balreal_direct_pyΒΆ

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:

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

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.

param A:

Plant Matrix
type A:

np.ndarray
param B:

Input Matrix
type B:

np.ndarray
param C:

Output Matrix
type C:

np.ndarray
param DLTI:

Discrete time state-space flag
type DLTI:

bool
param Schur:

Use Schur decomposition to solve the Lyapunov equations
type Schur:

bool
returns:
Tuple of the form (S, T, Tinv) containing:
  • Singular values in diagonal matrix (S)
  • Transformation matrix (T).
  • Inverse transformation matrix(Tinv).
rtype:

tuple of np.ndarrays

References

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