# Direct¶

class sharpy.rom.balanced.Direct[source]

Find balanced realisation of continuous (DLTI = False) and discrete (DLTI = True) time of LTI systems using scipy libraries.

The function proceeds to achieve balanced realisation of the state-space system by first solving the Lyapunov equations. They are solved using Barlets-Stewart algorithm for Sylvester equation, which is based on A matrix Schur decomposition.

$\begin{split}\mathbf{A\,W_c + W_c\,A^T + B\,B^T} &= 0 \\ \mathbf{A^T\,W_o + W_o\,A + C^T\,C} &= 0\end{split}$

to obtain the reachability and observability gramians, which are positive definite matrices.

Then, the gramians are decomposed into their Cholesky factors such that:

$\begin{split}\mathbf{W_c} &= \mathbf{Q_c\,Q_c^T} \\ \mathbf{W_o} &= \mathbf{Q_o\,Q_o^T}\end{split}$

A singular value decomposition (SVD) of the product of the Cholesky factors is performed

$(\mathbf{Q_o^T\,Q_c}) = \mathbf{U\,\Sigma\,V^*}$

The singular values are then used to build the transformation matrix $$\mathbf{T}$$

$\begin{split}\mathbf{T} &= \mathbf{Q_c\,V\,\Sigma}^{-1/2} \\ \mathbf{T}^{-1} &= \mathbf{\Sigma}^{-1/2}\,\mathbf{U^T\,Q_o^T}\end{split}$

The balanced system is therefore of the form:

$\begin{split}\mathbf{A_b} &= \mathbf{T^{-1}\,A\,T} \\ \mathbf{B_b} &= \mathbf{T^{-1}\,B} \\ \mathbf{C_b} &= \mathbf{C\,T} \\ \mathbf{D_b} &= \mathbf{D}\end{split}$

Warning

This function may be less computationally efficient than the balreal Matlab implementation and does not offer the option to bound the realisation in frequency and time.

Notes

• Lyapunov equations are solved using Barlets-Stewart algorithm for Sylvester equation, which is based on A matrix Schur decomposition.

• Notation above is consistent with Gawronski .

Parameters
• A (np.ndarray) – Plant Matrix

• B (np.ndarray) – Input Matrix

• C (np.ndarray) – Output Matrix

• DLTI (bool) – Discrete time state-space flag

• Schur (bool) – Use Schur decomposition to solve the Lyapunov equations

Returns

Tuple of the form (S, T, Tinv) containing:
• Singular values in diagonal matrix (S)

• Transformation matrix (T).

• Inverse transformation matrix(Tinv).

Return type

tuple of np.ndarrays

References

 Anthoulas, A.C.. Approximation of Large Scale Dynamical Systems. Chapter 7. Advances in Design and Control. SIAM. 2005.

 Gawronski, W.. Dynamics and control of structures. New York: Springer. 1998

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name

Type

Description

Default

Options

tune

bool

Tune ROM to specified tolerance

True

use_schur

bool

Use Schur decomposition during build

False

rom_tolerance

float

Absolute accuracy with respect to full order frequency response

0.01

rom_tune_freq_range

list(float)

Beginning and end of frequency range where to tune ROM

[0, 1]

convergence

str

ROM tuning convergence. If min attempts to find minimal number of states.If all it starts from larger size ROM until convergence to specified tolerance is found.

min

reduction_method

str

Desired reduction method

realisation

realisation, truncation