Detection of Sensor Attack and Resilient State Estimation for Uniformly Observable Nonlinear Systems having Redundant Sensors

This paper presents a detection algorithm for sensor attacks and a resilient state estimation scheme for a class of uniformly observable nonlinear systems. An adversary is supposed to corrupt a subset of sensors with the possibly unbounded signals, while the system has sensor redundancy. We design an individual high-gain observer for each measurement output so that only the observable portion of the system state is obtained. Then, a nonlinear error correcting problem is solved by collecting all the information from those partial observers and exploiting redundancy. A computationally efficient, on-line monitoring scheme is presented for attack detection. Based on the attack detection scheme, an algorithm for resilient state estimation is provided. The simulation results demonstrate the effectiveness of the proposed algorithm

A Multi-Observer Based Estimation Framework for Nonlinear Systems under Sensor Attacks

We address the problem of state estimation and attack isolation for general discrete-time nonlinear systems when sensors are corrupted by (potentially unbounded) attack signals. For a large class of nonlinear plants and observers, we provide a general estimation scheme, built around the idea of sensor redundancy and multi-observer, capable of reconstructing the system state in spite of sensor attacks and noise. This scheme has been proposed by others for linear systems/observers and here we propose a unifying framework for a much larger class of nonlinear systems/observers. Using the proposed estimator, we provide an isolation algorithm to pinpoint attacks on sensors during sliding time windows. Simulation results are presented to illustrate the performance of our tools.Comment: arXiv admin note: text overlap with arXiv:1806.0648

The observer error linearization problem via dynamic compensation

Linearization by output injection has played a key role in the observer design for nonlinear control systems for almost three decades. In this technical note, following some recent works, geometric necessary and sufficient conditions are derived for the existence of a dynamic compensator solving the problem under regular output transformation. An algorithm which computes a compensator of minimal order is given. © 2014 IEEE

축소 차원 동적 관측기 오차 선형화와 확장된 비선형 관측기 정준형을 통한 비선형 관측기 설계

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2014. 8. 서진헌.본 논문은 비선형 시스템에 대한 관측기 설계 문제를 다루고 있다. 관측기 설계 문제란 주어진 시스템의 입력과 출력 정보만을 활용하여 대상 시스템의 상태 변수를 추정할 수 있는 시스템을 설계하는 것이다. 선형 시스템의 경우에는 루엔버거 관측기(Luenberger observer)로 알려진 일반적인 해법이 존재하는 반면, 일반적인 비선형 시스템에 대해 관측기를 설계하는 방법에 대한 연구 결과는 현재까지 보고된 바가 없다. 다만, 특정한 형태의 비선형 시스템에 대해 관측기를 설계하는 문제에 대한 연구는 활발하게 진행되어 오고 있다. 관측기 오차 선형화(observer error linearization) 기법은 이 문제에 대한 가장 잘 알려진 방법론 중의 하나로서, 주어진 비선형 시스템을 좌표 변환을 통해 관측 가능한 선형 시스템과 출력주입(output injection) 부분들로 구성된 비선형 관측기 정준형(nonlinear observer canonical form)으로 변환시키는 문제이다. 비선형 관측기 정준형으로 변환 가능한 좌표계에서는 시스템의 모든 비선형성이 시스템의 입력과 출력의 함수로 이루어진 출력 주입 부분에 제한되므로, 이를 상쇄시킴으로써 선형 시스템의 경우와 비슷한 형태의 루엔버거형의 관측기(Luenberger-type observer)를 설계하는 것이 가능하고, 이에 따라 선형화된 관측기 오차 동역학(observer error dynamics)을 얻을 수 있다. 관측기 오차 선형화 기법의 출현 이래로, 이를 적용할 수 있는 시스템의 범위를 확장시키기 위한 여러 연구가 진행되어 왔다. 그 중 하나는 주어진 시스템을 보다 높은 차수의 비선형 관측기 정준형으로 변환시키는 방법이다. 이러한 방식에는 시스템 이머젼 기법과 동적 관측기 오차 선형화(dynamic observer error linearization) 기법이 있는데, 그 중에서도 동적 관측기 오차 선형화 기법의 특징은 다음과 같이 크게 두 가지로 요약될 수 있다. 첫째는 대상 시스템의 출력을 입력으로 하는 보조 동역학(auxiliary dynamics)을 설계하는 것이고, 둘째는 보조 동역학을 포함하는 확장된 시스템을 대상 시스템보다 높은 차수의 일반화된 비선형 관측기 정준형(generalized nonlinear observer canonical form)으로 변환하는 것이다. 동적 관측기 오차 선형화 기법에서 제안된 일반화된 비선형 관측기 정준형은 관측 가능한 선형 시스템과 일반화된 출력 주입(generalized output injection)으로 구성되어 있고, 일반화된 출력 주입은 대상 시스템의 출력 뿐만 아니라 보조동역학의 상태 변수에 대한 함수로 이루어져 있다는 차이점이 있다. 하지만, 이 방법론은 관측기의 차수가 대상 시스템의 차수보다 크다는 단점을 가지고 있다. 이러한 문제를 해결하기 위해, 최근에는 동적 관측기 오차 선형화의 변형된 기법으로서 축소 차원 동적 관측기 오차 선형화(reduced-order dynamic observer error linearization)란 기법이 단일 출력 시스템에 대해 새롭게 제안되었다. 축소 차원 동적 관측기 오차 선형화 기법 역시 보조 동역학을 설계하여 확장된 시스템을 일반화된 비선형 관측기 정준형으로 변환시킨다는 점에서 동적 관측기 오차 선형화 기법과 공통점을 갖지만, 변환된 일반화된 비선형 관측기 정준형의 차수가 대상 시스템의 차수와 같다는 차이점이 있다. 비록 축소 차원 동적 관측기 오차 선형화 기법이 적용 가능한 시스템의 범주는 동적 관측기 선형화 기법이 적용 가능한 시스템 범주를 벗어날 수는 없지만, 축소 차원 동적 관측기 오차 선형화 기법은 동적 관측기 선형화 기법에 비해 더 작은 차수의 관측기를 설계할 수 있다는 이점이 있고, 보조 동역학의 개념을 도입함으로써 관측기 오차 선형화 기법에 비해 더 넓은 범주의 시스템에 적용 가능하다는 장점을 지닌다. 뿐만 아니라, 축소 차원 동적 관측기 오차 선형화 기법의 개념 자체가 관측기 오차 선형화 기법의 개념과 매우 흡사하기 때문에 (보조 동역학을 고려하지 않은 축소 차원 동적 관측기 오차 선형화 문제는 관측기 오차 선형화 문제와 일치한다.) 축소 차원 동적 관측기 오차 선형화 기법에 대한 연구를 통해 기존의 관측기 오차 선형화 기법을 해석할 수도 있다. 이에 따라, 본 논문에서는 축소 차원 동적 관측기 오차 선형화 기법을 다중 출력 시스템에 대해 확장시키고, 이에 대한 연구를 수행하여 궁극적으로는 주어진 다중 출력 시스템이 이 기법에 의해 일반화된 비선형 관측기 정준형으로 변환될 수 있는 필요충분 조건을 제시한다. 이 결과는 현재까지 확립되지 않았던 일반적인 형태의 출력 변환까지 고려하였을 경우의 다중 출력 시스템에 대한 관측기 오차 선형화 문제의 필요충분 조건을 내포하고 있다. 또한, 본 논문에서는 비선형 관측기 정준형의 선형 부분 또한 시스템의 출력과 보조 동역학의 상태 변수에 의해 결정되는 확장된 비선형 관측기 정준형(extended nonlinear observer canonical form)을 제안하고, 축소 차원 동적 관측기 오차 선형화의 확장된 기법으로서 주어진 단일 출력 시스템을 보조 동역학을 설계하여 확장된 비선형 관측기 정준형으로 변환하는 문제를 제안하고 이에 대한 필요충분 조건을 제시한다. 또한 이 결과를 뢰슬러 시스템(Rossler system)에 적용시켜봄으로써 새롭게 제안된 방법론이 축소 차원 동적 관측기 오차 선형화에 비해 더 넓은 범주의 시스템에 적용될 수 있음을 예증한다.This dissertation contributes to the observer design problem for some classes of nonlinear systems. The observer design problem is to construct a dynamic system (called observer) that can estimate the state of a given dynamic system by using available signals which are commonly the input and the output of the given system. While a standard solution (called Luenberger observer) to the problem was solved for linear systems, there has not been a unified solution for general nonlinear systems. However, there have been significant research efforts on the problem of designing observers for special classes of nonlinear systems. Observer error linearization (OEL) is one of the well-known methods, and it is the problem of transforming a nonlinear system into a nonlinear observer canonical form (NOCF) that is an observable linear system modulo output injection. If a nonlinear system can be transformed into the NOCF, then all the nonlinearities of the system are restricted to the output injection term which is a vector-valued function of the system input and the system output. As a result, we can design a Luenberger-type observer that cancels out the output injection and thus has a linear observer error dynamics in the transformed coordinates. In order to extend the class of systems to which the OEL approach is applicable, a lot of attempts have been made in the past three decades. One of them is to transform a nonlinear system into a higher-dimensional NOCF: system immersion and dynamic observer error linearization (DOEL). In particular, the main idea of DOEL is twofold: the first is to introduce an auxiliary dynamics whose input is system output, and the second is to transform the extended system into a generalized nonlinear observer canonical form (GNOCF) that is an observable linear system modulo generalized output injection depending not only on the system output but also on the state of auxiliary dynamics. By introducing such an auxiliary dynamics, the DOEL problem can be solved for a larger class of systems compared with the (conventional) OEL problem. However, it has a drawback on the dimension of observer. That is, the dimension of observer designed by the DOEL approach is larger than that of the given system, because the dimension of GNOCF equals to the sum of dimensions of the given system and the auxiliary dynamics. Recently, inspired by this fact, a new approach called reduced-order dynamic observer error linearization (RDOEL) was proposed for single output nonlinear systems. In the framework of RDOEL, we also introduce an auxiliary dynamics and transform the extended system into GNOCF in a similar fashion to DOEL, but the coordinate transformation preserves the coordinates corresponding to the state of auxiliary dynamics so that the dimension of GNOCF equals to that of the given system. Although RDOEL is a special case of DOEL (that is, the class of systems to which the RDOEL approach can be applied is a subset of that of DOEL), the RDOEL approach offers a lower-dimensional observer compared to the DOEL approach, and it is also applicable to a larger class of systems compared to the (conventional) OEL approach. In addition, since the framework of RDOEL is coterminous with that of OEL (in fact, the OEL problem is identical to the RDOEL problem with no auxiliary dynamics), most of results for the RDOEL problem can be also used to analyze the OEL problem by slight modification. In this respect, one of the topics of this dissertation is to deal with the RDOEL problem for multi-output systems. We first formulate the framework of RDOEL for multi-output nonlinear systems and provide three necessary conditions. And then, by means of the necessary conditions, we derive a geometric necessary and sufficient condition in terms of Lie algebras of vector fields. Since the proposed RDOEL problem is a natural extension of the (conventional) OEL problem, the result can be easily translated into a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where an output transformation of general form is considered. The other topic of the dissertation is to introduce an extended nonlinear observer canonical form (ENOCF) whose linear part also depends on the system output and the state of auxiliary dynamics, and to deal with the problem of transforming a single output nonlinear system with an auxiliary dynamics into the ENOCF as an extension of the RDOEL problem. Since the proposed ENOCF admits a kind of high-gain observers, the solvability of the problem allows us to design observers for a class of single output nonlinear systems. We also first present two necessary conditions, and then derive a geometric necessary and sufficient condition for the problem. Furthermore, as a case study, we apply the results to the Rӧssler system in order to show that the proposed method enlarges the class of applicable systems compared with the RDOEL approach.ABSTRACT i List of Figures ix Notation and Acronyms x 1 Introduction 1 1.1 Research Background 1 1.2 Organization and Contributions of the Dissertation 5 2 Mathematical Preliminaries 7 2.1 Manifolds and Differentiable Structures 7 2.2 Vector Fields and Covector Fields 10 2.3 Lie Derivatives and Lie Brackets 13 2.4 Distributions and Codistributions 16 3 Review of Related Previous Works 21 3.1 Observability of Multi-Output Nonlinear Systems 21 3.2 Observer Error Linearization (OEL) 23 3.3 System Immersion 28 3.4 Dynamic Observer Error Linearization (DOEL) 30 3.5 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Single Output Systems 36 3.6 Inclusion Relation among OEL, System Immersion, DOEL, and RDOEL 39 4 Reduced-Order Dynamic Observer Error Linearization (RDOEL) for Multi-Output Systems 43 4.1 Problem Statement 43 4.2 Necessary Conditions 47 4.2.1 Observability 47 4.2.2 Inverse Output Transformation 52 4.2.3 System Dynamics 61 4.3 Necessary and Sufficient Conditions 65 4.3.1 Necessary and Sufficient Condition for RDOEL 65 4.3.2 Necessary and Sufficient Condition for OEL 80 4.3.3 Procedure to Solve OEL and RDOEL 81 4.4 Illustrative Examples 85 5 Extension of RDOEL: System into Extended Nonlinear Observer Canonical Form (ENOCF) 97 5.1 Problem Statement 99 5.2 Necessary Conditions 102 5.2.1 Output Transformation and Observability 102 5.2.2 System Dynamics 105 5.3 Necessary and Sufficient Condition 109 5.4 Case Study: Rӧssler System into ENOCF 117 6 Conclusions 125 BIBLIOGRAPHY 129 국문초록 139 감사의 글 143Docto

Complexity reduction for resilient state estimation of uniformly observable nonlinear systems

A resilient state estimation scheme for uniformly observable nonlinear systems, based on a method for local identification of sensor attacks, is presented. The estimation problem is combinatorial in nature, and so many methods require substantial computational and storage resources as the number of sensors increases. To reduce the complexity, the proposed method performs the attack identification with local subsets of the measurements, not with the set of all measurements. A condition for nonlinear attack identification is introduced as a relaxed version of existing redundant observability condition. It is shown that an attack identification can be performed even when the state cannot be recovered from the measurements. As a result, although a portion of measurements are compromised, they can be locally identified and excluded from the state estimation, and thus the true state can be recovered. Simulation results demonstrate the effectiveness of the proposed scheme.Comment: 12 pages, 4 figures, submitted to IEEE Transactions on Automatic Contro

Finite time observers: application to secure communication

International audienceIn this paper, control theory is used to formalize finite time chaos synchronization as a nonlinear finite time observer design issue. This paper introduces a finite time observer for nonlinear systems that can be put into a linear canonical form up to output injection. The finite time convergence relies on the homogeneity properties of nonlinear systems. The observer is then applied to the problem of secure data transmission based on finite time chaos synchronization and the two-channel transmission method

Observability analysis and observer design for controlled population dynamics

The diploma thesis studies the design of nonlinear observers with exactly linear error dynamics via transformation or immersion into an appropriate observer canonical form. A model for the dynamics of two interacting species, which was derived as a generalisation of the predator-prey model by Lotka and Volterra, is used as benchmark system for this design. In particular, an additional control input is modelled in three ways. As observability of the system is a necessary condition for observer design, the methods for an observability analysis are presented and applied to the model. After that, the theoretical basics of the observer design methods are described and used for the design of an observer with exactly linear error dynamics, with regard to the results from the observability analysis. The observers are designed as Luenberger observers with output injection for the uncontrolled system and with input/output injection for the controlled systems. Some new studies concern the invariance properties of the nonlinear observers on a state region which is relevant for the system. For this purpose, the notation of invariant observers is introduced, which guarantee a global observation of the system on the relevant region. Based on the considered observer canonical form, this notation helps to develop some general methods how to design such observers.Die Diplomarbeit untersucht den Entwurf nichtlinearer Beobachter mit exakt linearer Fehlerdynamik durch Transformation oder Immersion in eine geeignete Beobachternormalform. Als Benchmark-System für diesen Entwurf wird ein Modell zweier interagierender Populationen verwendet, das sich aus einer Verallgemeinerung des Rauber-Beute-Modells von Lotka und Volterra ergibt. Insbesondere werden drei Möglichkeiten betrachtet, einen Eingang zu diesem System hinzuzufügen. Da die Beobachtbarkeit des Systems eine notwendige Voraussetzung für den Entwurf eines Beobachters ist, werden zuerst die Methoden zur Beobachtbarkeitsanalyse vorgestellt und auf das Modell angewandt. Danach werden die theoretischen Grundlagen der Beobachterentwurfsverfahren erläutert, welche unter Berücksichtigung der Ergebnisse aus der Beobachtbarkeitsuntersuchung für den Entwurf eines Beobachters mit exakt linearer Fehlerdynamik verwendet werden. Die Beobachter werden als Luenberger-Beobachter mit Ausgangsaufschaltung für das autonome System und mit Eingangs-/Ausgangsaufschaltung für die Systeme mit Eingang entworfen. Neue Fragestellungen betreffen die Invarianzeigenschaften der entworfenen Beobachter auf einem für das System relevanten Zustandsbereich. Hierfür wird die Notation der invarianten Beobachter eingeführt, die eine globale Beobachtung des Systems im relevanten Bereich garantieren. Basierend auf der verwendeten Beobachternormalform werden mit Hilfe dieser Notation einige allgemeine Methoden entwickelt, mit denen solche Beobachter entwerfen werden können