254
I. A. Kudryavtseva and K. A. Rybakov
Fig. 17.1 The graph of μ(t k , X k , Z k )h for sample trajectories of random processes X (t) and Z (t)
MMSE criterion is chosen as a quality criterion. In the considered model, several
corrections in comparison with the model in [6] have been made.
Firstly, the matrix σ (t, x) has been restricted in the motion model. In [6], the
matrix σ (t, x) is the 7 × 7 matrix σ (t, x) = diag(0, σ 1 , 0, σ 1 , 0, σ 1 , σ 2 ). As it is
distinctly seen a number of its zero elements are superfluous that leads to extra
calculations related to drawing a pseudorandom sequence when simulating aircraft
trajectories. In the modified model the matrix σ (t, x) has a reduced dimension 7 × 4
and does not contain zero columns. Secondly, the initial condition has been changed
so that aircraft trajectories do not cross the plane ε = 0.
Figure 17.1 shows the graph μ(t k , X k , Z k )h as a function of the time moments
t k for trajectories of random processes X (t) and Z (t) that are obtained by Heun’s
method [33] with the integration step h = 0.01. The threshold value 706.893 indicating a level, where the overflow error appears, is also marked in Fig. 17.1. This
overflow error occurs at t = 17s. It should be emphasized that the overflow error
appears not only in the case of implementation in Mathcad but in any other application supporting the double-precision floating-point format. The single-precision
floating-point format is unusable here (see Fig. 17.1).
The standard normalization (see Algorithm 2) is ineffective in the latter case as
the overflow error appears in one step at a time but it is not caused by the gradual
increase of the weight coefficient.
Several ways of grappling with the discussed question can be offered. The first
way is to reduce the integration step. For example, in considered simulations, the
integration step should be decreased even to 0.003. The second one is to carry out
calculations with the extended precision floating point format. Finally, the last way
is to apply Algorithm 3.
I. A. Kudryavtseva and K. A. Rybakov
Fig. 17.1 The graph of μ(t k , X k , Z k )h for sample trajectories of random processes X (t) and Z (t)
MMSE criterion is chosen as a quality criterion. In the considered model, several
corrections in comparison with the model in [6] have been made.
Firstly, the matrix σ (t, x) has been restricted in the motion model. In [6], the
matrix σ (t, x) is the 7 × 7 matrix σ (t, x) = diag(0, σ 1 , 0, σ 1 , 0, σ 1 , σ 2 ). As it is
distinctly seen a number of its zero elements are superfluous that leads to extra
calculations related to drawing a pseudorandom sequence when simulating aircraft
trajectories. In the modified model the matrix σ (t, x) has a reduced dimension 7 × 4
and does not contain zero columns. Secondly, the initial condition has been changed
so that aircraft trajectories do not cross the plane ε = 0.
Figure 17.1 shows the graph μ(t k , X k , Z k )h as a function of the time moments
t k for trajectories of random processes X (t) and Z (t) that are obtained by Heun’s
method [33] with the integration step h = 0.01. The threshold value 706.893 indicating a level, where the overflow error appears, is also marked in Fig. 17.1. This
overflow error occurs at t = 17s. It should be emphasized that the overflow error
appears not only in the case of implementation in Mathcad but in any other application supporting the double-precision floating-point format. The single-precision
floating-point format is unusable here (see Fig. 17.1).
The standard normalization (see Algorithm 2) is ineffective in the latter case as
the overflow error appears in one step at a time but it is not caused by the gradual
increase of the weight coefficient.
Several ways of grappling with the discussed question can be offered. The first
way is to reduce the integration step. For example, in considered simulations, the
integration step should be decreased even to 0.003. The second one is to carry out
calculations with the extended precision floating point format. Finally, the last way
is to apply Algorithm 3.