Balanced Truncation (Linear)

MaRDI QIDQ6684551

powerful technique to reduce the state-space dimension of a dynamical system with linear input equation

Description

In the case of a linear control system, a useful property of balanced truncation is that it admits easy control of the approximation error when truncating states.

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List of mathematical expressions with quantities

Control System Matrix A (Reduced)	$\tilde{A}\equiv T^{-1}AT$
$A$ symbol represents Control System Matrix A
$T$ symbol represents MOR Transformation Matrix
$\tilde{A}$ symbol represents Control System Matrix A (Reduced)
Control System Matrix B (Reduced)	$\tilde{B}\equiv T^{-1}BT$
$B$ symbol represents Control System Matrix B
$T$ symbol represents MOR Transformation Matrix
$\tilde{B}$ symbol represents Control System Matrix B (Reduced)
Control System Matrix C (Reduced)	$\tilde{C}\equiv CT$
$C$ symbol represents Control System Matrix C
$T$ symbol represents MOR Transformation Matrix
$\tilde{C}$ symbol represents Control System Matrix C (Reduced)
Initial Control State (Reduced)	$\tilde{x}(t=0)\equiv {\tilde{x}}_0$
$\tilde{x}$ symbol represents Control System State (Reduced)
$t$ symbol represents Time
${\tilde{x}}_0$ symbol represents Initial Control State (Reduced)
Balancing Transformation	$T^{-1}{W}_c{\left(T^{-1}\right)}^{*}=T^{*}{W}_{o}T=\left(\begin{array}{lll}{\sigma }_1& & 0\\ & \ddots & \\ 0& & {\sigma }_n\end{array}\right)=\mathrm{\Sigma }$
$T$ symbol represents MOR Transformation Matrix
$W_c$ symbol represents Gramian Matrix Controllability
$W_o$ symbol represents Gramian Matrix Observability
$\sigma$ symbol represents Hankel Singular Value
Control System Output Linear (Reduced)	$y(t)=\tilde{C}\tilde{x}(t)$
$\tilde{C}$ symbol represents Control System Matrix C (Reduced)
$\tilde{x}$ symbol represents Control System State (Reduced)
$t$ symbol represents Time
$y$ symbol represents Control System Output
Initial Control State	$x(t=0)=x_0$
$t$ symbol represents Time
$x$ symbol represents Control System State
$x_0$ symbol represents Control System Initial
Control System Input Linear	$\dot{x}(t)=Ax(t)+Bu(t)$
$A$ symbol represents Control System Matrix A
$B$ symbol represents Control System Matrix B
$t$ symbol represents Time
$u$ symbol represents Control System Input
$x$ symbol represents Control System State
Control System Input Linear (Reduced)	$\dot{\tilde{x}}(t)=\tilde{A}\tilde{x}(t)+\tilde{B}u(t)$
$\tilde{A}$ symbol represents Control System Matrix A (Reduced)
$\tilde{B}$ symbol represents Control System Matrix B (Reduced)
$\tilde{x}$ symbol represents Control System State (Reduced)
$t$ symbol represents Time
$u$ symbol represents Control System Input
Control System Output Linear	$y(t)=Cx(t)$
$C$ symbol represents Control System Matrix C
$t$ symbol represents Time
$x$ symbol represents Control System State
$y$ symbol represents Control System Output
Lyapunov Equation Controllability	$A{W}_c+W_c{A}^{*}+B{B}^{*}=0$
$A$ symbol represents Control System Matrix A
$B$ symbol represents Control System Matrix B
$W_c$ symbol represents Gramian Matrix Controllability
Lyapunov Equation Observability	$A^{*}{W}_o+W_{o}A+C^{*}C=0$
$A$ symbol represents Control System Matrix A
$C$ symbol represents Control System Matrix C
$W_o$ symbol represents Gramian Matrix Observability

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