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Secure estimation and control for cyber-physical systems under adversarial attacks

Hamza Fawzi, Paulo Tabuada, Suhas Diggavi

arXiv:1205.5073v1math.OCcs.CRcs.ITeess.SY

TL;DR

The paper addresses estimation and control of linear systems when attackers corrupt sensors or actuators, a concern for increasingly exposed control systems. It characterizes attack resilience, proposes an efficient attack-resilient state decoder, and studies resilience improvement through parameter changes. For sensor attacks, it shows that stabilizing output-feedback design can be reduced to resilient state estimation.

  • Problem

    The paper asks how linear systems can be estimated and controlled when some sensors or actuators are corrupted by an attacker.

  • Method

    It characterizes resilience, uses a computationally feasible decoder inspired by compressed sensing and error-correction over the reals, and connects output-feedback design to resilient estimation.

  • Results

    The paper shows that resilience can be increased through state-feedback parameter changes and that a stabilizing output-feedback law implies a decoder resilient to the same number of attacks.

  • Takeaways & Limitations

    Resilient output-feedback stabilization can be designed by focusing on the corresponding resilient state-estimation problem.

Abstract

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The vast majority of today's critical infrastructure is supported by numerous feedback control loops and an attack on these control loops can have disastrous consequences. This is a major concern since modern control systems are becoming large and decentralized and thus more vulnerable to attacks. This paper is concerned with the estimation and control of linear systems when some of the sensors or actuators are corrupted by an attacker. In the first part we look at the estimation problem where we characterize the resilience of a system to attacks and study the possibility of increasing its resilience by a change of parameters. We then propose an efficient algorithm to estimate the state despite the attacks and we characterize its performance. Our approach is inspired from the areas of error-correction over the reals and compressed sensing. In the second part we consider the problem of designing output-feedback controllers that stabilize the system despite attacks. We show that a principle of separation between estimation and control holds and that the design of resilient output feedback controllers can be reduced to the design of resilient state estimators.

I. INTRODUCTION

Modern control systems are increasingly exposed to cyberattacks that can damage physical processes, creating a need for resilient estimation and control. The paper models arbitrary-magnitude sensor and actuator attacks as sparse corruptions and develops efficient estimation and resilient output-feedback control methods.

  • Motivation: Wireless and Internet-connected control systems are more exposed to cyberattacks that can cause physical-process faults and failures.Reported incidents include sewage flooding and attacks on industrial supervisory-control systems.
  • Limitations of prior approaches: Existing fault-detection methods generally require one residual signal per failure mode, which is impractical when many failure modes are possible.The paper contrasts this limitation with its setting, where the number of failure modes can be very large.
  • Problem formulation: Unlike robust and stochastic control formulations, the paper does not assume adversarial disturbances are bounded or follow a probabilistic model.The attack model permits arbitrary-magnitude attacks on captured nodes.
  • Estimation: The first part characterizes resilience and the maximum tolerable number of attacked nodes, then gives a computationally feasible decoder inspired by compressed sensing and error-correction over the reals.The results include extensions to actuator attacks.
  • Resilience improvement: Changing the dynamics matrix from A to A + BK can increase system resilience while preserving freedom to choose performance.This parameter change corresponds to implementing a state-feedback law.
  • Output-feedback control: For sensor attacks, if a stabilizing output-feedback law exists, the state can also be estimated despite attacks, making resilient stabilization reducible to resilient estimation.The paper describes this as a separation between estimation and control.

II. THE FORMAL SETTING AND NOTATIONS

The paper models linear control systems with sensor and actuator attacks represented as arbitrary real-valued disturbances on fixed attacked-node sets. It defines support-based notation to characterize how many attacks can be tolerated.

  • System model: The system evolves as x(t+1) = Ax(t) + B(K(t)(y(0), . . . , y(t)) + w(t)), with outputs y(t) and sensor attacks e(t).The feedback input depends on past measurements, while e(t) and w(t) denote sensor and actuator attacks.
  • Attack model: Attacks may affect sensors, actuators, nodes, or their communication links, and the paper treats these interpretations within the attacked-node model.The paper explicitly considers compromised communication links as an alternative interpretation of node attacks.
  • Attack model: The attacked sensor and actuator sets remain constant over time, although attack values may vary arbitrarily at each time step.The support of e(t) and w(t) is restricted to fixed sets K and L, while the attack magnitudes are not assigned a specific model.
  • Attack model: The model assumes fixed attacked sets because gaining control of a node may take longer than the estimation algorithm’s time scale.Allowing the attacker to switch abandoned nodes every time step is described as unrealistic in this setting.
  • Objective: The framework characterizes the number of attacks that can be tolerated while correctly estimating or controlling the plant.This attack-count perspective is the central quantitative objective of the formal setting.
  • Notation: The paper uses support, ℓ0 cardinality, projections, row support, and matrix row notation to quantify sparse corrupted components.These definitions provide the notation for counting attacked sensors, actuators, vector entries, and nonzero matrix rows.

III. THE ESTIMATION PROBLEM

The estimation problem is to reconstruct a plant’s initial state from sensor measurements corrupted on a fixed set of sensors. The paper gives exact resilience conditions based on the supports of time-evolved outputs.

  • Problem formulation: The estimator reconstructs x(0) from T corrupted observations, with sensor attacks as the main case and actuator attacks treated as an extension.Because A is known, reconstructing the initial state is theoretically equivalent to reconstructing the current state.
  • Problem formulation: A decoder corrects q errors after T steps when it recovers x(0) for every initial state and every attack sequence supported on at most q sensors.The definition permits arbitrary error values on the attacked sensor set.
  • Characterization: State recovery is impossible when two distinct initial states can explain the same observations using fewer than q corrupted sensors.The indistinguishability condition supplies the converse obstruction to attack correction.
  • Characterization: q errors are correctable after T steps exactly when every nonzero z has more than 2q sensor indices in the union of supports of Cz, CAz, ..., CA^(T−1)z.This condition is equivalent to decoder-based resilience against q attacks.
  • Consequences: No attacks can be corrected unless the corresponding observability matrix is injective, so ordinary observability is necessary even before adversarial corruption is considered.The paper notes that non-observability prevents reconstruction even with error-free measurements.
  • Consequences: The maximum number of correctable errors cannot increase beyond T = n measurements and is always less than p/2.For almost all pairs (A, C), the maximum after n steps is ⌈p/2 −1⌉.

2) Computing the number of correctable errors:

Computing the exact correction capacity for a given system is generally hard. A direct method searches sensor subsets for a stacked observability matrix with a nontrivial kernel.

  • Computational problem: For a given (A, C) and horizon T, determining the number of correctable errors is a hard problem in general.The paper distinguishes the generic maximal value from computing the capacity of an individual system.
  • Direct algorithm: A direct algorithm finds the smallest sensor subset K for which the associated stacked matrix has a nontrivial kernel.If the smallest such subset has cardinality s, the maximum correctable errors are ⌈s/2−1⌉.
  • Limitations: The direct algorithm is computationally expensive and may require rank computations for 2^p matrices in the worst case.The cited complexity result suggests that substantially more efficient exact computation is unlikely in general.

B. Increasing the number of correctable errors by state feedback

State feedback can increase resilience by changing the closed-loop dynamics while preserving desired pole locations. Under controllability, an appropriate feedback law achieves maximal correction capacity without compromising control performance.

  • Design objective: The design problem chooses K so that (A + BK, C) corrects as many attacks as possible while retaining freedom over the closed-loop eigenvalues.The local controller changes the dynamics seen by the supervisory estimator receiving vulnerable sensor measurements.
  • Main result: If (A, B) is controllable, K can make ⌈p/2 −1⌉ errors correctable while placing the eigenvalues of A + BK at essentially arbitrary prescribed locations.The eigenvalue choices must avoid a finite exceptional set and have distinct magnitudes.
  • Main result: The maximal resilience result applies after n steps and coincides with the generic maximum established for almost all system pairs.The construction preserves the ability to choose closed-loop poles rather than trading stability or pole placement for resilience.
  • Proof mechanism: With distinct eigenvalue magnitudes, q-error correction after n steps is equivalent to every eigenvector producing more than 2q nonzero sensor outputs.This reduces the dynamic support condition to an eigenvector support condition.
  • Scope: The distinct-magnitude condition is described as a mild restriction because its performance consequences are negligible, and the argument extends to multi-input systems.The formal proposition is stated for the single-input case.
  • Proof mechanism: The feedback construction selects eigenvalues so that each corresponding eigenvector (λI − A)^−1B has full sensor support.Controllability ensures the selected eigenvalues can be assigned, while avoiding finite zero sets ensures all sensor coordinates are nonzero.

C. Optimization formulation of the optimal decoder

The optimal decoder reconstructs the initial state by finding the state and smallest attacked-sensor set consistent with all observations. It corrects q errors exactly when the system's observability map satisfies the corresponding error-separation condition, although the optimization is generally NP-hard.

  • The decoder minimizes the number of attacked sensors needed to explain the observed sequence and returns the corresponding initial-state estimate.It solves for x̂ from the received measurements using the smallest feasible attack support.
  • q sensor errors are correctable after T steps when every nonzero state direction produces observations supported on more than 2q sensors.This condition makes distinct initial states with q-sparse attacks distinguishable.
  • The optimal decoder is best in error-correction capability: whenever any decoder can correct q errors, this decoder can also correct them.
  • The optimal ℓ0 formulation is generally NP-hard, already in the static T = 1 case.The paper therefore motivates replacing the ℓ0 objective with an efficiently solvable ℓ1 convex program.

D. The ℓ1 decoder: a relaxation of the optimal decoder

The paper relaxes the optimal sparsity decoder by minimizing summed row magnitudes through a mixed ℓ1/ℓr norm. It characterizes when this convex decoder corrects q errors and connects the condition to compressed sensing.

  • The ℓ1/ℓr decoder estimates the state by minimizing the sum of row ℓr norms of the measurement residual matrix.The optimization is convex and can therefore be solved efficiently.
  • The ℓ1/ℓr condition is a quantitative counterpart of the optimal decoder's support condition, requiring observability rows to remain sufficiently spread out.
  • For the cyclic example with p = n, the decoder corrects the maximal ⌈n/2 −1⌉ errors after T = n steps.After n steps, the rows of Φ(n)z have equal ℓr norms up to permutation.

E. Numerical simulations

The numerical section evaluates the proposed decoding algorithm first on a random linear system and then on a more realistic electric-power-network model.

  • The simulations test the proposed decoding algorithm on both a randomly generated system and an electric power network.
  • The random-system experiment uses the ℓ1/ℓ2 decoder on a system with n = 25 states and p = 20 sensors.

1) Random system:

Across random and IEEE 14-bus examples, the decoders recover states despite large sensor and actuator attacks when the attacked set is sufficiently small. Recovery becomes harder as attacked sensors increase, while the power-network case remains largely insensitive to attacked generators.

  • Random system: For q less than 6, all random-system initial conditions were recovered in fewer than T = 15 steps.
  • Random system: As attacked sensors increase, the random-system decoder needs more measurements or time steps to recover the state correctly.
  • Electric power network: In the IEEE 14-bus example, the ℓ1/ℓ∞ decoder achieved a 100% success rate for q ≤4 and exceeded 90% recovery for q ≤12 within 10 steps.
  • Attacks on actuators: With actuator attacks, the decoder still recovers the state when attacked sensors and actuators are sufficiently few, but sensor resilience decreases as actuator attacks increase.
  • Attacks on actuators: In the power-network actuator-attack experiment, recovery remained largely unaffected by the number of attacked generators when attacked sensors were few.

IV. THE CONTROL PROBLEM WITH OUTPUT-FEEDBACK

This section asks whether sensor attacks can be countered by an output-feedback law that stabilizes the system. It shows that resilient state decoding and standard state feedback provide the relevant control structure.

  • The control problem is to drive the state to the origin despite sensor attacks that can alter control inputs through corrupted measurements.
  • Stabilizing control is not generally possible under actuator attacks, so the section focuses on sensor attacks.
  • If q sensor errors are correctable, the controller can decode the state despite attacks and apply standard state feedback such as u = Kx.
  • The section’s main contribution is showing that resilient output feedback can be composed from a decoder and standard state feedback without losing resilience.

A. Some properties

The section defines correctability for output-feedback systems and shows that it is determined by the system’s sensing dynamics rather than the feedback law or input matrix. Earlier correctability conditions therefore apply directly.

  • Correctability after T steps means a decoder can recover the initial state from T output measurements for every initial state and allowed q-sensor attack sequence.
  • Because inputs depend only on measurements, correctability for a linear output-feedback system does not depend on the feedback law or B, but only on A and C.
  • The equivalence follows because indistinguishable outputs produce identical control inputs, reducing controlled-system ambiguity to correctability for (A, C).
  • Thus, correctability conditions developed for systems without inputs can be applied to systems with output-feedback control inputs.
  • For example, q errors are correctable exactly when the combined support of C z through C A^(T−1) z exceeds 2q for every nonzero z.

B. Main result: separation of estimation and control

The main theorem establishes separation of estimation and control: sufficiently fast resilient stabilization implies q-error correctability after n steps. A fast decay assumption is essential for this converse.

  • Main theorem: If a feedback strategy stabilizes every trajectory under q sensor attacks with decay rate α below the smallest eigenvalue magnitude of A, then q errors are correctable after n steps.
  • Main theorem: The proof uses two indistinguishable initial states and attack sequences to force identical outputs and inputs, contradicting the required fast state decay.
  • Caveat: The fast-decay assumption cannot be omitted: a stable plant may remain stable under arbitrary sensor attacks even when one error is not correctable.
  • Caveat: For A = 0.5I, B = I, and C = I, the zero controller yields stable dynamics independent of sensor outputs, while no decoder corrects even one error.

V. CONCLUSION

The paper characterizes attack tolerance, develops decoding methods, and studies resilient output-feedback stabilization for attacked linear systems. Its central control result is that resilient output feedback can be reduced to resilient state estimation, while several extensions remain open.

  • Estimation: For estimation, the paper characterizes how many attacks can be tolerated while still exactly recovering the system state.
  • Estimation: It shows that state-feedback design can increase system resilience while guaranteeing a specified performance.
  • Decoding: An explicit decoder corrects the maximal number of errors, and a computationally feasible relaxation appears to perform well in numerical simulations.
  • Output feedback: For control, the paper studies output-feedback laws that stabilize a linear plant despite sensor attacks.
  • Output feedback: A resilient output-feedback law implies a resilient decoder, so composing decoding with standard state feedback does not reduce resilience.
  • Open questions: Open questions include iterative estimation, exogenous-noise effects, and specializing the techniques to applications with structural sensor and actuator vulnerabilities.
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