Self-measurement-based scheme

The self-measurement-based scheme is the first trigger scheme reported in the liter- ature (see [Dimarogonas and Johansson, 2009a,b; Dimarogonas et al., 2012] for some early work) and has been widely developed in the studies of event-triggered consen- sus problem investigating different agent dynamics [Mu et al., 2015; Yang et al., 2016; Garcia et al., 2014; Liu et al., 2016d], and information constraints (e.g. graph topol- ogy, information quantization, communication delays) [Seyboth et al., 2013; Garcia et al., 2013; Yi et al., 2016; Mu et al., 2015; Zhang et al., 2015].

In this scheme, each agent is required to monitor its own state (xi(t)) continuously in a global coordinate system. The interactions among the agents rely on on-board wireless communication devices. Let the event time instants for agent ibe denoted asti₀=0,· · · ,ti_ki,· · ·. The state mismatch for agentiis defined as

Every time an event is triggered,ei(t)is reset to be equal to zero. The control input is described by ui(t) =−

∑

where kj _{= arg min}

l∈N : t ≥ tjl. Fort ∈ [tiki,ti_ki₊₁), t j

kj is the last event time of agent

j. Each agent takes into account the last update value of each of its neighbours in its control law (seeExample2). The control input for agentiis updated both at its own event times ti

0,ti1, . . ., as well as at the event times of its neighbourst

j

0,t

j

1, . . ., j∈ Ni. The closed loop form of the MAS with agent dynamics (2.1) and control input (2.11) can be written as ˙ x(t) =−L      x1(t1_k1) x2(t2_k2) .. . xn(tn_kn)     

Note that xi(ti_ki) = xi(t) +ei(t). By substituting this to the above closed-loop form, we obtain

˙

x(t) =−Lx(t)− Le(t) (2.12) This closed loop form is of great importance for the design of trigger conditions and represents the main difference of the self-measurement-based scheme from the other two triggering schemes. We now show how to design a basic, distributed trigger function by using Lyapunov analysis and the closed loop form (2.12).

Consider the following Lyapunov function

V(t) = 1 2x(t)

>_{Lx(t) (2.13)} Its time derivative along the solution of (2.12) is

˙

V(t) =x(t)>Lx˙(t)

=−x(t)>LLx(t)−x(t)>LLe(t)

SinceLis symmetric, we obtaing(t)>= x(t)>L. Then we can upper bound ˙V(t)as ˙

V(t)≤ −kg(t)k2+kLkkg(t)kke(t)k

To guarantee ˙V(t) < 0 ( ˙V(t) < 0 means limt→∞Lx(t) = 0, which implies consen-

sus), we must ensure that ke(t)k ≤ kg(t)k/kLk. For this purpose, we can design a distributed trigger function as follows:

whereβi >0. The trigger condition can be simply formulated as follows: an event for agentiis triggered as soon as the trigger function fi(t) =0 is satisfied. Note thatei(t) is reset at each event time, the trigger condition thus ensureskei(t)k ≤βikgi(t)k. By using a properly selected βi, the trigger condition can ensureke(t)k ≤ kg(t)k/kLk holds for allt ≥0, meaning consensus can be achieved.

Remark 2. The above design procedures follow the ideas proposed in [Dimarogonas and Jo- hansson, 2009a,b; Dimarogonas et al., 2012]. The trigger function(2.14)involves two terms, the error term ei and the comparison term gi, where gi can be calculated by using continuous

communication or measured by relative state sensors (e.g. cameras). To avoid continuous communication, [Dimarogonas and Johansson, 2009a,b; Dimarogonas et al., 2012] assumes each agent monitors the relative states using additional sensors. However, this assumption increases the cost for constructing the system since both wireless communication devices and relative state sensors are equipped. We note that the requirement of continuous relative measurement has been removed in [Garcia et al., 2013; Seyboth et al., 2013], by replacing the comparison term (gi) with state values measured at discrete-time instants and time-dependent

functions, respectively. The two ideas were then widely adopted in the follow-up work (e.g. [Yang et al., 2016; Liu et al., 2016d]) using agent-measurement-based scheme. According to this, we choose to say that, when using the self-measurement-based scheme, each agent only needs to monitor its self state continuously rather than measuring relative states continu- ously.

Now we record two features of the self-measurement-based scheme and present both its advantages and disadvantages.

• Each agent uses discrete-time state values at both its own and its neighbours’ event time instants to update the control input. For this purpose, each agent is assumed to be equipped with wireless communication devices to broadcast the discrete-time state values to its neighbours.

• Each agent monitors its own state continuously to calculate the state mismatch (2.10) and determine event time instants. The state measurement for each agent has to be with respect to a global coordinate system. For example, UAVs use Global Positioning System (GPS) to measure their own position information. Advantages: Each agent only needs to broadcast its discrete-time state values at specific time instants, and therefore continuous communications among the agents are avoided. On-board communication resources can be saved significantly.

Disadvantages: The control input (2.11) of each agent has to be updated at its own event times, as well as at the event times of its neighbours. This may result in a very frequent control input updates, especially when one agent has a large number of neighbours. Moreover, though measuring the self state in a continuous-time manner is simpler from the viewpoint of practical applications, the requirement of a global coordinate system restricts the application circumstances of the self-measurement- based scheme, for example, in a GPS-denied area.