Robust Stability under Asynchronous Sensing and Control Masashi Wakaiki 1 , Masaki Ogura 2 , and Jo˜ ao P. Hespanha 3 Abstract—We address the stability of networked control systems in which a sensor and an output-feedback controller operate asynchronously, which leads to uncertainty in the sampling instants. In addition, we also consider polytopic uncertainty in the plant model. The analysis is based on trans- forming the closed-loop system into an impulsive system, by considering an extended state variable that includes the states of the continuous-time plant and the discrete-time controller. We provide a sufficient condition for the robust stability of the closed-loop system in terms of linear matrix inequalities. This condition is based on the construction of a continuous-time Lyapunov functional that also incorporates the discrete-time state of the digital controller. We illustrate the obtained result with numerical simulations. I. I NTRODUCTION Parameter perturbation and disturbances/noises has been extensively studied in the robust control literature; see, e.g., [1], [2] and many references therein. While control systems also have uncertainty in the time domain [3], [4], relatively little work has been done on time-domain uncertainty. Our goal is to analyze how large uncertainty in both the parameter and time domains can be without compromising the closed- loop stability. In networked control systems, one of the major sources of time-domain uncertainties is a synchronization error between local subsystems. As surveyed in [5], [6], many synchro- nization algorithms have been developed, and easy access to global clocks such as GPS and radio clocks leads to high- precision synchronization in practical situations. However, there are fundamental limitations on clock synchronization due to variable delays [7]. Furthermore, the signals of GPS and radio clocks are not ubiquitously available, and it is reported in [8] that the GPS-based synchronization is vulnerable against attacks. Asynchronous dynamical systems have been investigated in various fields including engineering and biology. An observer-based control has been proposed for networked con- trolled systems under synchronization errors and parametric uncertainty in [9]. For systems with asynchronous sensing *This material is based upon work supported by the National Science Foundation under Grant No. CNS-1329650. Masashi Wakaiki acknowledges Murata Overseas Scholarship Foundation and The Telecommunications Advancement Foundation for their support of this work. 1 Masashi Wakaiki is with the Department of Electrical and Electronic Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan (email: wakaiki@chiba-u.jp). 2 Masaki Ogura is with the Department of Electrical and Systems En- gineering at the University of Pennsylvania, Philadelphia, PA 19104, USA (email: ogura@seas.upenn.edu). 3 Jo˜ ao P. Hespanha is with the Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa Barbara, CA 93106, USA (email: hespanha@ece.ucsb.edu). and control, stability analysis [10], L 2 -gain analysis [11], and limitations on the clock offset tolerable for stabilization [12], [13] have been studied. The time measure of the optimal controller in [14] is a stochastic process subject to noise. The authors in [15] have compensated clock offsets and skews for the timestamp-based synchronization of multiple plants over networks. The experimental results in [16], [17] indicate that human subjects potentially learn temporal uncertainty. In this paper, we study the robust stability of systems that have variable offsets between the clocks of the sensor and the digital controller. We assume that the system has polytopic uncertainty as, e.g., in [18], [19]. We provide a sufficient condition for the closed-loop stability via linear matrix inequalities (LMIs). The proposed method is illustrated with a numerical simulation by showing how large space and time- domain uncertainty would be allowed by a given controller. A significant challenge to the analysis of networked closed-loop systems stems from the fact that such systems have both continuous-time and discrete-time state variables. In the stability analysis of [9], a continuous-time controller is used with the input-delay approach [20]–[22], and hence the closed-loop system has only continuous-time states. The au- thors in [10] focus on discrete-time states by discretizing the closed-loop system. However, this discretization approach leads to a nonlinear term including both parametric and time- domain uncertainty, which brings conservativeness for robust stability analysis. In contrast to the references mentioned above, we repre- sent the closed-loop system as an impulsive system as done in [23]–[28] for systems with variable delays and aperiodic sampling. In this representation, space-domain uncertainty appears in an affine form, and therefore allows us to more ef- ficiently address polytopic uncertainty in the original system. We describe the state of the digital controller by a piecewise constant function and construct a Lyapunov functional that incorporates both continuous and discrete-time states. This paper is organized as follows. In Section II, we introduce the closed-loop system and basic assumptions, and then formulate our problem. Section III is devoted to the main result, and we provide its proof in Section IV. In Section V, we discuss the advantages and the disadvantages of a discretization of the closed-loop system, with respect to modeling the closed loop as an impulse system (as we have done here). We illustrate the proposed method with a numerical simulation in Section VI and give concluding remarks in Section VII. Notation and definitions: For a real matrix M , let M > denote its transpose. For a real square matrix Q, define He(Q) := Q + Q > . We denote the Euclidean norm of a