Discretization
In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Processing on a digital computer requires another process called quantization.
Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable of category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.
Contents
Discretization of linear state space models
Discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing.
The following continuous-time state space model
where v and w are continuous zero-mean white noise sources with covariances
can be discretized, assuming zero-order hold for the input u and continuous integration for the noise v, to
with covariances
where
- , if is nonsingular
and is the sample time, although is the transposed matrix of .
A clever trick to compute Ad and Bd in one step is by utilizing the following property, p. 215:^{1}
and then having
Discretization of process noise
Numerical evaluation of is a bit trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it (Van Loan, 1978):
The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G:
Derivation
Starting with the continuous model
we know that the matrix exponential is
and by premultiplying the model we get
which we recognize as
and by integrating..
which is an analytical solution to the continuous model.
Now we want to discretise the above expression. We assume that u is constant during each timestep.
We recognize the bracketed expression as , and the second term can be simplified by substituting . We also assume that is constant during the integral, which in turn yields
which is an exact solution to the discretization problem.
Approximations
Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps . The approximate solution then becomes:
Other possible approximations are and . Each of them have different stability properties. The last one is known as the bilinear transform, or Tustin transform, and preserves the (in)stability of the continuous-time system.
Discretization of continuous features
In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features. This can be useful when creating probability mass functions.
See also
- Discrete space
- Time-scale calculus
- Discrete event simulation
- Stochastic simulation
- Finite volume method for unsteady flow
- Properties of discretization schemes
References
- ^ Raymond DeCarlo: Linear Systems: A State Variable Approach with Numerical Implementation, Prentice Hall, NJ, 1989
- Robert Grover Brown & Patrick Y. C. Hwang. Introduction to random signals and applied Kalman filtering (3rd ed.). ISBN 978-0471128397.
- Chi-Tsong Chen (1984). Linear System Theory and Design. Philadelphia, PA, USA: Saunders College Publishing. ISBN 0030716918.
- C. Van Loan (Jun 1978). "Computing integrals involving the matrix exponential". IEEE Transactions on Automatic Control 23 (3): 395–404. doi:10.1109/TAC.1978.1101743.
- R.H. Middleton & G.C. Goodwin (1990). Digital control and estimation: a unified approach. p. 33f. ISBN 0132116650.
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