[1] I. Markovsky and G. Mercére. Subspace identification with constraints on the impulse response. Technical report, Vrije Univ. Brussel, 2016. [ bib | pdf | software ]
Subspace identification methods produce unreliable model estimates when a small number of noisy measurements are available. In such cases, the accuracy of the estimated parameters can be improved by using prior knowledge about the system. The prior knowledge considered in this paper is constraints on the impulse response, , steady-state gain, overshoot, and rise time. The method proposed has two steps: 1) estimation of the impulse response with linear equality and inequality constraints, and 2) realization of the estimated impulse response. The problem on step 1 is shown to be a convex quadratic programming problem. In the case of prior knowledge expressed as equality constraints, the problem on step 1 admits a closed form solution. In the general case of equality and inequality constraints, the solution is computed by standard numerical optimization methods. We illustrate the performance of the method on a mass-spring-damper system.

Keywords: system identification, subspace methods, prior knowledge, behavioral approach.
[2] I. Markovsky, Otto Debals, and Lieven De Lathauwer. Sum-of-exponentials modeling and common dynamics estimation using tensorlab. Technical report, Vrije Univ. Brussel, 2015. [ bib | pdf ]
Fitting a signal to a sum-of-exponentials model is a basic problem in signal processing. It can be posed and solved as a Hankel structured low-rank matrix approximation problem. Subsequently, local optimization, subspace, and convex relaxation methods can be used for the numerical solution. In this paper, we show another approach, based on the recently proposed concept of structured data fusion. Structured data fusion problems are solved in the Tensorlab toolbox by local optimization methods. The approach allows fitting of signals with missing samples and adding constraints on the model, such as fixed exponents and common dynamics in multi-channel estimation problems. These problems are non-trivial to solve by other existing methods.

Keywords: system identification; low-rank approximation; mosaic Hankel matrix; tensorlab; structured data fusion.
[3] I. Markovsky. A low-rank matrix completion approach to data-driven signal processing. Technical report, Vrije Univ. Brussel, 2015. [ bib | pdf | software ]
In signal processing as well as in control and other mathematical engineering areas, it is common to use a model-based approach, which splits the problem into two steps: 1) model identification and 2) model-based design. Despite its success the model-based approach has the shortcoming that the design objective is not taken into account at the identification step, i.e., the model is not optimized for its intended use. In this paper, we propose an approach for data processing, called data-driven signal processing, that combines the identification and the model-based design into one joint problem. The to-be-computed signal is modeled as a missing part of a trajectory of the (unknown) data generating system. Subsequently, the missing data estimation problem is reformulated as a mosaic-Hankel structured matrix low-rank approximation and completion problem. A local optimization methods, based on the variable projections principle, is used for the numerical solution of the matrix completion/approximation problem. The missing data estimation approach for data-driven signal processing and the local optimization method for its implementation in practice are illustrated on examples of state estimation, filtering/smoothing, and prediction. Development of fast algorithms with provable properties in the presence of measurement noise and disturbances will make the matrix completion approach for data-driven signal processing a practically feasible alternative to the model-based methods.

Keywords: data-driven, Kalman filtering, structured low-rank approximation, missing data, matrix completion.
[4] N. Guglielmi and I. Markovsky. Computing the distance to uncontrollability: the SISO case. Technical report, Vrije Univ. Brussel, 2014. [ bib | pdf ]
In this paper, the problem of computing the distance from a given linear time-invariant system to the nearest uncontrollable system is posed and solved in the behavioral setting. In the case of a system with two external variables, the problem is restated as a Sylvester structured distance to singularity problem. The structured distance to singularity problem is then solved by integrating a system of ordinary differential equations which describes the gradient associated to the cost functional. An advantage of the method with respect to other approaches is in its capability to include further constraints. Numerical simulations also show that the method is more robust to the initial approximation than the Newton-type methods.

Keywords: Sylvester matrix, structured pseudospectrum, structured low-rank approximation, ODEs on matrix manifolds, structured distance to singularity, distance to uncontrollability, behavioral approach.

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