6.3 Centralized control algorithm
6.3.1 Centralized algorithm design
We propose the following general form of event-triggered formation control system ˙
pi(t) =ui(t) =ui(th) (6.8) =
∑
j∈Ni
(pj(th)−pi(th))ek(th)
fort ∈[th,th+1), whereh =0, 1, 2,· · · andthis theh-th trigger time for updating new information in the control input. Thus, the control input takes piecewise constant values in each time interval. In a compact form, the above position system can be written as ˙ p(t) =−R(p(th))>e(th) (6.9) Denote a vector δi(t)as δi(t) =
∑
j∈Ni (pj(t)−pi(t))ek(t)−∑
j∈Ni (pj(th)−pi(th))ek(th) (6.10) fort∈ [th,th+1). Then (6.8) can be equivalently stated as˙
pi(t) =ui(th) =
∑
j∈Ni(pj(t)−pi(t))ek(t)−δi(t) (6.11)
Define a vectorδ(t) = [δ1(t)>,δ2(t)>,· · · ,δn(t)>]> ∈Rdn. Then there holds
δ(t) =R(th)>e(th)−R(t)>e(t) (6.12) which enables us to rewrite the compact form of the position system as
˙
p(t) =−R(t)>e(t)−δ(t) (6.13) To deal with the position system with the event-triggered control input (6.8), we instead analyse the distance error system. By noting that ˙e(t) = 2R(t)p˙(t), the distance error system can be derived as
˙
e(t) =2R(t)p˙(t)
=−2R(t)R(p(th))>e(th)) ∀t ∈[th,th+1) (6.14)
Note that all the entries ofR(t)ande(t)contain the real-time values of p(t), and all the entriesR(p(th))ande(th)contain the piecewise-constant valuesp(th)during the
time interval[th,th+1).
The new form of the position system (6.13) also implies that the compact form of the distance error system can be written as
˙
e(t) =−2R(t)(R(t)>e(t) +δ(t)) (6.15) Consider the function V = 14∑mk=1e2k as a Lyapunov-like function candidate for system (6.15). Similarly to the analysis in [Sun et al., 2015c], we define a sub-level set B(ρ) = {e : V(e) ≤ ρ}for some suitably small ρ, such that when e ∈ B(ρ)the
formation is infinitesimally minimally rigid andR(p(t))R(p(t))>is positive definite. Before giving the main proof, we record the following key result on the entries of the matrixR(p(t))R(p(t))>.
Lemma 11. When the formation shape is close to the desired one such that the distance error e is in the setB(ρ), the entries of the matrixR(p(t))R(p(t))>are continuously differentiable functions ofe.
This lemma enables one to discuss the self-contained distance error system (6.15) and thus a Lyapunov argument can be applied to show the convergence of the dis- tance errors. The proof ofLemma11 can be found in [Mou et al., 2015] or [Sun et al., 2015d] and will not be presented here. FromLemma11, one can show that
˙ V(t) = 1 2e(t) > ˙ e(t) =−e(t)>R(t)(R(t)>e(t) +δ(t)) =−e(t)>R(t)R(t)>e(t)−e(t)>R(t)δ(t) ≤ −kR(t)>e(t)k2+ke(t)>R(t)kk δ(t)k (6.16) If we enforce the norm of δ(t)to satisfy
kδ(t)k ≤γkR(t)>e(t)k (6.17)
and choose the parameterγto satisfy 0<γ<1, then we can guarantee that
˙
V(t)≤(γ−1)kR(t)>e(t)k2<0 (6.18)
This indicates that events are triggered when
f :=kδ(t)k −γkR(t)>e(t)k=0 (6.19)
The event time th is defined to satisfy f(th) = 0 for h = 0, 1,· · ·. For the time intervalt ∈ [th,th+1), the control input is chosen as u(t) = u(th)until the next event is triggered. Furthermore, every time an event is triggered, the event vectorδwill be
reset to zero.
We also show two key properties of the formation control system (6.8) with the above event function (6.19).
Lemma 12. The formation centroid remains constant under the control of (6.8) with the event function(6.19).
Proof. Denote by ¯p(t) ∈ Rd the centre of the mass of the formation, i.e., ¯p(t) =
1
n∑ n
i=1pi(t) = 1n(1n⊗Id)
>p(t). One can show ˙¯ p(t) = 1 n(1n⊗Id) > ˙ p(t) =−1 n(1n⊗Id) > R(p(th))>e(th) =−1 n Z(th)>H(1n¯ ⊗Id) > e(th) (6.20) Note thatker(H) = span{1n}and thereforeker(H) =¯ span{1n⊗Id}. Thus ˙¯p(t) =0, which indicates that the formation centroid remains constant.
The following lemma concerns the coordinate system requirement and enables each agent to use its local coordinate system to implement the control law, which is favourable for networked formation control systems in e.g. GPS denied environ- ments.
Lemma 13. To implement the controller(6.8)with the event-based control update condition in(6.19), each agent can use its own local coordinate system which does not need to be aligned with a global coordinate system.
The proof for the above lemma is omitted here, as it follows similar steps as in [Sun et al., 2015d, Lemma 4]. Note thatLemma13 implies the event-based formation system (6.8) guarantees the SE(N)invariance of the controller, which is a nice prop- erty to enable convenient implementation for networked control systems without coordinate alignment for each individual agent [Vasile et al., 2015].
We now arrive at the following main result of this section.
Theorem 10. Suppose the target formation is infinitesimally and minimally rigid and the initial formation shape is close to the target one. By using the above control input (6.8)
and the trigger function (6.19), all the agents will reach the desired formation shape locally exponentially fast.
Proof. The above analysis relating to Eq. (6.16)-(6.19) establishes boundedness of
e(t) since ˙V is non-positive. Now we show the exponential convergence of e(t) to zero will occur from a ball around the origin, which is equivalent to the desired formation shape being reached exponentially fast. According to Lemma 9, let ¯λmin
denote the smallest eigenvalue ofM(e):=R(p)R(p)>whene(p)is in the setB (i.e. ¯
λmin = min
e∈B λ(M(e)) > 0). Note that ¯λmin exists because the set B(ρ) is a compact set with respect toe and the eigenvalues of a matrix are continuous functions of the matrix elements. By recalling (6.18), there further holds
˙
V(t)≤(γ−1)λ¯minke(t)k2
Thus one concludes
with the exponential decaying rate no less thanκ=2(1−γ)λ¯min.
Note that the convergence of the inter-agent distance error of itself does not di- rectly guarantee the convergence of agents’ positions p to some fixed points, even though it does guarantee convergence to a correct formation shape. This is because that the desired equilibrium corresponding to the correct rigid shape is not a sin- gle point, but is a set of equilibrium points induced by rotational and translation invariance (for a detailed discussion to this subtle point, see [Dorfler, 2008, Chapter 5]). A sufficient condition for this strong convergence to a stationary formation is guaranteed by the exponential convergence as proved above. To sum up, one has the following Lemma on the convergence of the position system (6.9) as a consequence ofTheorem10.
Lemma 14. The event-triggered control law (6.8) and the event function (6.19)guarantee the convergence of p(t)to a fixed point.
Remark 15. We remark the above Theorem 10 (as well as the subsequent results in later sections) concerns a local convergence. This is because that rigid formation shape control system isnon-linearand exhibitsmultipleequilibria, which include the ones corresponding to correct shapes and those that do not correspond to the correct shape. It has been shown in [Anderson and Helmke, 2014] by using the tool of Morse Theory that multiple equilibria, including incorrect equilibria, are a consequence of any formation shape control algorithm which evolves in a steepest descent direction of a smooth cost function that is invariant under translations and rotations. A recent paper [Sun et al., 2015a] proves the instability of a set of degenerate equilibria that lives in a lower dimensional space. However, the stability property for more general equilibrium points is still unknown. It is in fact considered as a very challenging open problem to obtain an almost global convergence for general rigid formations, except for some special formation shapes such as 2-D triangular formation shape, or 2-D rectangular shape, or 3-D tetrahedral shape (see the review in [Oh et al., 2015]). We note that local convergence is still valuable in practice, if one assumes that initial shapes are close to the target ones (which is a very common assumption in most rigidity-based formation control works).