1 Constrained State Estimation - A Review Nesrine Amor, Ghulam Rasool and Nidhal C. Bouaynaya . Abstract—Increasingly for many real-world applications in The parametric techniques are based on the extended Kalman signal processing, nonlinearity, non-Gaussianity, and additional filter, unscented Kalman filter, ensemble Kalman filter and constraints are considered while handling dynamic state esti- moving horizon estimation. The non-parametric techniques are mation problems. This paper provides a critical review of the state of the art in constrained Bayesian state estimation for based on Sequential Monte Carlo as known particle filtering. linear and nonlinear state-space systems. Specifically, we provide This paper provides a critical review of constrained a review of unconstrained estimation using Kalman filters for Bayesian state estimation methods, i.e, Kalman filter, extended the linear system, and their extensions for nonlinear state-space Kalman filter, unscented Kalman filter, ensemble Kalman filter, system including extended Kalman filters, unscented Kalman moving horizon estimation and particles filters. filters and ensemble Kalman filters. In addition, we present the particle filters for non linear state space systems and discuss The paper is organized as follows: Section II presents recent advances. Next, we review constrained state estimation the problem statement. Section III reviews the unconstrained using all these filters where we highlighted the advantages and Bayesian state estimation framework. Section IV presents disadvantages of the different recent approaches. the literature available in constrained state estimation using Kalman filter, extended Kalman filter, unscented Kalman fil- I. INTRODUCTION ter, ensemble Kalman filter and moving horizon estimation. Due to physical laws, technological limitations, kinematic Section V introduces a large critical review of the constrained constraints, geometric considerations of many systems, such as particle filtering. Finally, Section VI summarizes the paper. material balance, bounds on actuators and plants, target speed constraints and road networks, the states of many dynamical II. PROBLEM STATEMENT systems are confined within constrained regions. Mathemati- cally, the constraints are given by a set of linear or nonlinear We consider a general state-space model defined by a state equalities or inequalities. In general, these constraints cannot transition and measurement models in a discrete form given be incorporated in the state-space model without a major by increase in the complexity of the model. Taking constraints x = f (x )+ w , (1) into account leads to physically meaningful and more accurate t+1 t t t h estimates. For instance, the exploitation of a known road yt = t(xt)+ vt, (2) network has been proven effective in tracking of ground where x Rnx and y Rny are, respectively, the hidden vehicles [1]. Similarly, in a maritime scenario, the knowledge t t state vector∈ with transition∈ probability density function (pdf) on shipping lanes and sea/land distinction can improve the p(x x − ), and the observation vector with conditional pdf tracking and detection performance [2]. t t 1 y |x at time instant . and are possibly nonlinear The general state space hidden Markov models provide p( t t) t ft ht state| transition and observation functions, respectively. an extremely flexible framework for modeling discrete-time nx and are state and output dimension. u and v are zero- dynamical systems. Dynamic systems are modeled using state ny t t mean state and observation white noise sequences with known arXiv:1807.03463v2 [eess.SP] 24 Dec 2018 evaluation and observation relations. The former captures the probability density functions (pdfs), respectively, u and evolution of the state with time and later provides noisy p( ) v . Both noise sequences are supposed to be uncorrelated measurement of a probably nonlinear function of the state. The p( ) with each other and the initial condition of the state x given great descriptive power of these models comes at the expense 0 by x . of intractability: it is impossible to obtain analytic solutions to p( 0) the inference problems with the exception of a small number of In the Bayesian context, an optimal state estimation of particulary simple cases. In the Bayesian framework, inference the state vector sequence xt given the available observations of the hidden state given all available observations at that history y1:t = [y1, ..., yt] up to time t, relies upon the posterior density p(x y ). Using Bayes rule and Chapman- time relies upon the posterior density function (pdf). For the t| 1:t linear and Gaussian estimation problems, optimal solution Kolmogorov equation, the posterior distribution can be com- can be obtained by the Kalman filter. For nonlinear and puted recursively using the following two-step formulas: non-Gaussian state-space models, there are two fundamental techniques have been emerging: parametric and nonparametric. N. Amor is with the National Superior School of Engineers of Tunis (EN- Prediction step SIT), University of Tunis, Tunis, Tunisia, e-mail: ([email protected] G. Rasool and N. C. Bouaynaya are with Department of Electrical and Computer Engineering, Rowan University, New Jersey, USA, e-mail: (ra- p(xt y1:t−1)= p(xt−1 y1:t−1) p(xt xt−1) dxt−1, (3) [email protected], [email protected]) | Z | | 2 Update step B. Extended Kalman Filter (EKF) p(yt xt) p(xt y1:t−1) p(xt y1:t)= | | , (4) For the nonlinear model, the extended Kalman filter (EKF) | p(y x ) p(x y − ) dx t t t 1:t 1 t was formulated by a linearization procedure of nonlinear R | | In the nonlinear case, unfortunately, these equations are functions ft(xt) and ht(xt), using the Taylor series expansion. only a conceptual solution because the defined integrals are The error covariance is propagated in time using the linearized generally intractable. However, closed-form solutions in some functions, whereas the state estimate are propagated using special cases may exist, e.g., the Kalman filter for linear nonlinear functions. Specifically, eqs. (5a) and (6b) will use dynamics system, linear observation models and Gaussian nonlinear functions f(x) and hx, while all other relations densities for the noise sequences. In other cases, we resort utilize linearized forms of these functions (first-order Taylor + to several approximations [3]. series expansion) evaluated at the estimated state xˆt [5], [4]. However, computational complexity for calculation of Hessian matrices may prohibit its use [4]. In addition, the linearization III. UNCONSTRAINED STATE ESTIMATION of nonlinear system and measurement models may induce Generally, for tracking problems with linear and Gaussian errors in the estimation of the state, and in the worst-case, models, an optimal solution can be obtained using the Kalman the filter may diverge especially for highly nonlinear function Filter. In fact, many real-world applications such as target [6]. tracking, electric power systems, navigation and biomedical engineering, these linear and Gaussian assumptions do not hold. Furthermore, approximations are required for these nonlinear and non-Gaussian tracking problems. Thus, many approaches were introduced to solve this problem such as C. Unscented Kalman Filter (UKF) Extended Kalman Filter (EKF), Moving Horizon estimation (MHE), Ensemble Kalman Filter (EnKF), Unscented Kalman Filter (UKF) and Particle Filters (PF). In this section, we The Unscented Kalman Filter (UKF) was proposed as a introduce these filters for linear and nonlinear systems when method based on a mathematical approach called the ‘Un- there are no constraints on the system. scented Transform’ (UT). The UKF approximates the prob- ability distribution based on UT which uses a deterministic set of samples, called sigma points, to propagate the mean A. Kalman Filter and covariance. The calculated sigma points are propagated When the state transition and observation models are linear through the nonlinear function. The statistics of transformed with additive Gaussian noise, the Kalman Filter (KF) is the points can be calculated to form an estimate of the nonlinearly minimum mean square error (MMSE) estimator for linear and transformed mean and covariance [7], [6], [8]. The sigma points X Rnx , j = 0, 1,..., 2n are chosen deterministi- Gaussian dynamic systems. The KF estimates the unknown j,t ∈ state by propagating the mean and the covariance at each time cally as opposite to the particle filters. Consider xt with mean step [4], [5]. The Kalman filter defined by the two steps: the xˆt and covariance Pt. Let the matrix of all the sigma points be := [ ,..., ]. prediction step and the update step. Xt X1,t X2n,t + 1/2 + 1/2 =x ˆ 1 × + √n + λ 0 × (P ) (P ) , Prediction step Xt t 1 (2n+1) n 1 t − t h (7)i 1/2 − + where (.) is the Cholesky square root, n is the dimension xˆ = F − xˆ (5a) t t 1 t−1 of the system and λ > n. Further, we define weights for − + T − Pt = Ft−1Pt−1Ft−1 + Qt (5b) all sigma points γ = [γ0,γ1,...,γ2n] with the condition − T 2n using the relation Pxy = Pt Ht (5c) j=0 γj =1 − T P Py = HtPt Ht + Rt (5d) − yˆ = H xˆ (5e) t t t λ 1 γ = , γ = , j =1, 2,..., 2n. (8) 0 n + λ j 2(n + λ) Update step where Eqs. (7) and (8) represent the unscented transforma- −1 tion ( ). In the following, we summarize the UKF algorithm. Kt = PxyPy (6a) UT + − xˆt =x ˆt + Kt(yt yˆt) (6b) − − P + = P P P 1P T (6c) t t − xy y xy 3 Update step where the last term is the arrival cost and for t = h, Πt−h = , i.e., the initial covariance of the state estimate at time − + + Π0 t = γ, t = (ˆxt−1, Pt−1,n,λ), (9a) 0. The purpose for development of the MHE was to formulate X UT − + = f( ), j =0, 1,..., 2n, (9b) a mathematical optimization problem where constraints can be Xj,t Xj,t 2n naturally incorporated into the estimation framework [14].