Model-Based Processing

7Stochastic Subspace Identification

7.1 Introduction

In this chapter, we discuss the stochastic realization problem from both the classical and the subspace perspectives as before in the previous chapter. We start with the classical problem that mimics the deterministic approach with covariance matrices replacing impulse response matrices 1–10. The underlying Hankel matrix, now populated with covariance rather than impulse response matrices, admits a factorization leading to the fundamental rank condition (as before); however, the problem expands somewhat with the set of additional relations, termed the Kalman–Szego‐Popov (KSP) equations, that must be solved to provide the desired identification 11–16. All of these results follow from the innovations (INV) model of Chapter 3 or equivalently the steady‐state Kalman filter of Chapter 4. We again begin the development of this model and discuss the underlying system theoretical properties that lead to a “stochastic” realization.

The subspace approach also follows in a development similar to the deterministic case. Starting with the multivariable output error state‐space (MOESP) formulation using orthogonal projection theory, both the “past input multivariable output error state‐space” (PI‐MOESP) and “past input/output multivariable output error state‐space” (PO‐MOESP) techniques evolve 17–19. Next the numerical algorithms for state‐space system identification (N4SID) approach follow based on oblique projection theory 19 –27. We concentrate ...