Recomendaciones para el Orientador

control problem

In this section, we propose a virtual vehicle approach to design a formation control scheme that removes the requirement of linear-velocity measurements in the presence of constant communication delays. The main idea in this approach is to associate a virtual vehicle to each aircraft with similar translational dynamics and an additional input. This input is designed so that the states of all virtual vehicles converge to the specified formation in the presence of communication delays. The advantage of this approach is that the states of the virtual vehicles, i.e., virtual positions and virtual velocities, are internally synthesized and hence available.

We propose the following intermediary control input F_i =₋k_ipχ(θ_i)₋k_idχ( ˙θ_i), (6.38) ¨ θ_i =F_i−u_i, (6.39) ¨ α_i =ui−φi− 2_Ti m_iR(Qi) T_S(¯q i)˜qi, (6.40) ui =−kivα˙i− n X j=1 kij(αi−αj(t−τij)−δij), (6.41)

where φ_i is an input vector to be designed latter and the control gains are defined as in Theorem 5.1. Note that the intermediary control structure is similar to (6.2)- (6.3) with (6.24) and the only difference is the design of the input u_i. In fact, the auxiliary system (6.40) in this scheme describes the translational dynamics of a virtual system, associated to theithaircraft. The inputuiis constructed based on the virtual

vehicle velocity and position, ˙α_i andα_i respectively, to guarantee that all the virtual vehicles converge to the desired formation in the presence of communication delays

i.e., (α_{i −}α_j) → δ_ij and ˙α_{i →} 0. The design of this input is motivated by the following preliminary result proved in Appendix A.5.4.

Lemma 6.1. Consider n-vehicles modeled as

¨ α_i =₋k_ivα˙_i− n X j=1 k_ij(α_i−α_j(t₋τ_ij)₋δ_ij) + ¯ε_i, (6.42)

for i ∈ N, where τij is a constant communication delay between the ith and jth

vehicles satisfying τij ≤ τ for all (i, j) ∈ E. Let the control gains kiv and kij satisfy

condition (6.14), for some ǫ > 0 and assume that the communication graph _G is connected. If the vector ε¯_i is bounded, such that kε¯_{ik ≤}ε¯b_i, for all t > 0 and i∈ N, and converges asymptotically to zero, then (α_i−α_j)andα˙_i are bounded andα˙_{i →}0,

(α_i−α_j)_→δ_ij, for all i, j _{∈ N}.

The above lemma states that if the inputφ_i and the input torque are designed such that ˜q_i and φ_i are guaranteed to be bounded and converge asymptotically to zero, the virtual vehicles will converge to the prescribed formation with zero virtual velocity in the presence of communication delays. Therefore, the intermediary con- trol design is reduced to determine an appropriate input φ_i, without linear-velocity measurements, such that each vehicle tracks the states of its corresponding virtual vehicle. To this end, we consider the following partial state feedback input

φ_i =₋Lp_iξ_i−Ld_i(ξ_i−ψ_i), (6.43) ˙

where Lp_i, Ld_i and Lψ_i are positive scalar gains, the vector ψ_{i ∈ R}3 is the output of the dynamic system (6.44) that can be initialized arbitrarily, and the error vector ξ_i

is defined in (6.25), i.e.,

ξ_i =pi−θi−αi , zi := ˙ξi. (6.45)

To complete the design of the input torque, notice first that the time-derivative of u_i in (6.41) can be obtained as ˙ u_i =₋kv_i u_i−φ_i−2Ti m_iR(Qi) T_S(¯q i)˜qi − n X j=1 kij α˙i−α˙j(t−τij), (6.46)

and is function of available signals. Therefore, the desired angular velocity and its time-derivative given in (5.9)-(5.10) with (6.8)-(6.9) are explicitly known. However, to implement the above control scheme, neighboring aircraft must communicate the position and velocity of their corresponding virtual vehicles, i.e., α_i and ˙α_i. Note also that the perturbation term in the translational dynamics has been compensated in the dynamics of the virtual system (6.40). Therefore, the input torque for each aircraft can be considered similar to the previous section, and the following theorem holds:

Theorem 6.4. Consider the VTOL-UAVs modeled as in (5.1)-(5.2). Let the thrust input _Ti and the desired attitude Qd_i be given, respectively, by (4.6) and (4.7), with

Fi given by (6.38)-(6.41) and (6.43)-(6.44). Let the input torque be given by (6.11)

with (6.28). Let the controller gains satisfy conditions (6.13) and (6.14) for some ǫ > 0 and τ_{ij ≤} τ, for all (i, j) _{∈ E}, and assume that the communication graph is connected. Then, starting from any initial conditions, the signals vi, (pi−pj) and

˜

ω_i are bounded andv_{i →}0, (p_i−p_j)_→δ_ij, q˜_{i →}0 and ω˜_i→0 for all i, j _{∈ N}.

Sketch of proof: Similar to the proof of Theorem 6.1, the thrust input and desired attitude for each aircraft can be extracted if condition (6.13) is satisfied. The transla- tional error dynamics can be obtained from (6.45) in view of (5.1), (6.39)-(6.40) and (6.43) is obtained as

˙

zi =−Lp_iξi−Ldi(ξi−ψi), (6.47)

Also, the attitude error dynamics are given similar to the proof of Theorem 6.1 in (6.16). The proof of the theorem consists of first showing that each vehicle converges to its corresponding virtual vehicle and the attitude tracking error converges to zero.

This is achieved using the following Lyapunov function

V =Vt₂+Va₁, (6.48)

where Va₁ is given in (6.19) and

Vt₂ = 1 2 n X i=1 zT_i zi+Lp_iξTi ξi+Ldi(ξi−ψi)T(ξi−ψi) , (6.49)

which leads to the negative semi-definite time-derivative

˙ V = n X i=1 −Ld_iLψ_i (ξ_i−ψ_i)T(ξ_i−ψ_i)₋kΩ_i ΩT_i Ωi−k_iqk_iβq˜Ti q˜i . (6.50)

Following similar steps as in the proof of Theorem 4.2, we show that ξ_{i →} 0, (ξ_i− ψ_i) _→ 0, zi → 0, ˜qi → 0 and ˜ωi → 0 for i ∈ N. Then, the dynamics of the

virtual system (6.40) with (6.41) can be rewritten as in (6.42) and the conditions of Lemma 6.1 are satisfied. As a result, the states of the virtual systems converge asymptotically to the predefined formation,i.e., α˙_{i →}0 and (α_i−α_j)_→δ_ij, for all

i, j ∈ N. Finally, we show using the results of Lemma 2.6 that θ_{i →}0 and ˙θ_{i →}0, leading to the results of the theorem in view of the error definition (6.45). Detailed proof of Theorem 6.4 is given in Appendix A.5.5.