Constrained State Estimation -- A Review

1 Constrained State Es timation - A Re vie w Nesrine Amor , Ghualm Rasool, and Nidhal C. Bouayn aya Abstract —The real-w orld applications in signal pr ocessing generally in volv e estimating the system state or parameters in nonlinear , non-Gaussian dynamic systems. The estimation problem may get ev en more challengin g when t here are physical constraints on th e system state. T his tutorial-style paper reviews the Bayesian state estimation for (non)linear state-space systems and introduces the formulation of constrained state estimation in such scenarios. Specifi cally , we start by providing a r eview of unconstrained state estimation using Kalman fi lters (KF) f or the linear systems a n d their extensions for nonlin ear st ate-sp ace systems, includin g ex ten ded Kalman filters (EKF), unscented Kalman filters (UKF), and ensemble Kalma n filters (EnKF). Next, we present particle filters (PF s) f or nonlin ear state-space systems. Finally , we review constrained sta te estimation u sing vario us fi l- tering techniqu es and highlight th e advantages and d isadvantages of the different co n strained state estimation approa ch es. I . I N T RO D U C T I O N The states of many dyn amical systems are confined with in constrained r egion s o w in g to relevant p hysical laws, geometric consideratio ns, and k in ematic limits, such as m a te r ial balance, maximum power rating , b ounds o n th e output of actuators and plants, and speed con straints in road networks [1], [ 2]. Th e mathematical formu lation of such co nstraints involv e s definin g a set of (non)line a r relationsh ips in the f orm of (in )equalities constraining the state estimates. One p ossible solution is to incorpo rate the co nstraints into th e state-space model of the system. Howe ver, generally , that is not possible withou t a significant incr ease in the co mplexity of the mod el. During th e state estimation pr ocess, tak ing constraints into account leads to accur ate and physically r elev ant estimates. For instance , the exploitation of a known road network has been pr oven effecti ve in the track in g of grou nd vehicles [3]. Similarly , in a maritime scenario, the knowledge o f shipping lanes and sea/land distinction can improve the tracking and detection perfor mance [4]. The state-space representatio n p rovides an extrem ely flexi- ble f ramework for modelin g d iscrete-time dy namical systems using two math e m atical relationsh ips, i.e. , the state tran sition and observation. The former captures the ev olution of the hidden state over time , an d th e latter pr ovides the noisy mea- surement of a (non )linear function of the state. The d escriptive power of the state-space rep resentation com e s at the expense of intractability . Generally , it is not p ossible to ob tain an analytic solution to the state estimation prob lem except for a small number o f cases, e. g., linear systems with Gaussian noise. In the Bayesian f r amew o rk, th e inf erence o f the hidden state given a ll a vailable ob servations relies u pon estimating N. Am or is with the Nation al Superior School of E nginee rs of Tunis (EN- SIT), Uni versity of Tunis, Tunis, T unisia, e-mail: (nisrine.amor@ho tmail.fr G. Rasool and N. C. Bouaynaya are with Department of Electrical and Computer Engineeri ng, Row an Uni versity , Ne w Jerse y , USA, e-mail: (ra- sool@ro wan.edu, bouaynaya@ro wan.edu) the posterior density functio n (pdf ). For the linear systems with Gau ssian noise, the clo sed-form o p timal so lu tion is g i ven by th e Kalman filter (KF). On the oth e r hand, for n onlinear and no n-Gaussian state-sp a ce mo dels, two funda m ental tech- niques h av e been emerging , param etric and non parametric [5]. The parametric methods include the exten d ed Kalman filter (EKF), u nscented Kalm an filter (UKF), ensemb le Kalman fil- ter (En KF) , and moving horizon estima tion. The n onpara m et- ric techniqu es are based o n seq uential Monte Carlo metho d s and inclu de particle filters (PF) [5]. This p aper pr ovides a tutorial-style re v iew of constrained state estimation m ethods f o r KF , EK F , UKF , EnKF , mov- ing ho rizon estimation , and PF . I n Section II, we start by presenting the pro b lem statement. Sectio n III provides de- tailed de scription of unconstrain e d Bayesian state estimation technique s. Section IV presents the literature available in constrained state estimatio n fo r KF , EKF , EnKF , and moving horizon estimation. In Sec tio n V, we formu late and revie w of the co nstrained PF problem. Finally , Section VI summ arizes and con cludes the pap er . I I . P R O B L E M S TA T E M E N T A. System De fi nition W e consider a g e neral state-spa ce repr esentation defined by the state tran sition and measu r ement mode ls for discrete-time systems, g iven b y : x t +1 = f t ( x t ) + w t , (1) y t = h t ( x t ) + v t , (2) where x t ∈ R n x and y t ∈ R n y represent the hidde n state and ob ser vation at time t , respectively , t ∈ N rep resents the time index and n x and n y are state and output dimen sions. In ou t settings, f t : R n x → R n x and h t : R n x → R n y are (non) linear mapp in gs, , and w t and v t are zero -mean process and measurement no ise with known PDFs p ( w ) and p ( v ) . Both noise sequences are un c orrelated with each other and the initial condition o f the state x 0 giv en by p ( x 0 ) . The mapp ing functions f t ( x t ) and h t ( x t ) can be defin ed in term of PDFs, i.e., p ( x t | x t − 1 ) = p w t ( x t − f t ( x t − 1 )) , (3) p ( y t | x t ) = p v t ( y t − h t ( x t )) . (4) B. Optima l State Estimatio n The state estimation prob lem aim s at finding the hidden state x t using all av ailable measuremen ts up to time t , y 1: t = [ y 1 , ..., y t ] . T h e solution to this problem is th e density of th e sy stem state conditioned on the measurem ents; either joint PDF p ( x 1 , . . . , x t | y 1: t ) or the marginal PDF p ( x t | y 1: t ) [5]. 2 W e make tw o assumptions here, (1) the sy stem state follows a first-order Markov process, i.e., p ( x t | x 1: t , y 1: t ) = p ( x t | x t − 1 ) a n d 2 ) mea su rements are condition ally indepen dent g i ven the sy stem state, i.e. , p ( y t | x 1 , ..., x t , y 1 , ..., y t − 1 ) = p ( y t | x t ) . In Bayesian estimation framework, a recursion can be de- fined to estimate po sterior PDF p ( x t | y 1: t ) using the prio r PDF p ( x t − 1 | y 1: t − 1 ) , tr a nsition PDF p ( x t | x t − 1 ) an d the likelihood PDF p ( y t | x t ) . W e co nsider the m arginal posterior, h owe ver, the sam e resu lts h old true for the joint cond itio nal PDF a lso . Using Bayes rule and Chapman -K olm ogorov e q uation, the posterior PDF can be co mputed recu rsiv ely u sing the following two-step relationship [5]: Prediction Step p ( x t | y 1: t − 1 ) = Z p ( x t − 1 | y 1: t − 1 ) p ( x t | x t − 1 ) d x t − 1 , (5) Update Step p ( x t | y 1: t ) = p ( y 1: t | x t ) p ( x t ) p ( y 1: t ) , = p ( y t | y 1: t − 1 , x t ) p ( y 1: t − 1 | x t ) p ( x t ) p ( y t | y 1: t − 1 ) p ( y 1: t − 1 ) , = p ( y t | x t ) p ( x t | y 1: t − 1 ) p ( y t | y 1: t − 1 ) , = p ( y t | x t ) p ( x t | y 1: t − 1 ) R p ( y t | x t ) p ( x t | y 1: t − 1 ) d x t . (6) For a general case with non linear state transition f t ( x t ) and ob servation fun c tions h t ( x t ) an d non -Gaussian noise sequences w t and v t , eq s. (5) and (6) ar e only a c onceptua l solution owing to the intractable integrals. For the case of linear systems with Gaussian no ise, a closed-fo rm optimal solution in the form of KF exists [6]. I I I . U N C O N S T R A I N E D S TA T E E S T I M AT I O N A. Kalman Fil ters for Line a r Systems with Ad ditive Gau ssian Noise The state tran sition f unction f t ( x t ) and the observation h t ( x t ) fun ction are both linear fun ctions and are rep resented by matrices F t ∈ R n x × n x and H t ∈ R n y × n x respectively . The noise sequen ces are both add iti ve an d Gau ssian, and defined as w t ∼ N ( 0 , Q t ) and v t ∼ N ( 0 , R t ) . Th e initial system state x 0 is also assumed to be kn own with a Gaussian d istribution N ( x 0 , P 0 ) . Th e Kalman filter, a minimum me an sq uare err o r (MMSE) estimator , is defin ed with two steps, th e prediction step and th e upd ate step. W e use ˆ to repr e sent the estimate, T for the matrix tran spose operation , and − and + symbols present quantities before an d af ter a state or observation upd ate is applied . Prediction step ˆ x − t = F t − 1 ˆ x + t − 1 , (7a) P − t = F t − 1 P + t − 1 F T t − 1 + Q t , (7b) P xy = P − t H T t , (7c) P y = H t P − t H T t + R t , (7d) Update step ˆ y t = H t ˆ x − t , (8a) K t = P xy P − 1 y , (8b) ˆ x + t = ˆ x − t + K t ( y t − ˆ y t ) , (8c) P + t = P − t − P xy P − 1 y P T xy , (8d) where K represents the Kalman gain matrix. W e can show that KF is th e best linear un biased estimator (BLUE) for all linear systems with additive noise. B. Exten d ed Kalman F ilter (EKF) The EKF assumes that th e no nlinear state tr ansition f t ( x t ) and/or ob servation h t ( x t ) functions can be linearized using the T aylo r series [ 7], [8]. F t = ∂ f t ( x t ) ∂ x t     ˆ x + t and H t = ∂ h t ( x t ) ∂ x t     ˆ x − t . (9) The error covariance P t is propag ated ( using eqs. 7b, 7c, and 7 d) with the lin earized fun ctions F t and H t from eq. (9). The state estimate ˆ x − t (eq. 7 a) and ˆ y t (eq. 8 a ) are calcu lated using no nlinear f unctions f t ( x t ) and h t ( x t ) as given below: ˆ x − t = f t ( ˆ x + t − 1 ) , (10) ˆ y t = h t ( ˆ x − t ) . (11) The co mputation al com plexity f or the calculation of F t and H t matrices may p rohibit the use of EKF in some applications [8]. Fur th ermore, the lin e arization of nonlinear systems may introdu c e errors in the estimation of the state, an d in the worst- case, the filter m ay diver g e especially for h ighly nonlinear systems [9]. C. Unscented Kalman Fil te r (UKF) UKF add resses the issues raising fr om the approximatio n operation s and limit the perfor mance of EKF [1 0]. UKF uses a set of carefully cho sen determin istic sam ples, called sigma points , to pr opagate the mean an d covariance o f th e posterior distribution of the state. The set of sigma points cap ture the true mean and covariance of the posterior for the case of Gaussian distribution when p r opagated thr ough the non linear system. Considering the system state x t with mean ˆ x t and co- variance P t , we can ch ose sigma p oints X j,t ∈ R n x , j = 0 , 1 , . . . , 2 n using following relation ships [9]–[1 2]: 3 X 0 ,t = ˆ x t , (12) X j,t = ˆ x t +  p ( n + λ ) P t  j j = 1 , . . . , n (13) X j,t = ˆ x t −  p ( n + λ ) P t  i − n j = n + 1 , . . . , 2 n (1 4) W ( m ) 0 = λ n + λ , (15) W ( c ) 0 = λ n + λ + (1 − α 2 + β ) , (16) W ( m ) j = W ( c ) j = 1 2( n + λ ) , j = 1 , . . . , 2 n, (17) where  p ( n + λ ) P t  i represents the j th row of the matr ix square root. λ = α 2 ( n + κ ) − n is a scalin g param e ter . α determines the spr ead of the sigma p oints aroun d th e mean ˆ x t and is set to a small value, e.g ., 1 e − 3 [10]. κ is secon dary scaling par ameter and is genera lly set to 0 . β en c o des the p rior knowledge abo u t the distribution of x and is set to 2 for the Gaussian distribution [1 0]. The sigma p oints [ X j,t ] 2 n j =0 (n-dime n sional vectors) are passed thro ugh the no nlinear fun ction, i.e. , Y j,t = g ( X j,t ) , j = 1 , . . . , 2 n, (18) where g is a nonlin e ar fun ction, e.g ., state transition f or o b- servation h . The m ean and covariance after the transfor mation can be calculated u sing: ˆ y ≈ 2 n X j =0 W ( m ) j Y j , (19) P y ≈ 2 n X j =0 W ( c ) j [ Y j − ˆ y ] [ Y j − ˆ y ] T . (20) The mathematica l r elations described in eq s. ( 12) - (20) are referred to as unscented tra n sformation (UT). UKF uses UT (eqs. 1 2 - 20) and KF re la tio ns (eqs. 7 and 8) to estimate the state mean an d covariance withou t having to lin earize the non linear functio n s f t and h t and calculating the Jacobia n matrice s. Howe ver, UKF ha s its own lim itations, e.g., it may n o t work well with systems that have nearly singular covariance matrices. Th is problem is linked to the matrix square roo t op eration perfor m ed on the covariance matrices using Cholesky d ecomposition , which , in tu r n, can be compu tationally deman d ing also. It is also impor ta n t to mention that all thr e e filter s describ ed above, i.e., KF , EKF and UKF assume that the p rior d istribution f u nction f ollows Gaussian distribution. The appr oximation s o btained with at least 2 n + 1 sampling points are accura te to th e 3 rd order for the add iti ve Gaussian noise a n d for all types o f no nlinear function s and at least to the 2 nd for non -Gaussian inputs [13]. D. Moving Ho rizon Estimation ( MHE) Moving horizon estimation ( MHE) is an optimization ap- proach that can be used to estimate th e unk nown system state [14], [15], [16], [17]. MHE em p loys an iterative p r ocedur e that relies on linear or non linear pr o grammin g to find th e desired solution, i.e., the system state. For maxim um a posteriori estimate, the state estimation p roblem can be expressed as: ˆ x t := a rgmax x t p ( x t | Y t ) . (21) For a movin g hor izon of size h ∈ { 0 , t } , we can determin e ˆ x t − h , . . . , ˆ x t = ar gmax x t − h ,...,x t p ( x t − h , . . . , x t | Y t ) . (22) Using Markovian assumption and Bayes rules, we get p ( x t − h , . . . , x t | Y t ) ∝ t Y j = t − h p ( y j | x j ) t − 1 Y j = t − h p ( x j +1 | x j ) p ( x t − h | Y t − h − 1 ) . (23) W ith Gaussian assump tion and tak in g logar ithm, we have argmin x t − h ,...,x t t − 1 X j = t − h k y j − h j ( x j ) k 2 R − 1 j + k x j +1 − f j ( x j ) k 2 Q − 1 j + k x t − h − ¯ x t − h k 2 Π − 1 t − h , (2 4) where the last term is the arriv a l cost and f or t = h, Π t − h = Π 0 , i.e., the initial covariance of the state estimate at time t = 0 . One motiv a tio n for th e development of MHE was to fo rmu- late a mathem a tical op timization problem where constraints on the system state can be naturally in corpor ated in to the estimation (n ow optimization ) f r amew o rk [1 8]. E. Ensemb le Kalman Fil ter (En KF) The ensemble Kalman filter (EnKF) belo n g to the a b r oader class of sequ ential Monte Carlo methods [19], [2 0], [21]. In the most gener al fo rm, an En KF is based on th e pr emise th at it is sufficient to estimate first two mom ents of th e p robability density fun ctions of in terest, i.e., p ( x t | Y t − 1 ) an d p ( x t | Y t ) , f or the time-u p date and me asurement-u pdate steps [ 22]. EnKF is initialized with N samples (or particles) tha t are sampled fr om a given probab ility distribution function . At e a ch subsequen t time step, N samp les are dr awn from pro cess and ob servation noise d istribution functions and pr opagated throug h system dynamics to compute a set o f transform e d particles. Giv en tha t we have N p articles { ˆ x i, + t − 1 } N i =1 , Prediction step ˆ x i, − t = f t − 1 ˆ x i, + t − 1 + w i t (25a) ˆ y i, − t = h t ( ˆ x i, − t ) + v i t (25b) ¯ x − t = 1 N N X i =1 ˆ x i, − t (25c) ¯ y − t = 1 N N X i =1 ˆ y i, − t (25d) P xy = 1 N − 1 N X i =1 [ ˆ x i, − t − ¯ x − t ][ ˆ y i, − t − ¯ y − t ] T (25e) P y = 1 N − 1 N X i =1 [ ˆ y i, − t − ¯ y − t ][ ˆ y i, − t − ¯ y − t ] T (25f) 4 Update step K t = P xy P − 1 y (26a) ˆ x i, + t = ˆ x i, − t + K t [ y t − ( h t ( ˆ x i, − t ) + v i t )] (26b) ¯ x i, + t = 1 N N X i =1 ˆ x i, + t (26c) P t = 1 N − 1 N X i =1 [ ˆ x i, + t − ¯ x i, + t ][ ˆ x i, + t − ¯ x i, + t ] T (26d) The estimate accuracy of the En KF d epends on th e numb er of samples N [23]. The difference between EnKF and particle filters is that E n KF makes the assump tion that a ll pro bability density function s are Gau ssian and can be represented only by the mean an d covariance. EnKF can be used for systems with non-Gau ssian pro bability distribution function s. Howe ver, as only first two m o ments a re being used in the prediction and the update steps, the estimatio n results may not be accu rate for systems that d o n ot follow the Gaussian assumption. V arious m e th ods, including E KF , UKF , MHE, and EnKF may not accur a tely estimate systems state when th e underlyin g system d ynamics are hig hly no n linear an d/or the noise in the system does not fo llow Gau ssian d istribution. particle filters (PFs) are able to han dle no n linear sy stem s with non-Ga u ssian noise. PFs are flexible and simp le simu lation-based n umerical approa c h es used for estimating the system state in a sequ e n tial manner . F . P article F ilters ( PFs) PFs solve th e op timal estimation p roblem in non linear and non-Gau ssian d ynamic sy stems by incor porating sequential Monte Carlo sampling within the Bayesian filtering f rame- work [24] [25] [26], [27] [28]. The go al is to estimate the posterior d ensity for th e state using Bayesian recursio n . PFs approx imate the posterior PDF using a gro u p o f samp les (also called particle s) x i t and their associated weights w i t ≥ 0 as: ˆ p ( x t | Y t ) = N X i =1 w ( i ) t δ ( x t − x ( i ) t ) , (2 7 ) where δ ( . ) rep resents the Dirac delta fun ction and N is the number o f particles. The con ditional mea n estimate of the state is given by the weighted mean of the par ticles as follows: b x t = E [ x t | Y t ] ≈ N X i =1 w ( i ) t x ( i ) t . (28) The particles are req uired to be sampled from the true posterior PDF , which is not av ailab le. Th erefore, ano ther PDF , usu a lly r e f erred to as th e importanc e distribution or the pr oposa l d istrib u tion q ( x t | x t − 1 ) , which is gen erally easy to sample fro m, is d efined [29]. The impo rtance weight o f every particle is given by: ˜ w ( i ) t = w ( i ) t − 1 p ( y t | x ( i ) t ) p ( x ( i ) t | x ( i ) t − 1 ) q ( x ( i ) t | x ( i ) t − 1 , y t ) , (29) w ( i ) t = ˜ w ( i ) t / N X j =1 w ( j ) t . (30) It has been shown tha t PFs co nverge asympto tically , as N → ∞ , towards th e op timal filter in the mean square er ror sense [24]. The selection of co rrect pr op osal distribution q ( x t | x t − 1 ) is an im portant step in using PFs. Some studies have prop osed to u se either EKF [30] [31] [24] or UKF [32] to generate the importan ce PDF . At each step, an EKF or UKF is run f o r each par ticle to gener a te the m ean and the covariance of the propo sal PDF . Later, th e par ticles are dr awn from the n ewly found PDFs. The obvious advantage is that EKF and UKF take into accoun t the most recent measuremen t while estimating the mean and the covariance. Further d iscussion on the prop o sal PDFs can be f o und in [24], [ 33], [34] an d [28]. Despite the selection of appro priate propo sal den sities, th e sequential im p ortance sampling algorithm may degener ate an d never c o n verge. The normalized we ig hts o f all but on e pa rticle degenerate to zero, referr ed to as samp le impoverishmen t . In order to av oid such a degeneracy p r oblem, re-samplin g is usually perfor med. Re-sampling will eliminate p articles with low we ight and mu ltip ly samples with h igh weigh ts. Re- sampling a lg orithms are discussed in [35], [36]. T he algor ithm 1 details the steps fo r particle filterin g. Algorithm 1 Particle filter Generate x ( j ) 0 ∼ q 0 ( x 0 ) , then calcu late w i 0 = p ( y 0 | x i 0 ) and normalize the weights. for t = 1 , 2 , · · · , T (wher e T : tim e length) do for i = 1 , 2 , · · · , N (where N is the numbe r of p articles) do Generate new samples fr om an accessible pro posal distribution x ( i ) t ∼ q t ( x t ) . Calculate the weights w ( i ) t of x ( i ) t ; then , normalize the weights. end for Re-sample to obtain equally weighted particles { x ( i ) t , 1 N } N i =1 . Compute the weighted me an ˆ x t = N P i =1 w ( i ) t x ( i ) t . end for 5 I V . C O N S T R A I N E D S T AT E E S T I M A T I O N Many en gineering applications, such as vision- based sys- tems, chemical pr ocesses, target track ing, biomedical systems, navigation and robotics, can be mo d eled using state-space framework. In most of th e se dynamical system, th e system state may be subject to constraints that arise from physical laws, natur al p henomen a, o r mo d el r estrictions [6], [3 7], [ 38]. There may not be an easy and v iable way to incor porate these constraints directly into th e state-space mode l or the estimatio n framework [39]–[4 1]. W e consider a set o f co nstraints C t , in c lu ding lin ear and nonlinear, defined as: a t ≤ φ t ( ˆ x t ) ≤ b t , (31) where φ t represents the con straint fu nction at time t . In the case wh e n φ is an identify tran sformation , eq. (31) reduces to a simple inter val con straint on th e mean of the state. The con straints C t can be hard or soft, wher e the estimation algorithm s req uired to satisfy soft constrain ts appro x imately [42]. A. Linear Systems Su bject to Constraints 1) Linea r Constraints: For line ar Gau ssian systems, linear constraints can b e incorp orated by directly within the KF framework as presented in eq s. 7 an d 8. The widely u sed methods a re based on mo del redu ction meth odolog y [43], [4 4], [45], p seudo-m e asurements, also called perfect measurement [46], [47], [4 8], [4 9], state estimate projec tio n [ 47], [ 44], [50], and gain proje c tion [47], [44]. 2) Non lin ear Constraints: For nonlinear constraints, a closed-for m solution m ay no t be po ssible, even for linear systems. The adopted method s r e ly on linear ap p roximatio n of the non linear con straints using T aylor series expansion [51], [50] or on dir ect num erical o ptimization of the nonlin e ar problem [43], [1 4]. Iterative ap p lication of the co nstraint linearization op eration at each measurem ent time is used to get closer to the con straint satisfaction with each iteration [52]. Furthermo re, PDF tru n cation m e th ods have also be e n p roposed that truncate Gau ssian PDFs estimated b y KF at the c o nstraint bound s [8], [53]. B. Nonlin e a r Systems S ubject to Co n strains Linear c o nstraints can be a d ded directly in various esti- mation fr amew o rks, i.e., EKF , UKF , o r En KF . The nonlin ear constraints can linearized and then Incorp orated in the filter . 1) Con strained State Estimation Usin g E KF: The system model (both dynamics and observation) and constrain ts can be linearized and used within the E KF fr a mew o r k for the constrained state estimation [54]. In some cases, th e con ver- gence pr operties of EKF can b e improved by a p plying the filter iteratively fo r enfor cing the con straint [55], [56]. For the e quality co nstraints, the m easuremen t- argumentation approa c h can be u sed [ 57], [58], [59], [60]. Th e measurem ent model can b e augmented using th e equ ality con stra in ts and then E KF ca n b e used for the linearized o bservation model. Obviously , m easurement- argumentation approach is limited to equality co nstraints only . Smoothing con stra in ed kalman filter (SCKF) can be used with the EKF [52]. The SCKF co n straints the system itera- ti vely an d the final co nstraint solu tion may n ot be accura te. This appr oach is on ly valid for eq uality constraints. A mo dified extended Ka lman filter h a s been intr oduced by Prakash et al. to hand le with the con straint imposed on the state estimation for a non -linear stochastic dy namic system [56]. Hence, Prakash et al. pro posed two schemes to mod- ify the prior and po sterior distributions based on generating samples from trunca ted m ultiv ariate nor mal d istribution. Recursiv e Nonlinear Dynamic Data Reconciliation (RNDDR) suitably combin e advantages of both EKF an d Nonlinear Dynamic Data Reconciliation (NDDR). The NDDR is a nonline a r optimiza tio n b ased strategy to estimate system par ameters a nd states. Du e to the optimization -based formu latio n o f th e prob lem, constraints o n states and u nknown parameters can be added in a natu ral way . Howe ver, the application of the NDDR for online estimation of states and parameters can be c o mputation ally pro hibitive due to at each time-step, a non linear co nstrained o ptimization problem is solved. Recently , Zixiao e t al. intr o duced a constrained du al ex- tended Kalman filter algorith m th at works in an alternating manner [61]. Th is algor ithm is b ased on a par a meter estima- tion and state prediction tec h nique. Th e inequ a lity co nstraints have been incorp orated using o ptimization procedur e. This propo sed alg o rithm is more d ifficult to imple m ent. Howe ver , it ha s the advantage of better co nvergence po tential and algorithm stability . In [62], the inequality constraints have been incorporated into the EK F u sing a gradient projection meth od. The per- forman ce of the prop osed techniqu e has been tested using synthetic mo del based on Gaussian functio n s. 2) Con strained Sta te E stima tio n with UKF: Many ap- proach e s h av e been proposed to incorp orate the c o nstraints within UKF . During the upda te step, sigma p oints can be projected onto the co n strained interval using sigm a point pro- jection ap proach [63], [ 64]. This a pproach perf orm p rojection after generatio n of sigma poin ts and then after passing sigma points thr o ugh system d y namics. Optimization in m easurement- u pdate has bee n prop osed to reform u late the measurem ent-upd ate step to integrate con - straints within the standar d UKF [6 5]. This technique propo ses to solve a quad ratic o r no n linear o p timization problem at each step of the alg o rithm. Unscented Recursive Non linear Dyn a mic Data Reconcilia- tion (URNDDR) is an extension of RNDDR where the EKF is replaced with the UKF [66], [67], [6 8]. All types of constraints are taken in to considera tio n b y solving th e con strained opti- mization proble m for each sig m a point. The algorithm may b e computatio nally expensive be cause the optimizatio n pro b lem is solved for a ll sigma poin ts. Recently Kadu, Mand ela et al. intr oduced improvements to the URNDDR algor ithm to address gener al constrain ts in the ge n eration pr ocess o f sigma points and comp utational issues [69]. Julier and LaV iola pr oposed a two-step appr oach fo r tack - ling nonlinea r equality constraints. Using the UKF a pproach , all the selected sigma points are projected onto the con stra in ed 6 surface individually in the first step, while in the second step the fina l estimate by the filter is a g ain p rojected o nto the co nstrained sur face [ 70]. Th e authors presented a details discussion o n the need for two-step pr ojection fo r nonlinear constraints both for samples and their statistics ( i.e., moments, which are expected value and covariance). T eixeira et al. extended th e UKF for different types con- straints. These appro aches includ e eq u ality co nstrained UKF , projected UKF , measureme n t-augme n ted UKF , c o nstrained UKF , co nstrained-in terval UKF , inter val UKF , sigma p oint UKF , tr uncated UKF , truncated interval UKF , p rojected- interval UKF [ 4 8], [71], [7 2], [73], [7 4]. PDF trunca tion approa c h applied b y T eixeira et al. is applicable for linear interval co nstraints only [ 8]. Straka et al. p roposed a truncated u nscented kalman filter (TUKF) algo rithm to solve the non-linear and no n gaussian system with constraints on th e state estimatio n [75]. The ma in idea of TUKF is to use th e PDF truncatio n ap proach . Recently , a tr uncated random ized unscen ted Kalma n filter (TR UKF) was p r esented in [76]. T h e TR UKF is based on th e random ized unscented Kalman filter algorith m (R UKF) and a pdf truncation techn ique. The main idea of T RUKF scheme is to intro duce a trun cation step within R UKF th at was pr oposed for th e unc o nstrained estimation pr oblem. Howe ver, the app li- cation of TR UKF on synthetic d ata results in comp utational costs com pared to UKF and TUKF ( tr uncated UKF) [7 6]. Alireza et al. p roposed a novel ap proach called the con - strained iterated unscented Kalm an filter (CIUKF) [77]. This approa c h com b ines th e advantage of iteration s and use o f constraints to provide an accu rate b ounded d ynamic state estimation. The co nstraints in the IUKF are in corpor ated b y projection the sigma poin ts that ar e outside th e feasible region to the b oundar y o f this region to obtain co n strained sigma points. Calabrese et a l. in troduced an approach to integrate the constraints within the UKF based on two ma in ly appr oaches [78]. In the first app roach, all sigma points that violate the constrained region are moved onto the feasib le region du ring the p rediction step. In the second app r oach, all tran sformed sigma p oints that violate th e co nstrained region a r e projected to co nstraints bou ndary only when the updated state estimate exceeds the bo undary in the cor rection step. V ar iants o f the algo rithms seek to imp rove th e p erform a nce and computation al issues of the original UKF-b ased meth od under add itional constraints [ 7 6], [79], [ 80]. 3) Con strained state estimation using EnKF: A constrained state estimation appr oach can also use the EnKF [81], [8 2], where an initial ensemble o f samples are dr awn fr om a truncated normal d istribution. For each iteration after th e prediction step, a transfor mation is ap plied to pro jec t the violating samples onto the bo undary . Parakash et al. presented co nstrained state estimatio n using the E nKF [22], [83]. The a u thors pr opose to gen e rate gro u p of initial samples fro m a trun cated norm al distribution. Late r, for each iteration af ter pre d iction step, a transforma tion is applied to p roject the v iolating samples on th e boun dary . A Constrained Dual Ensemble Kalm an Filter (dual C-EnKF) has emerged recently in [ 82]. The d ual C-EnKF algorith m combines the C-EnKF algorithm pro posed by Prak a sh in [81] for in corpor a tin g constraints within EnKF and the dual EnKF algorithm p roposed in [84] to r educe th e numb er of particles. Raghu et al. introduc e d two a lgorithms to incorp orate the constraints into th e EnKF [85]. The first algorithm uses th e projection -based me thod. The second algo r ithm r elies on the use of a technique fo r sof t constrained cov a riance localization. Simulations results showed that the second propo sed algor ith m provide better estimatio n of the unkn own states comp ared to the first pro posed alg orithm. 4) Con strained state estimation using MHE: It is evident from formu lation of the MHE ( 24) that constra ints c an be incorpo rated into its fra m ew or k in a n atural way [14], [15], [16], [ 8 6], [87]. Howe ver, there a re multip le issues with MHE framework, i.e., 1) the com putational effort especially fo r nonlinear o ptimization pro blem (constra in ts an d/or o b jectiv e function s) [15], [87], [8 8], [89]; 2 ) calculation o f the a r riv al cost [90]; 3) Gaussian assumption f o r d ensities th a t results in simplification o f th e re la tio ns, i. e ., f rom Eq . ( 23) to (2 4); and 4) selection of o ptimal horizon ( h ) size to balance perform a nce and com putationa l load [9 0]. Recently , Garcia Tirado et al. introdu ced an appr oach fo r constrained estimation pro blem depen ds on th e MHE and the game theor e tical ap proach to the H ∞ filtering with constraint handling [ 9 1]. The theor e tica l of the p roposed ap proach with constraints mainly based on a mod ified L yapu nov th eory f o r optimization -based systems. It is impor tant to emph asize tha t almost all m ethods th a t we present fo r th e co nstrained state estimation have an un derlying assumption of linearity o r Gaussianity , an unrealistic presump - tion in most real-world applica tio ns. Mo reover , the p r esence of constraints, such as bound s, on the states implies that the condition al state densities are non-Gau ssian. Furth ermore, the UKF and EnKF-ba sed meth o dologies systematically constrain all their sigma po ints and ensemble samples with no mathe- matical gro und or ju stification. V . C O N S T R A I N E D S TA T E E S T I M A T I O N U S I N G P A RT I C L E F I L T E R S Particle filters are wid ely used for laten t state e stima - tion/trackin g in d y namic systems wh ere systems dy namics or observation mod els are no n linear, or the system/observation noise are non - additive or fo llow non- Gaussian distributions [29]. The tech nique of PFs is b ased on powerful sampling that is aim ed to find an op timal estimate by explo iting a set of random weighte d sam ples called the particles. Th ese particles are used to appr oximate the posterio r den sity of the state and later fin d th e statistics of in terest [2 9]. Du e to the comp lex nature of computation s in PFs, it is not straigh tforward to incorpo rate constrains o n th e latent state. System atic efforts to incorpor ate constrain ts impo sed o n the unkn own state in PFs are limited an d heur istic in n a ture. A. Accep tance/Rejection A ppr oach An acceptance- rejection ap proach was prop osed for nonlin - ear inequ ality constraints [92], [93]. This appro ach focused on retain ing p articles tha t fell within the constrained inter val 7 and rejectin g all violating con straint region. Howe ver, the ir approa c h d oes no t make any assum ption on the d istributions and can gu arantee the validity of p a rticles, an d th us retain s the gene r al prop erties of the p article filter . Besides, in certain cases, the nu mber o f p articles may redu ce wh ich may further lead to a decrease in e stima tio n accu racy and compu tationally efficiency . In such cases, most of the p article may vio late the constrain t and the algorithm may fail. Also, uncon strained samp ling from followed by verification against constrain ts (especially nonlinear ) may be compu tatio nally mor e demandin g than sampling directly fro m the constrain e d region [90], [92], [ 93]. B. Optimization /Pr ojection Based Appr o a ch Shao et al. p resented a two stages ap p roach to d eal with constraints [94]. In the first stage, a set of par ticle s were drawn rand omly without consider ing state constraints. In the second stag e , all particles which did not satisfy c o nstraints, were pro jected into th e feasible region using an o ptimization formu latio n. However , by applying op timization m ethod to impose the particle to be within the con straining in terval, the obtained particles are no longer co nsidered as repr esentativ e samples of the p osterior distribution of the state. At e very sampling instant, solv ing m u ltiple optimization p roblems may make the algorithm co mputation ally expensiv e . Xiong et al. p roposed an adaptive c onstrained p article filter (A CPF) algorithm that uses all particles to accura te ly estimate the state [95]. Th e appro ach is based o n sample size testing that allows calculatio n of th e numb er of particles n eeded to obtain a desired state estimate. Xion g et al. c ombined th e sample size test with the g eneric p article filter to deal with constraints an d to ad dress the particle numb er pro blem in PF . The simulation results showed th at, in terms of ro ot m ean square error (RMSE) perfo rmance, convergence, an d ru nning time, th e A CPF approach exhib its a be tter estimation com pared to the constrain ed PF pro posed in [9 4] an d u nconstrain e d PF . Li et al. used a series of constrained optimizatio n to incorpo rate the inequ ality co nstraints into the auxiliary particle filter by modify ing the priori pdf [9 6]. The aux iliary particle filter con sists of resamp ling and sampling step s at each time step. In the resampling steps, it selected particles with a lower likelihood an d/or far from the f easible region. It then perfor med a series of c onstrained optimization to tran sform the ce n ter of transition distribution into a fe asible region. Recently , Hongwei et a l. introduc e d a constrain ed m ulti- ple mo d el particle filtering (CMMPF) method to solve the constrained hig h d imensional state estimation p roblems [9 7]. The p roposed ap proach divided the pro b lem of target tracking into two sub-pr o blems: i) motion model estimation an d m odel- condition ed state filtering as stated in the Rao–Blackwellised theorem; ii) Th e hidden state estimation is formulate d usin g the m ultiple switching d ynamic mod els in a jump Markov sys- tem framework. In order to inco rporate the constraints within the prop osed appr o ach, a mo d ified sequential importan ce resampling ( MSIR) method b ased on a series of o ptimizations is used to gener ate mod e l par ticles that can b e co nstrained in the f easible region. C. Con strained Imp ortance Distributions The pa rticles can b e drawn from a propo sal d istribution having suppor t on the constrained region only . T h ere a re different appr oaches to inc o rporatin g constraints in the PF using the pro p osal distribution presented in literatur e. Density truncation can be perfo rmed analytica lly in ca se of m ulti- variate Gaussian distribution [53], [98]. Perfect M o nte Carlo simulations can be used to estimate first two momen ts of the truncated prop osal distribution. Samp les ar e drawn from a distribution o f interest and a ll con straint-violatin g samples are rejected . Leftover samples are u sed to estimate m ean and covariance of the trun cated PDF . The impo rtance distribution c a n be c o nstrained also, i.e. , in th e samp ling step, particles ar e dr awn f rom an impor tance distribution which has its support on the co nstrained region only [ 99], [ 100]. Constrain ed-EKF , con stra in ed-UKF and or constrained -EnKF are used at each iteration to gen erate co n- strained pr oposal d istributions. Specifically , a co n strained-filter (EKF , UKF or En KF) is used f or each particle and an estimate of mean a nd covariance of con strained distribution is foun d. Particles are drawn from the newly found p roposal distribution. An an alytical solu tion f or PDF tru ncation is also p roposed [99], [1 0 0]. Straka et al. [ 75] prop osed a tru ncated unscen te d particle filter to inco rporate the co n straints in the un scented particle filter . Proposal density is generated using the UKF and b efore sampling particles fro m it, a trun cation p rocedu r e, in accor- dance with constraints on the system, is perfor m ed to f orm a truncated propo sal d ensity . Th e trunca te d pro posal density is for med using perfect Monte Carlo or impor tance-samplin g. The trun cation pr ocedure pr oduces mean and variance of the propo sal den sity , which is assumed to be Gaussian. On the other hand, Y u et al. p roposed a truncated unscented particle filter to handle n onlinear con straints [101]. Th eir technique com bines both PF based accep t/reject ap proach and th e un scented Kalman filter . Au th ors start by dr awing the imp ortance d istribution fo r sampling new particles u sing the u nscented Kalman filter and th en ap plied the tru n cated probab ility e stimation using accep t/reject app roach. Howe ver, this techniqu e also used the accept/reject ap proach and thus suffers fr om same lim itations as m entioned earlier . Similarly , Pirard et al. proposed two approa c h es to solve the pr oblem of target track ing in the presence o f constraints [102]. Th e first app roach was ba sed on the extensio n of the Rao-Blackwellized PF (RBPF) to handle hard constraints. The technique of RBPF o r m arginalized PF can be used only when the state can be divided into two parts such as a linear part and a non -linear pa r t, an d wh ere th e constraints only de p end on the non-lin ear part. This appro a ch used the accept/rejection technique that ma y le a d to a r eduction the estimation ac c uracy . The second approach was built on the propo sal distributions adapted to the constraints. This appro ach dr ew a prop o sal distribution which guar anteed that particles were drawn f rom constrained regions. Th is could be quite ineffecti ve when a substantial part of the dr awn p articles is located in o utside constrained region s. Ungarala et a l. pr oposed con strained Bayesian state es- 8 timation using a cell filter [103], where a Markov chain is constructed by sampling the dy namics over co n straints. Howe ver, this app roach is limited to low dimensiona l system s due to expon e n tially increasing memory req uirements of the state transition o perator with the state dim ension. In addition, Ungarala [ 1 04] pr oposed a direct samplin g pa r ticle filter for linear and nonlinear constraints, howev er, its applicab ility is limited to Gaussian assum p tion fo r all p dfs. Zhao et et al. [ 105] propo sed three strategies for con - strained state estima tio n using pa r ticle filters. The first strategy ensures that the samples are dr awn fro m the c onstrained region u sing a constrain ed in verse tran sform techn ique. Usin g bound s (interval con straint) on th e state vector , bou nds on th e process noise are fou n d. Proc e ss noise samples are drawn fr om the con stra in ed co mmutative d istribution f u nction (CDF). In the second strategy , an accep tan ce/rejection scheme is used after re-sam pling and all v iolating particles are rejected . The third strategy deals with par ticles after resamp lin g. V iolating particles are deleted an d non -violating particles ar e regenerated to ensure tha t there are more n o n violating par ticles in the final estimate. T h ere are a few prob lem s with th is ap proach . First, it may no t be possible to find the CDF of the n oise in the first strategy . Also, the constraints on the state ma y n ot be of the interval-type, i.e, m ay have a non linear fo r m, then the first strategy is not ap plicable. For the second strategy , the acceptance/r e jection sch eme may result in r ejection of most of the particles in the worst case, lead ing to failure of th e particle filter . Th e thir d strategy g enerates mu ltip le nu m ber of state estimates (o ne for each de letion/regeneratio n) and an optimization p roblem is solved to fin d th e right state estimate out of all available. Essentially , anoth er optimization p roblem has to be solved wh ich may be compu tationally expen si ve. In ad dition, Zh ao et al. p roposed three strategies for con- strained state estimation [106]. Th e first strategy focused o n constrained prior particles using inverse an d Gib b s samplin g. This strategy used first the Gib bs sampling to compu te the constraint in terval of each variable in the state vector, an d then generated valid p articles using constrain ed inverse tran sform sampling. Howev er, the generation of each particle required computatio nally expensive p rocesses of con strained region calculation and then an d then g eneration of particles. I n the second prop osed strategy , an accept/reject scheme is used to constraint th e po sterior par ticles. This scheme m a y re sult in the rejection of many particles in the worst c ase, lead ing to failure of the PF . T h e third p r oposed strategy imposed con straints o n the state by scaling th e weights of the particles. Howe ver, the scaling of weigh ts in this way contr adicts b asic PF theory . In su mmary co nstrained state estimation includes ac- cept/reject approach es, projectio n/optimizatio n based method s, and constrained importan ce sampling distribution. W e ref erred to these schemes as “POintwise DEnsity T r u ncation” (PoDeT) [107]. All PoDet methods impo se constra ints on all particles of the PF and thu s constra in the poster io r distribution of the state estimate rath er than its mean. This m ay lead to mo r e stringent cond itions th an actually de sire d an d may also result into possibly irrelevant co nditions than the origin al constraints [108]. D. Challenges in Constrained P article F iltering The particle filtering represen ts the state-of - the-art fo r es- timation in non linear/non -Gaussian dynam ical systems; how- ev er, incor poration of co nstraints on the hidden state (e.g., non-n egati v ity) is a ch allenging pr o blem. 1) On The Conver gence o f Constrained P article F ilter- ing: By constrainin g every particle in the PF , the PoDeT approa c h will always result in an estimation error of the posterior density un le ss th e density has a bou nded supp ort. I n particular, if the unconstrain ed distribution natur ally satisfies the c o nstraints and ha s unb ounde d supp ort, the PoDeT will fail in ren dering the unco nstrained density . An ev aluatio n of the Po DeT estimation error is d iscussed in [108], wh ere we d eriv ed th e o p timality boun ds of the PoDeT ap proach . W e have shown that the estimation erro r is b ounded by the a r ea o f the state posterior d ensity th at do es no t inclu de the constrain ing interval. Specifically , if the density is well localized, i. e . , most of th e un constrained posterio r density were inside the constrain ing inter val, then th e PoDeT error will be upper b ounde d. On anoth er hand, if th e den sity is not we ll- localization, i.e., the constrained interval was did n ot co ntained most of the posterior distribution, then the PoDeT error will be bound ed from b e low . In p articular, if the co nstraining in te r val covers a small ar e a of the den sity , then the Po DeT er ror will be large [1 08]. 2) Mean Density T ru ncation: Based on th e PoDeT es- timation er ror results, the mea n density trunca tion (MDT) method has bee n recently pro posed in [109]. MDT d eals with errors o f the PoDeT by proposing a n ew strategy to satisfy the constrain t on the c onditiona l mean estimate rather than the p osterior den sity itself [109]. Th e auth ors gener ated N uncon stra in ed particles from th e importance distribution as in th e standar d PF . If these N weighted p articles satisfied the constrain ts o n the me an, th e constraine d state estimate is considered op timal. If not, an ( N + 1) st pa r ticle was d rawn from a high p robability region to enforce constraint on th e weighted mean. In case if ( N + 1) st particle was not sufficient to g uarantee th e constraints on the cond itional mean , au thors generated add itional particles iteratively one by one ( 2 , then 3 , then 4 ..., etc.) until the co nstraints were satisfied. One drawback of this a p proach was e v ident whe n the proposal distribution was poor ly chosen. In that case, it may take a large number o f particles to move the mean o f the con ditional distribution into the feasible region . Conseque n tly , the iterative technique of g enerating pa rticles one by one may make the algorithm inefficient and co mputation a lly expensive. Later , au th ors in troduce d systematic and indu c ti ve proce- dure to ensur e that th e co nstraints were satisfied with th e generation o f N particles, referred to as the Iterative Mean Density T runca tio n (IMeDeT ) [ 107]. It is imp ortant to notice that, this perturbation ap proach remains very different from the PoDeT app r oach where the or ig inal c onstraint is imposed on every particle, wh ile the N -particle IMeDeT en forces the desirable constraint only o n the condition al mean estimate. Howe ver, the only drawback of IMeDe T is that in ductively choosing particles j = 1 , . . . , N to satisfy co n straints on the condition al mean at every time step m ay lead to co mputation - 9 ally expen si ve. Recently , we addressed the limitation s of MDT and I MeDeT by pro posing a new strategy based o n p erturbin g the un c o n- strained density with only one p a rticle [110]. W e referred to this techniq u e as Mean Den sity T runca tion (MeDeT ) . In MeDeT , we choose one p article in a special way to satisfy the constrain ts on the mean and construct a sequ ence of densities satisfying constraints [11 0]. W e start by gen e rat- ing N unconstra in ed p articles fr om the impor tance sampling distribution. If the conditional mea n estimate using th e N- order appr oximation satisfies the co nstraints, we retain these particles. Otherwise, after the resampling step, we re move a pa rticle with the lo west weight and located closest to the bound ary of the feasible region and replac e it with another particle that is drawn from the high pro b ability r egion. Th e process of remove/add one particle can b e v ie wed as the ”minimal p erturbatio n” of the u nconstrain ed d e nsity with using on ly one particle. Keeping in view the way constrained are im plemented during filtering, we can classify co n straints as soft constraints and hard constraints . In the techn iques of MDT , I M eDeT and MeDeT , the state satisfies the con straint on the conditiona l mean, so the con straints are considered soft constraints. On the o ther hand , for Po DeT , th e con straints are enforced on each par ticle and ar e con sidered h ard con straints. W e ho p e that this p aper encou rages fu r ther resear ch in the development of mor e algorithms that co n strain the state estimate ra th er than the den sity itself. V I . C O N C L U S I O N This p aper revie ws th e advances appro aches to incor porate the constra in ts within state estimation. W e p rovided a critical revie w of constrained Bayesian state estimation algorithm s using Kalman filter for linear systems an d extended Kalman filter , un scen ted K a lman filter , ensem ble Kalm an filter , moving horizon estimatio n and particles filters for n onlinear systems. A C K N OW L E D G M E N T This work was suppo rted by the National Scien ce Founda- tion u n der Grants NSF CCF-15278 22 and NSF AC I -1429 467. R E F E R E N C E S [1] N. Amor , G. Rasool, N. C. Bouaynaya , and R. Shterenber g, “Con- strained parti cle filtering for movement identifica tion in forearm pros- thesis, ” Signal Pr ocessing , vol . 161, pp. 25–35, 2019. [2] D. J . Albers, P .-A. Blanc quart, M. E. Le vine, E . E. Seyla bi, and A. Stu- art, “Ensemble kalman methods with constraint s , ” In verse Probl ems , vol. 35, no. 9, p. 095007, 2019. [3] C. Y ang, M. 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