Elsevier

Automatica

Volume 119, September 2020, 109043
Automatica

Technical communique
A multi-observer based estimation framework for nonlinear systems under sensor attacks

https://doi.org/10.1016/j.automatica.2020.109043Get rights and content

Abstract

For general discrete-time nonlinear systems, we address the state estimation problem under sensor attacks. We provide a general estimation scheme, built around the idea of sensor redundancy and multi-observers, capable of reconstructing the system state in spite of sensor attacks and noise, for a large class of nonlinear plants and observers. This scheme has been proposed by others for linear systems and here we propose a unifying framework for a much larger class of nonlinear systems.

Introduction

Security of Network Control Systems (NCSs) has become a concern as wireless communication networks increasingly serve as new access points for adversaries trying to disrupt the system dynamics. In Chong and Kuijper, 2016, Chong et al., 2015, Fawzi et al., 2014, Kafash et al., 2017, Murguia and Ruths, 2016, Murguia et al., 2016, Pasqualetti et al., 2013, Shoukry et al., 2015, Shoukry et al., 2017, Teixeira et al., 2012 and Vamvoudakis, Hespanha, Sinopoli, and Mo (2015), a range of problems related to security of control systems have been discussed. In general, they provide analysis tools for quantifying the performance degradation induced by different classes of attacks and propose reaction and prevention strategies to counter their effect on the system dynamics. Most of the existing work, however, has considered control systems with linear dynamics, although in many engineering applications the dynamics of the plants being monitored and controlled is highly nonlinear. There are only a few results addressing the problem of state estimation under attacks for some classes of nonlinear systems. The recent work in Kim, Lee, Shim, Eun, and Seo (2016) addresses the problem of sensor attack detection and state estimation for uniformly observable continuous-time nonlinear systems. For a class of power systems under sensor attacks, the authors in Hu, Fooladivanda, Chang, and Tomlin (2017) provide an estimator of the system state using compressed sensing techniques. In Shoukry et al. (2015), satisfiability modulo theory is used for state estimation for differentially flat systems with corrupted sensors. In our previous work (Yang et al., 2018a, Yang et al., 2018b), the problem of state estimation and attack isolation for a class of nonlinear systems with positive-slope nonlinearities is considered. We provided an observer-based estimation/isolation strategy, using a bank of circle-criterion observers.

The core of our estimation scheme is based on the work in Chong et al. (2015), where the problem of state estimation for continuous-time Linear Time Invariant (LTI) systems is addressed. The authors propose a multi-observer estimator, using a bank of Luenberger observers, which provides a robust estimate of the system state in spite of sensor attacks. In this manuscript, we extend the results in Chong et al. (2015) and Yang et al., 2018a, Yang et al., 2018b by considering systems with general nonlinear dynamics — thus considerably extending the applicability of multi-observer estimation techniques. We cast the multi-observer estimation problem in terms of the existence of a bank of (local and practical) nonlinear observers with Input-to-State Stable (ISS) (with respect to disturbances) estimation error dynamics. Following the results in Chong et al. (2015), we use a bank of observers to construct an estimator that provides a robust state estimate in the presence of false data injection attacks and noise.

The main idea behind the multi-observer estimator is the following: Each observer in the bank is driven by a different subset of sensors. Then, for every pair of observers in the bank, the estimator computes the difference between their estimates and selects the observers leading to the smallest difference. If there are attacks on some of the sensors, the observers driven by those sensors produce larger differences than the attack-free ones, in general, and thus they are not selected by the estimator. Assuming that the estimation error dynamics of each observer is ISS with respect to disturbances, in the attack-free case, our estimator provides estimates whose errors satisfy an ISS-like property with respect to disturbances and independent of the attack signals.

Notation: For any vector vRn, we denote vJ the stacking of all vi, iJ, J1,,n, |v|vv, and the support of v as supp(v)=i1,,n|vi0. For a sequence v(k)k=0, vsupk0|v(k)|, v(k)Rn. We say that a sequence v(k) belongs to l, v(k)l, if v<. A continuous function α:[0,a)[0,) is said to belong to class K, if it is strictly increasing and α(0)=0, Khalil (2002). Similarly, a continuous function β:[0,a)×[0,)[0,) is said to belong to class KL if, for fixed s, the mapping β(r,s) belongs to class K with respect to r and, for fixed r, the mapping β(r,s) is decreasing with respect to s and β(r,s)0 as s, Khalil (2002).

Section snippets

Robust multi-observer based estimator

The tools given in this section, generalize the results in Chong et al. (2015) and Yang et al. (2018a) by considering systems with general nonlinear dynamics, disturbances, and noise. Consider the system: x+=F(x,u,d),yi=gi(x,u,mi,ai),i1,,p,with state xRn, input uRnu, disturbance dRs, d(k)l, ith sensor measurement yiR, stacked measurements y(y1,,yp)Rp, attack signal aiR, stacked attack vector a(a1,,ap)Rp, measurement noise miR, mi(k)l, and nonlinear functions F:Rn×Rnu×RsRn

Example: Circle-criterion observers

Consider the system x+=Ax+Gf(Hx)+ρ(u,y),y=Cx+a+m,with state xRn, control uRnu, output yRp, measurement noise mRp, m(k)l, and matrices GRn×r and HRr×n. The term ρ(u,y) is a known arbitrary real-valued vector that depends on the system inputs and outputs. The state-dependent nonlinearity f(Hx) is an r-dimensional vector where each entry is a function of a linear combination of the states: fi=fij=1nHijxj,i=1,,r,where Hij are the entries of matrix H.

Assumption 3

For any i1,,r: fi(vi)fi(wi)viwi0,v

Conclusion

Built around the idea of multi-observers and exploiting sensor redundancy, we have proposed a general estimation framework applicable to a large class of nonlinear dynamical systems. We have proved that the proposed estimator provides robust ISS estimates of the system state provided that a sufficiently small subset of sensors are corrupted by attack signals. We have posed the estimator design in terms of the existence of a bank of (local and practical) nonlinear observers with ISS estimation

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    This work was partially supported by the Australian Research Council (ARC) under the Discovery Project DP170104099. The material in this paper was partially presented at The 57th IEEE Conference on Decision and Control, December 17–19, 2018, Miami Beach, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Daniele Astolfi under the direction of Editor André L. Tits.

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