# Design and Experimental Verification of Robust Motion Synchronization Control with Integral Action

Authors: Chen Peng; Bo Zhu; Keqi Wei
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

A robust attitude motion synchronization problem is investigated for multiple 3-degrees-of-freedom (3-DOF) helicopters with input disturbances. The communication topology among the helicopters is modeled by a directed graph, and each helicopter can only access the angular position measurements of itself and its neighbors. The desired trajectories are generated online and not accessible to all helicopters. The problem is solved by embedding in each helicopter some finite-time convergent (FTC) estimators and a distributed controller with integral action. The FTC estimators generate the estimates of desired angular acceleration and the derivative of the local neighborhood synchronization errors. The distributed controller stabilizes the tracking errors and attenuates the effects of input disturbances. The conditions under which the tracking error of each helicopter converges asymptotically to zero are identified, and, for the cases with nonzero tracking errors, some inequalities are derived to show the relationship between the ultimate bounds of tracking errors and the design parameters. Simulation and experimental results are presented to demonstrate the performance of the controllers.


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## Body

## 1. Introduction

In the field of multivehicle cooperative control, robust consensus tracking under model uncertainties and exogenous disturbances has received increasing attention in recent years, where the output (or state) of each vehicle is required to robustly track a common, desired trajectory. For instance, switching controllers or sliding-mode controllers were designed in works [1–3] to reject input disturbances; a sliding-mode disturbance observer is combined with a consensus tracking algorithm in recent work [4] to improve the robustness and control accuracy of a multimotor system. Alternative robust control approaches include the ones based on uncertainty and disturbance estimators as in works [5–7], adaptive control approach [8], and the output-regulation approach [9].In practice, integral control (IC) is widely used to attenuate disturbances in various (single) vehicle systems. This mainly owes to its structural simplicity and the well-known performance property that IC can asymptotically reject constant input disturbances. Noting these facts, many researchers begin to study IC-based robust control schemes for multivehicle systems (MVSs) as in [10–12]. In particular, the recent work [10] shows that PI controllers successfully attenuate constant disturbances in the network of multiple single-integrator dynamics or the network of multiple double-integrator dynamics.Another practical issue encountered in many control systems is the lack of sensors. As a result, state observers are often needed to generate the estimates of some necessary states. State observers can be roughly classified into two types: model-dependent ones and model-independent ones. As two representative model-dependent observers, Luenberger observer and Kalman filter suffer from the limitation that the estimation accuracy cannot be guaranteed when the system model suffers from severe uncertainties. To deal with this problem, many model-independent observers are proposed. For instance, some higher-order sliding modes (HOSM) observers (differentiators) were designed in work [13, 14] to ensure finite-time convergence even in the presence of input disturbances.The main objective of this paper is to use integral control to improve the robustness of a distributed control algorithm for consensus tracking without velocity measurements. The effectiveness of the approach is proved by showing that (a) the resulting tracking errors are ultimately bounded for any input disturbances satisfying a simple Lipschitz-constant condition, and (b) zero-error asymptotic tracking is achieved for a constant input disturbance. The key technical differences between this paper and work [10] are summarized as follows:(1)
The paper [10] considers the consensus problem without a common reference. In contrast, this paper considers a consensus tracking problem, where a common desired trajectory exists (and is supposed to be accessible only to partial vehicles in the group). In [10], velocity signals were used in the control design for second-order systems. We here assume that neither the velocity of leader nor the velocity of neighboring vehicles is accessible for control design.(2)
Concerning the robustness improvement owing to integral control, the discussion in work [10] is restricted to the rather special case with constant input disturbances. This is not the case in this paper. Actually, we will use the concepts of input-to-state stability to study the general cases where the disturbances are nonconstants and are not completely rejected.(3)
In work [10], the controller performance is verified by numerical simulation. In this paper, both numerical simulation results and experimental results on three 3-DOF helicopters are presented, to demonstrate the performance improvement owing to the use of integral control.The experimental platform of “three 3-DOF helicopters” used in this paper is shown in Figure1. The single laboratory 3-DOF helicopter with a so-called active disturbance system (ADS) is the same as that in [5] and is shown in Figure 2. This experimental apparatus was developed by Quanser Consulting Inc. for the purpose of control education and research [15–21].Figure 1
Experimental platform of “three 3-DOF helicopters” (see [15]).Figure 2
3-DOF helicopter with ADS (see [5]).The rest of this paper is organized as follows. In Section2, some preliminary knowledge is described and the problem is formulated. The robust distributed consensus tracking controllers are designed in Section 3. Numerical simulation and experimental results are presented in Section 4. Some concluding remarks are drawn in Section 5.


## 2. Preliminaries

### 2.1. Notation and Graph Theory

For matrixM,M-1 denotes its inversion and rankM denotes its rank. In∈Rn×n refers to the identity matrix. For vector X∈Rn×1 and matrix M=mij∈Rm×n, X2=∑i=1n‍xi21/2, X∞=max1≤i≤n⁡xi, M∞=max1≤i≤m⁡∑j=1n‍mij, and M2=λmaxMTM1/2, where λmax(·)=maxi⁡λi with λi being eigenvalues. 0n refers to n-dimensional column vector with all elements being 0. a denotes absolute value (modulus) of real number a.The notations related toith 3-DOF helicopter (see Figure 2), i∈I={1,…,n}, are listed in Notations.The communication networks ofn helicopters can be modeled by directed graph G(V,E,A), where V={υ1,υ2,…,υn}, E⊂V×V, and nonnegative matrix A=aij∈Rn×n denote the set of nodes, the set of edges, and the weighted adjacency matrix of G, respectively. Node υi (i=1,…,n) represents ith helicopter, and an edge (υj,υi)∈E denotes that ith helicopter can obtain information from jth helicopter; that is, υj is a neighbor of υi. Use Ni to denote all the neighbors of υi, and Ni⊂V. A directed path from υi to υj is a sequence of ordered edges of form (υi,υi1),(υi1,υi2),…,(υil,υj), with distinct nodes υik,k=1,2,…,l, and υj is said to be reachable from υi. A node is called the root node and G is said to have a directed spanning tree, if all the other nodes are reachable from this node.The elements of the weighted adjacency matrixA satisfy aii=0 and aij>0 (i,j=1,…,n and j≠i) if and only if (υj,υi)∈E. The Laplacian matrix of G is denoted by L=lij∈Rn×n, where lij=-aij, if i≠j, and lii=∑j∈Niaij. Note that the desired trajectory information is not accessible to all helicopters in this paper. Use constant matrix B-=diag(b1,…,bn)∈Rn×n, which is defined as bi>0 if ith helicopter can access the desired trajectory information and otherwise bi=0, to describe this fact by allowing bi=0 for some i∈I.


### 2.2. Problem Formulation

The elevation and pitch motions ofith helicopter can be modeled as follows (see [5] or [21]): (1)Jeiα¨it=Kfilaicos⁡βitVsit-miglaicos⁡αit+fαit,i∈I,Jpiβ¨it=KfilhiVdit+fβit,i∈I,where(2)Vsit=Vfit+Vbit,i∈I,Vdit=Vfit-Vbit,i∈I.The pitch angle is limited to within (-π/2,π/2) mechanically.We made the following assumption onfαit and fβit.Assumption 1.
For eachi∈I, the first-order derivatives of fαit and fβit with respect to t exist and are piecewise continuous and bounded for all t≥0.As shown in recent work [5], by applying the standard normalization and feedback linearization technique, the above nonlinear helicopter model can be reduced to the following 2-dimensional, disturbed double integrators:(3)y¨i=ui+di,i∈I,where yi=(αit,βit)T∈R2, ui=uαi,uβiT∈R2, and the new disturbances di=dαi,dβiT∈R2.Under Assumption1, d˙i=(d˙αi, d˙βi)T also exist and are bounded. Define(4)d-d=d-dα,d-dβT=max1≤i≤nsupt≥0⁡d˙αit,max1≤i≤nsupt≥0⁡d˙βitT,where d-dα and d-dβ are positive scalars.For eachi∈I, we define yi=(αi,βi)T∈R2 and use yd=(αd,βd)T∈R2 to denote the desired attitude trajectory for yi, which may be time-varying but are second-order differentiable with respect to t. Then, the objective of this paper is to design ui for (3) to achieve robust attitude synchronization; that is, yit→ydt for each i∈I as t→+∞.


## 2.1. Notation and Graph Theory

For matrixM,M-1 denotes its inversion and rankM denotes its rank. In∈Rn×n refers to the identity matrix. For vector X∈Rn×1 and matrix M=mij∈Rm×n, X2=∑i=1n‍xi21/2, X∞=max1≤i≤n⁡xi, M∞=max1≤i≤m⁡∑j=1n‍mij, and M2=λmaxMTM1/2, where λmax(·)=maxi⁡λi with λi being eigenvalues. 0n refers to n-dimensional column vector with all elements being 0. a denotes absolute value (modulus) of real number a.The notations related toith 3-DOF helicopter (see Figure 2), i∈I={1,…,n}, are listed in Notations.The communication networks ofn helicopters can be modeled by directed graph G(V,E,A), where V={υ1,υ2,…,υn}, E⊂V×V, and nonnegative matrix A=aij∈Rn×n denote the set of nodes, the set of edges, and the weighted adjacency matrix of G, respectively. Node υi (i=1,…,n) represents ith helicopter, and an edge (υj,υi)∈E denotes that ith helicopter can obtain information from jth helicopter; that is, υj is a neighbor of υi. Use Ni to denote all the neighbors of υi, and Ni⊂V. A directed path from υi to υj is a sequence of ordered edges of form (υi,υi1),(υi1,υi2),…,(υil,υj), with distinct nodes υik,k=1,2,…,l, and υj is said to be reachable from υi. A node is called the root node and G is said to have a directed spanning tree, if all the other nodes are reachable from this node.The elements of the weighted adjacency matrixA satisfy aii=0 and aij>0 (i,j=1,…,n and j≠i) if and only if (υj,υi)∈E. The Laplacian matrix of G is denoted by L=lij∈Rn×n, where lij=-aij, if i≠j, and lii=∑j∈Niaij. Note that the desired trajectory information is not accessible to all helicopters in this paper. Use constant matrix B-=diag(b1,…,bn)∈Rn×n, which is defined as bi>0 if ith helicopter can access the desired trajectory information and otherwise bi=0, to describe this fact by allowing bi=0 for some i∈I.


## 2.2. Problem Formulation

The elevation and pitch motions ofith helicopter can be modeled as follows (see [5] or [21]): (1)Jeiα¨it=Kfilaicos⁡βitVsit-miglaicos⁡αit+fαit,i∈I,Jpiβ¨it=KfilhiVdit+fβit,i∈I,where(2)Vsit=Vfit+Vbit,i∈I,Vdit=Vfit-Vbit,i∈I.The pitch angle is limited to within (-π/2,π/2) mechanically.We made the following assumption onfαit and fβit.Assumption 1.
For eachi∈I, the first-order derivatives of fαit and fβit with respect to t exist and are piecewise continuous and bounded for all t≥0.As shown in recent work [5], by applying the standard normalization and feedback linearization technique, the above nonlinear helicopter model can be reduced to the following 2-dimensional, disturbed double integrators:(3)y¨i=ui+di,i∈I,where yi=(αit,βit)T∈R2, ui=uαi,uβiT∈R2, and the new disturbances di=dαi,dβiT∈R2.Under Assumption1, d˙i=(d˙αi, d˙βi)T also exist and are bounded. Define(4)d-d=d-dα,d-dβT=max1≤i≤nsupt≥0⁡d˙αit,max1≤i≤nsupt≥0⁡d˙βitT,where d-dα and d-dβ are positive scalars.For eachi∈I, we define yi=(αi,βi)T∈R2 and use yd=(αd,βd)T∈R2 to denote the desired attitude trajectory for yi, which may be time-varying but are second-order differentiable with respect to t. Then, the objective of this paper is to design ui for (3) to achieve robust attitude synchronization; that is, yit→ydt for each i∈I as t→+∞.


## 3. Design of Distributed Controllers without Velocity Measurements

### 3.1. Controller Design

Noting thatuαi and uβi for (3) can be designed in the same way, we thus introduce new unified variable ρ∈{α,β} and define(5a)ρ~i=ρi-ρd,i∈I,(5b)ρ~=ρ~1,…,ρ~nT∈Rn,(5c)ρ~ij=ρi-ρj,i,j∈I,(5d)eρi=biρ~i+∑j∈Niaijρ~ij.In [8, 22], error eρi is called local neighborhood synchronization error (LNSE).For the considered helicopter, only angular position sensors (encoders) are equipped. Besides, we assume that the desired velocity is not accessible to any helicopter. To deal with this problem, we consider the following distributed controllers:(6)uρit=1ki∑j∈Niaijuρjt+biρ¨^dt-kρiPeρit-kρiDe˙^ρit-kρiI∫0teρiτdτ,where kρiP, kρiI, and kρiD are positive control gains, ki=bi+∑j∈Niaij, ρ¨^dt, and e˙^ρi(t) denote the estimates of ρ¨dt and e˙ρi(t), respectively. We construct the following systems to generate ρ¨^dt and e˙^ρi(t). Specifically, the third-order FTC estimators(7)ρ^˙d=-λ2L-ρd1/3ρ^d-ρd2/3sign⁡ρ^d-ρd-μ2ρ^d-ρd+ρ˙^d,ρ˙^˙d=-λ1L-ρd1/2ρ˙^d-ρ^˙d1/2sign⁡ρ˙^d-ρ^˙d-μ1ρ˙^d-ρ^˙d+ρ¨^d,ρ¨^˙d=-λ0L-ρdsign⁡ρ¨^d-ρ˙^˙d-μ0ρ¨^d-ρ˙^˙d,are for ρ¨^dt, and the following second-order FTC estimators are for e˙^(t):(8)e^˙ρi=-λ1L-e1/2e^ρi-eρi1/2sign⁡e^ρi-eρi-μ1e^ρi-eρi+e˙^ρi,e˙^˙ρi=-λ0L-esign⁡e˙^ρi-e^˙ρi-μ0e˙^ρi-e^˙ρi,i∈I,where ρ^d, ρ˙^d, and e^ρi are the estimates of ρd, ρ˙d, and eρi, respectively; ρd and eρi are consisting of a locally bounded Lebesgue-measurable noise with unknown features and an unknown base signal ρd0(t) and eρi0 whose second and first derivative have an known Lipschitz constant L-ρd>0 and L-e>0, respectively; λi and μi>0,i=0,1,2, are the design parameters of the FTC estimators.This estimation scheme is illustrated in Figure3.Figure 3
The way to obtainρ¨^dt and e˙^(t).We have the following result for the above two FTC estimators [14, Theorem  1].Lemma 2.
Letλi and μi in (7) and (8) be recursively chosen as in Theorem 1 of [13, 14]. Then, the estimation errors achieve zero in the absence of input noises after finite time tc of transient process. The convergence of the estimation errors is uniform in the sense that the convergence time is uniformly bounded by finite time tc which is a locally bounded function of initial estimation errors.Remark 3.
According to Lemma2, in the absence of input noises, after finite time tci(i∈I), the following equalities hold:(9)ρ˙^d-ρ˙d=0,ρ¨^d-ρ¨d=0,e˙ρi-e˙^ρi=0,i∈I.Remark 4.
Note that the neighbors’ control signalsuρj(t)(j∈Ni) are used in (6) and then a possible implementation loop issue arises in practical applications. As discussed in work [5], if the sample frequency is high enough, this problem can be resolved with using the neighbors’ control signals obtained during the previous sampling period, that is, uρjt-τ, where τ denotes the fixed sampling step and τ=0.001sec in the following numerical simulations and experiments.


### 3.2. Analysis of the Closed-Loop Stability

Before the closed-loop stability analysis, we need to establish and analyse the relationship between tracking errorsρ~i and LNSEs eρi,i∈I. Define(10)Eρ=eρ1,…,eρnT∈Rn,ρ∈α,β.From (5c), ρ~ij=ρi-ρj=ρ~i-ρ~j; then from (5d), eρi=biρ~i+∑j∈Niaijρ~i-ρ~j for each ρ∈{α,β}. Then, with L and B- as defined in Section 2.1, the following relationship equation between the tracking errors and LNSEs is derived:(11)Eρt=L+B-ρ~t,ρ∈α,β,where rank(L+B-) is determined by the communication topology which satisfies the following condition in this paper.Assumption 5.
The desired trajectory informationρd,ρ∈α,β, has directed paths to all helicopters.According to [23, Lemma  1.6], all eigenvalues of L+B- have positive real parts under Assumption 5. Then, the following lemma is obtained directly.Lemma 6 (see [23, Lemma  1.6]).
Under Assumption5, matrix L+B- is full rank, that is, rank(L+B-)=n.Thus, the relationship equation (11) implies that, under Assumption 5, the objective ρ~t→0n and ρ~˙t→0n as t→+∞, ρ∈α,β, can be achieved by driving Eρt and E˙ρt to zero as t→+∞, respectively.Applying (6) to (3) gives(12)kiρ¨it=∑j∈Niaijρ¨jt-dρjtt+biρ¨^dt-kρiPeρit-kρiDe˙^ρit-kρiI∫0teρiτdτ+kidρi.With (5d) and (12), we can further get(13)e⃛ρi+kρiDe¨ρi+kρiPe˙ρi+kρiIeρi=δ˙ρi,ρ∈α,β,i∈I,where δρi=bi(ρ¨^d-ρ¨d)+kρiD(e˙ρi-e˙^ρi)+Δρi and Δρi is defined as(14)Δρi=kidρi-∑j∈Niaijdρj.Define(15)δρ=δρ1,…,δρnT∈Rn,ρ∈α,β,(16)Δρ=Δρ1,…,ΔρnT∈Rn,ρ∈α,β,(17)dρ=dρ1,…,dρnT∈Rn,ρ∈α,β,(18)t-c=tc1,…,tcnT∈Rn.Then, the following result is clear based on Lemma 2.Lemma 7.
There existst-cmax=t-c∞ such that, in the absence of input noises,(19)δρ=Δρ,∀t≥t-cmax,ρ∈α,β.Therefore, after finite timet-cmax, (13) is the same as follows:(20)e⃛ρi+kρiDe¨ρi+kρiPe˙ρi+kρiIeρi=Δ˙ρi,ρ∈α,β,i∈I.Define(21)Aρi=010001-kρiI-kρiP-kρiD∈R3×3,ρ∈α,β,i∈I,(22)Bρi=001T∈R3×1,ρ∈α,β,i∈I.Then, (20) can be written as(23)e˙ρi,e¨ρi,e⃛ρiT=Aρieρi,e˙ρi,e¨ρiT+BρiΔ˙ρi,ρ∈α,β,i∈I.Furthermore, based on Lemma 7, by applying Routh-Hurwitz stability criterion and Lemma A.1 in Appendix to forced systems (13) with δ˙ρi as inputs, the following theorem is obtained.Theorem 8.
Consider systems (13) under Assumptions 1 and 5 and the following parameter condition:(24)kρiD,kρiP,kρiI>0,kρiDkρiP>kρiI.Then, for each ρ∈α,β, in the absence of noise,(1)
the system trajectoriesEρ, E˙ρ, ρ~, and ρ~˙ are globally bounded;(2)
Eρ→0, E˙ρ→0, ρ~→0, and ρ~˙→0 as t→+∞ if d˙ρt→0 as t→+∞;(3)
for the general cases with nonvanishingd˙ρt, there exists t-ρ>t-cmax such that, under any initial condition,(25)Eρt2≤bρ,∀t≥t-ρ,(26)ρ~t2≤L+B--12bρ,∀t≥t-ρ,where(27)bρ=2nL+B-∞d-dρθmax1≤i≤n⁡λmaxPρiλmaxPρiλminPρi,and 0<θ<1, Pρi are the solutions of the Lyapunov equations(28)PρiAρi+AρiTPρi=-I3,ρ∈α,β,with Aρi defined by (21).Proof.
Using (14), (16), and (17), the relationships between Δ˙ρi and d˙ρi can be written in a compact form as(29)Δ˙ρ=L+B-d˙ρ,ρ∈α,β.Then, with (4), it follows that(30)Δ˙ρt∞≤L+B-∞d˙ρt∞≤L+B-∞d-dρ,∀t≥0,ρ∈α,β.Moreover, for ρ∈α,β, based on Lemma 6, under Assumption 5, Δ˙ρ→0n as t→+∞ if and only if d˙ρ→0n as t→+∞. Thus, under Assumptions 1 and 5 and parameter conditions (24), it is straightforward to derive the points (1)-(2) with these results and Lemmas 6 and 7.
The detailed proof of point (3) is the same as [5, Theorem 2] and hence omitted.Remark 9.
As shown in Theorem8, by FTC differentiators (7) and (8) in the absence of input noises, for the case without angular velocity measurements, if condition limt→+∞d˙ρi(t)=0 is satisfied, the distributed controllers (6) are capable of achieving zero-error attitude-trajectory tracking for each helicopter in the group of helicopters. Thus, the distributed consensus attitude-trajectory zero-error tracking for the group of helicopters is achieved under this condition (limt→+∞d˙ρi(t)=0). If this condition is not satisfied, the ultimate bounds of attitude-trajectory tracking errors ρ~i resulting from controllers (6) are globally bounded for i∈I and ultimately bounded with ultimate bounds depending on parameters kρiP,kρiD,kρiI which can make the term max1≤i≤n⁡λmaxPρiλmaxPρi/λminPρi small enough. Future efforts will be devoted to the optimization problem of the three parameters such that the term is minimized.Remark 10.
If delayτ in neighbors’ control inputs uρjt-τ is systematically considered, the obtained error equations (13) are disturbed neutral delay systems as in [24]. For these kinds of delayed systems, [25, Lemma  1] can be used to examine the stability of the associated nominal delayed systems, where three easily testable conditions are included. We also note that these conditions ensure that the considered systems are delay-independent stable. However, the delay-independent stability is not the general case. Despite this fact, since the case with τ=0 is exponentially stable, we readily derive that the system with an enough small time-delay is also exponentially stable, which ensure that the correspondingly disturbed equations are input-to-state stable provided that the delay is sufficiently small. In our future work, we will investigate these problems and systematically assess the effect of the introduced delay on the system stability as well as the convergence speed of tracking errors (as in the work [26]).


## 3.1. Controller Design

Noting thatuαi and uβi for (3) can be designed in the same way, we thus introduce new unified variable ρ∈{α,β} and define(5a)ρ~i=ρi-ρd,i∈I,(5b)ρ~=ρ~1,…,ρ~nT∈Rn,(5c)ρ~ij=ρi-ρj,i,j∈I,(5d)eρi=biρ~i+∑j∈Niaijρ~ij.In [8, 22], error eρi is called local neighborhood synchronization error (LNSE).For the considered helicopter, only angular position sensors (encoders) are equipped. Besides, we assume that the desired velocity is not accessible to any helicopter. To deal with this problem, we consider the following distributed controllers:(6)uρit=1ki∑j∈Niaijuρjt+biρ¨^dt-kρiPeρit-kρiDe˙^ρit-kρiI∫0teρiτdτ,where kρiP, kρiI, and kρiD are positive control gains, ki=bi+∑j∈Niaij, ρ¨^dt, and e˙^ρi(t) denote the estimates of ρ¨dt and e˙ρi(t), respectively. We construct the following systems to generate ρ¨^dt and e˙^ρi(t). Specifically, the third-order FTC estimators(7)ρ^˙d=-λ2L-ρd1/3ρ^d-ρd2/3sign⁡ρ^d-ρd-μ2ρ^d-ρd+ρ˙^d,ρ˙^˙d=-λ1L-ρd1/2ρ˙^d-ρ^˙d1/2sign⁡ρ˙^d-ρ^˙d-μ1ρ˙^d-ρ^˙d+ρ¨^d,ρ¨^˙d=-λ0L-ρdsign⁡ρ¨^d-ρ˙^˙d-μ0ρ¨^d-ρ˙^˙d,are for ρ¨^dt, and the following second-order FTC estimators are for e˙^(t):(8)e^˙ρi=-λ1L-e1/2e^ρi-eρi1/2sign⁡e^ρi-eρi-μ1e^ρi-eρi+e˙^ρi,e˙^˙ρi=-λ0L-esign⁡e˙^ρi-e^˙ρi-μ0e˙^ρi-e^˙ρi,i∈I,where ρ^d, ρ˙^d, and e^ρi are the estimates of ρd, ρ˙d, and eρi, respectively; ρd and eρi are consisting of a locally bounded Lebesgue-measurable noise with unknown features and an unknown base signal ρd0(t) and eρi0 whose second and first derivative have an known Lipschitz constant L-ρd>0 and L-e>0, respectively; λi and μi>0,i=0,1,2, are the design parameters of the FTC estimators.This estimation scheme is illustrated in Figure3.Figure 3
The way to obtainρ¨^dt and e˙^(t).We have the following result for the above two FTC estimators [14, Theorem  1].Lemma 2.
Letλi and μi in (7) and (8) be recursively chosen as in Theorem 1 of [13, 14]. Then, the estimation errors achieve zero in the absence of input noises after finite time tc of transient process. The convergence of the estimation errors is uniform in the sense that the convergence time is uniformly bounded by finite time tc which is a locally bounded function of initial estimation errors.Remark 3.
According to Lemma2, in the absence of input noises, after finite time tci(i∈I), the following equalities hold:(9)ρ˙^d-ρ˙d=0,ρ¨^d-ρ¨d=0,e˙ρi-e˙^ρi=0,i∈I.Remark 4.
Note that the neighbors’ control signalsuρj(t)(j∈Ni) are used in (6) and then a possible implementation loop issue arises in practical applications. As discussed in work [5], if the sample frequency is high enough, this problem can be resolved with using the neighbors’ control signals obtained during the previous sampling period, that is, uρjt-τ, where τ denotes the fixed sampling step and τ=0.001sec in the following numerical simulations and experiments.


## 3.2. Analysis of the Closed-Loop Stability

Before the closed-loop stability analysis, we need to establish and analyse the relationship between tracking errorsρ~i and LNSEs eρi,i∈I. Define(10)Eρ=eρ1,…,eρnT∈Rn,ρ∈α,β.From (5c), ρ~ij=ρi-ρj=ρ~i-ρ~j; then from (5d), eρi=biρ~i+∑j∈Niaijρ~i-ρ~j for each ρ∈{α,β}. Then, with L and B- as defined in Section 2.1, the following relationship equation between the tracking errors and LNSEs is derived:(11)Eρt=L+B-ρ~t,ρ∈α,β,where rank(L+B-) is determined by the communication topology which satisfies the following condition in this paper.Assumption 5.
The desired trajectory informationρd,ρ∈α,β, has directed paths to all helicopters.According to [23, Lemma  1.6], all eigenvalues of L+B- have positive real parts under Assumption 5. Then, the following lemma is obtained directly.Lemma 6 (see [23, Lemma  1.6]).
Under Assumption5, matrix L+B- is full rank, that is, rank(L+B-)=n.Thus, the relationship equation (11) implies that, under Assumption 5, the objective ρ~t→0n and ρ~˙t→0n as t→+∞, ρ∈α,β, can be achieved by driving Eρt and E˙ρt to zero as t→+∞, respectively.Applying (6) to (3) gives(12)kiρ¨it=∑j∈Niaijρ¨jt-dρjtt+biρ¨^dt-kρiPeρit-kρiDe˙^ρit-kρiI∫0teρiτdτ+kidρi.With (5d) and (12), we can further get(13)e⃛ρi+kρiDe¨ρi+kρiPe˙ρi+kρiIeρi=δ˙ρi,ρ∈α,β,i∈I,where δρi=bi(ρ¨^d-ρ¨d)+kρiD(e˙ρi-e˙^ρi)+Δρi and Δρi is defined as(14)Δρi=kidρi-∑j∈Niaijdρj.Define(15)δρ=δρ1,…,δρnT∈Rn,ρ∈α,β,(16)Δρ=Δρ1,…,ΔρnT∈Rn,ρ∈α,β,(17)dρ=dρ1,…,dρnT∈Rn,ρ∈α,β,(18)t-c=tc1,…,tcnT∈Rn.Then, the following result is clear based on Lemma 2.Lemma 7.
There existst-cmax=t-c∞ such that, in the absence of input noises,(19)δρ=Δρ,∀t≥t-cmax,ρ∈α,β.Therefore, after finite timet-cmax, (13) is the same as follows:(20)e⃛ρi+kρiDe¨ρi+kρiPe˙ρi+kρiIeρi=Δ˙ρi,ρ∈α,β,i∈I.Define(21)Aρi=010001-kρiI-kρiP-kρiD∈R3×3,ρ∈α,β,i∈I,(22)Bρi=001T∈R3×1,ρ∈α,β,i∈I.Then, (20) can be written as(23)e˙ρi,e¨ρi,e⃛ρiT=Aρieρi,e˙ρi,e¨ρiT+BρiΔ˙ρi,ρ∈α,β,i∈I.Furthermore, based on Lemma 7, by applying Routh-Hurwitz stability criterion and Lemma A.1 in Appendix to forced systems (13) with δ˙ρi as inputs, the following theorem is obtained.Theorem 8.
Consider systems (13) under Assumptions 1 and 5 and the following parameter condition:(24)kρiD,kρiP,kρiI>0,kρiDkρiP>kρiI.Then, for each ρ∈α,β, in the absence of noise,(1)
the system trajectoriesEρ, E˙ρ, ρ~, and ρ~˙ are globally bounded;(2)
Eρ→0, E˙ρ→0, ρ~→0, and ρ~˙→0 as t→+∞ if d˙ρt→0 as t→+∞;(3)
for the general cases with nonvanishingd˙ρt, there exists t-ρ>t-cmax such that, under any initial condition,(25)Eρt2≤bρ,∀t≥t-ρ,(26)ρ~t2≤L+B--12bρ,∀t≥t-ρ,where(27)bρ=2nL+B-∞d-dρθmax1≤i≤n⁡λmaxPρiλmaxPρiλminPρi,and 0<θ<1, Pρi are the solutions of the Lyapunov equations(28)PρiAρi+AρiTPρi=-I3,ρ∈α,β,with Aρi defined by (21).Proof.
Using (14), (16), and (17), the relationships between Δ˙ρi and d˙ρi can be written in a compact form as(29)Δ˙ρ=L+B-d˙ρ,ρ∈α,β.Then, with (4), it follows that(30)Δ˙ρt∞≤L+B-∞d˙ρt∞≤L+B-∞d-dρ,∀t≥0,ρ∈α,β.Moreover, for ρ∈α,β, based on Lemma 6, under Assumption 5, Δ˙ρ→0n as t→+∞ if and only if d˙ρ→0n as t→+∞. Thus, under Assumptions 1 and 5 and parameter conditions (24), it is straightforward to derive the points (1)-(2) with these results and Lemmas 6 and 7.
The detailed proof of point (3) is the same as [5, Theorem 2] and hence omitted.Remark 9.
As shown in Theorem8, by FTC differentiators (7) and (8) in the absence of input noises, for the case without angular velocity measurements, if condition limt→+∞d˙ρi(t)=0 is satisfied, the distributed controllers (6) are capable of achieving zero-error attitude-trajectory tracking for each helicopter in the group of helicopters. Thus, the distributed consensus attitude-trajectory zero-error tracking for the group of helicopters is achieved under this condition (limt→+∞d˙ρi(t)=0). If this condition is not satisfied, the ultimate bounds of attitude-trajectory tracking errors ρ~i resulting from controllers (6) are globally bounded for i∈I and ultimately bounded with ultimate bounds depending on parameters kρiP,kρiD,kρiI which can make the term max1≤i≤n⁡λmaxPρiλmaxPρi/λminPρi small enough. Future efforts will be devoted to the optimization problem of the three parameters such that the term is minimized.Remark 10.
If delayτ in neighbors’ control inputs uρjt-τ is systematically considered, the obtained error equations (13) are disturbed neutral delay systems as in [24]. For these kinds of delayed systems, [25, Lemma  1] can be used to examine the stability of the associated nominal delayed systems, where three easily testable conditions are included. We also note that these conditions ensure that the considered systems are delay-independent stable. However, the delay-independent stability is not the general case. Despite this fact, since the case with τ=0 is exponentially stable, we readily derive that the system with an enough small time-delay is also exponentially stable, which ensure that the correspondingly disturbed equations are input-to-state stable provided that the delay is sufficiently small. In our future work, we will investigate these problems and systematically assess the effect of the introduced delay on the system stability as well as the convergence speed of tracking errors (as in the work [26]).


## 4. Numerical Simulation and Experimental Results

In this section, to demonstrate the effectiveness of the proposed scheme for robust motion synchronization, the numerical simulation results and experimental results of the attitude-trajectory consensus tracking of three 3-DOF helicopters which are labeled as H1 to H3 are presented. Note that we focus on the analysis of the results of elevation-trajectory consensus tracking hereafter. The results of pitch channel are similar to that of elevation channel and hence omitted.The nominal parameters of the four helicopters (including virtual helicopter H0) are the same, which are presented in Table1. Their initial states are specified as follows:(31)α00,α˙00=-20.0deg,0deg/s,α10,α˙10=-21.5deg,0deg/s,α20,α˙20=-27.0deg,0deg/s,α30,α˙30=-16.9deg,0deg/s.Table 1
Nominal parameters of the helicopters (i=0,1,2,3).
Parameter Value K f i 0.1188 N/V l a i 0.660 m l h i 0.178 m m i 0.094 kg J e i 1.034 kg⋅m2 J p i 0.045 kg⋅m2 g 9.81 m/s2All the results are obtained on the directed communication graph shown in Figure4. In this graph, helicopters 1 and 3 have access to desired trajectories αd(t) which are the responses of H0 to desired signal α-d(t), and Assumption 5 holds with b1=b3=a21=a23=a32=1 and all other entries of B- and An being 0. Then,(32)L=000-12-10-11,B-=100000001,L+B-=100-12-10-12.Figure 4
The considered communication topology.Without loss of generality, the following nonstep desired trajectories provided to H0 are adopted for numerical simulations and experiments:(33)α-dt=10sin⁡0.04πt-5,α-˙dt=0.4πcos⁡0.04πt,α-¨dt=-0.016π2sin⁡0.04πt,where all the angles are given in degrees. For the parameters of control signal uα0 which is determined by (6) with b0=1, a0j=0,j∈I, α¨^dt=α-¨dt, and e˙^α0(t)=e˙α0(t), choose(34)kα0P=1.5,kβ0D=2,kα0I=0,where H0 is a virtual helicopter without the acting of LUDs; hence the integral term in the control law (6) is not adopted by H0 (i.e., kα0I=0) for simplicity.The design parameters chosen for the FTC estimators (7) and (8) are shown in Table 2. The initial state values of the differentiators in (7) and (8) are set to zero; that is, ρ^d(0)=ρ˙^d(0)=ρ¨^d(0)=0 and e^ρi(0)=e˙^ρi(0)=0,i∈I.Table 2
Design parameters of the FTC estimators.
Parameter Equation (7) Equation (8) λ 0 1.1 5 λ 1 1.5 8 λ 2 2 N/A μ 0 3 8 μ 1 6 12 μ 2 8 N/A L - ρ d 1 N/A L - e N/A 1In the following, for comparison purposes, six cases (Cases1–6) are considered. In each case, uαi are determined by (6). In order to theoretically verify the results of Theorem 8, Cases 1–3 are designed for the numerical simulations. Specifically, to demonstrate that both tracking and synchronization performance without IC are not acceptable, IC is not adopted in Case 1; points (1)-(2) of Theorem 8 are verified by Case 2, where the helicopters with IC are subject to different constant disturbances; the helicopters with IC are subject to different time-varying disturbances in Case 3 to show the validity of points (1) and (3) of Theorem 8. The results of Theorem 8 are experimentally verified by Cases 4–6 on the experiment platform shown in Figure 1. IC is adopted in Cases 5-6 while, in Case 4, it is not. The ADSs of helicopters 2 and 3 are activated in Case 6 while in Cases 4-5 are not. Note that, since the disturbances are unknown in the experiments, the control parameters of uαi in experiments are not the same as those in numerical simulations, where the disturbances are specified and used to choose the control parameters.The detailed differences among Cases1–6 are as follows:(1) For Case 1, the numerical example with constant disturbance (dα1=5deg/s2,dα2=10deg/s2,dα3=15deg/s2, d˙αi=0,i=1,2,3) is simulated. The parameters of uαi are chosen as kαiP=3.2,kαiD=1.3,kαiI=0(i=1,2,3), which corresponds to the situation without IC.(2) Case 2 is the same as Case 1, except for kαiI=1(i=1,2,3). That is, IC is applied for the numerical example with constant disturbance.(3) Case 3 is the same as Case 2, except for time-varying dαi that are given as follows:(35)dαi=5sin⁡0.05iπt+5ideg/s2,i=1,2,3.Note the three points: (a) as shown in (35), the three helicopters are subject to different disturbances; (b) initial disturbances dαi(0)=5ideg/s2,i=1,2,3; (c) to verify the results of Theorem 8, the disturbances used in Case 2 satisfy conditions limt→+∞d˙αi=0 for each i=1,2,3, while those used in Case 3 do not.(4) For Case 4, uαi without IC are applied to the experiment platform shown in Figure 1. The parameters of uαi are chosen as kαiP=5,kαiD=8,kαiI=0(i=1,2,3).(5) Case 5 is the same as Case 4, except for kαiI=3(i=1,2,3). That is, IC is applied to the experiment platform.(6) For Case 6, in the experiments, the ADSs of helicopters 2 and 3 are activated and the same control laws as for Case 5 are applied.It is worthwhile noting that ADS has a dramatic effect on the motion of helicopter body, whether it is static or moving. This point is easily seen from the mechanical structure of helicopter as shown in Figure2 and is also verified by the following experiments.


### 4.1. Case 1: Numerical Simulation without Using IC

The numerical simulation results for this case are presented in Figure5. As shown in subfigures (a) and (b) therein, because IC is not adopted in control design to attenuate constant disturbances, neither tracking error nor synchronization error converges to a small neighborhood of zero. More specifically,(1)
for the elevation axis, the magnitude of tracking error (of each helicopter) is greater than5.5deg when t≥20sec and is steady as t increases. The synchronization error between any pair of the three helicopters is of a magnitude greater than 1deg when t≥20sec and is steady as t increases. The reason is that the constant disturbance of the helicopter is different from each other and is not attenuated by IC.(2)
as shown in subfigures (c) and (d) of Figure5, the voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage 24 V at the beginning (t≤0.8sec), has a magnitude smaller than 24 V for all t>0.8sec (smaller than 3 V for all t>2sec).(3)
as shown in subfigures (e) and (f), the attained accuracies of the FTC estimators are|α¨d-α¨^d|≤0.0074deg/s2 and max1≤i≤3e˙αi-e˙^αi≤0.003deg/s that corresponds to Lemma 2 for FTC estimators (7) and (8) with finite time tc (in this case, tc≤7sec and max1≤i≤3tci≤2.5sec for (7) and (8), resp.).Figure 5
Numerical simulations results for Case1 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)


### 4.2. Case 2: Numerical Simulation with Using IC for Constant Disturbances

In this case, IC is adopted in control design to attenuate constant disturbances. The corresponding simulation results are presented in Figure6, demonstrating that both tracking and synchronization performance are dramatically improved, compared with that achieved in Case 1. A detailed analysis of the results is given as follows:(1)
For the elevation axis, whent≥20sec, the magnitude of tracking error (of each helicopter) is smaller than 0.03deg and the synchronization error between any two helicopters has a magnitude smaller than 0.01deg. Points (1) and (2) of Theorem 8 are verified by these results.(2)
The voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤0.8sec), is of a magnitude less than 3 V for all t>2sec.(3)
The result of FTC estimator (7) is the same as that in Case 1 and hence omitted. The attained accuracies of FTC estimator (8) are max1≤i≤3e˙αi-e˙^αi≤0.009deg/s and the finite convergence time satisfies max1≤i≤3tci≤15sec.Figure 6
Numerical simulation results for Case2 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)


### 4.3. Case 3: Numerical Simulation with Using IC for Time-Varying Disturbances

In this case, IC is adopted in control design to attenuate time-varying disturbances. The corresponding simulation results are presented in Figure7. Although subject to the time-varying disturbances shown in (35), both tracking and synchronization performance are improved compared with that achieved in Case 1, where the disturbances are constants. However, compared with that achieved in Case 2, where the disturbances are constants and also attenuated by IC, both tracking and synchronization performance are degraded observably. A detailed analysis of the results is given as follows:Figure 7
Numerical simulation results for Case3 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)(1) For the elevation axis, when t≥20sec, the magnitude of tracking error (of each helicopter) is smaller than 1.0deg; that is,(36)α~it<1.0deg,∀t>20sec,i=1,2,3,and the synchronization error between any two helicopters has a magnitude smaller than 1.51deg.With parameterskαiP=3.2,kαiD=1.3,kαiI=1(i=1,2,3) and (21), the following matrix Pαi can be obtained by resolving the Lyapunov equations (28):(37)Pαi=2.231.520.501.523.100.630.500.630.87,λmaxPαi=4.43,λminPαi=0.68.With (4) and (35), we have d-dα=0.041deg/s3; then, along with (32) and (37), we get bα in (27) that satisfies bα≤6.41deg and the ultimate bound of the tracking errors in (26) satisfies α~t2≤9.16deg. Then, points (1) and (3) of Theorem 8 are verified by (36).(2) The voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage 24 V at the beginning (t≤0.8sec), is of a magnitude less than 3.8 V for all t>2sec.(3) The result of the FTC estimators is the same as that in Case 2 and hence omitted here.


### 4.4. Case 4: Experiment without Using IC

The experimental results for this case are presented in Figure8. It is seen that the tracking error (of each helicopter) does not converge to a small neighborhood of zero and the synchronization performance is better than tracking performance. A detailed analysis of the results is given as follows:(1)
For the elevation axis, the magnitude of tracking error (of each helicopter) is greater than3deg for all t>10sec; the synchronization error (between any two helicopters) is α1-α2≤1.11deg, α1-α3≤0.76deg, and α2-α3≤0.69deg for all t>10sec; the synchronization error between H2 and H3 has a smaller magnitude than that between H1 and H2 (or H1 and H3), which is an immediate consequence of the fact that H2 and H3 can obtain information from each other and both of them are equipped with ADSs, whereas H1 is not.(2)
The voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤6sec), has a magnitude smaller than 21 V for all t>6sec.Figure 8
Experimental results for Case4 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)


### 4.5. Case 5: Experiment with Using IC

The experimental results for this case are presented in Figure9. It is observed that, compared with Case 4, both tracking and synchronization performance are improved because the disturbances are attenuated by IC in this case. More specifically,(1)
for the elevation axis, the magnitude of tracking error (of each helicopter) is smaller than1.02deg for all t>10sec (α~1=1.01deg, α~2=0.78deg, and α~3=0.66deg); the synchronization error (between any two helicopters) is |α1-α2|≤0.58deg, |α1-α3|≤0.67deg, and |α2-α3|≤0.45deg for all t>10sec;(2)
the voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤5sec), has a magnitude smaller than 22 V for all t>5sec.Figure 9
Experimental results for Case5 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)


### 4.6. Case 6: With Activated ADS and the Same Controller as for Case5

The ADSs equipped on helicopters 2 and 3 are activated in this case, whose dynamic positions along the arms of helicopters are shown in Figure10, with the same initial position -0.14m (-0.14m is the farthest possible position from propellers). This is different from the previous Case 4 and Case 5 where they are both fixed at -0.14m. The experimental results for this case are presented in Figure 11. It is seen that motion of ADSs does not lead to dramatic performance degradation in the steady state of tracking and synchronization errors, compared with the performance achieved in Case 5. A detailed comparative analysis of the results is given as follows:(1)
For the elevation axis, the ultimate bound of either tracking error or synchronization error is nearly the same with that achieved in Case5; the transient performance is degraded a little (this degradation is relatively obvious for helicopter 2 because the effect of the moving ADS on helicopter 4 is delivered to helicopter 2 which has no access to the desired trajectory as shown in Figure 4).(2)
The voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤6sec), has a magnitude smaller than 23.5 V for all t>5sec.Figure 10
Positions of ADSs on helicopters 2 and 3: helicopter 2 (blue) and helicopter 3 (green).Figure 11
Experimental results for Case6 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)To demonstrate the different steady-state tracking performance clearly, the maximum magnitudes of tracking errors that appeared in each case during time intervalt∈20,30 are shown in Table 3 (Cases 1–3) and Table 4 (Cases 4–6). As shown in these tables, the tracking accuracy is improved in numerical simulations and experiments by IC.Table 3
Maximum magnitudes ofα~i, i=1,2,3, in simulations for t>20 s (deg).
Case H1 H2 H3 1 5.58 6.59 7.59 2 0.027 0.027 0.027 3 0.28 0.62 0.95Table 4
Maximum magnitudes ofα~i, i=1,2,3, in experiments for t>20 s (deg).
Case H1 H2 H3 4 2.99 3.40 3.12 5 1.0 0.78 0.66 6 0.89 0.69 0.94


## 4.1. Case 1: Numerical Simulation without Using IC

The numerical simulation results for this case are presented in Figure5. As shown in subfigures (a) and (b) therein, because IC is not adopted in control design to attenuate constant disturbances, neither tracking error nor synchronization error converges to a small neighborhood of zero. More specifically,(1)
for the elevation axis, the magnitude of tracking error (of each helicopter) is greater than5.5deg when t≥20sec and is steady as t increases. The synchronization error between any pair of the three helicopters is of a magnitude greater than 1deg when t≥20sec and is steady as t increases. The reason is that the constant disturbance of the helicopter is different from each other and is not attenuated by IC.(2)
as shown in subfigures (c) and (d) of Figure5, the voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage 24 V at the beginning (t≤0.8sec), has a magnitude smaller than 24 V for all t>0.8sec (smaller than 3 V for all t>2sec).(3)
as shown in subfigures (e) and (f), the attained accuracies of the FTC estimators are|α¨d-α¨^d|≤0.0074deg/s2 and max1≤i≤3e˙αi-e˙^αi≤0.003deg/s that corresponds to Lemma 2 for FTC estimators (7) and (8) with finite time tc (in this case, tc≤7sec and max1≤i≤3tci≤2.5sec for (7) and (8), resp.).Figure 5
Numerical simulations results for Case1 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)


## 4.2. Case 2: Numerical Simulation with Using IC for Constant Disturbances

In this case, IC is adopted in control design to attenuate constant disturbances. The corresponding simulation results are presented in Figure6, demonstrating that both tracking and synchronization performance are dramatically improved, compared with that achieved in Case 1. A detailed analysis of the results is given as follows:(1)
For the elevation axis, whent≥20sec, the magnitude of tracking error (of each helicopter) is smaller than 0.03deg and the synchronization error between any two helicopters has a magnitude smaller than 0.01deg. Points (1) and (2) of Theorem 8 are verified by these results.(2)
The voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤0.8sec), is of a magnitude less than 3 V for all t>2sec.(3)
The result of FTC estimator (7) is the same as that in Case 1 and hence omitted. The attained accuracies of FTC estimator (8) are max1≤i≤3e˙αi-e˙^αi≤0.009deg/s and the finite convergence time satisfies max1≤i≤3tci≤15sec.Figure 6
Numerical simulation results for Case2 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)


## 4.3. Case 3: Numerical Simulation with Using IC for Time-Varying Disturbances

In this case, IC is adopted in control design to attenuate time-varying disturbances. The corresponding simulation results are presented in Figure7. Although subject to the time-varying disturbances shown in (35), both tracking and synchronization performance are improved compared with that achieved in Case 1, where the disturbances are constants. However, compared with that achieved in Case 2, where the disturbances are constants and also attenuated by IC, both tracking and synchronization performance are degraded observably. A detailed analysis of the results is given as follows:Figure 7
Numerical simulation results for Case3 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, (d) voltage of the back motor, (e) estimation error of the elevation angular acceleration of H0, and (f) estimation errors of the first-order derivative of LNSE of H1–H3.
(a) (b) (c) (d) (e) (f)(1) For the elevation axis, when t≥20sec, the magnitude of tracking error (of each helicopter) is smaller than 1.0deg; that is,(36)α~it<1.0deg,∀t>20sec,i=1,2,3,and the synchronization error between any two helicopters has a magnitude smaller than 1.51deg.With parameterskαiP=3.2,kαiD=1.3,kαiI=1(i=1,2,3) and (21), the following matrix Pαi can be obtained by resolving the Lyapunov equations (28):(37)Pαi=2.231.520.501.523.100.630.500.630.87,λmaxPαi=4.43,λminPαi=0.68.With (4) and (35), we have d-dα=0.041deg/s3; then, along with (32) and (37), we get bα in (27) that satisfies bα≤6.41deg and the ultimate bound of the tracking errors in (26) satisfies α~t2≤9.16deg. Then, points (1) and (3) of Theorem 8 are verified by (36).(2) The voltage applied to either front motor or back motor, with a value equal to the maximum acceptable voltage 24 V at the beginning (t≤0.8sec), is of a magnitude less than 3.8 V for all t>2sec.(3) The result of the FTC estimators is the same as that in Case 2 and hence omitted here.


## 4.4. Case 4: Experiment without Using IC

The experimental results for this case are presented in Figure8. It is seen that the tracking error (of each helicopter) does not converge to a small neighborhood of zero and the synchronization performance is better than tracking performance. A detailed analysis of the results is given as follows:(1)
For the elevation axis, the magnitude of tracking error (of each helicopter) is greater than3deg for all t>10sec; the synchronization error (between any two helicopters) is α1-α2≤1.11deg, α1-α3≤0.76deg, and α2-α3≤0.69deg for all t>10sec; the synchronization error between H2 and H3 has a smaller magnitude than that between H1 and H2 (or H1 and H3), which is an immediate consequence of the fact that H2 and H3 can obtain information from each other and both of them are equipped with ADSs, whereas H1 is not.(2)
The voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤6sec), has a magnitude smaller than 21 V for all t>6sec.Figure 8
Experimental results for Case4 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)


## 4.5. Case 5: Experiment with Using IC

The experimental results for this case are presented in Figure9. It is observed that, compared with Case 4, both tracking and synchronization performance are improved because the disturbances are attenuated by IC in this case. More specifically,(1)
for the elevation axis, the magnitude of tracking error (of each helicopter) is smaller than1.02deg for all t>10sec (α~1=1.01deg, α~2=0.78deg, and α~3=0.66deg); the synchronization error (between any two helicopters) is |α1-α2|≤0.58deg, |α1-α3|≤0.67deg, and |α2-α3|≤0.45deg for all t>10sec;(2)
the voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤5sec), has a magnitude smaller than 22 V for all t>5sec.Figure 9
Experimental results for Case5 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)


## 4.6. Case 6: With Activated ADS and the Same Controller as for Case5

The ADSs equipped on helicopters 2 and 3 are activated in this case, whose dynamic positions along the arms of helicopters are shown in Figure10, with the same initial position -0.14m (-0.14m is the farthest possible position from propellers). This is different from the previous Case 4 and Case 5 where they are both fixed at -0.14m. The experimental results for this case are presented in Figure 11. It is seen that motion of ADSs does not lead to dramatic performance degradation in the steady state of tracking and synchronization errors, compared with the performance achieved in Case 5. A detailed comparative analysis of the results is given as follows:(1)
For the elevation axis, the ultimate bound of either tracking error or synchronization error is nearly the same with that achieved in Case5; the transient performance is degraded a little (this degradation is relatively obvious for helicopter 2 because the effect of the moving ADS on helicopter 4 is delivered to helicopter 2 which has no access to the desired trajectory as shown in Figure 4).(2)
The voltage applied to either the front or back motor, with a value equal to the maximum acceptable voltage24 V at the beginning (t≤6sec), has a magnitude smaller than 23.5 V for all t>5sec.Figure 10
Positions of ADSs on helicopters 2 and 3: helicopter 2 (blue) and helicopter 3 (green).Figure 11
Experimental results for Case6 (α-d-cyan, H0-black, H1-red, H2-blue, and H3-green): (a) elevation angular position trajectory, (b) elevation angle tracking error, (c) voltage of the front motor, and (d) voltage of the back motor.
(a) (b) (c) (d)To demonstrate the different steady-state tracking performance clearly, the maximum magnitudes of tracking errors that appeared in each case during time intervalt∈20,30 are shown in Table 3 (Cases 1–3) and Table 4 (Cases 4–6). As shown in these tables, the tracking accuracy is improved in numerical simulations and experiments by IC.Table 3
Maximum magnitudes ofα~i, i=1,2,3, in simulations for t>20 s (deg).
Case H1 H2 H3 1 5.58 6.59 7.59 2 0.027 0.027 0.027 3 0.28 0.62 0.95Table 4
Maximum magnitudes ofα~i, i=1,2,3, in experiments for t>20 s (deg).
Case H1 H2 H3 4 2.99 3.40 3.12 5 1.0 0.78 0.66 6 0.89 0.69 0.94


## 5. Concluding Remarks

The robust distributed consensus tracking controllers with integral action for multiple 3-DOF experimental helicopters without velocity measurements have been studied under the condition that only the desired angular position measurements are accessible to a small subset of the helicopters. Motivated by the effectiveness of the tracking controller with integral action in disturbances rejection for the single vehicle, the distributed controllers have been proposed by combining FTC estimators with distributed integral controllers. With using the FTC estimators, great accuracy and finite-time convergence have been achieved for the estimation of the lacking information. Meanwhile, the distributed controllers with integral action stabilized the tracking errors and rejected the input disturbances. Through analysing the closed-loop stability, the conditions ensuring zero-error tracking and the ultimate bound of errors for the general cases with nonzero error have been derived. It has been verified through the results of numerical simulations and experiments on platform of “three 3-DOF helicopters" that the tracking and synchronization accuracy have been improved by the proposed controllers with proper parameters.Future work will focus on the design and experimental verification of bounded distributed controller for the platform under the directed communication graph with time-delay. The control parameters in (6) also need to be optimized to get the minimized ultimate bound of tracking errors.



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*Source: 7324508-2016-10-31.xml*