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 arbitarily 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.Comment: 17 pages, 9 figure

Global optimization for low-dimensional switching linear regression and bounded-error estimation

The paper provides global optimization algorithms for two particularly difficult nonconvex problems raised by hybrid system identification: switching linear regression and bounded-error estimation. While most works focus on local optimization heuristics without global optimality guarantees or with guarantees valid only under restrictive conditions, the proposed approach always yields a solution with a certificate of global optimality. This approach relies on a branch-and-bound strategy for which we devise lower bounds that can be efficiently computed. In order to obtain scalable algorithms with respect to the number of data, we directly optimize the model parameters in a continuous optimization setting without involving integer variables. Numerical experiments show that the proposed algorithms offer a higher accuracy than convex relaxations with a reasonable computational burden for hybrid system identification. In addition, we discuss how bounded-error estimation is related to robust estimation in the presence of outliers and exact recovery under sparse noise, for which we also obtain promising numerical results

Identification of Piecewise Linear Models of Complex Dynamical Systems

The paper addresses the realization and identification problem or a subclass of piecewise-affine hybrid systems. The paper provides necessary and sufficient conditions for existence of a realization, a characterization of minimality, and an identification algorithm for this subclass of hybrid systems. The considered system class and the identification problem are motivated by applications in systems biology

Hybrid System Identification of Manual Tracking Submovements in Parkinson\u27s Disease

Seemingly smooth motions in manual tracking, (e.g., following a moving target with a joystick input) are actually sequences of submovements: short, open-loop motions that have been previously learned. In Parkinson\u27s disease, a neurodegenerative movement disorder, characterizations of motor performance can yield insight into underlying neurological mechanisms and therefore into potential treatment strategies. We focus on characterizing submovements through Hybrid System Identification, in which the dynamics of each submovement, the mode sequence and timing, and switching mechanisms are all unknown. We describe an initialization that provides a mode sequence and estimate of the dynamics of submovements, then apply hybrid optimization techniques based on embedding to solve a constrained nonlinear program. We also use the existing geometric approach for hybrid system identification to analyze our model and explain the deficits and advantages of each. These methods are applied to data gathered from subjects with Parkinson\u27s disease (on and off L-dopa medication) and from age-matched control subjects, and the results compared across groups demonstrating robust differences. Lastly, we develop a scheme to estimate the switching mechanism of the modeled hybrid system by using the principle of maximum margin separating hyperplane, which is a convex optimization problem, over the affine parameters describing the switching surface and provide a means o characterizing when too many or too few parameters are hypothesized to lie in the switching surface

Realization of multi-input/multi-output switched linear systems from Markov parameters

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the submodels have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, zero sets of LTV Hankel matrices where the realized system has a linear time-invariant (LTI) pulse response matching that of the original SLS are exploited to extract the submodels, up to arbitrary similarity transformations, by a clustering algorithm using a statistics that is invariant to similarity transformations. Recovery is shown to be complete if the dwell times are sufficiently long and some mild identifiability conditions are met. In the third stage, the switching sequence is estimated by three schemes. The first scheme is based on forward/backward corrections and works on the short segments. The second scheme matches Markov parameter estimates to the true parameters for LTV systems and works on the medium-to-long segments. The third scheme also matches Markov parameters, but for LTI systems only and works on the very short segments. In the fourth stage, the submodels estimated in Stage~2 are brought to a common basis by applying a novel basis transformation method which is necessary before performing output predictions to given inputs. A numerical example illustrates the properties of the realization algorithm. A key role in this algorithm is played by time-dependent switching sequences that partition the state-space according to time, unlike many other works in the literature in which partitioning is state and/or input dependent