TABLE OF CONTENTS The subspace identification problem is formulated for deterministic LTI systems, that is, LTI systems that are not disturbed by noise. Let such a system be given by
where x(k) ? 
n, u(k) ? m, and y(k) ? ?.
| Given a finite number of samples of the input signal u(k) and the output signal y(k) of the minimal (reachable and observable) system (9.1) (9.2), the goal is to determine the system matrices (A, B, C, D) and initial state vector up to a similarity transformation. |
An important and critical step prior to the design (and use) of subspace identification algorithms is to find an appropriate relationship between the measured data sequences and the matrices that define the model. This relation will be derived in Section 9.2.1. We proceed by describing subspace identification for an autonomous system (Section 9.2.2) and for the special case when the input is an impulse sequence (Section 9.2.3). Finally, we describe subspace identification for more general input sequences (Section 9.2.4).
In Section 3.4.2 we showed that the state of the system (9.1) (9.2) with initial state x(0) at time instant k is given by
By invoking Equation (9.2), we can specify the following relationship between the input data batch 
and the output data batch 
:
where s is some arbitrary positive integer. To use this relation in subspace identification, it is necessary to take s > n,
