Ivan Markovsky

Konstantin Usevich, Ivan Markovsky

The structured low-rank approximation problem for general affine structures, weighted 2-norms and fixed elements is considered. The variable projection principle is used to reduce the dimensionality of the optimization problem. Algorithms for evaluation of the cost function, the gradient and an approximation of the Hessian are developed. For $m \times n$ mosaic Hankel matrices the algorithms have complexity $O(m^2 n)$.

András Sasfi, Ivan Markovsky, Alberto Padoan et al.

We propose a modeling framework for stochastic systems, termed Gaussian behaviors, that describes finite-length trajectories of a system as a Gaussian process. The proposed model naturally quantifies the uncertainty in the trajectories, yet it is simple enough to allow for tractable formulations. We relate the proposed model to existing descriptions of dynamical systems including deterministic and stochastic behaviors, and linear time-invariant (LTI) state-space models with Gaussian noise. Gaussian behaviors can be estimated directly from observed data as the empirical sample covariance. The distribution of future outputs conditioned on inputs and past outputs provides a predictive model that can be incorporated in predictive control frameworks. We show that subspace predictive control is a certainty-equivalence control formulation with the estimated Gaussian behavior. Furthermore, the regularized data-enabled predictive control (DeePC) method is shown to be a distributionally optimistic formulation that optimistically accounts for uncertainty in the Gaussian behavior. To mitigate the excessive optimism of DeePC, we propose a novel distributionally robust control formulation, and provide a convex reformulation allowing for efficient implementation.

Farzan Kaviani, Ivan Markovsky, Hamid R. Ossareh

We revisit the problem of predicting the output of an LTI system directly using offline input-output data (and without the use of a parametric model) in the behavioral setting. Existing works calculate the output predictions by projecting the recent samples of the input and output signals onto the column span of a Hankel matrix consisting of the offline input-output data. However, if the offline data is corrupted by noise, the output prediction is no longer exact. While some prior works propose mitigating noisy data through matrix low-ranking approximation heuristics, such as truncated singular value decomposition, the ensuing prediction accuracy remains unquantified. This paper fills these gaps by introducing two upper bounds on the prediction error under the condition that the noise is sufficiently small relative to the offline data's magnitude. The first bound pertains to prediction using the raw offline data directly, while the second one applies to the case of low-ranking approximation heuristic. Notably, the bounds do not require the ground truth about the system output, relying solely on noisy measurements with a known noise level and system order. Extensive numerical simulations show that both bounds decrease monotonically (and linearly) as a function of the noise level. Furthermore, our results demonstrate that applying the de-noising heuristic in the output error setup does not generally lead to a better prediction accuracy as compared to using raw data directly, nor a smaller upper bound on the prediction error. However, it allows for a more general upper bound, as the first upper bound requires a specific condition on the partitioning of the Hankel matrix.

Konstantin Usevich, Ivan Markovsky

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.