LQR Performance Index Distribution with Uncertain Boundary Conditions Marcus J. Holzinger Graduate Research Assistant Aerospace Engineering Sciences University of Colorado at Boulder Daniel J. Scheeres Professor, Seebass Chair Aerospace Engineering Sciences University of Colorado at Boulder Abstract—The track assignment problem in applications with large gaps in tracking measurements and uncertain boundary conditions is addressed as a Two Point Boundary Value Problem (TPBVP) using Hamiltonian formalisms. An L2-norm analog Linear Quadratic Regulator (LQR) performance function metric is used to measure the trajec- tory cost, which may be interpreted as a control distance metric. Distributions of the performance function are determined by linearizing about the deterministic optimal nonlinear trajectory solution to the TPBVP and accounting for statistical variations in the boundary conditions. The performance function random variable under this treat- ment is found to have a quadratic form, and Pearson’s Approximation is used to model it as a chi-squared random variable. Stochastic dominance is borrowed from mathe- matical finance and is used to rank statistical distributions in a metric sense. Analytical results and approximations are validated and an example of the approach utility is given. Finally conclusions and future work are discussed. I. I NTRODUCTION AND BACKGROUND In recent years there has been a significant increase in trackable on-orbit objects due to new launches, col- lisions, and improved sensor capabilities [1], [2], [3]. Object correlation for objects with maneuver execution capability has become increasingly complex, and oper- ational methods to correlate objects have become more important, specifically for collision avoidance operations [4], [5]. Object track correlation has an extensive body of literature, particularly in continuous visible and radar tracking applications. Many approaches use statistical properties of sensors or known target qualities to mini- mize false alarms and the effects of clutter. Probabilistic Multi-Hypothesis Tracking (PMHT), Probabilistic Data Association Filter (PDAF), and Modified Gain Extended Kalman Filter (MGEKF) are representative of these algorithms (discussions and examples of their usage are in [6], [7], [8], [9]). Largely, these approaches update their correlations as new measurements are generated. An alternate class of problems to examine are those with large gaps in observation, such as on-orbit object tracking. In these scenarios the problem is to associate individual object tracks incorporating 5-15 minutes of observations (perhaps generated using some of the meth- ods mentioned above) separated by observation gaps on the order of tens of minutes to days [10], [11]. One way to support candidate object pairing associ- ation is to propagate the initial track uncertainty (also introducing process noise) and compute the Kullback- Leibler Distance (KL-D) [12], the Battacharya Distance (B-D) [13], or the Mahalanobis Distance (M-D) [14] from the newly generated object track state and uncer- tainty. Alternately, with an initial and final track for each candidate association pairing, each association may be addressed as a Two Point Boundary Value Problem (TPBVP) linking uncertain boundary conditions. In this approach the optimal connecting trajectory performance distribution may itself be used as a metric, and mutu- ally exclusive track association pairings may be ranked against one another in a metric sense. Specifically, LQR- type costs (quadratic in state or fuel deviations from a nominal homogeneous trajectory) directly measure the effect of active maneuvers. Note that for space applications, a subclass of this problem concerned only with quadratic cost in fuel usage can be used [15]. This paper examines the full LQR performance index to expand upon previous efforts. This proposed method is fundamentally different from computing the statistical KL-D, B-D, or M-D from the objects’ expected state distribution as it directly accounts for control usage. Computing the distribution of the LQR performance index has the additional advantage that the resulting probability distribution function may be used to support hypothesis testing to infer intentions or detect maneu- vers. The contribution of this paper is to generalize from quadratic control costs to full LQR trajectory costs. Effort is made to maintain applicability to general dynamical systems with LQR trajectory costs. Theory 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 ©2011 AACC 913