Measurements generated by any sensor will be noisy. The factors which control the level of noise in a radar system have been discussed in detail in the preceding Section. Once noisy measurements are available, whether modelled in the case of this research, or within a real world system, the next issue to address is how to estimate the state of the target at each point in time. A general approach known as stochastic methods can be used for the purposes of estimation. Stochastic methods estimate an unknown target state assuming that the measurement noise is either statistical in nature or can be modelled as such [47]. They often make use of state-space models.
An overview of the principle of state space models as well as important statistical concepts are discussed in the following sections.
4.5.1
State-space models
State-space models are a convenient mathematical notation for estimation and control problems. State-space models can be used to describe many processes in the world from biological systems to processes in economics and physics [48]. By deriving a mathematical system model to represent the process, the tools of mathematics can be used to obtain information about the process and potentially control it.
There are two types of system model, linear and non-linear. Most real pro- cesses are non-linear. However the mathematics required to describe a non-linear process are often highly complex. Therefore many non linear processes are often approximated as a linear system as the mathematics required to define a linear process are often simpler than that of non linear processes [48].
The state of a system is expressed as a state vector, with the variables within the state vector describing the dynamic state of the system at a particular instant in time. A state space model contains two equations consisting of the system and measurement equations. The system equation is a mathematical model which predicts the evolution of the system subject to a defined dynamic model and external influences such as system inputs. The system equation will normally be an incomplete characterisation of the system. Therefore, a process noise is also used to account for modelling inaccuracies.
The measurement equation describes how the variables in the state vector are related to the noisy observed measurements, where each measurement will be the true system state with some added unknown noise which is often due to inac- curacies in the sensor. Estimating the future state of a system using only noisy measurements will often be ineffective for most tracking problems. In order to produce a more reliable prediction of the future state of the system, predictive filters can be used.
Predictive filters estimate the optimal state of a system by using a mathematical model of the dynamics of the systems to propagate the state and the associated uncertainties [49]. The propagated state is then combined with the best informa- tion available from the measurements. There are numerous filter types available with each one being appropriate for a certain type of uncertainty representation and dynamic model.
The predictive filter used in this thesis is a Kalman filter which represents uncer- tainties as Gaussian random variables (justification of a Kalman filter is provided in Section 4.6). In order to understand how the filter works, the fundamental sta- tistical concepts related to the Kalman filter are firstly reviewed, with the general mathematical representation of a state space model then provided.
4.5.2
Statistical Concepts
There are four main statistical concepts which must be understood in order to fully understand how the Kalman filter works. These consist of the mean, stan- dard deviation and variance of a scalar random variable as well as the covariance of two scalar random variables which are defined as follows :
The expected value (E) [32] of a scalar random variable (x) which is more com- monly known as the mean ¯x is defined as
¯
x=E[x] =
Z ∞
−∞
xp(x)dx (4.24)
The second central moment [32] or variance (σ2) is then defined as
σ2x=E[(x−x¯2)] =
Z ∞
−∞
(x−x¯2)p(x)dx (4.25)
with the square root of the variance yielding the standard deviation (σx)
σx =
p σ2
x (4.26)
In a multivariate case i.e two scalar random variables (x1, x2) with respective
means of ( ¯x1,x¯2) a covariance matrix is obtained [32] as
cov(x1, x2) = E[(x−x¯1)(x−x¯2)] (4.27)
The Kalman filter describes uncertainties using Gaussian random variables. This means that the random variables has a probability density functionp(x) defined as : p(x) = √ 1 2πσx exp −(x−x¯) 2 2σ2 x (4.28)
The Gaussian distribution is one of the most historically popular probability dis- tributions used in modelling random systems. It is a special distribution because it appears many random processes which occur in nature appear to be Normally distributed or very close to it. Under some moderate conditions it can even be proved that a sum of random variables with any distribution will tend towards a normal distribution. This property is stated formally by the central limit theo- rem [50].
4.5.3
State-space model mathematical representation
The concept of state space can be explained by considering a moving object. The variables which represent the dynamical motion of the system i.e. position and
velocity are defined by a sequence of states which can be collated into a state vector x(k). The state vector can then be used to define the dynamical state of the system at each time step (k). The evolution of the system state over time is then predicted by thesystem equation. The system equations defined in this thesis are direct discrete-time models as such they have the following general form [32] :
x(k+ 1) =F(k)x(k) +G(k)u(k) + Γ(k)v(k) (4.29) where F(k) is state transition matrix and v(k) is a white process noise which enters through a noise gain Γ [32]. The process noise accounts for inaccuracies in the system model. It has an associated covariance matrix Q(k). u(k) represents a set of assumed known controls with an associated input gain G(k) [51].
The measurement equation describes the relationship between the predicted state vector x(k) obtained from the system equation and the observed measure- mentz(k). The observed measurement will be the true state of the system with the addition of a unknown noise w(k) which is defined mathematically as :
z(k) = H(k)x(k) +w(k) (4.30)
whereH(k) is measurement matrix used to selected the measured states from the state vector.
The system and measurement equations which define the state-space model form the basis of virtually all linear predictive filters.
The Kalman filter was chosen as the fundamental component of the tracking algorithm in this thesis because linear target dynamics are assumed and the mea- surement noise produced by the radar model is white and Gaussian distributed. The Kalman filter under these conditions will provide the optimal state esti- mate [48]. In cases where the process model is not strictly linear, and can not be successfully approximated as such, other filters exist such as the Extended Kalman Filter [52], Unscented Kalman Filter [53] and the Particle filter [54].