Abstract:
This invention relates to state estimation after processing time-delayed measurements with unknown biases that may vary arbitrarily in time within known physical bounds. These biased measurements are obtained from systems characterized by state variables and by multidimensional parameters, for which the latter are also unknown and may vary arbitrarily in time within known physical bounds. If a measurement is time-late, apply the measurements to an out-of-sequence filter using a mean square optimization criterion that accounts for all sources of uncertainty and delay time, to produce estimates of the true states of the system. If the measurement is not time-late, apply the measurements to an in-sequence filter using a mean square optimization criterion that accounts for all sources of uncertainty to produce estimates of the true states of the system. The estimates are applied to one of (a) making a decision, (b) operating a control system, and (c) controlling a process.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This patent application is related to copending applications “Reduced State Estimator for Systems with Physically Bounded Parameters” filed on Mar. 16, 2005 in the names of P. Mookerjee and F. Reifler, “REDUCED STATE ESTIMATION WITH MULTISENSOR FUSION AND OUT-OF-SEQUENCE MEASUREMENTS”, filed on Mar. 30, 2005, in the names of P. Mookerjee and F. Reifler, and “REDUCED STATE ESTIMATION WITH BIASED MEASUREMENTS”, filed on Apr. 22, 2005, in the names of P. Mookerjee and F. Reifler. 
   FIELD OF THE INVENTION 
   This invention relates generally to state estimation after processing measurements having unknown biases that may vary arbitrarily in time within known physical bounds and arriving with time delays from multiple sensors. These measurements are obtained from systems characterized by state variables and by multidimensional parameters, for which the latter are also unknown and may vary arbitrarily in time within known physical bounds. In a particular aspect, the invention relates to the tracking of moving targets by multiple sensors, with different measurement bias for each sensor and feeding these measurements into a central processing site, possibly with different time delays. The invention provides a computationally efficient recursive algorithm for optimally estimating the state of a system, using the criterion of minimizing the mean-square total error. 
   BACKGROUND OF THE INVENTION 
     FIG. 1  represents a simplified target tracking system  10 . System  10  tracks a target, illustrated as being an aircraft  12 , by the use of multiple radar systems  14 ,  16 . Radar system  14  includes a radar antenna  14   a , which transmits and receives radar signals illustrated by “lightning bolt” symbols  18 . Portions of the transmitted signals  18  are reflected by target  12  and return to the radar antenna  14   a . The returned signals allow the generation of measurements at an output port  14   o  of radar system  14 . Radar system  16  includes a radar antenna  16   a , which transmits and receives radar signals illustrated by “lightning bolt” symbols  20 . Portions of the transmitted signals  20  are reflected by target  12  and return to the radar antenna  16   a . The returned signals allow the generation of measurements at an output port  16   o  of radar system  16 . These measurements include values of at least target position, possibly in the form of range and angles from the radar systems  14  and  16 . A possible scenario is that radar systems  14  and  16  have measurements which are corrupted by unknown random measurement noises, characterized by a covariance and unknown time-varying biases with known bounds. The biases may be, for example, the result of sensor misalignment. Furthermore, radar system  16  provides data with time latency, but which are more accurate (less measurement noise) than those provided by radar system  14 . The measurements are applied to a processing arrangement  22 , which determines from the measurements various target parameters, which may include course (direction of motion), speed, and target type. The estimated position of the target, and possibly other information, is provided to a utilization apparatus or user, illustrated in this case as being a radar display  24 . The operator (or possibly automated decision-making equipment) can make decisions as to actions to be taken in response to the displayed information. It should be understood that the radar tracking system  10  of  FIG. 1  is only one embodiment of a general class of estimation systems for systems with distributed sensors such as nuclear, chemical, or manufacturing factories or facilities, control processes subject to external parameter changes, space station subject to vibrations, automobile subject to weather conditions, and the like. 
   State-of-the-art tracking systems utilize measurements fed with time delays from multiple sensors to a processing site. These sensors may have different measuring accuracies (i.e., random errors) and unknown measurement biases that may be time-varying within physical bounds. Furthermore, these sensors may be geographically dispersed over a region of interest. Availability of reliable high bandwidth communication media allows such a topology of distributed multiple sensors for real-time processing of the measurements. Therefore, latent measurements from a remote sensor could arrive after state estimation at the current time is complete. 
   In spite of today&#39;s high bandwidth and fast switching communication network, physical distances, path diversity and relays may result in different delays from various sensors to the processing site. Let a sensor S 1  measure a tracked object at time t 1  and a sensor S 2  measure that same object at time t 2  where t 2 &gt;t 1 . It is possible that the measurement from sensor S 1  may arrive many sampling intervals after the measurement from sensor S 2  has already been processed. A simple decision methodology is to throw out the late-arriving measurement from sensor S 1 , and not process it at all. However, if sensor S 1  is the more accurate sensor, this methodology does not make good use of the available sensor resources. 
   A difficulty is that accounting for measurements received out of sequence, as frequently happens in situations of multiple sensor tracking with variable communication delays between sensors, greatly complicates the design of a Kalman filter, particularly when more than one subsequent measurement is processed before an out-of-sequence measurement is received. A solution to this problem is discussed in a pending patent application entitled “REDUCED STATE ESTIMATION WITH MULTISENSOR FUSION AND OUT-OF-SEQUENCE MEASUREMENTS”, filed on Mar. 30, 2005, in the names of P. Mookerjee and F. Reifler. 
   Consider the problem of tracking an airplane whose trajectory in three dimensions is an arbitrary curve with bounded instantaneous turn rate and tangential acceleration. The parameters of this tracking problem are the turn rate ω (which can be related to the curvature of the trajectory) and the tangential acceleration α. Sensors, such as multiple radars, observe the position of this airplane. Each sensor is subject to alignment errors which cause it to be rotated by an unknown amount from its nominal alignment, possibly as a consequence of imperfect mechanical mounting, flexure of the array structure due to temperature effects etc. These small or infinitesimal rotations constitute a vector b. The parameters, ω, α, and b are neither exclusively constant nor strictly white noise stochastic processes, but vary arbitrarily in time within physical bounds. 
   This problem belongs to a more general problem of estimating the state of a system using biased measurements. A solution to this problem is discussed in a pending patent application entitled “REDUCED STATE ESTIMATION WITH BIASED MEASUREMENTS”, filed on Apr. 22, 2005, in the names of P. Mookerjee and F. Reifler. 
   The method of the prior art is to estimate the states of a system when measurements are either out-of-sequence or biased. Improved or alternative estimators are desired for coping with biased measurements which are also delayed. 
   SUMMARY OF THE INVENTION 
   A method according to the invention is for recursively estimating the state of a system having multidimensional parameters λ in addition to state variables x(k) at time t k  for k=0, 1, 2, . . . , where the parameters λ are unknown, arbitrarily time-varying, but bounded, and driven by the input function u(x(k), λ), which may be nonlinear, and expressed by the state equation
 
 x ( k+ 1)=Φ x ( k )+Γ u ( x ( k ),λ)  (1)
 
where Φ, Γ are system matrices dependent on the discrete time interval T=t k+1 −t k . The method comprises the steps of:
     (a) Measuring aspects of the state of the system to produce initial measurements expressed by the measurement equation
 
 z ( k )= Hx ( k )+ Jb+n ( k )  (2)
    for 1≦k≦k 0 , where, if no measurements are used in the initialization of the filter, k 0 =0. Variable b is an unknown, arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and where n(k) is the measurement noise with covariance N and measurement matrix H at time t k ;   (b) Initializing state estimates {circumflex over (x)}(k 0 |k 0 ) and matrices M(k 0 |k 0 ), D(k 0 |k 0 ), E(k 0 |k 0 ) associated with the initial state covariance and using a priori information and the initial measurements (D(k 0 |k 0 )=0 if the initial state estimates do not depend on the parameters λ; E(k 0 |k 0 )=0 if there are no initial measurements). In general,
       vector {circumflex over (x)}(t|k) is defined as the estimate of the state of the system at time t after processing k measurements z(i) for 1≦i≦k;   vector {circumflex over (x)}(t j |k) is denoted as {circumflex over (x)}(j|k) when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . ;   matrix M(t|k) is defined as the covariance of the state estimation errors at time t due only to the random errors in the k measurements z(i) for 1≦i≦k and a priori initial information that is independent of the parameter uncertainty and the measurement bias uncertainty. The matrix M(t j |k) is denoted as M(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . ;   matrix D(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the parameter errors, at time t (after processing k measurements z(i) for 1≦i≦k);   matrix D(t j |k) is denoted as D(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . ;   matrix E(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the sensors measurement bias, at time t (after processing k measurements z(i) for 1≦i≦k);   matrix E(t j |k) is denoted as E(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . .   
       (c) The time t k+1  of a new measurement and the time t when the filter was last updated are determined.   (d) The system transition matrices Φ and Γ are determined using the update interval T=t k+1 −t. Note that for the update interval T=t k+1 −t, the time t denotes the time when the filter was last updated, whereas t k+1  is the time of the new measurement, which can be prior to t due to delay. For this case T&lt;0. If there is no delay, then usually t=t k , the time of the previous measurement. For this case T≧0.   (e) The mean value  λ  of unknown but bounded parameters λ, and the input vector u({circumflex over (x)}(t|k),  λ ) are determined.   (f) Aspects of the state of the system are measured and expressed by the measurement equation
 
 z ( k )= Hx ( k )+ Jb+n ( k )  (3)
    where b is an unknown arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and where n(k) is the measurement noise with covariance N and measurement matrix H at time t k  for k=1, 2, 3, . . . .   (g) A determination is made as to whether the measurement is time-late by testing T&lt;0, If the measurement is time-late then
       1. Matrices F,G are determined by   
       

   
     
       
         
           
             
               
                 
                   
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           2. A parameter matrix Λ is generated, representing physical bounds on those parameters λ that are not state variables of the system. 
           3. The state estimates {circumflex over (x)}(t|k) and matrices M(t|k), D(t|k), E(t|k) are extrapolated to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), E(k+1|k) as in
 
{circumflex over ( x )}( k+ 1 |k )=Φ{circumflex over ( x )}( t|k )+Γ u ({circumflex over ( x )}( t|k ),  λ )  (6)
 
 M ( k+ 1 |k )= FM ( t|k ) F′   (7)
 
 D ( k+ 1 |k )= FD ( t|k )+ G   (8)
 
 E ( k+ 1 |k )= FE ( t|k )  (9)
 
            and P(k+1|k) is calculated, as in
 
 P ( k+ 1 |k )= M ( k+ 1 |k )+ D ( k+ 1 |k )Λ D ( k+ 1 |k )  (10)
 
           4. The covariance of the residual Q is determined, as in
 
 V=HE ( k+ 1 |k )+ J   (11)
 
 Q=HP ( k+ 1 |k ) H′+VBV′+N   (12)
 
           5. The filter gain matrix K is determined, as in
 
 A=M ( t|k ) F′H′+D ( t|k )Λ D ( k+ 1 |k )′H′+ E ( t|k ) BV′   (13)
 
 K=AQ   −1   (14)
 
           6. The matrix L is determined, as in
 
 L=I−KHF   (15)
 
           where I is the identity matrix 
           7. The state estimate {circumflex over (x)}(t|k) is updated as
 
{circumflex over ( x )}( t|k+ 1)={circumflex over ( x )}( t|k )+ K[z ( k+ 1)− H{circumflex over (x)} ( k+ 1 |k )]  (16)
 
           8. The matrices M(t|k), D(t|k), E(t|k) are updated to yield M(t|k+1), D(t|k+1), and E(t|k+1), as in
 
 M ( t|k+ 1)= LM ( t|k ) L′+KNK′   (17)
 
 D ( t|k+ 1)= D ( t|k )− KHD ( k+ 1 |k )  (18)
 
 E ( t|k+ 1)= E ( t|k )− KV   (19)
 
           respectively, and 
           9. The total mean square error S(t|k+1) is generated, as in
 
 S ( t|k+ 1)= M ( t|k+ 1)+ D ( t|k+ 1)Λ D ( t|k+ 1)′+ E ( t|k+ 1) BE ( t|k+ 1)′  (20)
 
         
       
       (h) Determining whether the measurement is time-late by testing if T&lt;0. If the measurement is not time-late then
       1. Matrices F,G are determined using   
     
     
  
   
     
       
         
             
           
             
               
                 
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           2. A parameter matrix Λ is generated, representing physical bounds on those parameters λ that are not state variables of the system; 
           3. The state estimates {circumflex over (x)}(k|k) and matrices M(k|k), D(k|k), E(k|k) are extrapolated to yield {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), E(k+1|k) as
 
{circumflex over ( x )}( k+ 1 |k )=Φ{circumflex over ( x )}( k|k )+Γ u ({circumflex over ( x )}( k|k ),  λ )  (23)
 
 M ( k+ 1 |k )= FM ( k|k ) F′   (24)
 
 D ( k+ 1 |k )= FD ( k|k )+ G   (25)
 
 E ( k+ 1 |k )= FE ( k|k )  (26)
 
           4. Filter matrix P(k+1|k), intermediate matrix V, and covariance of the residual Q are determined as
 
 P ( k+ 1 |k )= M ( k+ 1 |k )+ D ( k+ 1 |k )Λ D ( k+ 1 |k )′  (27)
 
 V=HE ( k+ 1 |k )+ J   (28)
 
 Q=HP ( k+ 1 |k ) H′+VBV′+N   (29)
 
           5. The filter gain matrix K is determined as
 
 A=P ( k+ 1 |k ) H′+E ( k+ 1 |k ) BV′   (30)
 
 K=AQ   −1   (31)
 
           6. The matrix L is determined as
 
 L=I−KH   (32)
 
            where I is the identity matrix 
           7. The state estimate {circumflex over (x)}(k+1|k) is updated as
 
{circumflex over ( x )}( k+ 1 |k+ 1)={circumflex over ( x )}( k+ 1 |k )+ K[z ( k+ 1)− H{circumflex over (x)} ( k+ 1 |k )]  (33)
 
           8. The matrices M(k+1|k), D(k+1|k), and E(k+1|k) are updated as
 
 M ( k+ 1 |k+ 1)= LM ( k+ 1 |k ) L′+KNK′   (34)
 
 D ( k+ 1 |k+ 1)= LD ( k+ 1 |k )  (35)
 
 E ( k+ 1 |k+ 1)= LE ( k+ 1 |k )− KJ   (36)
 
            and 
           9. The total mean square error S(k+1|k+1) is generated as follows
 
 S ( k+ 1 |k+ 1)= M ( k+ 1 |k+ 1)+ D ( k+ 1 |k+ 1)Λ D ( k+ 1 |k+ 1)′+ E ( k+ 1 |k+ 1) BE ( k+ 1 |k+ 1)′  (37)
 
         
       
     
  
   According to another aspect of the invention, a method comprises the steps of:
         (i) Observing a system having state variables and also having unknown, multidimensional, arbitrarily time-varying parameters, but which are subject to known bounded values;   (ii) Measuring certain aspects of the state of the system in the presence of sensor measurement biases and random errors to produce initial measurements;   (iii) Initializing state estimates and matrices using a priori information and the initial measurements;   (iv) Using the update interval in determining the system transition matrices and the mean value of unknown but bounded parameters and the input vector;   (v) Determining if the measurement is time-late by testing the sign of the update interval;   (vi) If the measurement is time-late, applying the measurements to an out-of-sequence estimating filter that explicitly uses a mean square optimization criterion that separately accounts for measurement biases and errors and the bounding values, as well as the delay time, to produce estimates of the true state of the system;   (vii) If the measurement is not time-late, applying the measurements to an in-sequence estimating filter that explicitly uses a mean square optimization criterion that separately accounts for measurement biases and errors and the bounding values, to produce estimates of the true state of the system;   (viii) Applying the estimates to one of (a) make a decision, (b) operate a control system, and (c) control a process.       

   
     BRIEF DESCRIPTION OF THE DRAWING 
       FIG. 1  is a simplified block diagram of a system for estimating the state of a target, which in this case is the track of an aircraft performing bounded maneuvers, after processing measurements with biases and random errors from multiple sensors; and 
       FIGS. 2 and 3  together constitute a simplified logic flow chart or diagram illustrating the recursive processing associated with a reduced-state estimator according to an aspect of the invention, which may be used in the arrangement of  FIG. 1 . 
   

   DESCRIPTION OF THE INVENTION 
   State estimation after processing multisensor measurements with time delays and with unknown biases that may vary arbitrarily in time within known physical bounds is considered. These measurements are obtained from systems characterized by state variables and by multidimensional parameters, for which the latter are unknown and may vary arbitrarily in time within known physical bounds. The solution of the problem requires a completely different method from the prior art. The simplified logic flow chart or diagram  200  of  FIG. 2  illustrates a method according to an aspect of the invention. In  FIG. 2 , the logic flow of the invention begins in a block  210  with initialization of the state estimate {circumflex over (x)}(k 0 |k 0 ) and the matrices M(k 0 |k 0 ), D(k 0 |k 0 ), E(k 0 |k 0 ). In general, a vector {circumflex over (x)}(t|k) is defined as the estimate of the state of the system at time t after processing k measurements z(i) for 1≦i≦k. The vector {circumflex over (x)}(t j |k) is denoted as {circumflex over (x)}(j|k) when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . . matrix M(t|k) is defined as the covariance of the state estimation errors at time t due only to the errors in the measurements z(i) for 1≦i≦k and a priori initial information that is independent of the parameter uncertainty and measurement bias uncertainty. The matrix M(t j |k) is denoted as M(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . . A matrix D(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the parameter errors, at time t (after processing k=0, 1, 2, . . . measurements). The matrix D(t j |k) is denoted as D(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . . A matrix E(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the sensor measurement bias, at time t (after processing k=0, 1, 2, . . . measurements). The matrix E(t j |k) is denoted as E(j|k), when the time t=t j  is the time of the j th  measurement for j=1, 2, 3, . . . . The use of the matrices M(j|k), D(j|k), and E(j|k) instead of S(j|k) is a difference of the current invention from the prior art. From block  210 , the logic of the invention of  FIG. 2  flows to a block  212 , which represents the incrementing of a time index k to k+1. 
   The logic of  FIG. 2  then flows to a further block  214 , which represents calculating the update interval according to
 
 T=t   k+1   −t   (38)
 
and determining the system transition matrices Φ, Γ, and the mean value  λ  of unknown but bounded parameters λ and also the input vector u({circumflex over (x)}(t|k),  λ ). Note that for the update interval T=t k+1 −t, the time t denotes the time when the filter was last updated, whereas t k+1  is the time of the new measurement, which can be from a time prior to t due to propagation delay and, therefore, T&lt;0. If there is no delay, then usually t=t k , the time of the previous measurement and in such a case T≧0.
 
   From block  214  of  FIG. 2 , the logic of  FIG. 2  flows to a block  216 , which represents the measurement z(k+1) of aspects of the state of the system. 
   The latency of the measurement is determined in decision block  218  by testing the sign of the update interval, i.e., T&lt;0. 
   If T&lt;0, the logic of the invention then flows from decision block  218  of  FIG. 2  by way of the YES output and a path  219  to a block  220 , in which the matrices F and G are calculated according to 
   
     
       
         
             
           
             
               
                 
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   From block  220 , the logic of  FIG. 2  flows to a block  222 , which accesses or inputs a parameter matrix Λ. The parameter matrix Λ represents the physical bounds on the unknown parameters, such as turn, rate and tangential acceleration, and is preferably selected based on a precise description of the physical system, and depends upon well-known physical phenomena and characteristics of real systems. Types of physical parameters, which might be considered for determination of Λ in the case of maneuvers, are the turn rate ω and the tangential acceleration α. This is a departure from the prior art, which uses a white process noise covariance W that does not correctly model the class of systems addressed by the invention herein. 
   From block  222  of  FIG. 2 , the logic flow of the invention proceeds to a block  224 , which represents the extrapolation of state estimates {circumflex over (x)}(t|k) and matrices M(t|k), D(t|k), E(t|k) to yield {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k) and E(k+1|k) according to
 
{circumflex over ( x )}( k+ 1 |k )=Φ{circumflex over ( x )}( t|k )+Γ u ({circumflex over ( x )}( t|k ),  λ )  (41)
 
 M ( k+ 1 |k )= FM ( t|k ) F′   (42)
 
 D ( k+ 1 |k )= FD ( t|k )+ G   (43)
 
 E ( k+ 1 |k )= FE ( t|k )  (44)
 
   From block  224  of  FIG. 2 , the logic flows to a block  226 , which represents the accessing or inputting of the measurement bias covariance B and measurement noise covariance N. 
   Block  228  of  FIG. 2  computes filter matrix, P(k+1|k), intermediate matrix V, and covariance of the residual Q according to
 
 P ( k+ 1 |k )= M ( k+ 1 |k )+ D ( k+ 1 |k )Λ D ( k+ 1 |k )  (45)
 
 V=HE ( k+ 1 |k )+ J   (46)
 
 Q=HP ( k+ 1 |k ) H′+VBV′+N   (47)
 
respectively. Thus, there are several differences between the invention herein as exemplified in  FIG. 2  and the prior art. First,  λ  in the input function u({circumflex over (x)}(t|k),  λ ) in formula (41) refers to the mean value of the parameters λ, whose bounds are represented by the matrix Λ in formula (45). Note that the input function u(x(k),λ) may be a nonlinear function of the state variables x(k) and the multidimensional parameters λ. Another difference is that the prior art Kalman filter extrapolates the state covariance, while the current invention extrapolates the matrices M(t|k), D(t|k), E(t|k) to M(k+1|k), D(k+1|k), E(k+1|k) respectively, as in formulas (42) and (43), and then forms the matrix P(k+1|k) from M(k+1|k), D(k+1|k), and Λ as in formula (45).
 
   Block  230  of  FIG. 2  computes filter matrix A, filter gain matrix K, and filter matrix L. The filter matrix A is calculated as
 
 A=M ( t|k ) F′H′+D ( t|k )Λ D ( k+ 1 |k )′ H′+E ( t|k ) BV′   (48)
 
The filter gain matrix K is calculated as
 
 K=AQ   −1   (49)
 
and the matrix L is calculated as
 
 L=I−KHF   (50)
 
where I is the identity matrix.
 
   The logic flows from block  230  of  FIG. 2  to a block  232 , which represents the updating of the state estimate and the associated matrices. More particularly, the state estimate {circumflex over (x)}(t|k+1) is calculated as
 
{circumflex over ( x )}( t|k+ 1)={circumflex over ( x )}( t|k )+ K[z ( k+ 1)− H{circumflex over (x)} ( k+ 1 |k )]  (51)
 
The matrices M(t|k+1), D(t|k+1), E(t|k+1) are calculated as
 
 M ( t|k+ 1)= LM ( t|k ) L′+KNK′   (52)
 
 D ( t|k+ 1)= D ( t|k )− KHD ( k+ 1 |k )  (53)
 
 E ( t|k+ 1)= E ( t|k )− KV   (54)
 
respectively. Finally, the matrix of the total covariance S(t|k+1) is calculated as
 
 S ( t|k+ 1)= M ( t|k+ 1)+ D ( t|k+ 1)Λ D ( t|k+ 1)′+ E ( t|k+ 1) BE ( t|k+ 1)′  (55)
 
and equation (55) represents a mean-square criterion which may be used for a root-mean-square determination. The logic of  FIG. 2  returns from block  232  to block  212  by way of a path  233 .
 
   Returning now to decision block  218  of  FIG. 2 , the logic may leave block  218  by the NO output if T≧0, and in that case the logic flows by way of a path  239  to block  234  of  FIG. 3 . Block  234  of  FIG. 3  represents calculation of the matrices F and G according to 
                       F   =     Φ   +     Γ   ⁢       ∂   u       ∂   x       ⁢            x   =       x   ^     ⁡     (     k   ❘   k     )         ,     λ   =     λ   _                       (   56   )               G   =     Γ   ⁢       ∂   u       ∂   λ       ⁢          x   =       x   ^     (       k   ⁢        k   )       ,     λ   =     λ   _                         (   57   )                 
These equations are different from those used in block  220  of  FIG. 2  upon the substitution indicated above.
 
   From block  234  of  FIG. 3 , the logic of the invention of  FIG. 3  flows to a block  236 , equivalent to block  222 , which accesses or inputs a parameter matrix Λ. The parameter matrix Λ represents the physical bounds on the unknown parameters, such as turn rate and tangential acceleration, and is preferably selected based on a description of the physical system, which should be as precise as possible or practicable. The parameter matrix Λ depends upon well-known physical phenomena and characteristics of real systems. Types of physical parameters, which might be considered for determination of Λ in the case of maneuvers, are the turn rate ω and the tangential acceleration α. As noted, this is a departure from the prior art, which uses a white process noise covariance W that does not correctly model the class of systems addressed by the invention herein. 
   From block  236  of  FIG. 3 , the logic flow proceeds to a block  238 , which represents the extrapolation of state estimates {circumflex over (x)}(k|k), and matrices M(k|k), D(k|k), and E(k|k) according to
 
{circumflex over ( x )}( k+ 1 |k )=Φ{circumflex over ( x )}( k|k )+Γ u ({circumflex over ( x )}( k|k ),  λ )  (58)
 
 M ( k+ 1 |k )= FM ( k|k ) F′   (59)
 
 D ( k+ 1 |k )= FD ( k|k )+ G   (60)
 
 E ( k+ 1 |k )= FE ( k|k )  (61)
 
respectively. Thus, another difference between the invention herein and the prior art is that the prior art Kalman filter extrapolates the state estimate and its covariance, while the current invention extrapolates the state estimate {circumflex over (x)}(k|k) and matrices M(k|k), D(k|k), E(k|k) to {circumflex over (x)}(k+1|k) and M(k+1|k), D(k+1|k), E(k+1|k) respectively.
 
   From block  238  of  FIG. 3 , the logic flows to a block  240 , which represents the accessing or inputting of the measurement bias covariance B and measurement noise covariance N. 
   Block  242  of  FIG. 3  computes filter matrix P(k+1|k), intermediate matrix V, and residual Q according to
 
 P ( k+ 1 |k )= M ( k+ 1 |k )+ D ( k+ 1 |k )Λ D ( k+ 1 |k )′  (62)
 
 V=HE ( k+ 1 |k )+ J   (63)
 
 Q=HP ( k+ 1 |k ) H′+VBV′+N   (64)
 
respectively.
 
   Block  244  of  FIG. 3  computes filter matrix A, filter gain matrix K, and filter matrix L. While generically similar to the prior art, the filter gain matrix K obtained in block  244  of the arrangement of  FIG. 3  is or are different from the filter gain matrix K obtained in the prior art because of the use of M(k+1|k), D(k+1|k), E(k+1|k) and Λ, B to formula Q in equation (64). The filter gain matrix K is calculated as
 
 A=P ( k+ 1 |k ) H′+E ( k+ 1 |k ) BV′   (65)
 
 K=AQ   −1   (66)
 
and the matrix L is calculated as
 
 L=I−KH   (67)
 
where I is the identity matrix.
 
   The logic flows from block  244  of  FIG. 3  to a block  246 , which represents the updating of the state estimate and the associated matrices. More particularly, the state estimate {circumflex over (x)}(k+1|k+1) is calculated as
 
{circumflex over ( x )}( k+ 1 |k+ 1)={circumflex over ( x )}( k+ 1 |k )+ K[z ( k+ 1)− H{circumflex over (x)} ( k+ 1 |k )]  (68)
 
Also in block  246 , the matrices M(k+1|k+1), D(k+1|k+1), E(k+1|k+1) are calculated as
 
 M ( k+ 1 |k+ 1)= LM ( k+ 1 |k ) L′+KNK′   (69)
 
 D ( k+ 1 |k+ 1)= LD ( k+ 1 |k )  (70)
 
 E ( k+ 1 |k+ 1)= LE ( k+ 1 |k )− KJ   (71)
 
respectively. Finally, the matrix of the total covariance S(k+1|k+1) is calculated as
 
 S ( k+ 1 |k+ 1)= M ( k+ 1 |k+ 1)+ D ( k+ 1 |k+ 1)Λ D ( k+ 1 |k+ 1)′+ E ( k+ 1 |k+ 1) BE ( k+ 1 |k+ 1)  (72)
 
The logic leaves block  246  of  FIG. 2  and returns to block  212  of  FIG. 2  by way of a logic path  247 . The above equations (38)–(72) are set forth in  FIGS. 2 and 3 .
 
   The calculations associated with blocks  238  and  246  of  FIG. 3  separate the state estimation error covariance S(j|k) into components, M(j|k), D(j|k)ΛD(j|k)′, E(j|k)BE(j|k)′ attributable to random measurement error, parameter uncertainty, and measurement biases, and separately propagates these covariances from one time index k to next time index k+1. In addition, the gain matrix K is computed based on these propagated covariances. The gains weight the measurements to form the state estimates. The values of these gains so computed are different from those computed by the prior art method. 
   A salient difference between the prior-art method and that of the invention is the introduction into the operative equations of the matrices M(j|k), D(j|k), and E(j|k). The matrix M(j|k) is defined as the covariance of the state estimation errors at time t j  due only to the errors in the measurements z(i) for 1≦i≦k and a priori initial information that is independent of the parameter uncertainty and measurement bias uncertainty. D(j|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the parameter errors, at time t j  for j=0, 1, 2, . . . after processing k measurements z(i) for 1≦i≦k. E(j|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the sensor measurement bias, at time t j  for j=0, 1, 2, . . . after processing k measurements z(i) for 1≦i≦k. 
   The invention uses a novel mean-square optimization criterion (equation (55)) which explicitly addresses the known physical bounds of the multidimensional system parameters and sensor biases, and incorporates analytical modeling of their bounds. The invention provides a computationally efficient recursive algorithm for optimally estimating the state of a system, using the criterion of minimizing the mean-square total error.