L 1 Adaptive Control of Event-Triggered Networked Systems Xiaofeng Wang and Naira Hovakimyan Abstract— This paper studies the implementation of L1 adap- tive controller over real-time networks using event-triggering. Event-triggering schedules the data transmission dependent upon errors exceeding certain threshold. We provide event- triggering schemes to characterize the time instants of data transmissions, while using L1 adaptive controller in the feed- back loop. Lower bounds on transmission periods are provided. We show that with the proposed event-triggering schemes the states and the input in the networked system can be arbitrarily close to those of a stable reference system by increasing the rate of adaptation and the transmission frequency. I. I NTRODUCTION With the progress in digital technology the feedback loops in a wide range of applications are closed via real- time communication networks for the advantages of lower cost, ease of maintenance and diagnosis, and great flex- ibility. The introduction of real-time networks raises new challenges regarding the impact that communication has on the system performance. Communication, especially wire- less communication, takes place over a digital network, which means that information is transmitted in discrete time rather than continuous-time. Moreover, because all real-time networks have limited bandwidth, the information transmission has to be scheduled in an appropriate manner for a proper operation of the control system. This paper studies the impact of a real-time communication network on the performance of L 1 adaptive controller [1] using an event-triggering technique. Event-triggering has the data transmitted only when “needed”. By “needed”, it means that some error exceeds certain threshold. Empirical evidence has suggested that event-triggering can largely reduce usage of computational/communication resources. Moreover, it can dynamically adjust the task periods in response to external disturbances [2], [3]. Event-triggering has been studied in [2], [3], [4], [5], [6], [7]. Among these, [2], [3], [4], [5] studied single processor systems and [6], [7], [8] consid- ered distributed implementation. These results are limited to event-triggered feedback in systems without uncertainties. This paper provides event-triggering schemes to character- ize the time instants of data transmissions in uncertain sys- tems with L 1 adaptive controller being in the feedback path. We show that with the proposed event-triggering schemes the states and the input in the networked system can be arbitrarily close to those of a stable reference system by increasing the rate of adaptation and the transmission frequency. We also Both authors are with the Department of Mechanical Science and En- gineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801; e-mail: wangx,nhovakim@illinois.edu. Research is supported by AFOSR under Contract No.FA9550-09-1-0265. provide guaranteed lower bounds on transmission periods generated by our scheme and verify those in simulations. This paper is organized as follows. Section II gives the problem formulation. Section III presents the event- triggering scheme that has guaranteed performance. Sim- ulations results are shown in Section IV. Finally, some discussion is stated in section V. II. PROBLEM FORMULATION Notation: We denote by R n the n-dimension real vector space and by R + the real positive numbers. We also use R + 0 = R + ∪{0}. ‖·‖ is the Euclidean norm of a vector. ‖·‖ L1 and ‖·‖ L∞ are the L 1 and L ∞ norm of a function, respectively. The truncated L ∞ norm of a function x : R + 0 → R n is defined as ‖x τ ‖ L∞  sup 0≤t≤τ ‖x(t)‖. We also use ∨ to denote the logical operator OR, where E 1 ∨E 2 is true when either E 1 or E 2 is true. · denotes the logical operator NOT, where E is true when E is false. The Laplace transform of a function x(t) is denoted by x(s). For a function x : R → R n , We denote x(t + ) = lim ρ→t + x(ρ) and x(t - ) = lim ρ→t - x(ρ). This paper considers an implementation of L 1 adaptive controller in networked systems using event-triggering tech- nique. As shown in Figure 1, in such a system, the plant transmits the state to the controller through a real-time network only when some event occurs. The time instants of transmitting the states can be characterized by a monotonic sequence {s k } ∞ k=1 , where s k ∈ R + is the kth time instant of transmitting the state from the plant to the controller. The control input is computed based on the received state using the L 1 adaptive control scheme. The updated control input is transmitted back to the plant only when another event occurs. We use another monotonic sequence {r j } ∞ j=1 to characterize the time instants when the input is transmitted back to the plant, where r j ∈ R + is the j th time instant of transmitting the updated input from the controller to the plant. The control input is held by a zero-order-hold until the next transmission of inputs happens. In Figure 1, solid lines represent continuous signals and dashed lines represent discrete signals. We assume that there is no delay in sampling, data transmission, and computation. The system under consideration is a single-input-single- output (SISO) system: ˙ x(t) = Ax(t)+ b(u(t) - θ ⊤ x(t)) y(t) = c ⊤ x(t), x(0) = x 0 (1) where x : R + 0 → R n is the state trajectory, u : R + 0 → R is the control input, y : R + 0 → R is the system output, A ∈ R n×n , b, c ∈ R n are known and θ ∈ R n is an unknown