SSconv

Convert a DLTI system with prediction and delay of the form:

\[\begin{split}\mathbf{x}_{n+1} &= \mathbf{A\,x}_n + \mathbf{B_0\,u}_n + \mathbf{B_1\,u}_{n+1} + \mathbf{B_{m1}\,u}_{n-1} \\ \mathbf{y}_n &= \mathbf{C\,x}_n + \mathbf{D\,u}_n\end{split}\]

into the state-space form:

\[\begin{split}\mathbf{h}_{n+1} &= \mathbf{A_h\,h}_n + \mathbf{B_h\,u}_n \\ \mathbf{y}_n &= \mathbf{C_h\,h}_n + \mathbf{D_h\,u}_n\end{split}\]

If \(\mathbf{B_{m1}}\) is None, the original state is retrieved through

\[\mathbf{x}_n = \mathbf{h}_n + \mathbf{B_1\,u}_n\]

and only the \(\mathbf{B}\) and \(\mathbf{D}\) matrices are modified.

If \(\mathbf{B_{m1}}\) is not None, the SS is augmented with the new state

\[\mathbf{g}_{n} = \mathbf{u}_{n-1}\]

or, equivalently, with the equation

\[\mathbf{g}_{n+1} = \mathbf{u}_n\]

leading to the new form

\[\begin{split}\mathbf{H}_{n+1} &= \mathbf{A_A\,H}_{n} + \mathbf{B_B\,u}_n \\ \mathbf{y}_n &= \mathbf{C_C\,H}_{n} + \mathbf{D_D\,u}_n\end{split}\]

where \(\mathbf{H} = (\mathbf{x},\,\mathbf{g})\).

param A:dynamics matrix type A:np.ndarray param B0:input matrix for input at current time step n. Set to None if this is zero. type B0:np.ndarray param B1:input matrix for input at time step n+1 (predictor term) type B1:np.ndarray param C:output matrix type C:np.ndarray param D:direct matrix type D:np.ndarray param Bm1:input matrix for input at time step n-1 (delay term) type Bm1:np.ndarray returns:tuple packed with the state-space matrices \(\mathbf{A},\,\mathbf{B},\,\mathbf{C}\) and \(\mathbf{D}\). rtype:tuple

References

Franklin, GF and Powell, JD. Digital Control of Dynamic Systems, Addison-Wesley Publishing Company, 1980

Warning

functions untested for delays (Bm1 != 0)