Estimation of Velocity of a Frictionless Motion of a Truck on an Infinitely Long Straight Rail Ogwola, Peter, Sullayman

This paper is concerned with estimation of velocity of a frictionless motion of a truck on an infinitely long straight rail. For simplicity assume that the Truck is controlled only by the throttle producing an accelerative force per unit mass. A discrete dynamic model of first order difference equation is to describe the system. Kalman filtering technique is applied to the discrete dynamic model to estimate the velocity of the Truck at any particular time. A computer programme is developed to simulate the system.


Introduction
In system analysis, a fundamental problem is to provide values for the unknown states or parameters of a system given noisy measurements which are some functions of these states and parameters.
According to Greg and Gary (2006), Kalman filter is defined as a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the square error. Kalman filter was originally developed for use in spacecraft navigation but turns out to be useful for many applications. Kalman (1960) published the discrete-time filter in a Mechanical Engineering Journal and Kalman and Bucy (1961), the continuous-time filter. In the meantime, Swerling (1959) had derived an equivalent formulation of the Kalman filter and applied it to the problem of estimating the trajectories of satellites using ground-based sensor. His results were published in an Astronomy journal the year before Kalman (1960) appeared.

Materials and Methods Kalman filter Wikipedia
The Kalman filter model assumes the true state at time k is evolved from the state at (k-1) as stated below.
Where,  is the state transition model which is applied to the previous state Xk; B is the control-input model which is applied to the control vector k U  k is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Q.
Where H is the observation model which maps the true state space into the observed space and k is the observation noise which is assumed to be zero mean Gaussian white noise with covariance R,  k~ N(0,R) The initial state, and the noise vectors at each , 1 are all assumed to be mutually independent. The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state and as such no history of observations and /or estimates is required. In what follows, the notation x which is the 1 step prediction represents the estimates of k x at time k given observations up to and including at time k-1.
The covariance matrix for the one step prediction error is given by and Pk|k-1 are the Predicted (a priori) state estimate and Predicted(a priori) estimate of covariance respectively and represent the initial values for the Kalman filter. The state of the filter is represented by two variables Pk|k,, the updated( a posterior )error covariance matrix ( variance of the estimation error) given by Pk|k = (I -KkH)Pk|k-1 (6) Where Kk is the Kalman (Filter) gain and given by Kk = Pk|k-1H T (H Pk|k-1H T +R ) -1 (7)

The Kalman filter loop The Kalman filter loop given below summarizes what is known as the Kalman filter.
Enter prior estimate  X k|k-1and its covariance Pk|k-1 Figure 1: The Kalman Filter Loop as in Robert and Patrick (1992) Once the loop is entered it can be continued for any N (N≥ 1) iterations, k = 0,1,……..,N-1.

Modelling and identification
Suppose a Truck is being driven along a straight rail, and let its distance from initial point be ) ( 1 t x at time t. For simplicity assume that the Truck is controlled only by the throttle, producing an accelerating force per unit mass. Ignoring friction, wind resistance etc. It is required to find the velocity of the Truck at time t = 1,2,3 ..... seconds.

The estimation problem
The estimation problem is stated as follows: