An Optimization Framework for Resilient Batch Estimation in Cyber-Physical Systems

Abstract

This paper proposes a class of resilient state estimators for LTV discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. The latter in addition to being arbitrary in its form, need not be strictly bounded. In this setting, we construct the estimator as the set-valued map which associates to the measurements, the minimizing set of some appropriate performance functions. We consider a family of such performance functions each of which yielding a specific instance of the proposed general estimation framework. It is then shown that the proposed class of estimators enjoys the property of resilience, that is, it induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Hence, the estimation error may be bounded while the measurement noise is virtually unbounded. Moreover, we provide several error bounds (in different configurations) whose expressions depend explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

On sparsity‐inducing methods in system identification and state estimation

“…In this section we illustrate how sparse optimization can be used to design resilient state estimators, see, e.g., 29,27,28 . For this purpose, let us consider a linear time-invariant system subject to disturbances…”

“…In this section we illustrate how sparse optimization can be used to design resilient state estimators, see, for example, References 27‐29. For this purpose, let us consider a linear time‐invariant system subject to disturbances where is the state of the system, is the output, and are the process and measurement noises respectively.…”

“…The current paper will focus essentially on the formal characterization, the properties and the implementation of the LAD estimator and some of its variants for robust regression, hybrid system identification and resilient state estimation. The robustness properties of the LAD estimator make it applicable to hybrid system identification 21,22,23,24 , subspace clustering 25,26 and secure state estimation 27,28,29,30,31,32 .…”

Analysis of the least sum-of-minimums estimator for switched systems

“…for all t ∈ T. The defining constraint (7) of the switching signal σA allows indeed for multiple choices of σA(t) whenever arg min i∈S yt − x ⊤ t ai ⊂ S is not a singleton. One simple choice to solve this issue would be to set arbitrarily σA(t) to be equal to the smallest element of arg min i∈S yt − x ⊤ t ai .…”

“…Hence, it is of interest to use the possible extra-degree of freedom offered by Eq. ( 7) to select σA so as to maximize min i∈S |Ii(A)| subject to the constraint (7). In case the maximizing σA is still not unique, we can make it unique for a given A by selecting the one which assigns to each t, the smallest admissible index i ∈ S. To sum up, given A ∈ R n×s , σA can be selected uniquely by following the process described above.…”