NationMaster - Encyclopedia: Discretization

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. In order to be processed on a digital computer another process named quantization is essential. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ...

Euler discretization
Zero order hold

Discretization is also somewhat connected to discrete mathematics. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...

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. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...

The following continuous state space model In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. ...

$dot{mathbf{x}}(t) = mathbf A mathbf{x}(t) + mathbf B mathbf{u}(t) + mathbf{v}(t)$

$mathbf{y}(t) = mathbf C mathbf{x}(t) + mathbf D mathbf{u}(t) + mathbf{w}(t)$

where v and w are continous zero-mean white noise sources with covariances

$mathbf{v}(t) sim N(0,mathbf Q)$

$mathbf{w}(t) sim N(0,mathbf R)$

can be discretized, assuming zero-order hold for the input u and continous integration for the noise v, to

$mathbf{x}[k+1] = mathbf A_d mathbf{x}[k] + mathbf B_d mathbf{u}[k] + mathbf{v}[k]$

$mathbf{y}[k] = mathbf C_d mathbf{x}[k] + mathbf D_d mathbf{u}[k] + mathbf{w}[k]$

with covariances

$mathbf{v}[k] sim N(0,mathbf Q_d)$

$mathbf{w}[k] sim N(0,mathbf R_d)$

where

$mathbf A_d = e^{mathbf A T} = mathcal{L}^{-1}{(smathbf I - mathbf A)^{-1}}_{t=T}$

$mathbf B_d = left( int_{tau=0}^{T}e^{mathbf A tau}dtau right) mathbf B = mathbf A^{-1}(mathbf A_d - I)mathbf B$ , if $mathbf A$ is nonsingular

$mathbf C_d = mathbf C$

$mathbf D_d = mathbf D$

$mathbf Q_d = int_{tau=0}^{T} e^{mathbf A tau} mathbf Q e^{mathbf A^T tau} dtau$

$mathbf R_d = mathbf F$

and $T$ is the sample time.

Derivation

Starting with the continuous model

$mathbfdot{x}(t) = mathbf Amathbf x(t) + mathbf B mathbf u(t)$

we know that the matrix exponential is In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...

$frac{d}{dt}e^{mathbf At} = mathbf A e^{mathbf At} = e^{mathbf At} mathbf A$

and by premultiplying the model we get

$e^{-mathbf At} mathbfdot{x}(t) = e^{-mathbf At} mathbf Amathbf x(t) + e^{-mathbf At} mathbf Bmathbf u(t)$

which we recognize as

$frac{d}{dt}(e^{-mathbf At}mathbf x(t)) = e^{-mathbf At} mathbf Bmathbf u(t)$

and by integrating..

$e^{-mathbf At}mathbf x(t) - e^0mathbf x(0) = int_0^t e^{-mathbf Atau}mathbf Bmathbf u(tau) dtau$

$mathbf x(t) = e^{mathbf At}mathbf x(0) + int_0^t e^{mathbf A(t-tau)} mathbf Bmathbf u(tau) d tau$

which is an analytical solution to the continuous model.

Now we want to discretize the above expression. We assume that u is constant during each timestep. A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...

$mathbf x[t] equiv mathbf x(kT)$

$mathbf x[kT] = e^{mathbf AkT}mathbf x(0) + int_0^{kT} e^{mathbf A(kT-tau)} mathbf Bmathbf u(tau) d tau$

$mathbf x[k+1] = e^{mathbf A(k+1)T}mathbf x(0) + int_0^{(k+1)T} e^{mathbf A((k+1)T-tau)} mathbf Bmathbf u(tau) d tau$

$mathbf x[k+1] = e^{mathbf AT} left[ e^{mathbf AkT}mathbf x(0) + int_0^{kT} e^{mathbf A(kT-tau)} mathbf Bmathbf u(tau) d tau right]+ int_{kT}^{(k+1)T} e^{mathbf A(kT+T-tau)} mathbf Bmathbf u(tau) d tau$

We recognize the bracketed expression as $mathbf x[k]$ , and the second term can be simplified by substituting $v = k T + T - τ$ . We also assume that $mathbf u$ is constant during the integral, which in turn yields Î¨ÅÄ›ÃŸÄŽÅÄÄ: This article deals with the concept of an integral in calculus. ...

$mathbf x[k+1] = e^{mathbf AT}mathbf x[k] + left( int_0^T e^{mathbf Av} dv right) mathbf Bmathbf u[k]$

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 far more easily to calculate an approximate discrete model, based on that for small timesteps $e^{mathbf AT} approx mathbf I + mathbf A T$ . The approximate solution then becomes:

$mathbf x[k+1] approx (mathbf I + mathbf AT) mathbf x[k] + (mathbf I T + frac{1}{2} mathbf A T^2 ) mathbf B mathbf u[k]$

which can further be approximated if $frac{1}{2} mathbf A T^2$ is small; yielding

$mathbf x[k+1] approx (mathbf I + mathbf AT) mathbf x[k] + Tmathbf B mathbf u[k]$

Categories: ‪Control theory‬ | ‪Numerical analysis‬

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29CFR541.202 - Discretion and independent judgment. (0 words)
	In general, the exercise of discretion and independent judgment involves the comparison and the evaluation of possible courses of conduct, and acting or making a decision after the various possibilities have been considered.
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	DISCRETION - When it is said that something is left to the discretion of a judge, it signifies that he ought to decide according to the rules of equity, and the nature of circumstances.
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Derivation

Approximations

Discretization of linear state space models