Identification of Shifted‐Affine Linear Parameter‐Varying Models in the Errors‐In‐Variables Setting

ABSTRACT This paper presents an identification method for linear parameter‐varying (LPV) models with affine dependence on the scheduling variables, referred to as shifted‐affine linear parameter‐varying (SALPV) models. Utilizing recent work extending the Fundamental Lemma by Willems et al. to LPV models, the identification problem is shown to be equivalent to a structured low‐rank approximation problem and is solved using a variable projection method. The proposed method exploits the Kronecker–Hankel structure, a block Hankel matrix where each block element is the Kronecker product of a known vector with an unknown vector, which is a structure not previously considered in related literature. We consider noise on both the input and output signal, and noise‐free scheduling variables, which corresponds to the errors‐in‐variables (EIV) setting in the linear time‐invariant (LTI) literature, a setting which is not typically considered in existing LPV identification methods. Under reasonable assumptions, the proposed method is guaranteed to converge to a stationary point of the cost function, which is proportional to the log‐likelihood under the considered noise setting, and has a runtime per iteration that scales linearly with the number of samples. Extensions of the proposed method are introduced to allow for additional freedom in the problem definition, including missing and fixed elements and a weighted cost function. The proposed method is compared to state‐of‐the‐art methods on a simulated mass‐spring‐damper system.