Combat System Filter Engineering Combat System Filter Engineering Scott A. Hays and Michael A. Fatemi ABSTRACT To develop a new combat system, the designer has to determine the optimal filter type, filter application point, and dynamic model assumption. The definition of an optimal filter will change depending on the filter’s purpose and can change drastically depending on its application within the plan–detect–control–engage sequence. Ideally, the designer could apply a single computationally efficient filter algorithm that adapts in real time to threat maneuver, system bias, and measurement noise level while maintaining an accurate estimate with a high level of confidence. In practice, however, several different filters (often of different types) are applied to a single combat system in separate parts of the plan–detect–control–engage sequence to ensure the best results for problems that include track consistency, association, filter errors, and correlation. There are several key factors in developing a robust filter with the flexibility for the technology upgrades that are required to keep up with threat evolution. This article describes a design methodology to provide robustness in the face of dynamic threat behavior, lack of a priori knowledge, and threat evolution. INTRODUCTION Air and missile defense systems can be general- the future, which requires an accurate dynamic model ized as combat systems with architecture that allows and state estimate. Additionally, some components in the fire control loop to be integrated into the plan– the plan–detect–control–engage sequence could pass detect–control–engage sequence. Air and missile tracks to the next sequence, which changes the track defense combat systems usually have tracking sensors picture. The new track picture results from filtered track (plan–detect sequences), a decision component (control estimate inputs rather than direct measurements. Early sequence), a control component (control sequence), and combat systems were forced to rely on simple filters such a guided missile (engage sequence). Optimal conditions as alpha-beta filters, alpha-beta-gamma filters, discrete required for the plan–detect–control–engage sequence Kalman filters, and lookup-based filters; these filters can differ based on needs; a specific example of optimal provided computationally efficient algorithms but little conditions can be a very accurate position estimate and system robustness. Increased computation capability tight covariance to ensure that the tracking sensor can allows for further expansion of filter types into nonlinear associate the different system tracks. A decision compo- filters—i.e., extended Kalman filters (EKFs), unscented nent needs to accurately extrapolate a track state into Kalman filters (UKFs), and, to an extent, particle filters Johns Hopkins APL Technical Digest, Volume 35, Number 1 (2020), www.jhuapl.edu/techdigest 107 S. A. Hays and M. A. Fatemi (PFs). This article introduces each of these nonlinear bounded uncertainty (also referred to as measurement filters and provides a background for application of process noise); f() is the nonlinear state transition func- advanced filter algorithms. tion; h() is the nonlinear measurement function; Rw is the covariance associated with model-bounded uncer- tainty assuming the noise follows a Gaussian distribu- NONLINEAR FILTERS tion with a zero mean; Rv is the covariance associated The increase in threat capability necessitates an with measurement-bounded uncertainty assuming the increase in the sensor and tracking platform’s ability noise follows a Gaussian distribution with a zero mean; to accurately predict future motion. Nonlinear filters and nx or N refers to the dimension of the state vector. provide a capability that extends beyond the linear Terminology used in this article is as follows: xk1|k 2 is assumptions to ensure system robustness. The expansion the state estimate at time k1 given measurements up to of filters from the linear to nonlinear realm inherently and including time k2 (where k2 ≤ k1); Pk1|k 2 is the esti- produces nonoptimal estimation, unlike that produced mated covariance matrix at time k1 given measurements by the Kalman filter, which is an optimal estimator for up to and including time k2 (where k2 ≤ k1); yk1 is the linear dynamic systems and linear measurement models. measurement at time k1; zk1 is the innovation (or mea- Nonlinear filters use different methodologies (Taylor surement residual) at time k1; and Kk1 is the gain matrix series expansion, unscented transformation [UT], at time k1. The initial estimation period in this article Monte Carlo–based sampling approaches, and dynamic is referred to as the a priori estimate, but in the litera- constrained optimization) to characterize and approxi- ture this is also referred to as the predictive step or time mate nonlinear motion. update state. The update estimation period is referred In this article, the linear/nonlinear dynamic and to as the a posteriori estimate, but in the literature this measurement equations are defined as: is also referred to as the corrective step, update step, or measurement update step. xk = Ak–1xk–1 + Bk–1uk–1 + wk–1, (1) yk = Ckxk + vk, (2) EXTENDED KALMAN FILTER The EKF is an extension of the Kalman filter to rep - x = f(x , u ) + w , and (3) k k–1 k–1 k–1 resent nonlinear motion by using a Taylor series expan- sion to linearize about a local point (Figure 1). The EKF y = h(x ) + v , (4) k k k is based on local linearization of the nonlinear equa- where xk is the state; u k is the control input; w k is a tions of motion around the prior state estimate. In the model-bounded uncertainty (also referred to as state or EKF, to represent the covariance prediction, a partial model process noise); Ak is the state transition matrix; derivative matrix (Jacobian) of the state estimate matrix Bk is the control input matrix; y k is the measurement; is computed and evaluated at each time step based on Ck is the observation matrix; v k is a measurement- the current predicted states. More information on EKF Initialization t x00,P A posteriori estimate Kalman gain A priori estimate T T –1 Prediction KPkk=+;;kk––11HHkkPHkkR␯ State estimate: 8 B tt xkk;;––11= fxkk– 1 Update ^ h State estimate: Covariance estimate: tt t T xKkk; =+xxkk;;– 1 kkyh– kk– 1 PF=+PFR ^ h kk;;––11kkk – 1 ––11k w Covariance 6estimate: @ PPkk;;= IK– kkH kk– 1 where the state Jacobian is dened as: 6 @ t 2fxkk––11; ^ h evaluated at xt where the measurement Jacobian is dened as: Fkx– 1 = 2 kk– 1 ; – 1 t 2h xkk; – 1 ^ h evaluated at xt Hk = 2x k ; k – 1 Figure 1. EKF flow diagram. 108 Johns Hopkins APL Technical Digest, Volume 35, Number 2 (2020), www.jhuapl.edu/techdigest Combat System Filter Engineering algorithms, derivation, and advances in research can be approximate a probability distribution than an arbitrary found in Refs. 1–5. nonlinear function. The UKF uses the UT, creating The EKF has limitations due to the calculation of the a related statistic set (mean and covariance informa- Jacobian matrix and the local linearization approxima- tion) for a predetermined number of transformed points tion. The EKF requires an accurate representation of the (Figure 2). The UKF approximates a probability distri- nonlinear equations of motion and an accurate estimate bution by a set of sample points; this results in a filter- of the partial derivative state equations. For highly non- ing method that does not require state linearization or linear systems, the equations of motion can be difficult calculation of a Jacobian matrix. State dynamic lin- to describe to a high degree, and the construction of the earization and calculation of Jacobian matrices can be Jacobian matrix can be difficult. Errors within equa- computationally intensive and require more accurate tions of motion, errors within Jacobian construction, or state estimation. a local/global motion mismatch can result in divergence The number of transformed points (sigma points) is in the EKF error propagation, resulting in an unstable 2nx + 1 where n x is the dimension of the state vector. filter. The errors that can occur while deriving the EKF The weights for the UT can be selected to provide con- can lead to computational complexity and overall poor ditions that are bundled closer to the mean or approxi- performance when tracking challenging targets. mate higher moments of the nonlinear function. In a Gaussian distribution, the weights are predetermined. UKF can use the nonlinear dynamic equations in the UNSCENTED KALMAN FILTER update process and can be tuned to provide robustness The UT is a method for calculating the statistics of to reduce instability issues. More information on UKF a random variable that undergoes a nonlinear transfor- algorithms derivation and advances in research can be mation and builds on the principle that it is easier to found in Refs. 6–12. Initialization t x00,P A priori estimate A posteriori estimate Sigma points calculation Update L Covariance estimate: X = xxtt, ++LP␭ xt ^kk––h11; kk––11;;kk––11 kk––11;;, kk––11 2n 8 ^ h PWyy, = x c YY––yyttT + R – LP+ ␭ ,,Ln=+01,,22f, 1 kk; – 1 /i = 0 i ik,,;;kk––11kikk;;––11kk v ^ h kk––11; x 6 @6 @ B x,y 2nx c ttT PWkk; – 1 = /i = 0 i Xik,,;;kk––11––x kiY kk;;––11ykk Prediction 6 @6 @ State sigma points: where the subscripts ik, k – 1 refer to the vector index of X X the sigma points. kk;;––11= ftkk– 1, ^ h State estimate: Kalman gain t 2nx m xy, y,y –1 xWkk;;––1 = /i = 0 i Xik, k 1 KP= P k kk;;––118 kk B Covariance estimate: 2nx c t T Update Pkk; – 1 = /i = 0 Wxi Xik, ; k – 1 – kk; – 1 X , ; – xt ; + Rw 6 @6 ik k – 1 kk– 1 State estimate: @ tt t where the subscripts ik, k – 1 refer to the vector index of xxkk;;=+kk––11Kykk– ykk; 6 @ the sigma points.