State-Space Models and Subspace Identification Method¶
Process form:
where: \(y_k \in \mathbb{R}^{n_y}\), \(x_k \in \mathbb{R}^{n}\), \(u_k \in \mathbb{R}^{n_u}\), \(w_k \in \mathbb{R}^{n}\) and \(v_k \in \mathbb{R}^{n_y}\) are the system output, state, input, state noise and output measurement noise respectively (the subscript "k" denotes the \(k-th\) sampling time); \(A \in \mathbb{R}^{n\times n}\), \(B \in \mathbb{R}^{n\times n_u}\),\(\: C \in \mathbb{R}^{n_y \times n}\), \(D \in \mathbb{R}^{n_y \times n_u}\) are the system matrices.
Innovation form:
Predictor form:
where the following relations hold:
where \(K\) is the steady-state Kalman filter gain, obtained from Algebraic Riccati Equation.
The user has to define the future and past horizons (\(f\) and \(p\) respectively).
For traditional methods, that is, N4SID, MOESP and CVA methods, the future and past horizons are equal, set by default \(f=20\) (integer number).
For parsimonious methods, that is, PARSIM-P, PARSIM-S and PARSIM-K methods, the future and past horizons can be set, by default: \(f=20\) , \(p=20\) (integer numbers).
After performing the singular value decomposition (SVD) scheduled for the identification, which allows building the suitable subspace from the original data space,