Masashi Wakaiki

Observability of linear systems under adversarial attacks

We address the problem of state estimation for multi-output continuous-time linear systems, for which an attacker may have control over some of the sensors and inject (potentially unbounded) additive noise into some of the measured outputs. To characterize the resilience of a system against such sensor attacks, we introduce a new notion of observability - termed “observability under attacks” - that addresses the question of whether or not it is possible to uniquely reconstruct the state of the system by observing its inputs and outputs over a period of time, with the understanding that some of the available systems outputs may have been corrupted by the opponent. We provide computationally efficient tests for observability under attacks that amount to testing the (standard) observability for an appropriate finite set of systems. In addition, we propose two state estimation algorithms that permit the state reconstruction in spite of the attacks. One of these algorithms uses observability Gramians and a finite window of measurements to reconstruct the initial state. The second algorithm takes the form of a switched observer that asymptotically converges to the correct state estimate in the absence of additive noise and disturbances, or to a neighborhood of the correct state estimate in the presence of bounded noise and disturbances.

SMT-based observer design for cyber-physical systems under sensor attacks

We introduce a scalable observer architecture to estimate the states of a discrete-time linear-time-invariant (LTI) system whose sensors can be manipulated by an attacker. Given the maximum number of attacked sensors, we build on previous results on necessary and sufficient conditions for state estimation, and propose a novel multi-modal Luenberger (MML) observer based on efficient Satisfiability Modulo Theory (SMT) solving. We present two techniques to reduce the complexity of the estimation problem. As a first strategy, instead of a bank of distinct observers, we use a family of filters sharing a single dynamical equation for the states, but different output equations, to generate estimates corresponding to different subsets of sensors. Such an architecture can reduce the memory usage of the observer from an exponential to a linear function of the number of sensors. We then develop an efficient SMT-based decision procedure that is able to reason about the estimates of the MML observer to detect at runtime which sets of sensors are attack-free, and use them to obtain a correct state estimate. We provide proofs of convergence for our algorithm and report simulation results to compare its runtime performance with alternative techniques. Our algorithm scales well for large systems (including up to 5000 sensors) for which many previously proposed algorithms are not implementable due to excessive memory and time requirements. Finally, we illustrate the effectiveness of our algorithm on the design of resilient power distribution systems.

Stability analysis of sampled-data switched systems with quantization

We propose a stability analysis method for sampled-data switched linear systems with finite-level static quantizers. In the closed-loop system, information on the active mode of the plant is transmitted to the controller only at each sampling time. This limitation of switching information leads to a mode mismatch between the plant and the controller, and the system may become unstable. A mode mismatch also makes it difficult to find an attractor set to which the state trajectory converges. A switching condition for stability is characterized by the total time when the modes of the plant and the controller are different. Under the condition, we derive an ultimate bound on the state trajectories by using a common Lyapunov function computed from a randomized algorithm. The switching condition can be reduced to a dwell-time condition.

Sensitivity Reduction by Strongly Stabilizing Controllers for MIMO Distributed Parameter Systems

This note investigates a sensitivity reduction problem by stable stabilizing controllers for a linear time-invariant multi-input multioutput distributed parameter system. The plant we consider has finitely many unstable zeros, which are simple and blocking, but can possess infinitely many unstable poles. We obtain a necessary condition and a sufficient condition for the solvability of the problem, using the matrix Nevanlinna-Pick interpolation with boundary conditions. We also develop a necessary and sufficient condition for the solvability of the interpolation problem, and show an algorithm to obtain the solutions. Our method to solve the interpolation problem is based on the Schur-Nevanlinna algorithm.

Stabilization of Switched Linear Systems With Quantized Output and Switching Delays

This paper addresses the problem of designing time-varying quantizers for the stabilization of switched linear systems with quantized output and switching delays. The detection delays of switches are assumed to be time-varying but bounded, and the dwell time of the switching signal is assumed to be larger than the maximum delay. Given a switching controller, we analyze reachable sets of the closed-loop state by using a common Lyapunov function and then construct a quantizer that guarantees asymptotic stability. A sufficient condition for the existence of such a quantizer is characterized by the maximum switching delay and the dwell time.

Real-time control under clock offsets between sensors and controllers

This paper studies the impact of clock mismatches in spatially distributed real-time control systems. We consider a configuration in which sensor measurements are collected by one processor that transmits the measurements to another control/actuation processor through a network, but the two processors do not have a common clock. Due to the clock mismatch, there will be an offset between the actual time at which a measurement is taken and the time reported by the sensor. Our goal is to discover fundamental limitations to the ability to stabilize the control loop arising from the clocks mismatch. We consider time-varying bounded offsets and derive limitations on the offset bound for the stability of the feedback system. For the case of a scalar linear process, there exists a critical limitation, which depends on the level of instability of the plant and the nominal sampling period. In contrast, for the vector linear processes, if the process dynamics has at least two distinct real eigenvalues, then there is no fundamental limitation on the offset bound.

Stabilization of networked control systems with clock offsets

We consider the stabilization of networked control systems with time-invariant clock offsets between the sensors and the controllers. Clock offsets are modeled as parametric uncertainty and we provide necessary and sufficient conditions for the existence of a single controller that is capable of stabilizing the closed loop for every clock offset in a given range. For scalar systems, we obtain the maximum length of the offset interval for which the system can be stabilized by a single linear time-invariant controller. We compare the offset bounds that would be allowed by specific classes of controllers for scalar systems.

Stable controllers for robust stabilization of systems with infinitely many unstable poles

H∞ control a b s t r a c t This paper studies the problem of robust stabilization by a stable controller for a linear time-invariant single-input single-output infinite dimensional system. We consider a class of plants having finitely many simple unstable zeros but possibly infinitely many unstable poles. First we show that the problem can be reduced to an interpolation–minimization by a unit element. Next, by the modified Nevanlinna–Pick interpolation, we obtain both lower and upper bounds on the multiplicative perturbation under which the plant can be stabilized by a stable controller. In addition, we find stable controllers to provide robust stability. We also present a numerical example to illustrate the results and apply the proposed method to a repetitive control system.

Control under clock offsets and actuator saturation

This paper studies the stability analysis and the stabilization problem for systems with asynchronous sensors and controllers, and actuators subject to saturation. We consider systems with parameter uncertainties caused by clock offsets. By using a polytopic overapproximation, we investigate how large clock offsets affect stability. In addition, we employ a sector characterization approach to address actuator saturation. We see from a numerical study that the range of allowable clock offset bounds drops if the saturation limit becomes smaller than a certain value.

Output feedback stabilization of switched linear systems with limited information

We propose an encoding and control strategy for the stabilization of switched systems with limited information, supposing the controller is given for each mode. Only the quantized output and the active mode of the plant at each sampling time are transmitted to the controller. Due to switching, the active mode of the plant may be different from that of the controller in the closed-loop system. Hence if switching occurs, the quantizer must recalculate a bounded set containing the estimation error for quantization at the next sampling time. We establish the global asymptotic stability under a slow-switching assumption on dwell time and average dwell time. To this end, we construct multiple discrete-time Lyapunov functions with respect to the estimated state and the size of the bounded set.