Properties of interconnected negative imaginary systems and extension to formation‐containment control of networked multi‐ UAV systems with experimental validation results

Properties


INTRODUCTION
Negative Imaginary (NI) systems theory, first presented in [26], was a new development in the broad area of robust control, parallel to Passivity theory [13].It was initially motivated by the principle of 'position control of inertial systems via positive feedback' [15], which is common in systems with collocated position sensors (or acceleration sensors) and force actuators [26,4].NI dynamics are highly correlated to the notion of Counter-clockwise Input-Output (I/O) systems [2], I/O Hamiltonian systems [43] and a new type of dissipative systems defined with respect to the input () and time-derivative of the system's output ( ẏ) [7,24].Various potential applications of the NI theory include large space structures [15], robotic manipulators [32], networked multiagent systems [46,37,19,6], nano-positioning systems [35], etc.
NI theory has become attractive due to its simple closedloop stability criterion that relies on the systems' gains only at  = 0. [26] first established that a positive feedback loop containing a stable NI and a Strictly NI (abbreviated as SNI) system remains asymptotically stable if the product of the systems' gains at  = 0 is smaller than one (defined as the DC loop gain criterion).Later, [48] showed that this stability criterion equally applies to the case when one of the systems has poles on the -axis except at  = 0.It has been further generalised to allow NI systems to have a maximum of two poles at  = 0 [17,32,16,25].
Over the last two decades, cooperative control of multiagent systems (MASs) has been a promising research domain within the control and robotics communities.This field encompasses different types of coordinated control problems, such as leader-following consensus, formation-tracking, containment, formation-containment, rendezvous, flocking, herding, etc.Among them, the formation-containment control (FCC) action involves formation-tracking of the leader agents and containment control of the follower agents [27,41].[31] did the pioneering research in developing an FCC scheme for MASs.After that, [10] proposed an adaptive formation-containment control scheme for networked Euler-Lagrange systems.Not only FCC but the entire cooperative control literature has been flourishing gracefully, with recent developments drawing significant attention.For instance, [11] investigated the leaderfollower affine formation manoeuvre control for high-order multi-agent systems; [30] proposed a bipartite consensus tracking methodology applying a fuzzy-based fault-tolerant control approach; [29] addressed the consensus control problem with power integrators using a neuro-adaptive approach; and [39] proposed a new robust adaptive formation controller for muti-UAV systems that can be applied to cooperative payload transportation missions.These advancements showcase the vibrant and dynamic nature of the cooperative control field.Since 2015, the NI theory has found practical applications in the development of cooperative control laws for MASs, including multi-robot and multi-UAV systems [40,38,1].The articles [46] and [45] established the theoretical foundation for NIbased cooperative control schemes.Later, V. P. Tran et al. exploited the results of [46] and [45] in developing various formation control configurations for multi-quadcopter systems (see the survey paper [42] and the references cited therein).In parallel, [37] designed an NI-based rendezvous control scheme for a group of feedback-linearised two-wheeled mobile robots.More recently, [19] and [6] have come up with NI-based cooperative control schemes for single-integrator MASs, with their theoretical proofs relying on the properties of characteristic loci (also known as eigenvalue loci) of NI systems [8].
Drawn by a consistent urge to expand the scope of NI theorybased cooperative control and improve its performance, this paper has designed a distributed FCC scheme for a class of multi-UAV systems utilising the notion of 'mixed' SNI + Strictly Passive system property and characteristic loci technique.Below, we summarise the main contributions of this paper and the salient features of the proposed FCC scheme: • This paper has introduced a two-stage and two-loop formation-containment control scheme for a class of multi-UAV systems whose translational dynamics can be modelled as a double-integrator MAS.Stage 1 and Stage 2 (see Fig. 2) represent formation-tracking and containment control, respectively.The two-loop control configuration (see Fig. 7b) consists of an inner loop that deploys a cascaded PID controller to ensure stable hovering and an outer loop that contains a distributed control protocol for achieving the FCC (formation-tracking and containment control) objectives; • A 'mixed' SNI + Strictly Passive FCC protocol is designed that offers a better dynamic and steady-state performance compared to the SNI-only scheme (e.g.[42,19,37,46]).Besides, the scheme works with both negative and positive feedback; • This FCC scheme relies on dynamic output feedback, making it a suitable choice for applications where fullstate feedback is unavailable.Moreover, this scheme applies to both single and double-integrator NI MASs; • The proposed FCC protocol does not use any nonlinear or switching functions, unlike many existing wellknown cooperative control schemes [18,44,27,28,34,23] that result in a discontinuous control action and, in turn, suffer from the chattering effect; • In contrast to the well-established Lyapunov theorybased FCC schemes [14,18,21,27,10,11,20], the proposed methodology exploits the characteristic loci technique to derive the theoretical proof.It helps to reduce the theoretical complexity (Theorem 5) and facilitates straightforward implementation; • The proposed FCC scheme also takes care of topology switching and network reconfiguration, offers robustness to any stable unstructured uncertainty in additive/multiplicative form and to a sudden loss of agents; • The paper has tested the feasibility and performance of the proposed FCC scheme through two indoor multi-UAV experiments.These experiments were conducted on a group of Crazyflie 2.1 nano quadcopters to achieve i) a formation-containment control mission and ii) a time-varying formation-tracking mission (refer to Subsections 6.3 and 6.4).
Apart from the main contribution to developing a two-stage and two-loop FCC scheme for multi-UAV systems, the paper has also discussed the properties of a positive feedback interconnection of NI systems allowing poles on the imaginary axis including the origin (refer to Section 3).

Notation and symbols
The notation and acronyms are standard throughout.ℝ and ℂ denote the sets of real and complex numbers respectively.The set of all ( × ) real matrices are denoted by ℝ × . * denotes the complex conjugate transpose of a matrix A while  ⊤ indicates the transpose of a matrix.The symbol (  −1 ) * is used as a shorthand for  − * when exists.(⋅) ′ represents the transpose of a vector.For a matrix  having only real eigenvalues,  max () stands for its maximum eigenvalue. ∼ () indicates the adjoint of a transfer function matrix () given by  ⊤ ( s) where s denotes the complex-conjugate of .() * = (−) ⊤ is noted for a transfer function matrix ().RH × ∞ is the space containing all proper, real, rational and stable transfer function matrices with  ×  dimension.() =

𝐴 𝐵 𝐶 𝐷
] signifies a minimal state-space description of a proper, real, rational transfer function matrix ().Let   be the column vector with all  entries equal to 1.The Kronecker product of two matrices  and  is denoted by  ⊗ .‖.‖ expresses the 2-norm of a vector or a matrix.

Preliminaries of NI theory
This subsection reviews the frequency-domain definitions of NI and SNI systems.Afterwards, it presents a state-space characterisation for NI systems without imposing any minimality condition.
We will now present a state-space characterisation for NI systems with possible poles on the  axis, including the origin.Lemma 1. [5] Let  be a square, LTI, dynamical system with a proper, real, rational transfer function matrix () and a statespace description We now recall the following Lemma, which gives a criterion for a system to become NI with poles on the  axis excluding  = 0. Lemma 2. [25] Suppose () is NI.Choose a negative definite matrix Ψ that satisfies  max [(∞)Ψ] < 1.Then,  1 () = ()[ − Ψ()] −1 is NI without any pole at  = 0. Lemma 2 states that, under certain technical assumptions, the closed-loop system interconnection comprised of () and Ψ, via positive feedback, exhibits the NI property.
With reference to Lemma 2, the next Lemma shows the existence of a positive definite matrix P that can be described in terms of () and Ψ.In essence, Lemma 3 extends Lemma 2. Lemma 3. [5] Suppose () is an NI system with a minimal state-space description Then, in Lemma 4, we show that the positive definite matrix P is a solution of the LMI conditions given in (2).
To this end, we will present the closed-loop stability theorem for an NI-SNI interconnection without imposing any restrictions on the systems' gains at  = ∞, which in turn makes the loop strictly proper.
Theorem 1. (NI-SNI stability theorem without poles at the origin) [25] Consider an NI system () without poles at  = 0 and an SNI system ().Then the positive feedback closedloop system comprised of () and (), shown in Fig. 1, is asymptotically stable if and only if The readers are referred to [25] for results when () contains poles at  = 0.

Passivity and 'mixed' NI+Passive systems
After discussing the NI theory, we will now review the definitions of Passive and strictly Passive systems.Definition 3. (Passive System) [13] A system () ∈ R × with no RHP poles is called Passive if () + () * ≥ 0 ∀ ∈ ℝ except those  0 ∈  where  =  0 is a pole of ().The multiplicity of the pole  =  0 cannot be more than one and the residue matrix Ultimately, we would like to put forward a new class of LTI systems that exhibits a 'mixed' SNI and Strictly Passive property.Originally, [36] proposed the notion of 'mixed' NI + Finite-gain system property and later, [12] defined a class of 'mixed' NI + Finite-gain + Passive systems along the direction of [36].According to [36] and [12], a system is called 'mixed' SNI + Strictly Passive if it exhibits the SNI property in some frequency intervals and the Strictly Passive property in others.For example, 1 For detailed classification of Strictly Passive systems, please refer to [13].

Graph theory
A group of networked agents exchanges information with each other via a communication topology.In this work, we used a weighted undirected graph G = {V , E , A } to describe the communication topology among each agent.Here, V = {1, ..., } denotes the node set, E ⊂ V ×V represents the edge set, and A = [  ] ∈ ℝ × is the associated adjacency matrix.
The edge   = (  ,   ) ∈ E indicates that information is transmitted from node  to node .The weight   corresponds to the weight of   , and   > 0 if   ∈ E .The in-degree matrix is defined as If the  th agent is connected to a virtual leader or target (labelled as ''), an edge   is said to exist between them with a pinning gain   > 0.

Properties of multi-agent NI systems
We will now mention an important property that needs to be satisfied by a networked multi-agent NI system.Property 1.The communication topology among the  homogeneous NI agents is described by an undirected and connected graph G .A root node (the virtual target) always exists that directly provides reference trajectory to one or multiple connected agents.
Owing to Property 1, the graph Laplacian matrix enjoys the property (ℒ + ℙ) > 0 in the homogeneous case, where ℙ = diag{ 1 ,  2 , ⋯ ,   } > 0 is the pinning-gain matrix.The element   for  ∈ {1, 2, … , } in ℙ represents the weight of the interaction edge connecting a root node and the  th agent.The following Lemma proves that a homogeneous multi-agent NI (or SNI) system satisfying Property 1 retains the NI (or SNI) property.The concept was first proved in [46] and has since been used in many other papers.Lemma 5. [46] Consider a homogeneous multi-agent NI (or SNI) system with Property 1.Then, Lemma 6 shows that a network of all homogeneous stable NI (including SNI) systems retains the same sign definiteness of its DC-gain matrix when the corresponding communication topology satisfies Property 1.

Characteristic Loci theory
The concept of characteristic loci and its application in determining the closed-loop asymptotic stability of LTI MIMO systems were introduced by MacFarlane and Belletrutti during the period of 1969-1973 [3], [33].This concept is analogous to a multi-loop Nyquist criterion that offers a pretty convenient graphical stability analysis tool for MIMO systems.The characteristic loci   (), where  ∈ {1, 2, … , }, of any square LTI system () ∈ R × are obtained through a conformal mapping of the complex function det[()] into another complex plane when  follows the standard -plane -contour in a clockwise (CW) direction (see Fig. 3a).Theorem 2. [3], [33] A necessary and sufficient criterion for a closed-loop interconnection of two LTI systems () and   (), connected via negative feedback, to maintain asymptotic stability is that the total number of the counter-clockwise (CCW) encirclements about the (−1 + 0) point by the characteristic loci   () of () ≜ ()  () ∀ ∈ {1, 2, … , } should be equal to the number of right-half plane (RHP) poles of ().When () ∈ RH × ∞ , none of the characteristic loci   () should encircle the (−1 + 0) point.

Problem statement
Given a multi-UAV system whose translational dynamics can be modelled as a double integrator multi-agent system, the primary objective is to design a two-stage and two-loop Formation-Containment Control (FCC) scheme (see Fig. 7b) of which the inner loop applies a cascaded PID control block to ensure a stable hovering of the UAVs and the outer loop deploys a distributed dynamic output feedback formationcontainment controller exploiting a 'mixed' NI plus passivity approach.The second objective is to validate the proposed FCC scheme on a group of real-world quadcopter UAVs.

PROPERTIES OF INTERCONNECTED NI SYSTEMS
This section presents the results from our previous work in [5], which will serve as the basis for developing a new distributed dynamic output feedback FCC scheme in Section 4.

Properties of a positive feedback NI interconnection allowing poles at the origin
Then, the inputoutput transfer function mapping from , designated by Σ(), is NI and has no poles at  = 0 (resp.

Properties of an NI interconnection with possible 𝑗𝜔-axis poles excluding 𝑠 = 0
Theorem 4 gives a set of 'if and only if' criteria for a positivefeedback interconnection of two NI systems (refer to Fig. 1), allowing poles on the imaginary axis (excluding the origin) to maintain the NI property in closed-loop.Theorem 4. [5] Suppose () and () are NI systems with no poles at  = 0.Then, the input-output transfer function mapping from , designated by Σ(), is NI and has no pole(s) at  = 0 A formation-containment control scheme for a networked multi-agent/multi-UAV system involving distributed 'mixed' SNI + Strictly Passive controllers.

A 'MIXED' SNI PLUS STRICT PASSIVITY-BASED FORMATION-CONTAINMENT CONTROL SCHEME
This section presents the key developments of this paper.A distributed dynamic output feedback formation-containment control (FCC) scheme is built for a class of multi-agent systems, with a specific focus on applications to multi-UAV systems.The dynamics/kinematics of these systems can be approximated by a group of double integrator NI agents connected via an undirected graph.The proposed scheme exploits a new strategy of 'mixed' SNI + Strictly Passive cooperative control law instead of the 'NI only' cooperative control laws used in the existing literature [46,45,37,42,19,6].Furthermore, to the best of our knowledge, this is the first paper in the NI literature that addresses the formation-tracking and containment control problem while also introducing the concept of a 'mixed' SNI + Strictly Passive cooperative control scheme for a class of multi-agent/multi-UAV systems.
We consider a homogeneous multi-UAV system consisting of   leaders and   followers.This paper aims to achieve two primary control objectives: i) Formation (static/time-varying) tracking of the leader agents surrounding a virtual target; and ii) Containment of the follower agents.Although the present work addresses a single group formation-tracking problem, it can be readily extended to a multi-group formation-tracking problem, considering multiple targets to track.The closedloop translational dynamics of the UAVs can be approximated by a decoupled three-input-three-output double integrator system that, by default, exhibits the NI property with a double pole at  = 0.This fact inspires us to apply NI theory to develop a distributed dynamic output feedback FCC scheme for multi-UAV systems.A two-loop control configuration is employed in practical implementation, as depicted in Fig. 7b.The inner-loop controller, which can be a single/cascaded PID controller, a sliding-mode controller, or a back-stepping controller, is responsible for maintaining stable hovering of the UAVs.The distributed FCC protocol serves as the outer-loop controller, enabling coordinated formation-containment control of the UAVs.In the present case, the graph Laplacian matrix of the overall network can be expressed in the par- where We will now mention a few crucial properties that the network topology G of the multi-UAV system should satisfy.Property 2. In the case of an undirected graph topology, the leaders should be well-connected.At least one leader should be directly connected to the virtual target (treated as the root node).In the case of a directed graph, there must exist a spanning tree from the root node.Property 3. The followers are also well-connected, and at least one leader must exist for each follower that has a directed path to that follower.Property 4. The leaders cannot receive information from the followers.Leaders and followers rely on the output information of their neighbours only, but not on the information of all agents.
As a consequence of Property 2, for an undirected graph, (ℒ  + ℙ) > 0 and for directed cases, (ℒ  + ℙ) qualifies as an -matrix.Owing to Property 3, the rows of the matrix −ℒ −1  ℒ   have a row-sum equal to 1.This result is wellknown in matrix theory and is related to algebraic graphs [27].We are now ready to state the main Theorem of this paper, which provides the theoretical basis of the projected 'mixed' SNI + Strictly Passive FCC scheme for networked multi-UAV systems.
Theorem 5. Given a homogeneous NI multi-UAV system consisting of   leaders and   followers whose simplified closed-loop translational dynamics (in each channel) are given by () = 1  2 .Suppose the associated network topology G satisfies Properties 2-4.Choose a 'mixed' SNI + Strictly Passive controller   () ∈ RH ∞ with   (0) > 0 for the scheme shown in Fig. 2. Let  =     denote the position of the virtual target and  = is the desired formation configuration vector.Then, there always exists a finite  ∈ (0,  ⋆ ] such that the multi-UAV system achieves the formation-containment objectives (i.e., formation-tracking and containment control) by the following distributed dynamic output feedback control law and   =   () Proof.The proof has been divided into two parts -Part I is dedicated to the formation-tracking and Part II establishes the containment control technique.
To proceed with the proof, three sets Ψ 0 , Ψ ±ℑ and Ψ ∞ of the Laplace variable  are defined corresponding to three specific regions marked (as shown in Fig. 3a) along the standard -plane -contour.To prove the asymptotic stability of the proposed formation-tracking scheme, we will exploit the characteristic loci theorem (Theorem 2).This proof consists of the following three cases.

Case I: For the subset 𝑠 ∈ Ψ 0
We can approximately express the characteristic loci λ () where   simply denotes the  th eigenvalue of a real matrix.
Case III: For the subset  ∈ Ψ ∞ Similar to Case I, for this subset also, we can write Following the same way, we can easily compute the counterpart λ (−∞).This then follows that each λ () joins the infinitefrequency points λ (+∞) and λ (−∞) in the CCW direction via a semicircular arc of radius 1  → 0 as illustrated in Fig. 3b.The above three cases can be combined to guarantee that all the characteristic loci λ () of   () remain inside the Pink-coloured area marked in Fig. 3b.At the same time, it also ensures that there always exists a finite  ⋆ > 0 such that the critical point (− 1  ⋆ + 0) is never encircled by any λ ().This proves the asymptotic stability of the proposed formation-tracking scheme.
We will now establish the asymptotic convergence of the formation-tracking error  = [   1   2 ⋯     ] ⊤ .The formation-tracking error dynamics can be obtained from the block diagram in Fig. 2 as The expression of the time-domain steady-state error can be derived as since   (0) > 0, (ℒ  +ℙ) > 0 and () and () all are bounded signals for all  ≥ 0. This hence implies  → ( + ) at the steady-state.This completes the proof.

Part II: Containment of 𝑁 𝑓 follower UAVs
Proceeding along the same direction as exercised in Part I, the asymptotic stability of the containment control scheme (the lower block diagram in Fig. 2) for the followers can be readily established along with lim →∞ () = 0 where  denotes the containment error.This implies lim ] ⊤ and ∑   =1   = 1 with   > 0 since each row sum of the matrix −ℒ −1  ℒ   is equal to 1.This means that all followers will be driven inside a convex hull spanned by the coordinates of the leaders.Note that ℒ −1  exists because ℒ  > 0 via Property 3.This completes the proof.■ Remark 1.In contrast to many existing FCC schemes that rely on the Lyapunov stability approach [27,10,11,20,14], our method exploits the characteristic loci property of networked NI and SNI systems to prove the convergence of the formation-tracking and containment errors.Therefore, the proposed methodology does not need to search for an appropriate Lyapunov candidate function and is free from a complicated and long chain of proof.In addition, our method relies only on output feedback, making it a better choice when full-state measurement is not possible.Besides, the proposed 'mixed' SNI + Strictly Passive control law does not use any nonlinear control term, unlike the literature [18,44,27,28,34,23], etc.Hence, the latter does not suffer any chattering effect.
Remark 2. The proposed FCC scheme introduces a novel approach for implementing formation-tracking among leader agents in both static and time-varying formation configurations.Moreover, the proposed scheme provides a straightforward extension to tackle the complexities of a multi-group formation-tracking problem by incorporating multiple targets for tracking and accommodating multiple subgroups within the communication topology.

MATLAB SIMULATION RESULTS
The Matlab simulation case study presented in this section compares the performance of the proposed FCC scheme, which utilises a distributed 'mixed' SNI + Strictly Passive controller, with that of an SNI-only FCC scheme adopted from [19,6].We considered a 3D formation-containment control problem for a group of UAV agents whose translational dynamics can be modelled as a networked double integrator agent.The group consists of eight leaders and eight followers.Fig. 4 describes the interactions among the leader and follower agents.In this figure, Label '0' denotes the virtual target; Labels 1-8 indicate the leaders, and Labels 9-16 denote the followers.The main control objective was to attain a 3D cubic formation by the leaders, while the followers were expected to converge inside a convex hull spanned by the leaders.We selected a 'mixed' SNI + Strictly Passive controller   () = +0.01 2 +50+500 ( + 40) to be fit into the outer loop of the proposed scheme.At the same time, an SNI-only controller () = − +1 +100 was taken for a comparative study.

FIGURE 4
The communication topology used in the simulation case study.
Fig. 5a shows the initial orientation of all the participating agents.Fig. 5b shows that the eight leaders have successfully achieved the desired 3D cubic formation, and the followers have converged inside the convex hull spanned by the leaders under the influence of the proposed 'mixed' SNI and Strictly Passive FCC scheme.In these figures, The red stars mark the positions of the virtual target.Fig. 6 compares the 2-norms of the formation-tracking errors of the leaders [in Fig. 6(a)] and containment errors of the followers [in Fig. 6(b)], simulated by applying respectively the proposed FCC scheme and an SNIonly FCC scheme adopted from [19,6].In the case of the proposed FCC scheme, we observe that the formation-tracking and containment errors decayed to zero within 5 seconds only  (b) The pure SNI controller from articles [19,6].

FIGURE 6 [Performance comparison study]
Comparison of the 2-norms of the i) formation-tracking errors ‖  ()‖ of the leaders and ii) the containment errors ‖  ()‖ of the followers, simulated by applying the proposed FCC scheme and an SNIonly FCC scheme adopted from [19,6].
[see Fig. 6(a)].But in the case of the SNI-only FCC scheme, the errors decayed to zero after 20 seconds with some inaccuracies, as reported in Fig. 6(b).The comparative study reveals that the proposed 'mixed' SNI + Strictly Passive FCC scheme offers a better steady-state and dynamic performance than the SNI-only FCC scheme adopted from [19,6].

EXPERIMENTAL VALIDATION RESULTS
The proposed dynamic output feedback FCC scheme was implemented on a fleet of Crazyflie 2.1 quadcopters [9] for validation purposes.A 'mixed' SNI + Strictly Passive controller transfer function was chosen to conduct two indoor multi-UAV experiments aiming for a formation-containment operative and a time-varying formation-tracking mission.The experimental results demonstrate the feasibility of the scheme and show its potential applications.A recorded video clip of the experiments can be found in the supplementary material and at https://youtu.be/grq0LWp6b98.

Experimental setup
Fig. 7a shows the components of a Crazyflie 2.1 nano quadcopter with some expansion decks.The Crazyflie is a smallsize (with a diagonal length of 92 mm from motor to motor) and lightweight (about 27 g) quadcopter developed as an open-source UAV-test platform by Bitcraze Pvt. Ltd. [9].The Crazyflie quadcopters can be equipped with additional decks, such as Flow Deck v2 and Loco-Positioning System (LPS) Deck, to enable autonomous flight experiments.The LPS provides the absolute position coordinates of a flying UAV with an accuracy of 0.1 m.Fig. 7b depicts the two-loop control configuration implemented in the flight experiments for each UAV to achieve the formation-containment control objective.The inner loop employs a cascaded PID attitude controller to render the closed-loop translational dynamics of a UAV into a double integrator system.At the same time, a 'mixed' SNI + Strictly Passive distributed controller operates on the inner-closedloop double integrator dynamics and plays the role of FCC.The position of each quadcopter is transmitted to the base station, and the control command is generated through the base station and sent to each quadcopter via Crazyradio dongles [9].However, if each quadcopter can directly measure the relative positions of others or share its position with its neighbours via Bluetooth or Wi-Fi, the base station can be easily removed.

Modelling of the UAVs
The dynamics of small and lightweight quadcopter UAVs, such as Crazyflie nano quadcopter, can be represented by the following Newton-Euler equations: where  = [, , ] ⊤ and  = [, , ] ⊤ are the position vector in the earth frame and the angular velocity vector in the body frame. ∈ ℝ and  ∈ ℝ 3×3 are the mass and the inertia matrix of a quadcopter UAV.  = [0, 0,  ] ⊤ and   are the total force vector and the total drag torque vector acting on the quadcopter UAV with respect to the body frame. is the total thrust produced by the four rotors. is the gravity constant, and   = [0, 0, 1] ⊤ is the unit vector with respect to the earth frame.Following the  −  −  Euler rotation sequence, the rotation matrix for transforming a vector from the body frame to the earth (inertial) frame is given by where  = [, , ] ⊤ are the Euler angles,  = cos  and  = sin .The relation between the angular velocity and the derivatives of the Euler angles can be expressed as: Remark 3. Due to the different time scales of translational and attitude dynamics of a quadcopter UAV, hovering and manoeuvring can be controlled separately by a two-loop control configuration shown in Fig. 7b (see [22,41] and the references therein).To achieve the formation-containment and tracking behaviours, a distributed 'mixed' SNI + Strictly Passive controller is implemented in the outer loop and generates the virtual control input  = [  ,   ,   ] ⊤ .The virtual control input is then transformed into the desired thrust (  ), roll angle (  ), and pitch angle (  ), that is ) , (10) where (  ) is the desired yaw angle predefined by the user.
According to Remark 3, the closed-loop translational dynamics of a quadcopter UAV can be approximated by a decoupled three-input-three-output double integrator system p =   , where   = [  ,   ,   ] ⊤ and   = [   ,    ,    ] ⊤ denote the position and virtual control input vector of the  th quadcopter UAV.

Experiment 1: A formation-containment mission
In Experiment 1, we deploy a group of six Crazyflie quadcopters connected via a network of four leaders and two followers to perform a formation-containment mission.In the beginning, the leaders were scattered at the centre of the flight arena, while the followers were placed at the top right corner (far from the convex hull spanned by those leaders), as shown in Fig. 9a.The objective of the leader agents was to achieve a diamond-shaped formation, and the followers were supposed to converge into a convex hull spanned by the leaders.We selected a 'mixed' SNI + Strictly Passive controller   () = +1 ( 2 +10+200)(+40) , with  = 200, to implement as the outer-loop controllers of the proposed FCC scheme.The communication topology between the leader and follower quadcopters is described by Fig. 8a.Label '0' denotes the virtual target; Labels 1-4 denote the leaders, and Labels 5 & 6 indicate the followers.
Fig. 9 shows the snapshots of an indoor multi-UAV experiment on achieving a formation-containment mission.The time-evolvement of the trajectories is also shown on the X-Y plane.Fig. 9b depicts that four leader agents attained the desired diamond-shaped formation and, at the same time, the two followers entered and converged into the convex hull spanned by the leaders' positions.In addition, Fig. 10a and Fig. 10b portray the 2-norms of the formation-tracking and containment errors of the agents, revealing that the errors -   decayed almost to zero within a few seconds only.The X and Y components of the demanded control efforts by the UAVs, computed from ( 4) and ( 5), are reported in Fig. 11a and Fig. 11b respectively.All these experimental validation results conclude that the group of six Crazyflie 2.1 nano quadcopter UAVs accomplished the formation-containment objective under the influence of the proposed dynamic output feedback FCC scheme that implements a distributed 'mixed' SNI + Strictly Passive controller Σ  ().

Experiment 2: A circular time-varying formation-tracking mission
In Experiment 2, four Crazyflie quadcopters, connected via a graph, were deployed to perform a time-varying formationtracking mission.During the first 25 sec, the quadcopters would attain a circular time-varying formation surrounding a static virtual target.In between 25-40 sec, the whole formation assembly of the quadcopters must undergo a translational manoeuvre to track the moving virtual target without disturbing the formation shape.In Experiment 2, the virtual target had a velocity of 0.2 m/sec along the X-axis.The formation configuration vectors   for the agents to obtain a circular time-varying formation are noted below:  trajectories of all the UAV agents and the virtual target.The initial and final positions of the agents were marked by circles and triangles, respectively.Fig. 13b reports that the 2-norms of the formation-tracking errors of the UAVs converged nearly to zero within a few seconds only.Note that the positional accuracy of the LPS is around 0.1 m.Fig. 14a and Fig. 14b portray the X and Y components of the control efforts demanded by the UAV agents during the formation-tracking mission.These experimental validation results confirm the feasibility and effectiveness of the proposed distributed 'mixed' SNI + Strictly Passive FCC scheme.

CONCLUSIONS
The article dealt with the properties of interconnected Negative Imaginary (NI) systems and extended the results to networked multi-agent systems (MASs) that exhibit NI/SNI properties.A dynamic output feedback formation-containment control ) demonstrates the feasibility and performance of the proposed FCC scheme.However, it is noted that the proposed FCC scheme is only applicable to single/double integrator NI agents, and selecting the appropriate controller for the scheme may necessitate the expertise and experience of a control engineer.In the future, obstacle-avoidance controllers could be designed using improved Artificial Potential Function algorithms [47].

Theorem 3
proposes a set of 'if and only if' criteria required for an NI interconnection, constructed via positive feedback, allowing poles on the  axis, including  = 0 to preserve the NI property.Theorem 3.[5] Suppose () and () are NI systems.Choose Ψ 1 < 0 and

( a )FIGURE 5 [
FIGURE 5 [Simulation results] (a) Initial orientation of the UAV agents at  = 0 sec; (b) At  = 5 sec, the leaders attained the desired cubic formation, and the followers entered into the convex hull spanned by the leaders.

FIGURE 7
FIGURE 7 (a) A Crazyflie 2.1 nano quadcopter; (b) The proposed two-loop control configuration implemented in the experiments for each of the UAVs to achieve formation/containment.

FIGURE 8
FIGURE 8 (a) The communication topology used in Experiment 1.(b) The topology used in Experiment 2.

FIGURE 9 [
FIGURE 9 [Results for Experiment 1:] (a) Initial orientation of the leader and follower UAV agents at  = 0 sec; (b) At  = 15 sec, a diamond-shaped formation was attained by the leaders and the followers converged into the convex hull spanned by those leaders.In the snapshot (left ones), the Blue and Red circles mark the leaders and followers, respectively.

FIGURE 10 [
FIGURE 10 [Results for Experiment 1:] (a) The 2-norm of the formation-tracking errors ‖  ()‖ of the leaders; (b) The 2norm of the containment errors ‖  ()‖ of the followers.Note that the position sensing accuracy of the LPS is around 0.1 m.

FIGURE 11 [
FIGURE 11 [Results for Experiment 1:] (a) The X-axis component of the demanded control effort by each UAV; (b) The Y-axis component of the demanded control effort.
, 2, ..., 4}.A 'mixed' SNI + Strictly Passive controller   () = +1 ( 2 +10+200)(+40) , with  = 200, was chosen to be implemented in the outer-loop control.The communication topology among the leader and follower UAV agents, including the virtual target, has been portrayed in Fig.8b.Fig.12depicts the snapshots of the second multi-UAV experiment on a time-varying formation-tracking mission.Four UAV agents participated in this experiment and attained a desired circular time-varying formation while tracking a moving virtual target (marked by a Red star).Fig.13aplots the

FIGURE 12 [FIGURE 13 [
FIGURE 12 [Results for Experiment 2:] (a) At 20 sec, all four UAV agents achieved a circular time-varying formation surrounding a static virtual target (marked by a Red star); (b) The figure shows (at  = 35 sec) the time-varying formationtracking was maintained even when the virtual target moved from its initial location.

FIGURE 14 [
FIGURE 14 [Results for Experiment 2:] (a) The X-axis components of the control efforts demanded by the UAVs; (b) The Y-axis components of the demanded control efforts.