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Dynamic Attack Detection in Cyber-Physical Systems with Side Initial State Information

Yuan Chen, Soummya Kar, Jose' M. F. Moura

arXiv:1503.07125v2math.OCcs.CRcs.SYeess.SY

TL;DR

Dynamic attack detection asks whether manipulated data can produce outputs consistent with an unattacked cyber-physical system, and how side information about the initial state changes detectability. The paper gives conditions for undetectability and its persistence, identifies zero-state-inducing attacks, and designs a detector for detectable attacks.

  • Problem

    Attacks can manipulate cyber-physical systems while producing outputs consistent with an unattacked system, and prior work has focused on detectability rather than its sensitivity to initial-state information and time horizons.

    Source: §I · p. 1

  • Method

    The paper analyzes dynamic attack detectability using a linear function of the initial state that the attacker cannot alter, deriving subspace conditions and designing a consistent detector.

    Source: §I · p. 1, D. Attack Detection With Side… · p. 4

  • Results

    Undetectability is characterized by the intersection of the weakly unobservable subspace and the side-information null space, while indefinite extensions and arbitrarily long zero-state-inducing attacks have stated subspace conditions.

    Source: A. Initial State Information and… · p. 3, B. Extensions of Undetectable Attacks · p. 4, C. Zero State Inducing Attack · p. 4

  • Takeaways & Limitations

    The designed detector has no false alarms and misses only undetectable attacks, including zero-state-inducing attacks that remain undetectable regardless of side information.

    Source: D. Attack Detection With Side… · p. 4

    The analysis assumes the detector’s linear function of the initial state is uncorrupted and cannot be altered by the attacker.

    Source: §I · p. 1

Abstract

from arXiv · show

This paper studies the impact of side initial state information on the detectability of data deception attacks against cyber-physical systems. We assume the attack detector has access to a linear function of the initial system state that cannot be altered by an attacker. First, we provide a necessary and sufficient condition for an attack to be undetectable by any dynamic attack detector under each specific side information pattern. Second, we characterize attacks that can be sustained for arbitrarily long periods without being detected. Third, we define the zero state inducing attack, the only type of attack that remains dynamically undetectable regardless of the side initial state information available to the attack detector. Finally, we design a dynamic attack detector that detects detectable attacks.

I. INTRODUCTION

Dynamic attack detection matters because attacks can manipulate cyber-physical systems while producing outputs consistent with an unattacked system. This paper studies how side information about the initial state and the time horizon affect detectability, using a model in which an attacker can alter data sent through system channels.

I. INTRODUCTION

  • Dynamic detectors use system dynamics and histories of actuator inputs and sensor outputs, unlike static checks that cannot detect actuator attacks.Undetectable attacks can make system outputs appear as if the system were not under attack.Source: §I · p. 1
  • The paper derives conditions for undetectability and indefinite persistence, identifies zero-state-inducing attacks, and designs a detector that detects every detectable attack.The detector uses side information, has no false alarms, and misses only undetectable attacks.Source: §I · p. 1

A. System Model

  • The model represents state and output evolution with known inputs and attacks entering through the system matrices; the analysis sets the known input to zero without loss of generality.The model assumes the state-output pair is observable.Source: A. System Model · pp. 1–2
  • The detector receives a linear function of the initial state that does not rely on sensor measurements, so the attacker cannot modify it.A full-column-rank side information matrix gives full information about the initial state, while a zero matrix gives none.Source: A. System Model · p. 2

B. Extended System Subspaces

The paper uses weakly unobservable and output-nulling reachable subspaces to characterize states and attacks associated with zero output trajectories.

B. Extended System Subspaces

  • The weakly unobservable subspace contains initial states for which some input sequence produces a zero output trajectory over n−1 steps.The paper gives an equivalent condition using the extended input-output and observability matrices for any horizon T.Source: B. Extended System Subspaces · p. 2
  • The paper relies on established methods to calculate a basis for the weakly unobservable subspace.The cited approaches are,,, and.Source: B. Extended System Subspaces · p. 2
  • The output-nulling reachable subspace over k steps contains states reachable from zero by an attack sequence that produces zero output.Equivalently, the extended controllability matrix maps an attack sequence to the state while the input-output matrix maps it to zero output.Source: B. Extended System Subspaces · p. 2

C. Dynamic Attack Detection: Preliminaries

The preliminaries define the detector and attack classes used to analyze dynamic detectability. They distinguish attacks that are undetectable at a given horizon from those that can remain undetectable as new measurements arrive.

C. Dynamic Attack Detection: Preliminaries

  • The detector knows the system matrices A and C and the side information matrix, but not B, D, or the initial state; the attacker knows these quantities and cannot alter the side information.The attacker is assumed to know A, B, C, D, Ω, and the initial state.Source: C. Dynamic Attack Detection:… · pp. 2–3
  • A consistent detector avoids false alarms, while a sound consistent detector detects all possible attacks without violating consistency.Consistency requires reporting no attack for outputs corresponding to an unattacked system state.Source: C. Dynamic Attack Detection:… · p. 2
  • An attack is undetectable when every consistent detector reports no attack for its output and side information.The set of undetectable attacks also includes the zero attack.Source: C. Dynamic Attack Detection:… · p. 3
  • An undetectable attack may become detectable as new measurements arrive; the paper characterizes when it has undetectable extensions to every later horizon.If the extension condition fails, all extensions become detectable at some later time.Source: C. Dynamic Attack Detection:… · p. 3
  • A zero-state-inducing attack satisfies M_T E(T) = 0 and remains undetectable regardless of the detector’s side information, including when it has full rank.Its output change equals the response of the system with zero initial state.Source: C. Dynamic Attack Detection:… · p. 3

D. Problem Statement

The problem statement organizes the paper around four questions about attacks and dynamic detection under side initial-state information.

D. Problem Statement

  • The first problem is to find the set of all attacks undetectable over a given time interval.The set is denoted U_Ω,T.Source: D. Problem Statement · p. 3
  • The paper asks which undetectable attacks can be extended indefinitely and whether an arbitrarily long zero-state-inducing attack exists.Both questions concern persistence beyond a finite observation horizon.Source: D. Problem Statement · p. 3
  • The final problem is to design a consistent detector using side information that detects every detectable attack.Source: D. Problem Statement · p. 3

III. MAIN RESULTS

The section characterizes undetectable attacks and their possible continuations under side initial-state information. It also identifies conditions under which added state information restricts stealth, while some attacks can remain undetectable at every later horizon.

A. Initial State Information and Undetectable Attacks

  • An attack is undetectable exactly when its output contribution cancels the unforced output from a state in V(Σ) ∩ N(Ω).With full-column-rank Ω, this reduces to M_T E(T) = 0; the only undetectable attacks in that case are zero state inducing attacks.Source: A. Initial State Information and… · pp. 3–4
  • Nontrivial side information restricts the states that can mask an attack to N(Ω) ∩ V(Σ), potentially exposing attacks that would otherwise be undetectable.With no initial-state information, the condition reduces to θ ∈ V(Σ).Source: A. Initial State Information and… · p. 3

B. Extensions of Undetectable Attacks

  • An undetectable attack extends undetectably to every later horizon exactly when its attack-induced state change plus the subsequent zero-input response lies in V(Σ).When this condition fails, every extension becomes detectable at some later time as new measurements arrive.Source: B. Extensions of Undetectable Attacks · p. 4

C. Zero State Inducing Attack

The section gives a necessary and sufficient condition for zero state inducing attacks of arbitrary length, then describes a sequential detector that uses side information and finite measurement windows. The detector is consistent and sound for a sufficiently long window.

C. Zero State Inducing Attack

  • An arbitrarily long zero state inducing attack beginning at time 0 exists exactly when W1 ∩ V(Σ) is nonzero.W1 is the output-nulling reachable subspace over one time step.Source: C. Zero State Inducing Attack · p. 4

D. Attack Detection With Side Information

  • The sequential detector uses a finite window of sensor measurements and detects attacks outside UΩ,T after an initialization period.Its finite window avoids using the entire measurement history.Source: D. Attack Detection With Side… · p. 4
  • For a sufficiently long window, ψ is consistent and sound, with a no-attack decision requiring the measurements and side information to match a shared initial state.The paper states the window condition as l ≥ n + 1, where n is the state-space dimension.Source: D. Attack Detection With Side… · p. 5

IV. PROOF OF MAIN RESULTS

The proofs establish the undetectability criterion by matching attack outputs with outputs from alternative initial states that agree on side information. For extensions, observability identifies the induced state, and the remaining condition determines whether suitable continuations exist at every later horizon.

A. Proof of Theorem 1

  • Theorem 1 follows by showing that undetectability is equivalent to an alternative initial state matching both the attacked output and the side information.The proof sets the initial-state difference to θ, which must lie in N(Ω) ∩ V(Σ).Source: A. Proof of Theorem 1 · p. 5

B. Proof of Theorem 2

  • In the necessity proof for Theorem 2, observability makes O_T injective, so matching the first block of outputs forces the extended attack to retain the same induced state θ.The proof then applies the extension condition at T′ = T + n.Source: B. Proof of Theorem 2 · p. 5
  • For sufficiency, the proof constructs each longer undetectable attack by appending an attack sequence that satisfies the required later-horizon condition.The appended sequence may be zero.Source: B. Proof of Theorem 2 · p. 5

C. Proof of Theorem 3

The proof establishes the arbitrary-length attack condition in both directions. A nonzero one-step reachable state in the weakly unobservable subspace enables zero-output attack sequences of any length; conversely, an arbitrary-length attack yields such a nonzero state.

C. Proof of Theorem 3

  • If W1 ∩ V(Σ) is nonzero, the proof chooses a nonzero first attack that produces zero initial output and reaches a state in V(Σ).The initial system state can be taken as zero because it does not affect the extended subspaces used in the argument.Source: C. Proof of Theorem 3 · p. 5
  • From that reachable state, an attack sequence can maintain zero output for any horizon, producing M_T E(T) = 0.The constructed attack begins with a(0) ≠ 0.Source: C. Proof of Theorem 3 · p. 5
  • Conversely, an arbitrary-length zero state inducing attack implies D a(0) = 0 and a nonzero state x(1) = B a(0) that lies in W1 ∩ V(Σ).The remaining attack inputs keep the output zero over the following n steps, placing x(1) in V(Σ).Source: C. Proof of Theorem 3 · p. 5

D. Proof of Theorem 4

The proof proceeds by induction in both directions to establish the stated trajectory condition. It uses observability to show that an observability matrix is injective and identify the corresponding initial state.

D. Proof of Theorem 4

  • The proof establishes the claim in both directions by induction, using a base case and an induction step.The supplied proof passages show induction for both the “If” and “Only If” directions.Source: D. Proof of Theorem 4 · pp. 5–6
  • Because (A, C) is observable and l ≥ n + 1, the proof treats O_{l−2} as injective to identify x′(0) from the output history.The proof then substitutes the identified initial state into the output relation.Source: D. Proof of Theorem 4 · p. 6
  • The induction establishes that the output history equals O_T x(0) while the side-information value equals Ωx(0).The passages state this conclusion for the induction argument’s trajectory.Source: D. Proof of Theorem 4 · p. 6

V. NUMERICAL EXAMPLE

The numerical example uses a remotely piloted aircraft model to compare windowed detectors with and without initial horizontal-velocity information. The detector with side information detects the zero dynamics attack, while the detector without it does not.

V. NUMERICAL EXAMPLE

  • The example uses a four-state longitudinal aircraft model with two actuators and three sensors, and the (A, C) pair is observable.The states represent horizontal velocity, vertical velocity, pitch rate, and pitch angle; the sensors measure horizontal velocity, vertical velocity, and pitch angle.Source: §V · p. 6
  • The attacker can target both actuators and the horizontal- and vertical-velocity sensors, and a zero dynamics attack exists for this system.The attack is constructed against the remotely piloted aircraft model.Source: §V · p. 6
  • The detectors are implementations of the windowed detector, differing only in whether the detector knows the initial horizontal velocity x1(0).The no-side-information detector uses Ω = 0.Source: §V · p. 6
  • At t = 3, the detector with side information detects the zero dynamics attack, whereas the detector without side information reports “No Attack” throughout.Figure 2 presents detector performance without side information in the top panel and with side information in the bottom panel.Source: §V · pp. 6–7, Figure 2 · p. 7

VI. CONCLUSION

The conclusion characterizes undetectable attacks through subspace conditions, including when they can be extended or sustained arbitrarily long. It also states that the designed detector detects attacks that are not undetectable.

VI. CONCLUSION

  • An attack is undetectable if and only if its induced state lies in V(Σ) ∩ N(Ω), the intersection of the weakly unobservable subspace and the side-information matrix’s null space.Source: §VI · p. 7
  • An undetectable attack E(T) has an undetectable extension to any T′ > T if and only if CT E(T) + A^{T+1}θ lies in V(Σ).Source: §VI · p. 7
  • An arbitrarily long zero state inducing attack exists if and only if V(Σ) ∩ W1 is nonzero, where W1 is the one-step output-nulling reachable subspace.Source: §VI · p. 7
  • The paper designs an attack detector that uses side information and detects all attacks that are not undetectable.Source: §VI · p. 7

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