Using Maximum Likelihood (or Prediction Error) methods to identify linear state space model is a prime technique. The likelihood function is a nonconvex function and care must be exercised in the numerical maximization. Here the focus will be on affine parameterizations which allow some special techniques and algorithms. Three approaches to formulate and perform the maximization are described in this contribution: (1) The standard and well known Gauss-Newton iterative search, (2) a scheme based on the EM (expectation-maximization) technique, which becomes especially simple in the affine parameterization case, and (3) a new approach based on lifting the problem to a higher dimension in the parameter space and introducing rank constraints.@en

In this paper, a unified identification framework called constrained subspace method for structured state-space models (COSMOS) is presented, where the structure is defined by a user specified linear or polynomial parametrization. The new approach operates directly from the input and output data, which differs from the traditional two-step method that first obtains a state-space realization followed by the systemparameter estimation. The new identification framework relies on a subspace inspired linear regression problem which may not yield a consistent estimate in the presence of process noise. To alleviate this problem, the linear regression formulation is imposed by structured and low rank constraints in terms of a finite set of system Markov parameters and the user specified model parameters. The non-convex nature of the constrained optimization problem is dealt with by transforming the problem into a difference-of-convex optimization problem, which is then handled by the sequential convex programming strategy. Numerical simulation examples show that the proposed identification method is more robust than the classical prediction-error method (PEM) initialized by random initial values in converging to local minima, but at the cost of heavier computational burden.@en