Finite impulse response (FIR) state estimation algorithms have been much discussed in literature lately. It is well known that they allow overcoming the Kalman filter divergence caused by modeling uncertainties. In this paper, new receding horizon unbiased FIR filters ignoring noise statistics for time-varying discrete state-space models are proposed. They have the following advantages. First, the proposed filters use only known means of state vector components at starting points of sliding windows. This allows us to take into account priory statistical information (on average) about specified movements of the system. Second, the iterative version of the filter has a Kalman-like form. Besides, its initialization does not include a training cycle in a batch form. Such filters may have a wide range of applications. In this paper, position and speed estimation of sea targets using angle measurements in azimuth and elevation is considered as an example.
Ministry of Education and Science of the Russian FederationRFMEFI57818X02641. Introduction
Finite impulse response (FIR) state estimation algorithms can be considered as an alternative to the Kalman filter (KF). These algorithms allow us to overcome its divergence caused by model inaccuracy (incorrectly chosen models, temporary perturbations, and errors in the setting of the noise statistics) [1–3]. Divergence prevention is achieved by reducing the impact of the data or rejection the data out of the sliding window. In this context, the KF can be considered as an estimation algorithm with infinite impulse response (IIR).
Quite a number of methods have been offered to construct FIR algorithms. In [4], the FIR filter for a linear homogeneous system is proposed. The optimal unbiased FIR (OUFIR) and receding horizon algorithms are proposed in [2, 3, 5–7]. Initial conditions at starting points of sliding windows are assumed to be diffuse random variables or unknown arbitrary values. In [7], the receding horizon optimal unbiased FIR (RHOUFIR) filter which obtains information about the window initial conditions from finite observations is suggested. Another approach ensuring optimality and unbiasedness in a finite number of steps is described in [8, 9]. The RHOUFIR filter suggested by the authors uses known statistical information for parts of state vector components at starting points of sliding windows. Within the framework of covariance analysis this allows us to take into account priory statistical information about random biases, trends, and specified movements of the system. Training tasks of neural and neurofuzzy networks are another example of possible application of this filter [9].
In [10–14], the unbiased FIR (UFIR) filters ignoring the noise statistics of the process, measurements, and the statistics of state vector initial conditions are developed. Computer modeling shows that the UFIR filters can be more robust than the KF or the OUFIR filter if the size of the sliding window is properly fitted. There are several general approaches to select N based on a heuristic choice, via the mean square errors, via real measurements, and via the use of bandlimited property of signals [13]. However, the proposed recursive filters have two disadvantages. First, a butch form of the algorithm is needed for the initialization of the recursive UFIR filter during the learning cycle which may be inconvenient or even problematic in some cases (e.g., in case of gaps in the observations, estimation parameters of nonlinear systems, and nonstationary processes). Second, these filters do not use priorly known information for initialization at starting points of sliding windows.
In this paper, the receding horizon UFIR (RHUFIR) filters using only known means of state vector components at starting points of sliding windows are suggested. They allow us to take into account (on average) priory statistical information about the system. Furthermore, for the iterative filter initialization no training cycle in a batch form is required. While the implementation of the RHOUFIR filter described in [8, 9] may be difficult due to the lack of sufficiently accurate statistical information about system, the algorithms under consideration may produce quite acceptable estimates. Theoretically this can be justified by a selection of the optimal or sub-optimal values of N using the algorithms described in [13]. Another important feature is that the RHUFIR filter is computationally less demanding in comparison with the RHOUFIR. This is important, for example, in the tasks of neural and neurofuzzy networks training.
As a possible application of the proposed filters, the position and speed estimation of sea targets using angle measurements in azimuth and elevation from the observer-ship’s video camera is considered. The general approach to solve this problem is to use Kalman filtering methods [15]. The difficulties arising due to the specificity of both the problem itself and the methods commonly used are well known. First, measurement models may be highly nonlinear near the state estimate [16]. Second, the object state may not be observable if the observer does not maneuver in a special way [17, 18]. Third, there is a large initial uncertainty about the position, speed, and acceleration of the observed object. All this can lead to a divergence of the KF and its various modifications [19–21].
The work is organized as follows. The RHUFIR state estimation problem is formulated in Section 2. Sections 3 and 4 concern the batch and iterative RHUFIR filters, respectively. Section 5 contains a comparative analysis of the RHUFIR and RHOUFIR filters. A diffuse approach to FIR estimation and its connection with the RHOUFIR filter is considered in Section 6. The position and speed estimation of sea objects is considered in Section 7. The conclusions are presented in Section 8.
2. Problem Statement
Let us consider a linear discrete time-variant system of the form(1)xt+1=Atxt+Btwt,yt=Ctxt+Dtξt,where xt∈Rn is the state vector, yt∈Rm is the measured output vector, wt∈Rr and ξt∈Rl are random processes with zero means, i.e., E(wt)=0, E(ξt)=0, At, Bt, Ct, Dt are known matrices of appropriate dimensions, t∈{0,1,…}.
Consider the discrete intervals [t-N,t-1], t∈TN={N,N+1,…} (the sliding windows), where N is the horizon length. Let us assume that the following conditions are met.
A1: Means for arbitrary q∈{1,2,…,n-1} components of the state vector at the starting points xt-N of the sliding windows are known. A priori information on remaining n-q components of xt-N, t∈TN is absent and they are either unknown constants or random variables statistical characteristics of which are unknown. The value q=0 is used further if a priori information about all components of xt-N is absent.
A2: If q≠0 then without loss of generality it is assumed that the state vector elements are arranged so that means mi,t-N=E(xi,t-N), i=1,2,…,q, are known, where xt=(x1,t,…,xq,t,xq+1,t,…,xn,t)T, t∈TN={N,N+1,…}.
It is required to find the linear state estimate of the convolution-based batch form(2)x^t=Ωt,t-Nmt-N+∑i=t-Nt-1Yt,iyi,t∈TNfrom measurements yt-N,yt-N+1,…,yt-1 and its recursive representation for x^t calculation, where the matrices Yt,i∈Rn×m and Ωt,t-N∈Rn×q do not depend on a priori information about the unknown vector xt-N, mt-N=(m1,t-N,m2,t-N…,mq,t-N)T, q∈{1,2,…,n-1}, and Ωt,t-N=0, mt-N=0 if q=0. The estimate is found under the conditions that it is unbiased(3)Ex^t=Ext,t∈TN.
The first component in (2) allows to take into account (on average) a priori information about the system (1) at starting points of sliding windows xt-N, t∈TN. This may be necessary if the pair {At,Ct} is not observable but the estimate must be unbiased after processing the finite number of observations. Typical examples are models including random biases, trends and specified movements of the system (see the example in Section 7.3). The relation mi,t-N=const, t∈TN, i=1,2,…,q, is the simplest case describing a random bias with a known mean. Note that the inference of the UFIR filter in [11] for q=0 is based on the assumption that the pair{At,Ct} is observable. Note also that this assumption is not required for the KF. The need for the first component in (3) may also arise in the tasks of neural and neurofuzzy networks training which have a separable structure [22]. It is assumed that the linearly incoming parameters are unknown and in addition to the training set a priori information only about the nonlinearly incoming parameters is given [9]. This information is obtained from the distribution of a generating sample, a training set, or some linguistic information. The RHUFIR filter is used to estimate the parameters by linearizing networks relations in the vicinity of the last estimate.
3. Batch Receding Horizon UFIR Filter
The RHUFIR filter in a butch form is specified by the following theorem.
Theorem 1.
Let the condition (4)Mt=∑i=t-Nt-1RiTCiTCiRi>0,t∈TN be fulfilled. Then the unbiased estimate of the system state (1) in the batch form is determined by the expression (2), where(5)Yt,i=RtMt-1RiTCiT+Θt,iT,(6)Ωt,t-N=Zt-N,tTe1,e2,…,eq,q∈1,2,…,n-10,q=0,(7)Zt-N,t=Φt,t-NT-∑i=t-Nt-1Φi,t-NTYi,t-NT,(8)Φt+1,t-N=AtΦt,t-N,Φt-N,t-N=In,(9)Ri=Φi,t-NΛq,(10)Θt-N,t,Θt-N+1,t,…,Θt-1,t=INn-q-Ω+ΩΞ,(11)Λq=eq+1,eq+2,…,en,q∈0,1,…,n-1,Ω=(Rt-NTCt-NT,Rt-N+1TCt-N+1T,…,Rt-1TCt-1T)∈R(n-q)×Nm, Ξ is arbitrary (n-q)N×(n-q)N matrix, Ω+ is the Moore Penrose inversion of Ω, ei∈Rn is ith unit vector, and In is the identity matrix of the size n.
Proof.
First, we show that the estimation problem is equivalent to a dual control problem. Consider an auxiliary linear system of the form(12)Zi,t=AiTZi+1,t-CiTUi,t,Zt,t=In,i∈t-N,t-1,t∈TN,where Zi,t∈Rn×n is the state matric and Ui,t∈Rm×n is a control. Suppose that there is a control Ui,t bringing the system (12) in a state satisfying the condition ΛqTZt-N,t=0 for any t∈TN. Let us show that then there is an unbiased state estimate of the system (1) of the form (2), where (13)Yt,i=Ui,tT and Ωt,t-N is determined by (6).
Using the identity(14)xt=Zt-N,tTxt-N+∑i=t-Nt-1Zi+1,tTxi+1-Zi,tTxi,(1) and (12) give(15)xt=Zt-N,tTxt-N+∑i=t-Nt-1Zi+1,tTBiwi+Ui,tTCixi.
Taking into account this expression, the estimation error at the moment t can be presented in the form (16)et=xt-x^t=Zt-N,tTq1+q2+∑i=t-Nt-1Zi+1,tTBiwi-Ui,tTDiξi,where (17)q1=h1,h2,…,hq,01×n-qT,q∈1,2,…,n-1∈Rn,0,q=0q2=01×q,xq+1,t-N,xq+2,t-N,…,xn,t-NT,q∈1,2,…,n-1∈Rn,xt-N,q=0hi=xi,t-N-mi,t-N,i=1,2,…,q, and 01×q is 1×q the vector row with zero elements.
Averaging the left and right sides of this expression and using the condition ΛqTZt-N,t=0 get (18)Eet=Zt-N,tTEq2=Zt-N,tTΛqExq+1,t-N,Exq+2,t-N,…,Exn,t-NT=0.
Now we find the control providing a solution to the dual problem. Iterating (12) gives(19)Zi,t=Gi,tZt,t-∑j=it-1Gi,jCjTUj,t,i∈t-N,t-1,t∈TN,where the matrix Gi,s is determined by the system (20)Gi,s=AiTGi+1,s,Gs,s=In,s≥i.
Using the boundary condition ΛTZt-N,t=0, we obtain with help of (19) the linear equations system (21)0=ΛqTGt-N,tZt,t-∑j=t-Nt-1ΛqTGt-N,jCjTUj,t.We find a solution of this system in the form (22)Uj,t=CjGt-N,jTΛqL+Ωj,t,where L∈R(n-q)×n is a constant unknown matrix and Θj,t is arbitrary m×n matric function satisfying the condition (23)∑j=t-Nt-1ΛqTGt-N,jCjTΘj,t=0.Substituting (22) into (21) and taking in account (23) gives L=Wt-1ΛqTGt-N,tZt,t provided Wt>0, where (24)Wt=∑j=t-Nt-1ΛqTGt-N,jCjTCjGt-N,jTΛq.Using the expressions(25)Gj,t=AjTAj+1T…At-2TAt-1T,Φt,j=At-1At-2⋯Aj-1Aj,Gj,tT=Φt,j,Mt=Wt,we obtain(26)Uj,t=CjGt-N,jTΛqWt-1ΛqTGt-N,jZt,t+Θj,t=CjΦj,t-NΛqMt-1ΛqTΦt,t-NTZt,t+Θj,t=CjRjMt-1RtTZt,t+Θj,t.But as Zt,t=In and (10) is the general solution of the linear homogeneous system (23) then it implies (5) and (6).
Comment 1.
The condition {Ai,Ci} is observable for i∈{t-N,t-N+1,…,t-1}, and t∈TN={N,N+1,…} implies (3).
Comment 2.
The filter takes a particularly simple form for q=0 and Θt,i=0. We find from Theorem 1 that(27)x^t=Φt,t-NMt-1∑i=t-Nt-1Φi,t-NTCiTyi,t∈TN,where (28)Mt=∑i=t-Nt-1Φi,t-NTCiTCiΦi,t-N>0.These relations are equal to the algorithm from [11] up to the notation used if to ignore the one-step prediction.
4. Iterative Form of Batch Receding Horizon UFIR Filter
We will need the following assertion.
Lemma 2.
Let the condition (4) be true. Then the control bringing the system (12) in a state satisfying the condition ΛqTZt-N,t=0 can be presented in the form (29)Ui,tf=KiTZi+1,tf,i∈t-N,t-1,t∈TN,where(30)Ki=Ri+1Mi+1+RiTCiT,(31)Zi,tf=AiT-CiTKiTZi+1,tf,Zt,tf=In,i∈t-N,t-1,t∈TN.
Proof.
Let us show that the control (29) indeed solves our problem. Consider the system (12) under the action of the controls (22) with Θj,t=0. We show that Zi,t=Zi,tf. Substituting Ui,t with Θi,t=0 in (12) gives (32)Zi,t=AiTZi+1,t-CiTCiRiMt-1RtT,Zt,t=In,i∈t-N,t-1,t∈TN.From comparison of right parts of (31) and (32), it follows that the solutions of the systems really coincide if Ui,tf=Ui,t, i.e.,(33)KiTZi+1,t=CiRiMt-1RtT.Iterating the system (32), we successively find(34)Zi,t=Φt,iT-∑j=it-1Φj,iTCjTCjRjMt-1RtT,(35)KiTZi+1,t=CiRiMi+1+Ri+1TΦt,i+1T-∑j=i+1t-1Φj,i+1TCjTCjRjMt-1RtT=CiRiMi+1+In-∑j=i+1t-1RjTCjTCjRjMt-1RtT=CiRiMi+1+Mi+1Mt-1RtT.The expressions (34) and (35) follow from the identities(36)Ri+1=Φi+1,t-NΛq,Φt,i+1Φi+1,t-N=Φt,t-N,Φj,i+1Ri+1=Φj,i+1Φi+1,t-NΛq=Φj,t-NΛq=Rj.
Let us transform (35) using the orthogonal decomposition Mi+1=ViViT, where (37)Vi=Rt-NTCt-NT,Rt-N+1TCt-N+1T…,RiTCiT∈Rn-q×mN.Let l1i,l2i,…,lk(i),i be linearly independent columns of the matrix Vi. Using skeletal decomposition [23] yields Vi=LiΓi, where Li=(l1i,l2i,…,lk(i),i)∈Rk(i)(n-q)×k(i), Γi∈Rk(i)k(i)×mN, and Rrk×l are a set of k×l matrices that have rank r. Since Γ~i=ΓiΓiT>0 is the Gram matrix generated by linearly independent rows of the matrix Γi and rank(Li)=rank(Γ~iLiT) then(38)Mi+1=LiΓ~iLiT,Mi+1+=LiΓ~iLiT+=LiT+Γ~i-1Li+.
Substituting these expressions in (35) and using the representation Li+=(LiTLi)-1LiT, we find(39)KiTZi+1,t=CiRiLiT+Γ~i-1Li+LiΓ~iLiTMt-1RtT=CiRiLiLiTLi-1LiTMt-1RtT.It follows from (37) that RiTCiT=LiΓ1i, where Γ1i is some rectangular matrix. Substituting this expression in (39) gives(40)KiTZi+1,t=Γ1iTLiTLiLiTLi-1LiTRiTMt-1RtT=CiRiMt-1RtT.
The RHUFIR filter is specified by the following theorem.
Theorem 3.
The state of the system (41)x^i+1=Aix^i+Kiyi-Cix^i,i∈t-N,t-1,t∈TN,x^t-N=m~t-N=mt-NT,01×n-qT,is the unbiased estimate of the system state (1) for i≥tr, tr=mini{i:Mi>0,i=t-N,t-N+1,…}, where (42)Ki=AiRiMi+RiTCiTCiRi+RiTCiT,(43)Ri+1=AiRi,Rt-N=eq+1,eq+2,…,en,(44)Mi+1=Mi+RiTCiTCiRi,Mt-N=0.
Proof.
Put in (2) (45)Yt,i=Ui,tT=RiMi-1RtTCtT,Ωt,t-N=Zt-N,tTeq+1,eq+2,…,en,t≥tr,where the matric functions are identified in Theorem 1. This implies the unbiasedness of the estimate at the moment i≥tr and the relation (46)x^i=∑s=t-Ni-1Us,tTys+Zt-N,iTm~t-N.From Lemma 2 it follows that (47)Us,t=Us,tf=KsTZs+1,tf.
In view of this (48)x^i=∑s=t-Ni-1UsfTys+Zt-N,tfTm~t-N=∑s=t-Ni-1Hs+1,iTKsys+Ht-N,iTm~t-N,where (49)Hs,i=AsT-CsTKsTHs+1,i,Hi,i=In,s≤i.
By (48) we find(50)x^i+1-x^i=Kiyi+∑s=i-NiHs+1,i+1T-Hs+1,iTKsys+Ht-N,i+1T-Ht-N,iTm~t-N.Using the identities(51)Hs+1,i+1=Hs+1,iHi,i+1,Ht-N,i+1=Ht-N,iHi,i+1,Hi,i+1=AiT-CiTKiT,we transform this expression to the form (41) (52)x^i+1-x^i=Kiyi+Hi,i+1T-In∑s=t-Ni-1Hs+1,iTKsys+Hi,i+1T-InHt-N,iTm~t-N=Kiyi+Ai-KiCi-In·∑s=t-Ni-1Hs+1,iTKsys+Ht-N,iTm~t-N=Kiyi+Ai-KiCi-Inx^i.Putting in (48) i=t-N gives x^t-N=m~t-N.
Comment 1.
The filter is equal to the algorithm from [11] up to the used notation for q=0 if to ignore the one-step prediction and to use Theorem 1 to initialize it:(53)x^i+1=Aix^i+Kiyi-Cix^i,i∈t-N,t-1,t∈2N,2N+1,…,x^N=ΦN,0MN-1∑i=0N-1Φi,0TCiTyi,where (54)Ki=AiRiMi+RiTCiTCiRi-1RiTCiT,Ri+1=AiRi,Rt-N=In,Mi+1=Mi+RiTCiTCiRi,Mt-N=0.
Consider the system for the state transition matrix of the homogeneous part of the UFIR filter (41)(55)Xi+1,s=Ai-KiCiXi,s=A~iXi,s,Xs,s=In,i≥s.
Consequence 1.
The following representations are true:(56)Xi,s=Φi,s-RiMi-1∑j=si-1RjTCjTCjΦj,s,i≥tr,s<i,(57)KsTXi,s+1T=CsRsMi-1RiT,i≥tr,s<i.
Since(58)Hs,i=Zs,if=A~sTA~s+1T…A~i-1T,Xi,s=Hs,iT=A~t-1A~t-2⋯A~sthe statement follows from (34) and (35).
Consequence 2.
Let us show using (56) that the estimate x^i is unbiased indeed. Consider the system of equations for the expectation of the estimation error ei=E(xi-x^i)(59)ei+1=Ai-KiCiei,et-N=01×q,xqT,i≥tr,where xq=(xq+1,t-N,xq+2,t-N,…,xn,t-N)T is the arbitrary vector. We find from (56) (60)ei=Xi,t-Net-N=Φi,t-N-RiMi-1∑j=t-Ni-1RjTCjTCjΦj,t-NRt-Nxq=0,i≥trfor q∈{1,2,…,n-1}. Similarly, it is verified that ei=0 for i≥tr, q=0, and et-N=xt-N, where xt-N is the arbitrary vector.
Consequence 3.
Let the system (1) be time-invariant, q=0, and detA≠0. Then the representation for the state transition matrix Xi,s has the form(61)Xi,s=0,s=0,i>trAi-t+NMi-1MsA-s+t-N,i>tr,s<i,s≠0.The statement can be obtained from (56) with the help of elementary calculations.
5. Comparison of RHUFIR and RHOUFIR Filters
Let us assume that the following conditions are held.
B1: wt and ξt are the uncorrelated random processes with zero means and known covariance matrices E(wtwtT)=Vt,E(ξtξtT)=Σt, At,Bt,Ct,Dt are known matrices of appropriate dimensions.
B2: Means and covariances are known for arbitrary q∈{0,1,…,n-1} components of the state vector at the starting points xt-N of the sliding windows. A priori information on remaining n-q components of xt-N, t∈TN is absent and they are either unknown constants or random variables the statistical characteristics of which are unknown. The value q=0 is used further if a priori information about all components of xt-N is absent.
B3: If q≠0 then without loss of generality it is assumed that the state vector elements are arranged so that means and covariances(62)mi,t-N=Exi,t-N,s-ij,t-N=Exi,t-N-mi,t-Nxj,t-N-mj,t-N,i,j=1,2,…,qare known, where xt=(x1,t,…,xq,t,xq+1,t,…,xn,t)T, t∈TN={N,N+1,…}.
B4: If xi,t-N,i=1,2,…,q are random variables then they are uncorrelated with xi,t-N, i=q+1,q+2,…,n, wt and ξt for t∈T.
It is required to find the unbiased linear state estimate of (1) minimizing the criterion E[(xt-x^t)T(xt-x^t)]. This estimate is called the optimal estimate of the system state (1).
(2) The estimation error covariance matrix for i≥ttr is given by the expression (71)Pi=Si-+RiMi-1RiT.
Comment 1.
Formally, the RHUFIR filter follows from the RHOUFIR filter by specifying Vt=Im, St-N+=0, and Σt=0. At the same time, the statement of the problem and the derivation of the UFIR filter do not rely on these assumptions.
Comment 2.
The relation (71) is the decomposition of the error covariance matrix into two terms. The first one takes into account the known statistical information about initial conditions, process, and measurement noises covariance matrices on the estimate accuracy. The second one additionally reflects the impact of unknown initial conditions at starting points of sliding windows.
Comment 3.
Similarly to the covariance matrix, the decomposition is true for the UOFIR filter gain. Indeed, it follows from (64) that Ki=KiKF+KiUKF, where the term KiKF=AiSi-CiTNi-1 coincides with the KF gain taking in account the known statistical information and the term KiUKF=A1iRiMi+1+RiTCiTNi-1 reflects the additional effect connected with unbiasedness of the RHOUFIR filter.
6. Receding Horizon FIR Filter with Diffuse Initialization
Consider an alternative approach to constructing the FIR estimator which is important in view of possible applications. Now, we assume that the condition B1 from the previous section and the following conditions are met.
C1: Means and covariances are known for arbitrary q∈{0,1,…,n-1} components of the state vector at the starting points xt-N of the sliding windows.
C2: If q=0 then xi,t-N,i=1,2,…,n are treated as random variables with zero mean and covariance matrix proportional to the large parameter μ>0, i.e., E(xt-N)=0, E(xt-Nxt-NT)=μS~t-N, where S~t-N is an arbitrary positive definite matrix.
C3: If q∈{1,2,…,n-1} then without loss of generality it is assumed that the state vector elements are arranged so that means and covariances(72)mi,t-N=Exi,t-N,s-ij,t-N=Exi,t-N-mi,t-Nxj,t-N-mj,t-N,i,j=1,2,…,qare known. A priori information on remaining n-q components of xt-N, t∈TN, is absent and they are treated as random variables with zero mean and covariance matrix proportional to the large parameter μ>0, i.e., (73)Exi,t-N=0,Exi,t-Nxj,t-N=μs~ij,t-N,i,j=q+1,q+2,…,n,where S~t-N=||s~ij,t-N||q+1n>0, μ>0 is a large parameter.
C4: The random vector xt-N is not correlated with wt and ξt for [t-N,t-1], t∈TN={N,N+1,…}.
C5: The random variables xi,t-N for i=1,2,…,q,q∈{1,2,…,n-1}, are uncorrelated with variables xi,t-N, i=q+1,q+2,…,n.
It is required to find the limit relations for the KF as μ→∞ and to study their properties. In [8, 9], it was shown that as μ→∞ these relations are coincide with (63)–(70).
The conditions C1–C5 formalize the standard approach to the implementation of the KF for large initial uncertainty of the system state. At the same time, it is well known [9, 24] that the large values of μ can lead to a divergence of the KF. In [8, 9], the influence of the large values of μ on the KF divergence was studied. In particular, let δMi+ be the error connected with calculations of Mi+. Then (74)Ki=RtS~t-NIn-q-MiMi++δMi+RiTCiTNi-1μ+O1,μ→∞.For δMi+≠0 the matrix Kt becomes proportional to the large parameter μ and so divergence is possible even if the continuity condition of the matrix pseudoinversion finding is performed. Thus, the use of the limiting relations for the filter allows avoiding divergence of the FIR filters with the diffuse initialization.
Note that, for the special case N=0, the relations (63)–(70) (the IIR filter) can be used for the state estimation in absence or incompleteness of the statistical information about the initial conditions of (1). Following [8, 9], we call them the KF with diffuse initial conditions (DKF).
7. Application of Receding Horizon FIR Filters to Sea Target Tracking Problem7.1. Models of Objects and Measurements
To illustrate the capability of the proposed FIR algorithms, we use the following scenario. With the help of the camera installed on the ship, the bearing (α), and elevation (β) angles to the target (Figure 1) are measured.
Ship and target positions.
The expressions for α and β taking into account measurement errors have the following form:(75)αt=atan-etxsinψ+etycosψetxcosψ+etysinψ+ξ1t,(76)βt=atanhtetx2+ety2+ξ2t,where etx=xtt-xto and ety=ytt-yto,xtt,ytt are the positions of the target, ht is altitude of the camera, ξ1,t and ξ2,t are the centered uncorrelated white noises with the variances σ12 and σ22, respectively, and atan is the 4-quadrant arctangent.
It is assumed that the variability of ht is caused by a rolling with the dominant frequency and amplitude. More exactly, it is assumed that ht=h+δt,δt=psin(ωΔt), where h is approximately known and p,ω are unknown values to the observer and|δt|<<h, Δ is the sampling step. The function δt is interpreted as an uncontrolled disturbance in the elevation angle measurement channel.
The ship and the measurements model are described by a nonlinear discrete system of the form (77)qt+1o=foqto,ut,wto,q0o=qo,(78)ηt=goqto+ξt,t=0,1,…,where qto∈Rn is the state vector, ηt∈Rm is the measured output vector, wto∈Rr and ξt∈Rm are random processes, ut is the control, and fo·,· and go· are known vector functions.
The movement of the target to the observer is unknown and we rely on the common approach to describe its behavior consisting in the following [15, 24]. It is assumed that the target movement can be specified by means of a stochastic system of the form (79)qt+1t=ftqtt,wtt,q0t=qt,where qtt∈Rl is the state vector and wtt∈Rd is the random process. A priori information about the initial position of the target is absent and the right-hand parts of the system (79) and the state vector qtt and the random process wtt are chosen using the specifics of the problem under consideration.
Special cases of models (77), (78), and (79) are as follows [15, 24].
The ship moves at a constant fixed velocity in accordance with (80)xt+1o=xto+vocosψΔ,yt+1o=yto+vosinψΔ,where the linear speed (vo) and the heading angle (ψ) of the ship are known values and Δ is the sampling period. The coordinates of the ship in the horizontal plane (xto,yto) are measured and the observation models have the form(81)z3,t=xto+ξ3t,z4,t=yto+ξ4t,t=0,1,…,where ξ3t and ξ4t are the centered uncorrelated white noises with the known variances σ32 and σ42, respectively, Ex0o=0, Ey0o=0, cov(x0o)=σ32, and cov(y0o)=σ42.
The motion of the target is described by(82)xt+1t=xtt+Δvtt,x+Δ22wtx,vt+1t,x=vtt,x+Δwtx,(83)yt+1t=ytt+Δvtt,y+Δ22wty,vt+1t,y=vtt,y+Δwty,t=0,1,…,where vtt,x and vtt,y are the target velocity projections on coordinates axes x and y, wtx, and wtyare the accelerations projections (the centered uncorrelated white noises with the variances σx2,σy2, respectively).
7.2. Models of Pseudo Measurements
Let us find expressions for the pseudo measurements model using the relations (75) and (76). The general idea [25] is to represent the nonlinear measurement model y=h(x)+w in the next pseudo linear form z(y)=H(y)x+wz(x,y), where z(y) is the pseudo measurement vector, H(y) is the known function of the true measurements ofy, and wz(x,y) is the pseudo measurement errors. The KF is used with z(y), H(y) and cov[wz(x⌢,y)], where x⌢ is the predicted state value of x.
Rewrite (75) and (76) in the following form: (84)tanαt-ξ1tcosψ+sinψetx+tanαt-ξ1tsinψ-cosψety=0,(85)etx2+ety2=httanβt-ξ2t2.We find from (84)(86)etx=cosψ-tanαt-ξ1tsinψtanαt-ξ1tcosψ+sinψety=kety.Substitution of this expression into (85) gives(87)ety=httanβt-ξ2t1+k21/2and using (86) and (87), we get(88)etx=khttanβt-ξ2t1+k21/2.As (89)1+k21/2=1sinαt+ψ-ξ1t,k1+k21/2=cosαt+ψ-ξ1tthen(90)etx=htcosαt+ψ-ξ1ttanβt-ξ2t,ety=htsinαt+ψ-ξ1ttanβt-ξ2t.Linearizing the right-hand parts of (90) in a neighborhood of the points ξ1t=0, ξ2t=0, we find(91)etx=htcosαt+ψtanβt-ζt,ety=htsinαt+ψtanβt-ηt,where ζt and ηt are correlated random processes defined by expressions(92)ςt=sinαt+ψtanβtξ1t+cosαt+ψsin2βtξ2t,ηt=-cosαt+ψtanβtξ1t+sinαt+ψsin2βtξ2t. It follows the relations for the pseudo measurements model (93)z1tαt,βt=cosαt+ψtanβt=-xto+xtth+δt+ζt,(94)z2tαt,βt=sinαt+ψtanβt=-yto+ytth+δt+ηt,where(95)σς2=sinαt+ψtanβt2σ12+cosαt+ψsin2βt2σ22,(96)ση2=cosαt+ψtanβt2σ12+sinαt+ψsin2βt2σ22.Linearizing the right-hand sides of (93) and (94) in a neighborhood of the point δt=0, we also get(97)z1tαt,βt=cosαt+ψtanβt=qt-xto+xtt+ζt,(98)z2tαt,βt=sinαt+ψtanβt=qt-yto+ytt+ηt,where qt=(1-δt/h)/h.
7.3. Implementations of Filters
The models (80)–(83) are used to demonstrate the proposed FIR algorithms implementations. First of all, note that the filters inputs receive measurements defined by the expressions (75) and (76). Two close implementation schemes are developed. The first one is as follows.
Taking in account that δt is an unknown quantity and the relation |δt|<<h, the pseudo measurements model has form(99)z1tαt,βt=cosαt+ψtanβt=-xto+xtth+ζt,(100)z2tαt,βt=sinαt+ψtanβt=-yto+ytth+ηt.
In accordance with the relations (80)–(83), (99), and (100), the following linear system to construct the RHOUFIR filter and the DKF is used:(101)xt+1=Axt+wt+ft,yt=Cxt+μt,where (102)xt=xto,yto,xtt,vtt,x,xtt,vtt,yT,ft=vocosψΔ,vosinψΔ,0,0,0,0T,wt=0,0,Δ22wtx,Δwtx,Δ22wty,ΔwtyT,μt=ξ3t,ξ4t,ζt,ηtT,A=100000010000001Δ0000010000001Δ00000Δ,C=100000010000-1/h0-1/h0000-1/h00-1/h0,Exi,t-No=0,covxi,t-No=σi2,i=3,4,Ewtx=0,covwtx=σx2,Ewty=0,covwty=σy2,q=2.
It is easy to verify that the pair {A,C} is not observable but the condition (4) is fulfilled.
As the RHUFIR filter does not depend on statistics noises we get systems of the form (101) with q=1, where(103)xt=xto,xtt,vtt,xT,ft=vocosψΔT,wt=0,Δ22wtx,ΔwtxT,μt=ξ3t,ζtT,xt=yto,ytt,vtt,yT,ft=vosinψΔT,wt=0,Δ22wtx,ΔwtxT,μt=ξ4t,ηtT,A=10001Δ001,C=10001/h0,
The second implementation scheme of the RHUFIR filter is determined by the pseudo measurements model:(104)z1tαt,βt=cosαt+ψtanβt+x0o+vocosψΔth=xtth+ζt+ξ3t,(105)z2tαt,βt=sinαt+ψtanβt+y0o+vosinψΔth=ytth+ηt+ξ4t and the system (101), where(106)xt=xtt,vtt,xT,ft=0,wt=Δ22wtx,ΔwtxT,μt=ζt+ξ3t,yt=ytt,vtt,yT,ft=0,wt=Δ22wy,ΔwtyT,μt=ηt+ξ4t,A=1Δ01,C=1/h0,q=0.
7.4. Simulation
In this subsection, the behaviors of the DKF, the RHUFIR, and RHOUFIR filters, under the joint action of uncontrolled perturbations, are compared: rolling (δt), impulse perturbations of the bearing angle, and the target acceleration.
Input data for the simulation are listed in Table 1. We set N=130 (a heuristic choice) for the RHUFIR and RHOUFIR filters and N=0 for the DKF (the IIR filter).
Input data for the simulation.
Components
Parameters
of the system
Camera
Full HD (1920×1080), α=700(angle of view), σ1=0.88′,
σ2=1.56′, h=3m, ω=2π/T, T=10s, p=0.3m
Ship
x0o=y0o=0 m, σ3=σ4=1m, vo,x=2.55 m/s, vo,y=4.33
m/s, ψ=600, Δ=0.3s
Target
x0t=400m, y0t=600m, v0t,y=3.12m/s, v0t,x=2.98m/s,
σx=σy=0.1m/s2, Δ=0.3s, ψ=600
First, we compare the joint action effect of the rolling and the signal temporary uncertainty on position and velocity estimation errors of the DKF, the RHUFIR and RHOUFIR filters. We set αt+λt, where λt=1o when 60≤t≤75s and λt=0 otherwise. Estimation errors (ex=xtt-xto, ey=ytt-yto, evx=vtt,x-vto,x, and evy=vtt,y-vto,y) are shown in Figure 2. It is seen that the RHUFIR filter provides much more robust estimates compared with the DKF and the RHOUFIR filter and the RHOUFIR filter superiors the DKF. This is due to the following reasons. Firstly, the RHUFIR gain does not depend on the bearing (αt) and elevation (βt) angles. Secondly, the matrix functions Mi, Ni, and AiSi-+A1iRiMi+1+ in the expression (64) of the RHOUFIR filter gain are ill-conditioned with the condition number more than 105. This may be a consequence of the weak observability of the system with angular observations [17, 18]. At the same time, it seems that the RHOUFIR filter and the DKF characteristics can be improved by square root implementation.
Position and velocity estimation errors caused by the signal temporary uncertainty.
Compare the joint effect in the inaccurate setting of the target noise statistics and the rolling on position and velocity estimation errors of the RHUFIR and RHOUFIR filters and the DKF. We set σx=σy=0.13m/s2 for 60≤t≤150s and σx=σy=0.1m/s2 otherwise. The Root Mean Square (RMS) position and velocity errors for 100 Monte Carlo runs are shown in Figure 3. It is seen that the RHUFIR filter provides much more robust estimates compared with the DKF and the RHOUFIR filter. The dependencies of position and velocity estimation errors of the RHUFIR filter on the minimum distance (Dmin) between the ship and the target for Monte Carlo runs are shown in Figure 4. It is seen that ex and ey do not exceed 11 m and evx and evy − 0.8 m/s if Dmin≤300 m.
RMS position and velocity errors.
Dependencies of position and velocity estimation errors on the minimum distance.
8. Conclusions
The main contributions of this paper are as follows. First, new receding horizon unbiased FIR filters for linear discrete state-space models ignoring the noise statistics of the process and measurements were proposed. The possibility to use only known means of any state vector components at starting points of sliding windows is believed to be one of these filters’ vantage points. It allows us to take into account priory statistical information (on average) about random biases, trends, and specified movements of the system. Second, the proposed RHUFIR filter has a Kalman-like form, and it does not require a training cycle in a batch form and has significantly lower complexity compared to the RHOUFIR one. Third, the RHUFIR filter can provide more robust estimation of the position and speed of objects on the sea surface when exposed to uncontrolled disturbances in comparison with the RHOUFIR filter and the DFK (IIR filter). This effect is achieved by the structure of the RHUFIR fitter, the averaging, and the careful selection of the horizon length.
Data Availability
The data used to support the findings of this study are included within the article. More precisely, all the necessary data for the simulation are given in Table 1.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Studies were supported by the Ministry of Education and Science of the Russian Federation (Project RFMEFI57818X0264).
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