Laurent Bako

Analysis of a nonsmooth optimization approach to robust estimation

In this paper, we consider the problem of identifying a linear map from measurements which are subject to intermittent and arbitrarily large errors. This is a fundamental problem in many estimation-related applications such as fault detection, state estimation in lossy networks, hybrid system identification, robust estimation, etc. The problem is hard because it exhibits some intrinsic combinatorial features. Therefore, obtaining an effective solution necessitates relaxations that are both solvable at a reasonable cost and effective in the sense that they can return the true parameter vector. The current paper discusses a nonsmooth convex optimization approach and provides a new analysis of its behavior. In particular, it is shown that under appropriate conditions on the data, an exact estimate can be recovered from data corrupted by a large (even infinite) number of gross errors.

State and Parameter Estimation: A Nonlinear Luenberger Observer Approach

The design of a nonlinear Luenberger observer for an extended nonlinear system resulting from a parameterized linear SISO (single-input single-output) system is studied. From an observability assumption of the system, the existence of such an observer is concluded. In a second step, a novel algorithm for the identification of such a system is suggested. Compared to the adaptive observers available in the literature, it has the advantage to be of low dimension and to admit a strict Lyapunov function.

Adaptive identification of linear systems subject to gross errors

In this note, we investigate the convergence of a robust recursive identifier for linear models subject to impulsive disturbances. Under the assumption that the disturbance is unknown and can be of arbitrarily large magnitude, the analyzed algorithm attempts to minimize online the sum of absolute errors so as to achieve a sparse prediction error sequence. It is proved that the identifier converges exponentially fast into an euclidean ball whose size is determined by the richness properties of the estimation data, the frequency of occurrence of impulsive errors and the parameters of the algorithm.

Identification of linear systems with nonlinear Luenberger Observers

The design of a nonlinear Luenberger observer for a parametrized linear system is studied. From an observability assumption of the system, the existence of such an observer is concluded. In a second step, a constructive novel algorithm for the identification of multi-input multi-output linear systems is suggested and is implemented on a second order system.

On a Class of Optimization-Based Robust Estimators

In this paper, we consider the problem of estimating a parameter matrix from observations which are affected by two types of noise components: (i) a sparse noise sequence which, whenever nonzero can have arbitrarily large amplitude (ii) and a dense and bounded noise sequence of “moderate” amount. This is termed a robust regression problem. To tackle it, a quite general optimization-based framework is proposed and analyzed. When only the sparse noise is present, a sufficient bound is derived on the number of nonzero elements in the sparse noise sequence that can be accommodated by the estimator while still returning the true parameter matrix. While almost all the restricted isometry-based bounds from the literature are not verifiable, our bound can be easily computed through solving a convex optimization problem. Moreover, empirical evidence tends to suggest that it is generally tight. If in addition to the sparse noise sequence, the training data are affected by a bounded dense noise, we derive an upper bound on the estimation error.

Incremental stability of Lur’e systems through piecewise-affine approximations

Abstract Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur’e system can then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for asymptotic incremental stability are provided. These conditions are expressed as linear matrix inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples.

Adaptive identification of continuous-time MIMO state-space models

This paper studies the problem of identifying linear continuous-time state-space models from input-output measurements. An adaptive identifier is developed for the online estimation of both the state and the model parameters in a deterministic framework. Our approach is a non parametric one in the sense that it provides an arbitrary realization of the system. It relies on ideas from the subspace identification literature and adaptive observer.