalue will be iden- tically zero. What we must do in this case is compare the relative magnitudes of the normalized eigenvalues of Co. The normalized eigen- values are defined as e0 = eO/e2 (256) el = el/e2 (257) where eo 1.5 1.0 *rd cd J 0.5.-- + + -4-- 20 30 time (seconds) Figure 10. Elevation angle measurement versus time. I I I r~ 68 If at least one of the normalized eigenvalues is significantly less than unity we conclude that one or more deterministic relationships exist in the data. Displayed in figure 11 are the normalized eigenvalues versus time for the noisy trajectory data. We see that prior to the initiation of the maneuver (t<10 seconds) that e0< that there are two directions in which the velocity components are zero. We readily understand this result since the trajectory is rectilinear for t<10 seconds. We also note from figure 11 that el increases rapidly for 10 1 .01 1 1 I 0 U) > 0. 1) to - 0. 0 6 4 0.0 0 20 30 time (seconds) Figure 11. Normalized eigenvalues of CO versus time. e 69 relatively small. This means that for t>10 seconds only one direction can be identified in which the velocity component is zero. This direc- tion is obviously normal to the maneuver plane. The direction cosines of the unit vector normal to the maneuver plane are the components of V0, the eigenvector associated with e0. Plots of the components of V0 are shown in figures 12 through 14. The true values shown in figures 12 through 14 are the deterministic values given by equation (220). 1. 0. 0 o >1 CMA 0 0 0. U) o u r- F- 0. 0 8 - 6 true value 4 2 0 2 -------------------- 20 30 time (seconds) Figure 12. First component of V0 versus time. u 70 To do the model identification and tracking in real-time we must update the sample autocorrelation sequence with the addition of each position measurement. The autocorrelation sequence is updated using the following recursion: Cj(k) = (l/k)[(k-l)Cj(k-1) + (ARt)k(ARt)'k-.jl, j=0,l,...,m where (ARt)k = (t)k (t)k-l k>O (260) and m is the number of lags required to model the time series. -1.0 -0.8 0 o >1 -0.6 0 -A tu av 0 u a -0.4 0 O C) true ----- value -0.2 0.0 0 10 20 30 40 50 time (seconds) Figure 13. Second component of Vo versus time. 71 Once we have the updated autocorrelation sequence we can compute eO, e1 and M. If the time series is degenerate we transform the autocorrela- tion sequence to the new basis using MI, thus reducing the dimension of the time series to be modeled. Performing the transformation we obtain Cj = ML'Cj M j=0,l,...,m-k-l. (261) The model found by using the transformed autocorrelation sequence will be valid for the time series written in the new basis. We must now transform the AR model to the original basis because the measurements 1.0 true value o0.8 -----f---- ------ --- 0.8 o c>i 0 S0.6 0.2 0.2 L, 0.0 0 10 20 30 40 50 time (seconds) Figure 14. Third component of VO versus time. 72 are in this coordinate frame. The transformation matrix relating the two coordinate frames is M. Therefore we can write (A*it)k = M' (Rt)k k>O (262) and since M is orthogonal (At)k = M (ARt*)k k>0. (263) By using the Cj sequence in the modeling process, we obtain a model for the time series (Axt* Ayt*)'k = M,' (At)k k>0. (264) The finite order AR(n) model found can be written Axt n A = TYi M' (ARt)k-i k>l. (265) AYt k i=l Using equation (263) and the fact that (At*)k=O (k>O) we can write n I Ti M,' (A-Rt)k-i i=l (At)k = M k>n. (266) 0 Estimates of (Rt)k are obtained from equation (245). Maneuvering Target Tracking Algorithm Using the equations developed in the preceding sections we can for- mulate the following target tracking algorithm: 1. Measure azimuth, elevation and range at discrete times k=0, k=l and k=2. 73 2. Convert measurements to (Rt).k Use the inverse transfor- mation given by equations (253) through (255). 3. Compute (ARt)k = (.t)k (t)k-l' 4. Update Cj(k). Use equation (259). 5. Compute eg and el using equations (256) and (257) and com- pute M. Since we have defined the zt -axis as being normal to the maneuver plane we must arrange the columns of M so that M = [ V 1 1ol.0 I I 6. If e0< 7. If el< 8. Compute C (k) = M 'Cj(k)M. 9. Compute Yi (i=l,2,...,n) using the Levinson algorithm and the Cj(k) sequence. Alternatively we can find the AR(n) model coefficients using the Burg algorithm and the data sequence (ARt*)k = M_'(ARt)k. A A 10. Compute (ARt)k and (Rt)k+1. Use equations (266) and (245). 11. Increment k. 12. Measure azimuth, elevation and range at time k. 13. Go to step 2. Numerical Results and Predictor Performance Using the algorithm just presented with the noisy measurements displayed in figures 8 through 10, we can adaptively change the tracker model and thereby more accurately predict target inertial position. Models computed for a few discrete-time points along the trajectory 74 and the prediction errors resulting from use of the tracker are pre- sented in this section. When the autocorrelation sequence found after each measurement update is used in the Levinson algorithm, we find that an AR(2) model fits the data well. This determination is made by using the ABMA model identification algorithm presented earlier. Therefore, we use n=2 in equation (266) and in step 9 of the tracking algorithm. Referring again to figure 11, we note that for k<40 (t<10 seconds) e0<<1 and el< the data and the remaining stochastic variable to model is a scalar. From the tracking algorithm we find for k=40 Yl = 0.60829 (267) Y2 = 0.34687 (268) M = y2'= (-0.50420 0.86358 -0.00054) (269) and -0.14210 +0.85181 [V11 IV = +0.08235 -0.49743 (270) +0.98642 +0.16423 For k>50 only e0< dimensional vector. The model parameters for k=80 are +0.39993 -0.02337] 8i = 0(271) -0.04890 +0.23703_ +0.56281 -0.005351 Y2 = .1186 +.63050 +0.11486 +0.63050 S-0.50092 M' = +0.72606 Vo' = ( +0.47109 0.5 O -o 1-) A 0.0 r-0.5 0- -0.5 -1.o -0.86524 -0.43331 +0.02102 -0.53393 -0.25219 +0.84527 ). 10 20 30 40 time (seconds) Figure 15. Diagonal elements of yl. (272) (273) (274) For k=120 tre parameters found are +0.40771 Y1 = -0 .01610 2 +0.55403 Y2 =9444 +0.09444 Y2(1,1) Y2(1,1) -0.47800 +0.74339 0.0k i -1.0 0 -0.05620 I, +0.27461_ -0.02270 +0.63815] -0.87836 -0.40657 +0.00334 -0.53111 20 30 time (seconds) Figure 16. Diagonal elements of Y2. (275) (276) (277) i and V' = ( 0.46786 -0.25138 +0.84730 ). Finally, for k=160 the parameters are S= [+0.39404 Y1 = _-0.00785 S+0.56883 +0.06489 -0.50719 - M =+ +_0.71938 - -0.05639 +0.25850j +0.02654 +0.64127] ).86123 0.44380 +0.00325 -0.53435 (282) v_' = ( +0.47461 -0.24765 +0.84464 ). We see from figures 15 and 16 and from equations (271) through (282) that the model changes very little in the last 20 to 25 seconds of the trajectory. The transient phase which starts with maneuver initiation is relatively short in duration; lasting approximately 5 seconds. At a 4 Hz sample rate this represents 20 samples. Model parameters are quite dyna- mic during the transient period. Plotted in figures 17 through 19 are the inertial position predic- tion errors versus time. The errors are found from (- A (it)k = (Rt)k (t)k (283) where (Rft)k is the position vector derived from the angle and range (278) (279) (280) (281) and 78 measurements. The first 10 measurements were used only to compute the autocorrelations to get the model identification started. No tracking was done during this period. The tracking algorithm was started at k=10 (t=2.5 seconds). We observe from the error plots that the tracker is working well until maneuver onset. At that time the errors begin to increase and there is a noticable bias in the error sequence, par- ticularly in the x-axis and y-axis. As the tracker adapts the model parameters in response to the maneuver, the errors once again decrease and the mean is once again near zero. 100 50 0 -50 0 -50 -100 ----- 0 10 20 30 O0 50 time (seconds) Figure 17. Error in predicted position versus time (x-axis) 79 The error sequence bias which began at maneuver initiation results from the same phenomenon that Bullock and Sangsuk-Iam [71 used to detect the occurrence of maneuvers. The non-maneuvering target model used to do the tracking prior to maneuver onset is no longer valid after the maneuver starts and we observe a bias in the innovations sequence. The difference is, of course, that here we are not applying any statistical tests to determine if a maneuver has occurred. The important point to note here is that the tracker adjusted the model parameters in response to an unanticipated abrupt maneuver. There 100 50 0 S_ rl S-50 -100 0 10 20 30 40 50 time (seconds) Figure 18. Error in predicted position versus time (y-axis). 80 is no explicit maneuver detection scheme required for the algorithm. However, by using information already needed to do the tracking (for example, the noticable increase in el and the accompanying change in VO) or by monitoring the mean of the error sequence, we should be able to detect maneuvers with a reasonable degree of confidence. Of course, the objective of this research was not maneuver detection but the close relationship between detecting maneuvers and tracking highly agile eva- sive targets makes the maneuver detection function an attractive poten- tial by-product. 100 50 0 00 -I cd N -50 -100 0 10 20 30 40 50 time (seconds) Figure 19. Error in predicted position versus time (z-axis). 81 It is reasonable to assume that maneuver detection will enhance tracking performance. This is because we can reinitialize the tracker and include only the measurements made subsequent to the detected maneuver. We therefore remove the premaneuver measurements and totally delete the influence of these data from the model. Implementation is rather straightforward. An auxiliary, constant length data file of posi- tion vectors is maintained to perform the reinitialization. Prior to maneuver detection the oldest position vector in the auxiliary file is deleted each time a new measurement is added. This is typically referred to as the "sliding window" concept. The length of the window depends on the number of points required to detect the maneuver after the maneuver is actually initiated. Subsequent to maneuver detection the tracking algorithm is reinitialized using only the auxiliary file and then con- tinues to operate as described earlier. We can use the maneuver detection/reinitialization idea on the example trajectory considered here and determine if the tracker perfor- mance is actually improved. To do so let us assume that 10 postmaneuver measurements are required to detect the maneuver. The auxiliary file is therefore 10 measurements long. Since the maneuver began at k=40, the auxiliary file contains data for k=40 to k=49. The performance of the two methods is compared using the sample mean and sample standard deviation of the position errors in each of the three inertial axes. For the non-maneuver detection case (refer to figures 17 through 19), we find for k=50 to k=160 (time= 12.5 seconds to 40.0 seconds) [ Px Py VUz = -1.1621 1.0116 0.5674 1 m. (284) and Sxx yy zz = [ 20.724 18.092 20.358 1 m. (285) where P and a are the sample mean and sample standard deviation, respectively. For the maneuver detection/reinitialization case we find [ Px Py 1z 1 = [ -3.5220 1.4499 2.4523 ] m. (286) and Soxx uyy Uzz 1 = [ 24.200 20.632 22.700 1 m. (287) The performance of the non-maneuver detection tracker is slightly better than the maneuver detection/reinitialization tracker. Tais result is counterintuitive but can be attributed to the small number of samples used to reinitialize the tracker. Ten postmaneuver points are not enough data to obtain a good sample autocorrelation estimate. All of the dif- ferences in the models and in the error sequences for the two cases were in the first few seconds following maneuver detection. By the end of the trajectory the model parameters (AR coefficients and M) and prediction errors for both cases were essentially indistinguishable. This is, of course, because the first 40 measurements (deleted in the maneuver detection/reinitialization case) became an ever diminishing percentage of the total number of points. The effect of discarding the premaneuver points soon became negligible. There is apparently a trade-off which can be made between the length of the sliding window and tracker perfor- mance. Too many premaneuver points in the window will lead to a sluggish tracker and too few will not be enough to provide a good estimate of the autocorrelation sequence. SUMMARY AND RECOMMENDATIONS Summary In this research several theoretical issues have been investigated. A major result is the development of an algorithm for finding the coef- ficients and order of the minimum order ARMA model of a multivariable system from the output autocorrelation sequence. The model order is found by examining the ranks of submatrices within a block Toeplitz of output autocorrelation matrices. Once the order is known, the AR coef- ficients are determined in the usual way by solving the modified, or extended, Yule-Walker equations. The MA coefficients are then found from a set of linear equations involving the coefficients of the AR model equivalent to the ARMA model and the coefficients of the AR portion of the ARMA model. The algorithm is a new result. The ARMA model identified using the new algorithm is a linear mini- mum variance one-step-ahead predictor. This result follows from applica- tion of the projection theorem. The ARMA model found is equivalent to the Kalman filter innovations representation of the system modeled. This fact was discussed and demonstrated through a numerical example. The equivalence is significant because all equations necessary to realize the system using the ARMA model are linear and can therefore be solved analytically. There is no Riccati-type equation solution required in the ARMA model but there is in the innovations representation. 83 84 It has been shown that degenerate vector-valued time series can be modeled by identifying deterministic relationships in the data, thereby reducing the dimension of the stochastic model. This results in system models with fewer white noise inputs than outputs. The deterministic relationships are found by transforming the data to the null space of the error covariance matrix. The AR model which fits the data at the lag the degeneracy is encountered is used to find the deterministic relationships. Not only does this technique identify the presence of deterministic processes in the data, but also provides the equations describing the deterministic relationships. Other solutions to the dege- nerate time series modeling problem have been reported in the literature but none suggest the identification of deterministic relationships and dimension reduction of the stochastic model as a solution. It has been shown that the theory is applicable to the specific problem of tracking a highly maneuverable aircraft target. Measurement vectors consisting of range, azimuth angle and elevation angle, typi- cally available in radar tracking systems, are used to construct a time series of target position vectors in an inertial reference frame. The position data are nonstationary and the first difference must be formed. The difference data is a time series of target inertial velocity com- ponents multiplied by the measurement update time. This time series is degenerate. This is a particularly interesting result since it has a physical interpretation. It means that a linear combination of the com- ponents of the velocity vector is zero. Therefore, along some direction in inertial space, the velocity component is zero. This direction is normal to the plane which contains all the notion, that is, the maneuver plane. The direction cosines of the vector normal to the maneuver plane 85 are found from the eigenvectors and eigenvalues of the time series autocorrelation. This is a new and significant development in target tracking technology. The ARMA modeling technique can also be used for adaptive imdeling and tracking. This is accomplished by updating the model after each new measurement is made. The model thus contains the influence of the latest measurement. In this research an algorithm has been developed which adaptively models the dynamics of a maneuvering target in response to abrupt, unanticipated motion. Traditional tracking filters are usually constructed under the assumption that the target dynamic model characteristics are constant. Severe performance degradations can occur if insufficient bandwidth is provided. The adaptive method developed in this work circumvents this problem by changing the bandwidth when required. The bandwidth variation is a natural result of the modeling technique and is done automatically without the aid of any special detection techniques. However, the adaptive nature of the tracker also makes it useful as a device to detect maneuvers. Drastic changes in target dynamic behavior can be detected by monitoring the prediction residuals. When residual biases are detected it is assumed that the model being used for prediction is no longer valid due to some unan- ticipated change in dynamics. Detection is not a requisite for adaptive tracking -- the tracker adapts to maneuvers automatically -- but the maneuver detection function can perhaps be used to enhance tracking performance. This is also a new development in the tracking area. Recommendations for Further Research It is recommended that further research be done to refine the adap- tive tracking procedure. The sliding window concept should be investi- gated further to determine the minimum amount of data required for accurate identification and adequate response. Concepts along the lines of tracker reinitialization should be pursued in connection with adap- tive modeling to enhance performance. The adaptive tracking method pro- posed here should be considered as a maneuver detection device and com- bined with a rigorous statistical test to improve maneuver detection. REFERENCES 1. Robert A. Singer, "Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-6, no. 4, pp.473-482, July 1970. 2. John M. Fitts, "Aided Tracking as Applied to High Accuracy Pointing Systems," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-9, no. 3, pp. 350-368, May 1973. 3. R.J. McAuley and E. Denlinger, "A Decision-Directed Adaptive Tracker," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-9, no. 2, pp. 229-236, March 1973. 4. Norman H. Gholson and Richard L. Moose, "Maneuvering Target Tracking Using Adaptive State Estimation," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-13, no. 3, pp. 310-317, May 1977. 5. R.L. Moose, H.F. Vanlandingham and D.H. McCabe, "Modeling and Estimation for Tracking Maneuvering Targets," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-15, no. 3, pp. 448-456, May 1979. 6. J.D. Kendrick, P.S. Maybeck and J.G. Reid, "Estimation of Aircraft Target Motion Using Orientation Measurements," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-17, no. 2, pp. 254-259, March 1981. 7. Thomas E. Bullock and S. Sangsuk-Iam, "Maneuver Detection and Tracking with a Nonlinear Target Model," Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, Nevada, December 1984. 8. D.G. Hull, P.C. Kite and J.L. Speyer, "New Target Models for Homing Missile Guidance," Proceedings of the AIAA Guidance and Control Conference, Gatlinburg, Tennessee, August 1983. 9. P.L. Vergez and R.K. Liefer, "Target Acceleration Modeling for Tactical Missile Guidance," Journal of Guidance, Control and Dynamics, vol. 7, pp. 315-321, May-June 1984. 10. C.F. Lin and Marc W. Shafroth, "A Comparative Evaluation of Some Maneuvering Target Tracking Algorithms," Proceedings of the AIAA Guidance and Control Conference, Gatlinburg, Tennessee, August 1983. 11. John B. Pearson and Edwin B. Stear, "Kalman Filter Applications In Airborne Radar Tracking," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-10, no. 3, pp. 319-329, May 1974. 12. K. Spingarn and H.L. Weidemann, "Linear Regression Filtering and Prediction for Tracking Maneuvering Aircraft Targets," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-8, no. 6, pp. 800-810, November 1972. 13. James S. Thorp, "Optimal Tracking of Maneuvering Targets," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-9, no. 4, pp. 512-518, July 1973. 14. Y.T. Chan, J.B. Plant and J.R.T. Bottomley, "A Kalman Tracker with a Simple Input Estimator," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-18, no. 2, pp. 235-240, March 1982. 15. Y.T. Chan, A.G.C. Hu and J.B. Plant, "A Kalman Filter Based Tracking Scheme with Input Estimation," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-15, no. 2, pp. 237-244, March 1979. 16. Peter S. Maybeck, Stochastic Models, Estimation and Control, Vol. 1, Academic Press, Orlando, Florida, 1979. 17. D. Graupe, D.J. Krause and J.B. Moore, "Identification of Autoregressive Moving-Average Parameters in Time Series," IEEE Transactions on Automatic Control, vol. AC-20, no. 1, pp. 104-107, February 1975. 18. Joseph C. Chow, "On Estimating the Orders of an Autoregressive Moving-Average Process with Uncertain Observations," IEEE Transactions on Automatic Control, vol. AC-17, no. 5, pp. 707-709, October 1972. 19. Will Gersch, "Estimation of the Autoregressive Parameters of a Mixed Autoregressive Moving-Average Time Series," IEEE Transactions on Automatic Control, vol. AC-15, no. 5, pp. 583-588, October 1970. 20. M.B. Priestley, Spectral Analysis and Time Series, Academic Press, London, 1981. 21. E.J. Hannan, "The Identification of Vector Mixed Autoregressive Moving-Average Systems," Biometrika, 56, pp. 223-225, 1969. 22. John Makhoul, "Linear Prediction: A Tutorial Review," Proceedings of the IEEE, vol. 63, no. 4, pp. 561-580, April 1975. 23. Steven M. Kay and Stanley L. Marple, "Spectrum Analysis A Modern Perspective," Proceedings of the IEEE, vol. 69, pp. 1380-1419, November 1981. 24. Martin Morf, Augusto Vieira, Daniel Lee and Thomas Kailath, "Recursive Multichannel Maximum Entropy Spectral Estimation," IEEE Transactions on Geoscience Electronics, vol. GE-16, pp. 85-94, April 1978. 25. Raman K. Mehra, "On-line Identification of Linear Dynamic Systems with Applications to Kalman Filtering," IEEE Transactions on Automatic Control, vol. AC-16, no. 1, pp. 12-21, February 1971. 26. G.E.P. Box and G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1976. 27. B.D.O. Anderson and J.B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, New Jersey, 1979. 28. Ralph A. Wiggins and Enders A. Robinson, "Recursive Solution to the Multichannel Filtering Problem," Journal of Geophysical Research, vol. 70, 1965. 29. Norbert Levinson, "The Weiner rms (root-mean-square) Error Criterion in Filter Design and Prediction," Journal of Mathematical Physics, vol. 25, pp. 261-278, January 1947. 30. Yujiro Inouye, "Autoregressive Model Fitting and Levinson Algorithm in the Multichannel Degenerate Case," IEEE Transactions on Automatic Control, vol. AC-28, no. 1, pp. 94-95, January 1983. 31. Otto. N. Strand, "Multichannel Complex Maximum Entropy (Autoregressive) Spectral Analysis," IEEE Transactions on Automatic Control, vol. AC-22, no. 4, pp. 634-640, August 1977. 32. John Makhoul, "Stable and Efficient Lattice Methods for Linear Prediction," IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-25, no. 5, pp. 423-428, October 1977. 33. Tad J. Ulrych and Thomas N. Bishop, "Maximum Entropy Spectral Analysis and Autoregressive Decomposition," in Modern Spectrum Analysis, edited by Donald G. Childers, pp. 54-71, IEEE Press, New York, 1978. 34. M.D. Srinath and M.M. Viswanathan, "Sequential Algorithm for Identification of Parameters of an Autoregressive Process," IEEE Transactions on Automatic Control, vol. AC-20, no. 4, pp. 542-546, August 1975. 35. John P. Burg, "Maximum Entropy Spectral Analysis," in Modern Spectrum Analysis, edited by Donald G. Childers, pp. 34-41, IEEE Press, New York, 1978. 36. John P. Burg, "Maximum Entropy Spectral Analysis," Ph.D. Dissertation, Department of Geophysics, Stanford University, Stanford, California, May 1975. 37. G.J. Nazaroff, "An Optimal Terminal Guidance Law," IEEE Transactions on Automatic Control, vol. AC-21, no. 3, pp. 365-377, June 1976. BIOGRAPHICAL SKETCH Norman 0. Speakman, son of Paul O. Speakman and Norma D. (Pittman) Speakman, was born in Cullman, Alabama, on February 10, 1949. He attended the public schools of Birmingham, Alabama, and graduated from West End High School in 1967. He entered the University of Alabama in Birmingham in September, 1967, where he completed one year of pre-engineering. In September, 1968, he transferred to Auburn University where he participated in the Cooperative Education Program and was employed by Arnold Research Organization in Tullahoma, Tennessee. He earned the degrees of Bachelor of Aerospace Engineering and Master of Science in Aerospace Engineering in June, 1972, and June, 1973, respectively. In June, 1973, he was commissioned a second lieutenant in the U.S. Air Force. He served a four year tour of active duty at Eglin Air Force Base, Florida, from 1973 to 1977. Upon leaving the Air Force, he accepted an aerospace engineering position at the Air Force Armament Laboratory. In August, 1981, he entered the University of Florida and began doc- toral work in electrical engineering. After completing residency requirements in 1982, he returned to Eglin Air Force Base. He is currently chief of the Advanced Guidance Concepts Section at the Armament Laboratory. He married Donna Marie, daughter of Don and Marie (Deerman) Caldwell, in December, 1970. They have three children, Erik, Timothy and Amy. He is a member of Sigma Gamma Tau, Tau Beta Pi, Phi Kappa Phi and the American Institute of Aeronautics and Astronautics. 90 I certify that I have read this study and that in my opinion it con- forms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas E. Bullock, Chairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it con- forms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles V. Shaffer Professor of Electrical Engineering I certify that I have read this study and that in my opinion it con- forms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. P aj LJ2A- G seppe esile Professor of Electrical Engineering I certify that I have read this study and that in my opinion it con- forms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Edward W. Kamen Professor of Electrical Engineering I certify that I have read this study and that in my opinion it con- forms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. P.V. Rao Professor of Statistics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1985 Dean, College of Engineering Dean, Graduate School Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Speakman, Norman TITLE: Autoregressive Moving-Average (ARMA) model Identification for Degenerate Time PUBLICATION DATE: 1985 I, Norman Owen Speakman as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-based versions as appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder Norman Owen Speakman Printed or Typed Name of Copyright Holder/Licensee Personal InformationBlurred 23 April 2008 Date of Signature |

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degenerate time series with application to maneuvering target trackingdc:creator Speakman, Norman Owendc:publisher Norman Owen Speakmandc:date 1985dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082439&v=0000114281429 (oclc)000863402 (alephbibnum)dc:source University of Floridadc:language English |