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| [[Image:Finite element solution.svg|right|thumb|A solution to a discretized partial differential equation, obtained with the [[finite element method]].]]
| | My name is Keri and I am studying Directing and Industrial and Labor Relations at Lacchiarella / Italy.<br><br>my blog post - Luxury Way of Living ([http://klein.ga/258274 look at more info]) |
| In [[mathematics]], '''discretization''' concerns the process of transferring [[continuous function|continuous]] models and equations into [[wiktionary:discrete|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 (signal processing)|quantization]].
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| * [[Euler–Maruyama method]]
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| * [[Zero-order hold]]
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| 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.
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| == Discretization of linear state space models ==
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| Discretization is also concerned with the transformation of continuous [[differential equation]]s into discrete [[difference equations]], suitable for [[Numerical analysis|numerical computing]].
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| The following continuous-time [[State space (controls)|state space model]]
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| :<math>\dot{\mathbf{x}}(t) = \mathbf A \mathbf{x}(t) + \mathbf B \mathbf{u}(t) + \mathbf{w}(t)</math>
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| :<math>\mathbf{y}(t) = \mathbf C \mathbf{x}(t) + \mathbf D \mathbf{u}(t) + \mathbf{v}(t)</math>
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| where ''v'' and ''w'' are continuous zero-mean [[white noise]] sources with [[covariance]]s
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| :<math>\mathbf{w}(t) \sim N(0,\mathbf Q)</math>
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| :<math>\mathbf{v}(t) \sim N(0,\mathbf R)</math>
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| can be discretized, assuming [[zero-order hold]] for the input ''u'' and continuous integration for the noise ''v'', to
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| :<math>\mathbf{x}[k+1] = \mathbf A_d \mathbf{x}[k] + \mathbf B_d \mathbf{u}[k] + \mathbf{w}[k]</math>
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| :<math>\mathbf{y}[k] = \mathbf C_d \mathbf{x}[k] + \mathbf D_d \mathbf{u}[k] + \mathbf{v}[k]</math>
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| with covariances
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| :<math>\mathbf{w}[k] \sim N(0,\mathbf Q_d)</math>
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| :<math>\mathbf{v}[k] \sim N(0,\mathbf R_d)</math>
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| where
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| :<math>\mathbf A_d = e^{\mathbf A T} = \mathcal{L}^{-1}\{(s\mathbf I - \mathbf A)^{-1}\}_{t=T} </math>
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| :<math>\mathbf B_d = \left( \int_{\tau=0}^{T}e^{\mathbf A \tau}d\tau \right) \mathbf B = \mathbf A^{-1}(\mathbf A_d - I)\mathbf B </math>, if <math>\mathbf A</math> is [[Invertible matrix|nonsingular]]
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| :<math>\mathbf C_d = \mathbf C </math>
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| :<math>\mathbf D_d = \mathbf D </math>
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| :<math>\mathbf Q_d = \int_{\tau=0}^{T} e^{\mathbf A \tau} \mathbf Q e^{\mathbf A^T \tau} d\tau </math>
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| :<math>\mathbf R_d = \mathbf R </math>
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| and <math>T</math> is the sample time, although <math>\mathbf A^T</math> is the transposed matrix of <math>\mathbf A</math>. | |
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| A clever trick to compute ''Ad'' and ''Bd'' in one step is by utilizing the following property, p. 215:<ref>Raymond DeCarlo: ''Linear Systems: A State Variable Approach with Numerical Implementation'', Prentice Hall, NJ, 1989</ref>
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| :<math>e^{\begin{bmatrix} \mathbf{A} & \mathbf{B} \\
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| \mathbf{0} & \mathbf{0} \end{bmatrix} T} = \begin{bmatrix} \mathbf{M_{11}} & \mathbf{M_{12}} \\
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| \mathbf{0} & \mathbf{I} \end{bmatrix}</math>
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| and then having | |
| :<math>\mathbf A_d = \mathbf M_{11}</math>
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| :<math>\mathbf B_d = \mathbf M_{12}</math>
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| === Discretization of process noise ===
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| Numerical evaluation of <math>\mathbf{Q}_d</math> 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):
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| :<math> \mathbf{F} =
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| \begin{bmatrix} -\mathbf{A} & \mathbf{Q} \\
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| \mathbf{0} & \mathbf{A}^T \end{bmatrix} T</math>
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| :<math> \mathbf{G} = e^\mathbf{F} =
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| \begin{bmatrix} \dots & \mathbf{A}_d^{-1}\mathbf{Q}_d \\
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| \mathbf{0} & \mathbf{A}_d^T \end{bmatrix}.</math>
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| 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''':
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| :<math>\mathbf{Q}_d = (\mathbf{A}_d^T)^T (\mathbf{A}_d^{-1}\mathbf{Q}_d). </math>
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| === Derivation ===
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| Starting with the continuous model
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| :<math>\mathbf{\dot{x}}(t) = \mathbf A\mathbf x(t) + \mathbf B \mathbf u(t)</math>
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| we know that the [[matrix exponential]] is
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| :<math>\frac{d}{dt}e^{\mathbf At} = \mathbf A e^{\mathbf At} = e^{\mathbf At} \mathbf A</math>
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| and by premultiplying the model we get
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| :<math>e^{-\mathbf At} \mathbf{\dot{x}}(t) = e^{-\mathbf At} \mathbf A\mathbf x(t) + e^{-\mathbf At} \mathbf B\mathbf u(t)</math>
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| which we recognize as
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| :<math>\frac{d}{dt}(e^{-\mathbf At}\mathbf x(t)) = e^{-\mathbf At} \mathbf B\mathbf u(t)</math>
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| and by integrating..
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| :<math>e^{-\mathbf At}\mathbf x(t) - e^0\mathbf x(0) = \int_0^t e^{-\mathbf A\tau}\mathbf B\mathbf u(\tau) d\tau</math>
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| :<math>\mathbf x(t) = e^{\mathbf At}\mathbf x(0) + \int_0^t e^{\mathbf A(t-\tau)} \mathbf B\mathbf u(\tau) d \tau</math>
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| which is an analytical solution to the continuous model.
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| Now we want to discretise the above expression. We assume that u is [[mathematical constant|constant]] during each timestep.
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| :<math>\mathbf x[k] \ \stackrel{\mathrm{def}}{=}\ \mathbf x(kT)</math>
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| :<math>\mathbf x[k] = e^{\mathbf AkT}\mathbf x(0) + \int_0^{kT} e^{\mathbf A(kT-\tau)} \mathbf B\mathbf u(\tau) d \tau</math>
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| :<math>\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 B\mathbf u(\tau) d \tau</math>
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| :<math>\mathbf x[k+1] = e^{\mathbf AT} \left[ e^{\mathbf AkT}\mathbf x(0) + \int_0^{kT} e^{\mathbf A(kT-\tau)} \mathbf B\mathbf u(\tau) d \tau \right]+ \int_{kT}^{(k+1)T} e^{\mathbf A(kT+T-\tau)} \mathbf B\mathbf u(\tau) d \tau</math> | |
| We recognize the bracketed expression as <math>\mathbf x[k]</math>, and the second term can be simplified by substituting <math>v = kT + T - \tau</math>. We also assume that <math>\mathbf u</math> is constant during the [[integral]], which in turn yields
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| :<math>\mathbf x[k+1] = e^{\mathbf AT}\mathbf x[k] + \left( \int_0^T e^{\mathbf Av} dv \right) \mathbf B\mathbf u[k]=e^{\mathbf AT}\mathbf x[k] + \mathbf A^{-1}\left(e^{\mathbf AT}-\mathbf I \right) \mathbf B\mathbf u[k]</math>
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| which is an exact solution to the discretization problem.
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| === Approximations ===
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| 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 <math>e^{\mathbf AT} \approx \mathbf I + \mathbf A T</math>. The approximate solution then becomes:
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| :<math>\mathbf x[k+1] \approx (\mathbf I + \mathbf AT) \mathbf x[k] + T\mathbf B \mathbf u[k] </math>
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| Other possible approximations are <math>e^{\mathbf AT} \approx \left( \mathbf I - \mathbf A T \right)^{-1}</math> and <math>e^{\mathbf AT} \approx \left( \mathbf I +\frac{1}{2} \mathbf A T \right) \left( \mathbf I - \frac{1}{2} \mathbf A T \right)^{-1}</math>. 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.
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| == Discretization of continuous features ==
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| {{Main|Discretization of continuous features}}
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| 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.
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| ==See also==
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| *[[Discrete space]]
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| *[[Time-scale calculus]]
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| *[[Discrete event simulation]]
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| *[[Stochastic simulation]]
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| *[[Finite volume method for unsteady flow]]
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| *[[Properties of discretization schemes]]
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| == References ==
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| <references/>
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| * {{cite book|author=Robert Grover Brown & Patrick Y. C. Hwang|title=Introduction to random signals and applied Kalman filtering|edition=3rd|isbn=978-0471128397}}
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| * {{cite book|publisher=Saunders College Publishing|location=Philadelphia, PA, USA|year=1984|author=Chi-Tsong Chen|title=Linear System Theory and Design|isbn=0030716918}}
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| * {{cite journal|author=C. Van Loan|title=Computing integrals involving the matrix exponential|doi=10.1109/TAC.1978.1101743|journal=IEEE Transactions on Automatic Control|volume=23|issue=3|pages=395–404|date=Jun 1978}}
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| ==External links==
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| {{sisterlinks}}
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| [[Category:Numerical analysis]]
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| [[Category:Applied mathematics]]
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| [[Category:Functional analysis]]
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| [[Category:Iterative methods]]
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| [[Category:Control theory]]
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