2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010

ThC15.2

Synchronized Altitude Tracking Control of Multiple Unmanned Helicopters Rongxin Cui, Shuzhi Sam Ge∗ , Beibei Ren Abstract— In this paper, we consider the synchronized altitude tracking control for multiple helicopters, when the desired trajectory is only available to the leaders. By using the weighted average of the neighbors’ states as the reference signal, the adaptive neural network (NN) tracking control is designed for each helicopter. It is shown that, the output tracking error of each helicopter converges to an adjustable neighborhood of origin under the proposed NN control, although some of them do not access the desired tracking trajectory directly. Simulation results are provided to demonstrate the effectiveness of the approach presented.

I. I NTRODUCTION Cooperative control of multi-agent systems has drawn significant research interest in recent years [1]–[3]. Multiple helicopters cooperation can accomplish complex tasks such as surveillance, large area search and rescue. One fundamental problem in multi-helicopter cooperation is formation control, in which the helicopters keep a desired formation configuration and at the same time complete the assigned tasks. Various approaches have been proposed for formation control, including behavioral [4], virtual structure [5], [6], queues and artificial potential trenches [2], [3], and leaderfollowing [7]. In the control of helicopters, one of the main challenges is how to deal with the unknown perturbations to the nominal model, in the presence of parametric and functional uncertainties, unmodeled dynamics, and disturbances from the environment [8]. Much work has been done on this problem: Direct neural dynamic programming for helicopter trimming and tracking was proposed [9]. Adaptive neural network control of helicopters in vertical flight was presented in [8]. Approximation-based control was proposed for uncertain helicopter dynamics in [10], in which both altitude and yaw angle tracking were considered. Robust tracking control was proposed for a single helicopter in [11]. For the formation of helicopters, a non-iterative nonlinear model predictive control was applied to each leader-follower pair [12]. In this paper, we study the synchronized tracking problem of helicopters in vertical flight, in which multiple helicopters track the same desired trajectory, while the desired trajectory is not accessible to all the helicopters in the team. Since the R. Cui is with the Centre for Offshore Research & Engineering, National University of Singapore, Singapore 117576 [email protected] S. S. Ge and B. Ren are with the Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576

{samge,renbeibei}@nus.edu.sg ∗

To whom all correspondences should be addressed.

978-1-4244-7425-7/10/$26.00 ©2010 AACC

coupling between longitudinal and lateral-directional equations in this flight regime is week, it can be represented by SISO models with zero-dynamics [8], [13]. In the formation group, the desired trajectory is not available to all the helicopters in the team. Then the synchronized tracking control is designed for each helicopter by using the information exchange with its neighbors. The main contributions of this work include: (i) The extended formation graph Laplacian, which contains a spanning tree, with the desired trajectory only available to the root helicopter, is proved to be positive definite, and (ii) The neural approximation based control is designed for each helicopter by using the weighted average of its neighbors’ states for the purpose of synchronized tracking. The remainder of the paper is organized as follows: In Section II, the problem formulation for synchronized tracking of helicopters is presented. In Section III, synchronized tracking is designed by using the neighbors states. Simulation results are presented in Section IV. The paper is concluded in Section V. II. P ROBLEM F ORMULATION A. Helicopter Dynamics Consider the following SISO helicopter systems: x˙ j x˙ ρ η˙ y

= = = =

xj+1 , j = 1, . . . , ρ − 1 f (η, x) + g(x, η)(u + d) q(x, η) x1

(1)

where x = [x1 , . . . , xρ ]T ∈ Rρ and η ∈ Rn−ρ are the states; u, y ∈ R the input and output, respectively; f : Rn → R an unknown smooth function; and q : Rn → R a partially unknown vector field satisfying certain properties; g : Rn → R is an unknown function with certain properties, and d is the external disturbance in the input channel. Assumption 1: The zero dynamics of system (1), given by η˙ = q(x, η), are exponentially stable. In addition, q(x, η) is Lipschitz in x, i.e., there exists positive constants aq and ax such that kq(x, η) − q(0, η)k ≤ ax kxk + aq ∀(x, η) ∈ Rn (2) Under the assumption that the zero dynamics are stable, by the converse Lyapunov theorem, there exists a Lyapunov function V0 (η) which satisfies the following inequalities for ∂V0 2 (x, η) ∈ Rn : (i) γ1 kηk ° ≤° V0 (η) ≤ γ2 kηk, (ii) ∂η q(0, η) ≤ ° 0° −λa kηk2 , and (iii) ° ∂V ∂η ° ≤ λb kηk, where γ1 , γ2 , λa , and λb are positive constants.

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Assumption 2: The external disturbance d is uncertain bounded functions d ∈ L∞ . That is, there exists unknown positive constants ̺ such that |d(t)| ≤ ̺ < ∞, where ̺ can be arbitrarily large. Assumption 3: There exist smooth functions g¯(x, η) and a positive constant g > 0, such that g¯(x, η) ≥ g(x, η, u) > g > 0, ∀(x, η) ∈ Rn . Without loss of generality, it is further assumed that the sign of g(x, η, u) is positive ∀(x, η) ∈ Rn . Assumption 4: There exists a positive function g0 (x, η) satisfying |g(x, ˙ η)/2g(x, η)| ≤ g0 (x, η), ∀(x, η) ∈ Rn . Remark 1: The general nonlinear SISO helicopter model described as x˙ = f (x, u), y = h(x) with some assumptions such as it can be input-output linearizable with strong relative degree ρ < n, can be described as (1) [8]. B. Formation of Helicopters We associate the helicopters with nodes in a graph and information exchange with its edges. The following definitions are useful for describing the formation. Definition 1: [1] A directed graph G ′ consists of a nonempty finite set V ′ of elements called nodes and a finite set E ′ ⊂ V ′2 of ordered pairs of nodes called arcs, where e = (vi , vj ) ∈ E ′ and vi , vj ∈ V ′ . The neighbors set of vertical vi is defined as Ni′ = {vj ∈ V ′ | (vj , vi ) ∈ E ′ }. For the multiple agents tracking problem, we introduce a virtual agent v0 , whose motion follows the desired trajectory restrictively, and define a non-empty set V0 ⊂ V ′ , in which the elements can access the desired trajectory, i.e., v0 ∈ Nj , iff vj ∈ V0 . Then the extended formation graph can be described as G = {V, E}, where V = V ′ ∪ {v0 }, and E = E ′ ∪ {(v0 , vj )|v0 ∈ Nj }. For all agents vj ∈ V0 , Nj = Nj′ ∪ {v0 }. Definition 2: The weighted adjacency matrix of the extended formation graph G, denote A∗ (G), is a square matrix of size |V|, with its elements A∗ij > 0 if (vj , vi ) ∈ G, and is zero otherwise.PDefine a diagonal matrix ∆(G) with its elements ∆jj = k A∗jk , and the normalized Laplacian of the graph is defined as L = I − A with A = ∆−1 A∗ . Property 1: [14] For a diagonally dominant matrix with nonzero elements chain L = (ℓij ), we have following properties: (i) L is a nonsingular matrix; (ii) If B = I − D−1 L, where D = diag{ℓ11 , . . . , ℓnn }, ℓii 6= 0, then ρ(B) < 1, where ρ(B) is the spectrum of B; and (iii) If L is real and ℓij ≤ 0, ℓii > 0, then L is an M -matrix. Theorem 1: Consider the multiple agent synchronized tracking problem. If the formation graph G ′ contains a spanning tree with its root vj ∈ V0 , then the normalized adjacent matrix A of the extended formation graph G is substochastic, and L = I − A is positive definite. Proof: We know that all the elements of the first row of A are zero. Since G ′ has a spanning tree and vj ∈ {V0 } is the root, it means that each agent has at least one neighbor, then the sum of any other row of A equals to 1, i.e., A is a sub-stochastic matrix. It is clear that all the diagonal elements of L are 1, and all the row sum of A is 1 except the first row, that means L is a diagonal dominant matrix with the J = {0} (Since

the virtual agent is added, we start the row number from 0 corresponding to the label of agents). Let us revisit that G ′ has a spanning tree with vj ∈ V0 as the root, it also means that there is a path from v0 to any agent vi ∈ V, then in the matrix L, for every element i 6= 0, there exists a sequence of nonzero elements form ℓii1 , ℓi1 i2 , . . . , ℓis 0 . Then L is a diagonally dominant matrix with nonzero elements chain. Since L is real and ℓij < 0, i 6= j, ℓii = 1, According to Property 1, L is a nonsingular M -matrix [14], i.e., L is positive definite. This completes the proof. Example 1: To explain Theorem 1 clearly, take the sample graph shown in Fig. 1 as an example. Both v1 and v3 can access the desired trajectory and G ′ contains a spanning tree with node 1 as its root. Take node 5 as an example, we can find that in the laplacian matrix L, there exists a sequence ℓ54 , ℓ43 , ℓ32 , ℓ21 , ℓ10 6= 0.  1 0 0 0 0   −1 1 0 0 0   0 − 2 3 1 − 1 3 0  L = − 3 4 0 −1 4 1 0  0 0 −1 1  0  0 0 0 −1  0  0 0 −1 2 −1 2  0

Spanning Tree of G '

Graph G

Fig. 1.

0 0 0  0 0 1  0 0 2 0 0 3  0 0 4  1 0 5  0 1 6

Matrix L

Sample graph and its Laplacian.

In this paper, we studied the synchronized tracking problem of multiple unmanned helicopters as follows: Considering a group of helicopters, the desired trajectory of the team, yd (t), and its derivations up to ρ-th order is bounded, and is only available to the helicopters vj ∈ V0 . For each helicopter, we design a control by using its own states and its neighbors’ states, such that lim |yi (t) − yd (t)| = ε¯,

t→∞

i = 1, . . . , N

(3)

where ε¯ is a small positive constant. The desired trajectory yd (t) is generated by the following reference model: x˙ dj x˙ dρ

= x˙ di+1 , i = 1, . . . , ρ − 1 = fd (xd , t)

where ρ ≥ 2, xd = [xd1 , . . . , xdρ ]T ∈ Rρ are the states of reference system, yd = xd1 ∈ R is the system output. Assumption 5: The reference trajectory yd (t) and its ρ-th derivatives remain bounded, i.e., xd ∈ Ωd ⊂ Rρ , ∀t ≥ 0. Assumption 6: The formation graph G ′ has a spanning tree with the root helicopter can access the desired trajectory. The following lemma is useful for analysis the internal dynamics of the helicopter. Lemma 1: [15] Denote positive constants a1 = (λb ax )/λa and a2 = (λb aq )/λa . If Assumptions 1 and 5 satisfied, there exists a positive constant T0 such that the trajectories η(t) of the internal dynamics satisfy kηk ≤ a1 kx(t)k + a2 .

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III. S YNCHRONIZED T RACKING C ONTROL D ESIGN In this section, we design the tracking control for each helicopter using the information of itself and its neighbors. Since not all the helicopters can access the information of the desired trajectory, the tracking control is designed based on the relative states with its neighbors. Define the following error variables for the helicopters: zi,1 = yi,1 − yir , · · · ,

zi,ρ :=

(ρ) zi,1

= xi,ρ −

(ρ) yir

(4)

with X

yir (t) =

X

(k)

aij yj (t), yir (t) =

(k)

aij yj (t)

(5)

ui = −θˆiT ψi − ki si − ϕˆi si /2,

where k = 1, . . . , ρ − 1, aij is the element of the normalized adjacent matrix A of the extended formation graph G. For each helicopter, we define vectors z¯i = [zi,1 , . . . , zi,ρ ]T ∈ Rρ , Zi = [zi,1 , . . . , zi,ρ−1 ]T ∈ Rρ−1 , and the filtered tracking error as si = [ΛT 1]¯ zi

(6)

T

where Λ = [λ1 , λ2 , . . . , λρ−1 ] is an appropriately chosen coefficient vector so that zi,ρ → 0 as si → 0, i.e., pρ−1 + λρ−1 pρ−2 + . . . + λ1 is Hurwitz. Then we have Z˙ i = Ap Zi + bsi

where Ap

 =  

Remark 2: The NN is constructed to approximate (ρ) f (x ,η )+[0 ΛT ]¯ zi −yir +gi g0 si (ρ) on a f¯i (xi , ηi , z¯i , yir ) = i i i gi whole, which avoids the possible singularity of the direct approximation of gi . Select the following control ui for each helicopter

j∈Ni

j∈Ni



ϕ∗i = (̺i + ε¯i )2 , respectively. Then we have ¶ µ 1 ˙ g˙ i ˙ Vi = − g0 + 2 s2i + si (ui + di ) + θ˜i θ˜i 2gi γ2 +si [θi∗T ψi (Zi ) + ε¯i ] + ϕ˜i ϕ˜˙ i /γ1 (12)

0 .. . 0 −λ1

1 .. . 0 −λ2

··· .. . ... ···

[0, . . . , 0, 1]T . | {z }

(7) 

0 ..  , and b = .  1 −λρ−1

The update law of parameters are designed as ¡ ¢ ˙ ¡ ¢ ϕˆ˙ i = γ1 s2i /2 − σ1 ϕˆi , θˆi = γ2 ψi si − σ2 θˆi

i

Considering Vsi =

1 2 2gi si ,

(8)

we have

¸ · g˙ i s2 si s˙ i g˙ i V˙ si = − 2i + = − g0 + 2 s2i + si (ui + di ) 2gi gi 2gi © ª (ρ) + si fi (xi , ηi ) + [0 ΛT ]¯ zi − yir + gi g0 si /gi

(9)

Due to the existence of the uncertain items, we use the parameter linearized NN [16] to approximate the unknown non© (ρ) linear function f¯i (xi , ηi , z¯i , yir ) = fi (xi , ηi ) + [0 ΛT ]¯ zi − ª (ρ) yir + gi g0 si /gi , which can be described as f¯i (Zi ) = θi∗T ϕi (Zi ) + ε¯i (Zi )

(10)

where ε¯i (Zi ) is the approximation error which is bounded over the compact set, i.e., |εi (Zi )| ≤ ε∗i , ∀Zi ∈ Ωxi with ε∗i > 0 is an unknown constant, θ∗ is the ideal weighted (ρ) vector, and Zi = [xi , ηi , z¯i , yir ]T . Considering the Lyapunov function candidate Vi = Vsi + 0.5θ˜iT θ˜i /γ2 + 0.5ϕ˜2i /γ1

(11) θˆi − θi∗ ,

and where γ1 and γ2 are the positive constants, θ˜i = ϕ˜i = ϕˆ − ϕ∗i , are the estimated errors of parameters and the error bound, where θˆi and ϕˆi are the estimation of θi∗ and

(14)

i

+ σ1 ϕ∗2 i /2 + 1/2. Now define n ¯ o p ¯ Ωsi = si ¯ |si | ≤ c2i /ki ¯ o n p p ¯ Ωθi = (θ˜i , ϕ˜i ) ¯kθ˜i k ≤ 2c2i /σ2 , |ϕ˜i | ≤ 2c2i /σ1 ¯ ½ ¾ ¯ 2 1 1 T˜ 2 ˜ ˜ ¯ Ωei = (si , θi , ϕ˜i ) ¯ki si + σ2 θi θi + σ1 ϕ˜i ≤ c2i 2 2

where c2i =

ρ−2

(ρ)

(13)

where σ1 , and σ2 are the designed positive constants. By using the Young’s inequality, we have −σ2 θ˜iT θˆi ≤ −σ2 kθ˜i k2 /2+σ2 kθi∗ k2 /2, −σ1 ϕ˜i ϕˆi ≤ −σ1 ϕ˜2i /2+σ1 ϕ∗2 i /2, and (̺i + ε¯i )si ≤ 1/2 + s2i ϕ∗i /2. Considering (13) and (14), the time derivation of Vi in the closed-loop trajectory can be written as 1 g˙ i εi + di − ϕˆi ) V˙ i = −(g0 + 2 )s2i − ki s2i + si (¯ 2gi 2 (15) − σ2 θ˜iT θˆi + si ϕ˜i si /2 − σ1 ϕ˜i ϕˆi ≤ −ki s2 − σ1 ϕ˜2 /2 − σ2 kθ˜i k2 /2 + c2i

The dynamics of si is written as s˙ i = fi (xi , ηi ) + gi (ui + di ) + [0 ΛT ]¯ zi − yir

i = 1, . . . , N

σ2 kθi∗ k2 /2

Since c1i , σ1 , σ2 , and ki are positive constants, we know that Ωsi , Ωθi and Ωei are compact sets. Eq. (15) shows that V˙ i ≤ 0 once the errors are outside the compact set Ωei . According to the standard Lyapunov theorem, we conclude that si , θ˜i , and ϕ˜i are bounded. From (15) and (III), it can be seen that Vi is strictly negative as long as si is outside the compact set Ωsi . Therefore, there exsits a constant T1 such that for t > T1 , the filtered tracking error si converges topΩsi , that is to say, si ≤ βsi (ki , γ1 , γ2 , σ1 , σ2 , θi∗ , ϕ∗i , ε∗i ) = c2i /ki . Now, we will show that all the helicopters will track the desired trajectory although only some of them can access the desired trajectory. Define the error between i-th helicopter and the desired trajectory as y˜i (t) = yi (t) − yd (t) = yi (t) − y0 (t), and the auxiliary states of each helicopter ξi (t) = (1) (ρ−1) T [ΛT 1]Yi with Yi = [yi , yi , . . . , yi ] . The filtered error is denoted as ξ˜i (t) = ξi (t) − ξ (t) = ξ (t) − ξ0 (t). Using the i P d fact that si (t) = ξi (t) − j∈Ni aij ξj (t), we have X aij ξj + si − ξ0 , i = 1, . . . , N (16) ξ˜i = ξi − ξ0 =

4435

j∈Ni

and in the vector form

We have (22) as the following:

ξ˜ = Aξ + s − ξ0 1

λ0 T 1

e ˜ ˜ kY(t)k ≤ k0 e−λ0 t kY(0)k + k0 e−λ0 t

(17)

where 1 = [1, . . . , 1]T , s = [s0 , s1 , . . . , sN ]T , and A is the normalized adjacency matrix of the extended formation graph. Note [0, 1, . . . , 1]T = A[0, 1, . . . , 1]T , we have ξ˜ = A(ξ˜ + ξ0 1) + s + [1, 0 . . . , 0]T ξ0 − ξ0 1 = Aξ˜ + [0, 1, . . . , 1]T ξ0 + s + [1, 0 . . . , 0]T ξ0 − ξ0 1 (18) = Aξ˜ + s Under the Assumption 6, we know that that L = (I − A) is an invertible matrix, we have ξ˜ = L−1 s

(19)

Define Y = [Y0T , . . . , YNT ]T , Y˜ = [Y˜0T , . . . , Y˜NT ]T , X = T ˜ = [X ˜0, . . . , X ˜ ρ−1 ]T , where Xj = [X0T , . . . , Xρ−1 ]T , X T ˜ j = Xj − Xjd = Xj − y (j) 1, [X0,j , X1,j , . . . , XN,j ] , X 0 Y˜i = Yi − Yd = Yi − Y0 . Then we have (20) Y˜˙ = A¯ Y˜ + ¯bξ˜ p

where A¯p = IN +1 ⊗ Ap and ¯b = IN +1 ⊗ b. Symbol “⊗” stands for the Kronecker Product of the matrices. Considering (18), the error dynamics can be written as Y˜˙ = A¯p Y˜ + ¯bξ˜ = A¯p Y˜ + ¯bL−1 s

(21)

Lemma 2: For some time constant T1 , define si,max = sup0≤τ ≤t |si (t)|, βsi = supt>T1 |si (t)| , and smax,i (t) = maxi sup0≤τ ≤t |si (t)|, then the following equations hold: ¤ k0 £ ˜ ˜ kY(t)k ≤ k0 e−λ0 t kY(0)k + N λmax (L−1 ) smax,i (t) λ0 ¶ µ k0 eλ0 T1 −λ0 t ˜ ˜ βs (T1 ) + βsT kY(t)k ≤ k0 e kY(0)k + λ0 λ0 where βs (t) = N λmax (L−1 )smax,i (t) and βsT = N λmax (L−1 ) supT1 ≤t smax,i (t) with constants λ0 > 0 and k0 > 0. Proof: From (21) and theR fact that Ap is Hurwiz, t ¯ ¯p t A ˜ ˜ we have Y(t) = Y(0)e + 0 eAp (t−τ )¯bL−1 s dτ , and ¯ keAp t k ≤ k0 e−λ0 t . Then Z

t

° ° e−λ0 (t−τ ) °¯bL−1 s° dτ 0 Z ¤ t λ0 τ −λ0 t ˜ −λ0 t £ e dτ ≤k0 e kY(0)k + k0 e N λmax (L−1 )smax,i (t) ˜ ˜ kY(t)k ≤ k0 e−λ0 t kY(0)k +

0

£ ¤ eλ0 t − 1 ˜ ≤k0 e kY(0)k + k0 e−λ0 t N λmax (L−1 ) smax,i (t) λ0 k0 £ −λ0 t ˜ −1 ¤ ≤k0 e kY(0)k + N λmax (L ) smax,i (t) λ0 (22) −λ0 t

where λmax (·) is the maximum eigenvalue of the matrix. Noting the above equation R t −λ (t−τ ) ¯b(L−1 s)kdτ 0 and the fact that e k = 0R R T1 −λ (t−τ ) ¯b(L−1 s)kdτ + t e−λ0 (t−τ ) k¯b(L−1 s)kdτ . 0 e k 0 T1

λ0 t0

λ0

−1

βs (T1 )

λ0 T 1

e −e + k0 e−λ0 t βs T λ0 · ¸ λ0 T 1 e k0 ˜ + βs (T1 ) + βs ≤ k0 e−λ0 t kY(0)k λ0 λ0 T

(23)

This completes the proof. Now we will show that for a proper choice of the control parameters, the trajectories of each vehicle do remain in the ˜¯ compact set. From the fact that L−1 s = ([ΛT 1] ⊗ IN +1 )X, ˜¯ = [X ˜T x where X ˜Tρ ]T , we can see that x ˜ρ = L−1 s−(ΛT ⊗ ˜ ˜ Therefore, kXk ¯ ≤ kXk ˜ + k˜ ˜ + IN )X. xρ k ≤ (1 + kΛk)kXk ¡ −1 ˜ kL−1 kksk ≤ 1+kΛk)kYk+λ (L )ksk. It follows from max (23) and the fact that si will converge to Ωsi , we know that ˜¯ ≤ k kY(0)k ˜ kXk + kb βsT + kc , ∀t ≥ T1 , with ka = (1 + a kΛk)k0 , kb = (ka /λ0 ) + 1 and kc = ka (eλ0 T1 /λ0 )βs (T1 ), Hence ∀t ≥ T1 ¯ ¯˜ ¯ (t)1k ≤ k kY(0)k ˜ kX(t)k ≤ kX(t)k + kX +k β +k +c d

a

b sT

c

(24) ¯ ∈ ΩX¯ , We now provide the conditions which guarantee X ∀t ≥ 0. Define the compact set ¯ n ¯ ¯ ¯ ¯ ˜ X(0) Ω0 := ⊂ ΩX¯ , ¯{X|kX(t)k < ka kY(0)k} ª (25) λmax (L−1 )ks(0)k < βsT

the positive constant n ¯n ¯ ¯ ¯ ˜ c∗ := sup c ¯ X|k Xk < ka kY(0)k + kc + c, + c∈R ª ª ¯ (26) X(0) ∈ Ω0 ⊂ ΩX¯

We summarize our results in the following theorem. Theorem 2: Consider a group of helicopters dynamics (1) and the communication graph containing a spanning tree, where the root helicopter can access the desired trajectory, under Assumptions 1–5, the control (13) and parameters update law (14) for each helicopter. For initial conditions ¯ X(0), η(0), θ˜i (0) and ϕ˜i (0) starting in any compact set, and the desired trajectory with its derivations up to ρ-th is bounded, all closed signals of the system are semiglobal uniform bound, and the total tracking error of the helicopters ˜¯ converges to a neighborhood of the origin. X Proof: From (24), we know that the overall system ¯ state X(t) will stay in ΩX¯ for all time. Furthermore, since the NN weights have been proven bounded for any bounded θˆi (0) and ϕˆi (0), and from Lemma 1, it can be seen that ηi is bound if xi is bounded. As a result, the states of the internal dynamics of the helicopter will p converges to the compact set Ωηi = {ηi ∈ Rp | kηi k ≤ a1 ( 2c2 /c1 + kXd k) + a2 }, where a1 = λb ax /λa and a2 = λb aq /λa are positive constants. Because the control signal ui (t) is a function of the weights θˆi and ϕˆi , the states ηi , xi , and the filtered tracking error si , we know that it is also bounded. Therefore, we have all the closed-loop signals are semiglobal uniform bound. This completes the proof.

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7.5

IV. S IMULATION S TUDY In this section, we consider the synchronized altitude tracking of six X-cell 50 helicopters which communication graph shown in Fig. 1. The dynamics of the helicopter can be written as follows [17]

altitude of helicopters [ft]

ζ˙1 = ζ2 p ζ˙2 = a0 + a1 ζ2 + a2 ζ22 + (a3 + a4 ζ4 − a5 + a6 ζ4 )ζ32 ζ˙3 = a7 + a8 ζ3 + (a9 sin ζ4 + a10 )ζ32 + ath

y6 7

(27)

ζ˙4 = ζ5 ζ˙5 = a11 + a12 ζ4 + a13 ζ32 sin ζ4 + a14 ζ5 − K1 u

y5 y4

6.5 6 5.5

y3 yd

5

y2

4.5

y1 4 0

10

15

time [s]

Fig. 2.

Altitude synchronization of all helicopters.

[5.9, 0, 95.4, 0.22, 0]T , ζ4 (0) = [6.2, 0, 95.3567, 0.3, 0]T , ζ5 (0) = [6.8, 0, 95.3567, 0.22, 0]T , and ζ6 (0) = [7.4, 0, 95.4, 0.21, 0]T , and θˆi (0) = 0, ϕˆi (0) = 0, i = 1, . . . , 6. From Fig. 2, we can see that good tracking 300

200

2

0

1

control input u [m rad]

control input u [m rad]

400

x˙ i = xi+1 , i = 1, 2, 3 (28) x˙ 4 = b(x) + g(x)u ³ ´ 6 where g(x) = −K2 ζ32 a4 − 2√a a+a . 5 6 ζ4 The derivation of b(x) in (28) is omitted, and we proceed to verify that the system indeed satisfies the assumptions supported in the control design. It is clear that the system has a strong relative degree ρ = 4. To verify Assumption 3, we first note, from a practical standpoint, that the collective pitch angle, ζ4 , is stricter with a range, typically from 0 to 0.44 rad [13]. It can be verified that the bracketed terms in g(x) are virtually constant: they take values in the range [1.4, 1.5]×10−3 . Hence the control coefficient g(x) is always negative. Together with the fact that rotor speed, ζ3 , is nonzero during flight, it can be concluded that there does not exist any control singularities or zero crossings of g(x). Therefore, the first part of Assumption 3 is satisfied. Remark 3: The second part of the assumption, g(x) > 0, does not correspond to this example, there is no less of applicability of the results. The control is still valid as we only need to satisfy g(x) 6= 0. Lastly, it is not difficult to verify the existence of a function

−200 −400 −600

0

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g0 (x) = 2 (|a8 | + |a9 sin ζ4 + a10 ||ζ3 |+ ¶ a6 (a5 + a6 ζ4 )−1.5 > 0, ∀ζ3 > 0, ζ4 ∈ [0, 0.44] 8 [a4 − 0.5a6 (a5 + a6 ζ4 )−0.5 ] which fulfill Assumption 4 for the case g(x) < 0. Note that this function need not to be known; we only need to show its existence. The control parameters are chosen as Λ = [64, 48, 12]T , ki = 3, while the NN parameters for each helicopter are chosen as σ1 = 0.05, γ1 = 1, σ2 = 0.01, γ2 = 100. The saturation limits of the control are ±400mrad. The initial conditions are ζ1 (0) = [4.3, 0, 95.3567, 0.222, 0]T , ζ2 (0) = [4.8, 0, 95.3567, 0.3, 0]T , ζ3 (0) =

5

control input u6 [m rad]

The physical meaning of the symbols and the nominal values for constants K1 and ai are given in [17]. Let y be the altitude ζ1 . By restricting the throttle input to be constant, we obtain a SISO in which u is the only input variable forcing the output y to track a desired trajectory yd , which is generated by yd = 150.056 = s4 +12.6s3 +64.19s2 +154.35s+150.056 href , where href (t) 5.5 − 0.5 sin t. It can be shown that the system has strong relative degree 4, with the x system given by:

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(e) Control input of helicopter u5 . Fig. 3.

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(f) Control input of helicopter u6 .

Control input of helicopters under the proposed control.

performance is achieved. The initial errors of all helicopters are sufficiently reduced and the altitude trajectories lie in close proximity of the desired sinusoidal trajectory. Meanwhile, the internal dynamics and the NN weights are all bounded, as shown in Fig. 5 and Fig. 4. From Fig. 3, it can be seen that the control inputs of the helicopters are bounded as well.

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Fig. 4.

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(f) Norm of neural weights, kθˆ6 k.

Norm of neural weights under the proposed control.

R EFERENCES [1] J. A. Fax, R. M. Murray, N. G. E. Syst, and C. A. Woodland Hills, “Information flow and cooperative control of vehicle formations,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465– 1476, 2004. [2] C. H. Fua, S. S. Ge, K. D. Do, and K. W. Lim, “Multirobot formations based on the Queue-formation scheme with limited communication,” IEEE Transactions on Robotics, vol. 23, no. 6, pp. 1160–1169, 2007. [3] S. S. Ge and C. H. Fua, “Queues and artificial potential trenches for multirobot formations,” IEEE Transactions on Robotics, vol. 21, no. 4, pp. 646–656, 2004. [4] T. Balch and R. C. Arkin, “Behavior-based formation control for multirobot teams,” IEEE Transactions on Robotics and Automation, vol. 14, no. 6, pp. 926–939, 1998. [5] M. A. Lewis and K. H. Tan, “High precision formation control of mobile robots using virtual structures,” Autonomous Robots, vol. 4, no. 4, pp. 387–403, 1997.

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time [s]

10

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(c) Internal state of helicopter 3.

(d) Internal state of helicopter 4.

0.1

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0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25

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(b) Internal state of helicopter 2.

0.01

time [s]

10 time [s]

(a) Internal state of helicopter 1. 0

time [s]

(e) Norm of neural weights, kθˆ5 k.

0.6

−0.2

internal state response [rad/s]

norm of NN weights of helicopter 5

(c) Norm of neural weights, kθˆ3 k. (d) Norm of neural weights, kθˆ4 k.

0.1

0.8

time [s] internal state response [rad/s]

norm of NN weights of helicopter 3

(a) Norm of neural weights, kθˆ1 k. (b) Norm of neural weights, kθˆ2 k.

internal state response [rad/s]

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800

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internal state response [rad/s]

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norm of NN weights of helicopter 2

norm of NN weights of helicopter 1

In this paper, we have studied the synchronized tracking problem of multiple helicopters in vertical flight mode. With the condition that the Laplacian matrix of the extended formation graph is positive definite, by using the weighted average of neighbors’ states as its reference signal, the adaptive NN tracking control law has been designed for each helicopter. It has been shown that the tracking error of each helicopter converges to an adjustable neighborhood of origin. Simulation results have been obtained to show the effectiveness of the proposed methods.

[6] R. Cui, D. Xu, and W. Yan, “Formation Control of Autonomous Underwater Vehicles under Fixed Topology,” in IEEE International Conference on Control and Automation, 2007, pp. 2913–2918. [7] R. Cui, S. S. Ge, V. E. B. How, and Y. S. Choo, “Leader-follower formation control of underactuated auvs with leader position measurement,” in Proc. of 2009 IEEE International Conference on Robotics and Automation, vol. 1, Kobe, Japan, 2009, pp. 979–984. [8] K. P. Tee, S. S. Ge, and F. E. H. Tay, “Adaptive neural network control for helicopters in vertical flight,” IEEE Transactions on Control Systems Technology, vol. 16, no. 4, pp. 753–762, 2008. [9] R. Enns and J. Si, “Helicopter trimming and tracking control using direct neural dynamic programming,” IEEE Transactions on Neural Networks, vol. 14, no. 4, pp. 929–939, 2003. [10] S. S. Ge, B. Ren, K. P. Tee, and T. H. Lee, “Approximation based control of uncertain helicopter dynamics,” IET Control Theory & Applications, vol. 3, no. 9, pp. 941–956, 2009. [11] L. Marconi and R. Naldi, “Robust full degree-of-freedom tracking control of a helicopter,” Automatica, vol. 43, no. 11, pp. 1909–1920, 2007. [12] M. Saffarian and F. Fahimi, “Non-Iterative nonlinear model predictive approach applied to the control of helicopters’ group formation,” Robotics and Autonomous system, vol. 57, no. 6–7, pp. 749–757, 2009. [13] R. W. Prouty, Helicopter performance, stability, and control. Malabar, FL: Robert E. Krieger, 1990. [14] P. N. Shivakumar and K. H. Chew, “A sufficient condition for nonvanishing of determinants,” Proceedings of the American Mathematical Society, pp. 63–66, 1974. [15] S. S. Ge and J. Zhang, “Neural network control of nonaffine nonlinear pure-feedback system with zero dynamics by state and output feedback,” IEEE Transactions on Neural Networks, vol. 14, no. 4, pp. 900–908, 2003. [16] S. S. Ge, C. C. Hang, T. T. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Boston, MA: Kluwer Academic, 2002. [17] J. Kaloust, C. Ham, Z. Qu, L. Syst, and T. Dallas, “Nonlinear autopilot control design for a 2-DOF helicopter model,” IEE ProceedingsControl Theory and Applications, vol. 144, no. 6, pp. 612–616, 1997.

internal state response [rad/s]

V. C ONCLUSION

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(e) Internal state of helicopter 5. Fig. 5.

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(f) Internal state of helicopter 6.

Internal state response under the proposed control.

15