Comparison of GPS Tracking Loop Performance in High Dynamic Condition with Nonlinear Filtering Techniques

The conventional GPS tracking loop is optimal in Maximum likelihood Estimation (MLE), respectively. It well works in normal signal to noise ratio (SNR) and signal dynamics within the tracking loop bandwidth. But, when the receiver operates in high dynamic environment, discriminator linearity doesn t maintain and tracking loop error increase. In the previous rearch, several algorithms were proposed to overcome these problems such as EKF based tracking loop, grid method. In the previous paper [Gee, ENC 2005], LQG based GPS receiver tracking loop is developed using EKF and Linear Quadratic Regulator (LQR). The EKF estimate the range rate, code phase, carrier phase error and navigation bit from inphase and quadrature measurement. And LQR calculate the optimal DCO input using pre-calculated steady state feedback gain. It had good tracking performance than conventional tracking loop in normal condition because it is designed to consider correlation of code and carrier tracking loop. But there is problem about nonlinearity of measurement model as ever. 2351 ION GNSS 21st. International Technical Meeting of the Satellite Division, 16-19, September 2008, Savannah, GA. In this paper, LQG based GPS tracking loop is implemented using nonlinear filtering techniques, i.e. Unscented Kalman Filter (UKF) and Particle Filter (PF). Also, the implemented algorithm performance is evaluated and compared. In the EKF, the measurement equations are linearized to the first order Taylor series in order to apply the Kalman filter, which is supposed to linear Gaussian systems. Instead of truncating the nonlinear measurement equation the UKF and PF approximate the distribution of the state deterministically and randomly, with a finite set of samples, and then propagate these points or particles through the original nonlinear functions, respectively. Because the nonlinear functions are used without approximation, it provides the better performance. INTRODUCTION The code and the carrier tracking loop are the essential parts of the GNSS receiver. Usually the code tracking is implemented by the Delay Locked Loop (DLL) and the carrier tracking is implemented by the Phase Locked Loop (PLL) and/or the Frequency Locked Loop (FLL). The tracking loop consists of three parts in general: discriminator, loop filter and VCO. The loop filter is the main part of the tracking loop design to ensure good tracking performance. Although the carrier Doppler and code Doppler are generated by the same relative movement between the satellite and the user, often, the loop filters for the each tracking loop are designed separately and independently. Common design parameters for the loop filters are the loop bandwidth and the loop damping ratio. A usual choice for the damping ratio is 0.707. The loop filter order is often chosen by the expected dynamics of the Doppler. For examples, the 2nd or 3rd order loop filter is used for high dynamic environments [1, 2]. Sometimes, they are used in combined manner such as carrier aided code tracking, FLL, etc. One way of carrier aiding is to feed the carrier DCO value calculated by the FLL to the DLL with appropriate scaling of code carrier / f f . For better GNSS signal tracking performance, we need to design the FLL/DLL altogether optimally. In this paper we deal with the GPS C/A tracking loop optimization. First, it is shown that the DLL/FLL design can be shown as an optimal controller design problem. Specially, the well known 2nd order DLL/FLL is shown to be equivalent to the special case of the LQG controller, that is the combination of the Kalman filter and the output state feedback controller. Based on this fact, we can consider the code and carrier tracking problem as the two inputs two outputs control problem from the view point of control theory. The control objective is to minimize the total tracking error in carrier phase and code phase using two inputs, code and carrier DCO. The sate feedback control from the LQG design that is indeed a regulator will drive code and carrier DCO to track the incoming satellite signal with minimum mean squared tracking error. The tracking loop performance optimization could be done by the LQG design parameters tuning. First, the LQG control problem is introduced. The conventional design method for code and phase tracking loop is also reviewed. The tracking loop state space model will be derived and a LQG control problem for C/A code receiver tracking loop design is formulated and solved. COMBINED ESTIMATION AND CONTROL : LQG CONTROL Linear quadratic Gaussian (LQG) control refers to an optimal control problem where the plant model is linear, the cost function is quadratic, and the test conditions consist of random initial conditions, a white noise disturbance input, and a white measurement noise [3]. The system is described by the following linear state space model: 1 , ~ (0, ) , ~ (0, ) k k k k k f k k k k f N N + = + + = + x Fx Bu Gw w Q z Hx v v R (1) where k u is the control input and k w is a random disturbance input known as process noise. The measurements available for feedback are k z and k v is a random signal known as measurement noise. The process noise and measurement noise are assumed to be both white Gaussian and uncorrelated. Note that the description of the system and measurement noises is equivalent to that used in the Kalman filter problem. The system output to be controlled is k k = y Cx (2) A quadratic cost function including both this output and the control input is ( ) 1 ( , ) N T T k k k y k k c k k J E = ⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ ∑ x u y Q y u R u (3) where y Q , c R are positive definite. This cost function is frequently written directly in terms of the state: ( ) 1 ( , ) N T T k k k c k k c k k J E = ⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ ∑ x u x Q x u R u (4) where T c y = Q C Q C and c Q is positive semi-definite. 2352 ION GNSS 21st. International Technical Meeting of the Satellite Division, 16-19, September 2008, Savannah, GA. Fig 1. A LQG optimal control system A reasonable controller design for the LQG control problem can be obtained by using the linear quadratic regulator feedback gain matrix operating on the state estimation generated by the Kalman filter: ˆ k k = − u Kx (5) A block diagram of this feedback control system is given in Fig. 1. This control law is, in fact, the optimal solution to the LQG optimal control problem. The optimality of the control law (5) is called “the separation principle” in control theory. The state equations for the Kalman filter estimations are given by