Project topics and materials
Get Complete Project Material File(s) Now! »
Why time-varying formation-containment control
Various cooperative control problems of MASs (e.g., consensus, formation or containment) have attracted much research interest in the past decades. As one of the most important issues, formation control has been paid much attention due to its broad potential applications such as target enclosing (López-Nicolás, Aranda & Mezouar, 2017, Zhang & Liu, 2016), surveillance (Nigam et al., 2012), cooperative localization (Hurtado et al., 2004), load transportation (Bai & Wen, 2010), sea testing by heterogeneous autonomous marine vehicles (Abreu et al., 2015) and so on. Figure 1.2 (a) shows the promising effort to develop formation-flying satellites consisted of the Future, Fast, Flexible, Fractionated Free-Flying Spacecrafts, or System F6 by DARPA (2006). The time-invariant formation control problems have been studied substantially in Brinón-Arranz et al. (2014a), Lin et al. (2005), Liu & Jiang (2013), Peng et al. (2013), Tanner (2004), Wang & Xin (2013). Time-varying formation (TVF) control, which can be interpreted that the MAS/MRS is able to change its formation shapes in certain circumstances and keep being stable simultaneously, was studied by Antonelli et al. (2014), Rahimi et al. (2014) and Dong & Hu (2016) and other researchers recently. Changing formation shapes can be necessary for two reasons: covering the larger area or avoiding collisions with obstacles.
Containment control (see Fig. 3.7 as an example), whose objective is to drive the states of multiple followers into the convex hull spanned by multiple leaders, has been investigated a lot in recent years (Haghshenas et al., 2015, Liu et al., 2012, Wen et al., 2016). The motivation comes from some natural phenomena with many potential and important applications. For instance, a small portion of agents equipped with expensive sensors can be introduced as leaders to deal with the complicated surroundings and to form a safe region at the same time, such that followers without those expensive sensors can move inside that region. Then the whole system can move safely and the total cost is not very expensive. This kind of control method is especially important when the number of followers is very large. Figure 1.2 (b) gives an example for the leaders and followers to arrive at the destination with avoiding the obstacles and mines by adopting the containment control method.
Above all, combining the containment control and TVF control together to form the time-varying formation-containment (FC) control, whose objective is to make leaders achieve the predefined formation shapes and drive followers into the convex hull spanned by leaders, is very interesting and challenging.
Why heterogeneity
At the beginning, some of the earliest works related to MASs/MRSs deal with the large scale of homogeneous agents/robots, called swarms which obtain inspira-tions from biological societies (particularly ants, bees, fishes and birds) to develop similar behaviors to accomplish impressive group tasks (see Fig. 1.3). In multiple mobile robot systems, the homogeneous linear dynamics can be described as x˙i(t) = Axi(t) + Bui(t), yi(t) = Cxi(t), i ∈ I[1, N] (1.3).
where xi(t) ∈ Rn , ui(t) ∈ Rp and yi(t) ∈ Rq are the state, control input and measured output, respectively. A ∈ Rn×n, B ∈ Rn×p and C ∈ Rq×n are constant matrices. N is the number of agents. In swarm systems, individual robots are usually unaware of the actions of other robots, other than information on proximity. Dong (2015) finished a good thesis about formation and containment control of high-order LTI swarm systems. However, all the controllers inside his work need the information of Laplacian matrix L. The point is that when the number of agents is not large, it is not expensive for each agent to know the whole communication topology to calculate L. But when the number becomes very large as in the swarm system, it is nearly impossible for each agent to know the whole topology. The cost will be very high to calculate L, especially if there exists a small change in the communication topology, as we have stated out in detail in Section 1.1.2. In a word, the work in Dong (2015) is not fully distributed for swarm systems. In contrast, heterogeneous agents/robots in which team members may vary significantly in their dynamics and capabilities, gain researchers’ attention more.
Overview of formation/containment control
Distributed cooperative control has been researched for decades. The motiva-tion is clearly stated out in the Sections 1.1.1 and 1.1.2. As one of the fundamental problems in the cooperative control of MASs, consensus control has been investi-gated extensively in the literature (Jadbabaie et al., 2003, Li et al., 2010, Olfati-Saber & Murray, 2004, Ren, 2007, Ren & Beard, 2005). Specifically, Olfati-Saber & Murray (2004) introduced the theoretical framework of posing and solving the consensus problem for networked dynamic systems and then, Ren (2007) showed that many existing virtual structure, leader–follower and behavior-based formation control methods could be unified in the framework of consensus-based methods. A unified framework to extend conventional observers to distributed observers by exchanging estimated state information was proposed by Li et al. (2010) to solve the consensus problem and the synchronization problem of complex networks. De-tailed information about the recent study of consensus designing for MAS can be found in the survey paper Cao et al. (2013) and references therein.
As consensus control usually deals with at most one leader, containment con-trol, whose main objective is to drive the states of the followers into the convex hull spanned by multi-leaders, has been investigated a lot in recent years. The mo-tivation comes from some natural phenomena with many potential and important applications as stated out in Section 1.1.3. The stationary and dynamic leader cases were considered respectively for containment control of mobile agents with first-order dynamics under undirected communication topology by Ji et al. (2008). Then the results were improved to multiple stationary or dynamic leaders in fixed and switching directed networks for single-integrator dynamics in Cao et al. (2012). Distributed containment control for double-integrator dynamics in the presence of both stationary and dynamic leaders was investigated in Cao et al. (2011) where the communication topology is directed when leaders are static or have the same velocity, and it will become undirected among followers when leaders move with different velocities. The similar result was presented in Liu et al. (2012) with the control protocol being not fully distributed due to the requirement of eigenvalue information of Laplacian matrix L which is a piece of global information for each.
Overview of methods dealing with delays
Fig 1.5 shows how the dead time (delay) influences the control performance. The objective is to maintain temperature in the tank by adjusting the flow rate of hot/cold liquid entering the pipe. Because of this delay (time of the hot/cold liquid traveling through the pipe), there will be large oscillations in temperature around the set point. This problem was firstly successfully solved by Smith by proposing a frequency approach named Smith predictor in Smith (1957) and Smith (1959.
Table of contents :
Acknowledgement
List of Figures
List of Tables
Abbreviations and notations
1 General introduction
1.1 Motivations
1.1.1 Why cooperative control
1.1.2 Why fully distributed control
1.1.3 Why time-varying formation-containment control
1.1.4 Why heterogeneity
1.1.5 Why input/output delay
1.2 Overview of formation/containment control
1.3 Overview of methods dealing with delays
1.4 The structure of thesis
I Fully distributed formation/containment control
1.5 Preliminaries
1.5.1 Graph theory
1.5.2 Mathematical knowledge
1.6 Time-varying formation shape
1.6.1 TVF shape for homogeneous systems
1.6.1.1 Example
1.6.2 TVF shape for heterogeneous systems
1.6.3 Summary
2 A unified framework of time-varying formation
2.1 Undirected formation tracking
2.2 Directed formation tracking with full access to leader
2.3 Directed formation stabilization
2.4 Directed formation tracking with partial access to leader
2.5 Directed formation tracking with bounded leader input
2.6 Simulations
2.7 Summary
3 Heterogeneous formation-containment
3.1 Heterogeneous TVF control
3.2 Heterogeneous time-varying FC control
3.3 Simulations
3.3.1 Convergence rate analysis
3.3.2 Application to multi-robot systems
3.4 Summary
II Cooperative control with delays and disturbances
4 Constant input delay & matched disturbances
4.1 Homogeneous consensus tracking control without u0(t)
4.2 Homogeneous consensus tracking control with u0(t)
4.3 Simulations
4.4 Summary
5 Heterogeneous consensus with input delay
5.1 Consensus with the input delay
5.1.1 Observer v1,i(t) to estimate the leader’s state x0(t)
5.1.2 The state predictor
5.1.2.1 The derivative for the IF i, i 2 I[1,M]
5.1.2.2 The derivative for the UF i, i 2 I[M + 1,N]
5.1.3 The design of control inputs ui, i 2 I[1,N]
5.1.4 The thinking behind the control input designing
5.2 Consensus with the input delay and disturbances
5.3 Simulations
5.4 Summary
6 Constant input & time-varying output delay
6.1 Observer with time-varying output delay
6.1.1 Lyapunov-Krasovskii functional approach
6.1.2 LKF with descriptor approach
6.1.3 Comparisons
6.2 Heterogeneous consensus control
6.2.1 Simulations
6.3 Heterogeneous TVF tracking control
6.4 Heterogeneous time-varying FC control
6.4.1 Simulations
6.5 Summary
A Time-varying delays & mismatched disturbances
A.1 Model transformation
A.2 Predictive ESO design
A.3 Stability analysis
A.4 Simulations
A.5 Summary
Conclusions and future works
Bibliography
