The proof is thus completed.
7.4
A special quantizer: formation control with binary dis-
tance information
7.4.1 Rigid formation control with coarse measurements
In this section we consider the special case in which each agent uses very coarse distance measurements, in the sense that it only needs to detect whether the current distance to each of its neighbors is greater or smaller than the desired distance. This gives rise to a special quantizer defined by the followingsignumfunction:
sign(x) = 1 whenx >0; 0 whenx =0; −1 whenx <0.
Accordingly, we obtain the following rigid formation control system with binary distance measurements: ˙ pi =− m
∑
k=1 biksign(||zk|| −dk)zˆk (7.17)Remark 17. Formation control using the signum function has been discussed in several
previous papers. In [Zhao et al., 2014], a finite-time convergence was established for stabi- lization of cyclic formations using binary bearing-only measurements. The paper [Liu et al., 2014] studied the stabilization control of a cyclic triangular formation with the controller (7.17). Here we extend such controllers to stabilize general undirected formation which are minimally and infinitesimally rigid. The above controller (7.17)can also be seen as a high- dimensional extension of the one-dimensional formation controller studied in [De Persis et al., 2010]. Also, note that the right-hand side of (7.17)is composed of the sum of a unit vector multiplied by a signum function. This implies that the formation controller(7.17)is of special interest in practice since the control action is explicitly upper bounded by the cardinality of the set of neighbors for each agent i, which prevents potential implementation problems due to saturation.
Again, we consider the Filippov solution to the formation control system (7.17). The differential inclusionF(sign(ek))can be calculated as
F(sign(ek)) = 1 kzkk>dk, [−1, 1], kzkk=dk, −1 kzkk<dk.
In a compact form, the rigid formation system (7.17) can be rewritten as ˙
where sign(e)is defined in a component-wise way.
Note that the right-hand side of (7.18) is measurable and essentially bounded at any non-collocated and finite point p, and the existence of a local Filippov solution to (7.18) is guaranteed from such an initial point p(0). In the following analysis we will also show that the solutions are bounded and complete.
Similar to the analysis in deriving the distance error system shown in Section 7.3.2, the distance error system with binary distance information can be obtained as
˙
e∈ F[−Dz˜R(z)R>(z)Dz˜sign(e)], a. e. (7.19) Again, similar to the analysis for (7.11), one can also show that (7.19) is a self- contained system whenetakes values locally around the origin.
7.4.2 Convergence analysis
The main result in this section is stated in the following convergence theorem for the formation controller (7.18) with binary distance information.
Theorem 12. Suppose the target formation is infinitesimally and minimally rigid, the initial
formation shape is close to the target formation shape, and the formation controller (7.17) with binary distance information is applied.
• The formation converges locally to a static target formation shape;
• The convergence is achieved within a finite time upper bounded by T∗ = ke(0)k1
¯
λmin where ¯
λmin is defined in the proof.
Proof. Part of the proof for this theorem is similar to the proof of Theorem 11. Choose the Lyapunov function defined asV=∑mk=1Vk(ek)withVk(ek) =|ek|for the distance error system (7.19). Note that V is a convex and regular function of e. Also V is locally Lipschitz at e = 0 and is continuously differentiable at all other points. The generalized derivative ofVk(ek)can be calculated as
∂Vk = 1, ek >0; [−1, 1], ek =0; −1, ek <0.
and the generalized derivative ofV can be calculated similarly via the product rule (see [Cortés, 2008]). We define a sub-level setB(ρ) ={e :V(e)≤ρ}for some suitably smallρ, such that whene∈ B(ρ)the formation is infinitesimally minimally rigid and R(z)R>(z)andDz˜ are positive definite. Now the matrix Q(e):= Dz˜R(z)R>(z)Dz˜ is also positive definite whene∈ B(ρ). Let ¯λmindenote the smallest eigenvalue ofQ(e) whene(p)is in the compact setB (i.e. ¯λmin=min
e∈B λ(Q(e))>0).
In the following, we calculate the set-valued derivative of V along the trajectory described by the differential inclusion (7.19). The argument follows similarly to the
§7.4 A special quantizer: formation control with binary distance information 103
analysis in the proof of Theorem 11. By applying (7.30), the set-valued derivative is described by
˙
V(e)(7.19)∈ L˜(7.19)V(e) ={a∈R|∃v∈e˙(7.19),
such that ζ>v= a,∀ζ ∈∂V(e)}. (7.20) If the set ˜L(7.19)V(e)is not empty, there exists v ∈ −Q(e)sign(e) such that ζ>v = a for all ζ ∈ ∂V(e). A natural choice of v is to set v ∈ −Q(e)ζ, with which one can obtaina= −sign>(e)Q(e)sign(e). Then one can further show
max(L˜(7.19)V(e))≤ −λ¯minsign(e)>sign(e), (7.21) if the set is not empty, while if it is empty we adopt the convention max(L˜(7.19)V(e)) =
−∞. Note that this implies thatV is non-increasing, and consequently the Filippov solution e(t) is bounded. Thus, all solutions to (7.19) (as well as the solutions to (7.18)) are complete and can be extended tot =∞(i.e., there is no finite escape time). It can be seen that max(L˜(7.19)V(e)) ≤ 0 for all e ∈ B(ρ) and 0 ∈ max(L˜(7.19)V(e)) if and only if e = 0. According to the nonsmooth invariance principle shown in Theorem 15, the asymptotic convergence is proved.
We then prove the stronger convergence result, i.e., the finite-time convergence. From the definition of thesignfunction in (7.17), there holds sign(e)>sign(e)>1 for anye6=0, which implies
max(L˜(7.19)V(e))≤ −λ¯min (7.22) for anye6=0. Thus, by applying Theorem 17 (in the Appendix), any solution starting at e(0) ∈ B(ρ)reaches the origin in finite time, and the convergence time is upper bounded byT∗= V(e(0))/ ¯λmin= ke(0)k1/ ¯λmin.
Remark 18. (Dealing with chattering) In the controller(7.17)the sign function is used,
which may cause chattering when the formation is very close to the desired one (i.e. when e is very close to the origin). Possible solutions to eliminate the chattering include the following:
• Add deadzone to the sign function around the origin (similar to the case of uniform quantizers; see Part 1 of Theorem 11). This will give rise to a trade-off in the conver- gence, i.e., the distance error does not converge to the origin but to a bounded set, the size of which depends on how large the deadzone parameter is chosen;
• Use the hysteresis principle in the quantization function design; • Use the self-triggering principle, as in [Persis and Frasca, 2013].
(a) (b)
Figure 7.2: (a) Symmetric uniform quantizer function, defined in (7.2). (b) Asymmet- ric uniform quantizer function, defined in (7.23).