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 [2].
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

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

[2] 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