False Data Injection Attacks in Control Systems

Transcription

1 False Data Injection Attacks in Control Systems Yilin Mo, Bruno Sinopoli Department of Electrical and Computer Engineering, Carnegie Mellon University First Workshop on Secure Control Systems Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 1 / 17

2 Introduction Control Systems Control Systems are ubiquitous. Typical applications of control systems include aerospace, chemical processes, civil infrastructure, energy and manufacturing. Many of them are safety-critical. Advances in computation and communication technology have greatly increased the capability of control systems. But new challenges arise as the systems become more and more complicated. Our goal: analysis and design of secure control systems. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 2 / 17

3 System Description System Model We consider the control system is monitoring the following LTI(Linear Time-Invariant) system System Description x k+1 = Ax k + Bu k + w k, y k = Cx k + v k. (1) x k R n is the state vector. y k R m is the measurements from the sensors. u k R p is the control inputs. w k, v k, x 0 are independent Gaussian random variables, and x 0 N ( x 0, Σ), w k N (0, Q) and v k N (0, R). Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 3 / 17

4 System Description Kalman Filter and LQG Controller Kalman filter (Assume already in steady state) ˆx 0 1 = x 0, ˆx k+1 k = Aˆx k k +Bu k, ˆx k+1 k+1 = ˆx k+1 k +K(y k+1 C ˆx k+1 k ). The LQG controller minimizes the following cost [ J = min lim E 1 T 1 ] (xk T T T Wx k + uk T Uu k). k=0 The solution is a fixed gain controller u k = (BT SB + U) 1 B T SAˆx k k = Lˆx k k, where S = A T SA + W A T SB(B T SB + U) 1 B T SA. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 4 / 17

5 χ 2 Failure Detector System Description The innovation of Kalman filter z k y k C ˆx k k 1 is i.i.d. Gaussian distributed with zero mean. χ 2 Detector The χ 2 detector triggers an alarm based on the following event: g k = (y k C ˆx k k 1 ) T P 1 (y k C ˆx k k 1 ) > threshold. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 5 / 17

6 False Data Injection Attack Attack Model We assume the following: 1 The attacker knows matrices A, C, K. 2 The attacker can control the readings of a subset of sensors. Hence, the measurement received by the Kalman filter can be written as y k = Cx k + v k + Γy a k, where y a k is the bias introduced by the attacker, Γ = diag(γ 1,..., γ m ) is the sensor selection matrix. γ i = 1 if the attacker can control the readings of sensor i. γ i = 0 otherwise. 3 The attack begins at time 0. 4 The sequence of attacker s inputs (y0 a,..., y k a ) is chosen before the attack. Hence, yk a is independent of w k, v k. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 6 / 17

7 False Data Injection Attack System Diagram Actuator Plant Sensor u k Attacker z 1 Γy a u k 1 k y k Controller ˆx k Estimator Detector Figure: System Diagram Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 7 / 17

8 False Data Injection Attack Healthy System v.s. Compromised System Healthy System Compromised System x k+1 = Ax k + Bu k + w k y k = Cx k + v k z k+1 = y k+1 C(Aˆx k + Bu k ) ˆx k+1 = Aˆx k + Bu k + Kz k+1 u k = Lˆx k x k+1 = Ax k + Bu k + w k y k = Cx k + v k + Γy a k z k+1 = y k+1 C(Aˆx k + Bu k ) ˆx k+1 = Aˆx k + Bu k + Kz k+1 u k = Lˆx k Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 8 / 17

9 False Data Injection Attack Difference between the Compromised System and Healthy System Dynamics of the Difference x k+1 = A x k + Bu k, z k+1 = y k+1 C(A ˆx k + B u k ), y k = C x k + Γy a k, ˆx k+1 =A ˆx k + B u k + K z k+1, u k = L ˆx k. Since y a k is independent of w k, v k, we can actually prove that x k is Gaussian and E(x k ) = x k, Cov(x k ) = Cov(x k). Similar statement is also true for y k, z k, ˆx k, u k. Hence, to characterize the performance of control systems under false data injection attacks, we only need to focus on x k, y k, z k, ˆx k, u k. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 9 / 17

10 False Data Injection Attack Successful Attack Definition A sequence of attacker s input (y0 a,..., y N a ) is called α-feasible if during the attack, D(z k z k) = z T k P 1 z k /2 α, for k = 0,..., N, where D(z k z k) is the KL distance between z k and z k. 1 It can be proved that the probability of triggering an alarm at time k is an increasing function of D(z k z k). 2 If α goes to 0, then the compromised system and healthy system are undistinguishable by the χ 2 detector. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 10 / 17

11 False Data Injection Attack Constrained Control Problem 1 Under the requirement that zk T P 1 z k /2 α, the action of the attacker can be formulated as a constrained control problem, where yk a is the input from the attacker. 2 To characterize the resilience of control system, we need to compute the reachable region R k of x k. 3 In this talk, we will focus on finding a necessary and sufficient condition under which the union of all R k is unbounded, i.e. there exists an α-feasible attack sequence that can push x k arbitrarily far away from 0. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 11 / 17

12 Main Result False Data Injection Attack Theorem k=1 R k is unbounded if and only if A has an unstable eigenvalue and the corresponding eigenvector v satisfies: 1 Cv span(γ), where span(γ) is the column space of Γ. 2 v is in the reachable space of the pair (A KCA, K). 1 To check the resilience of control system, one can find all the unstable eigenvector of A and compute Cv. 2 If Cv is sparse, then the attacker only need to compromise a few sensors to launch an attack along the direction v. 3 To improve the resilience, the defender could add redundant sensors to measure every unstable mode. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 12 / 17

13 Illustrative Example Simulation We consider a vehicle moving along the x-axis, which is monitored by a position sensor and velocity sensor. System Description [ ẋk+1 x k+1 ] = [ y k,1 = ẋ k + v k,1, y k,1 = x k + v k,2. ] [ ẋk x k ] [ ] u k + w k, We assume that Q = R = W = I 2, U = 1. The Kalman gain and LQG control gain are [ ] K =, L = [ ] Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 13 / 17

14 Simulation Illustrative Example It is easy to check the only unstable eigenvector is v = [0, 1] T. If the position sensor is compromised, then the attacker could push the state x k to infinity. If only the velocity sensor is compromised, then k=1 R k is bounded. Here we use an ellipsoidal approximation to compute the inner and outer approximation of k=1 R k. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 14 / 17

15 Simulation Position Sensor is Compromised Figure: Evolution of ẋ k, x k and D(z k z k) Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 15 / 17

16 Simulation Velocity Sensor is Compromised Figure: Inner and Outer Approximation of Reachable Region k=1 R k under Constraint D(z k z k) 1 Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 16 / 17

17 Conclusion Conclusion In this presentation, we consider the false data injection attacks in control systems. We define the false data injection attack model. We formulate the action of the attacker as a constrained control problem. We prove an algebraic condition under which the attacker could successfully destabilize the system. We give a design criterion to improve the resilience of control systems against such kind of attacks. Bruno Sinopoli (Carnegie Mellon University) False Data Injection 1st SCS 17 / 17