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This chapter discusses the stochastic realization problem from both the classical and the subspace perspectives. It starts with the classical problem that mimics the deterministic approach with covariance matrices replacing impulse response matrices. The underlying Hankel matrix, now populated with covariance rather than impulse response matrices, admits a factorization leading to the fundamental rank condition. The subspace approach also follows in a development similar to the deterministic case. Starting with the multivariable output error state‐space formulation using orthogonal projection theory, both the “past input multivariable output error state‐space” and “past input/output multivariable output error state‐space” techniques evolve. Next, the numerical algorithms for state‐space system identification approach follow based on oblique projection theory. The chapter concentrates primarily on a solution to the “combined problem” that is the identification of both the deterministic and stochastic systems directly (without separation) through the estimated state vector embedded in the innovation representation of the system. |
