Coordinated control of multiagent systems has widespread application in engineering such as unmanned aerial vehicles and unmanned ground vehicles. Due to the fact that input saturation can lead a control system to deterioration and instability, a lot of efforts have been devoted to investigating this subject of great importance. The present article offers a survey of recent developments on coordinated control of multiagents systems subject to input saturation. Some preliminaries about graph theory, stability theory and input saturation are first provided, followed by some important results in the area, which are categorized into semiglobal and global coordinated controls. Future research topics are finally discussed.
Multiagent system ; Coordinated control ; Consensus tracking ; Input saturation ; Lowandhighgain feedback
Coordinated control of multiagent systems is an active research field in both science and engineering, seemingly originated from distributed algorithms (Lynch, 1996 ) and decisionmaking (Degroot, 1974 ) the like. In 1987, Reynolds (1987) suggested a distributed behavioural model, known as the Biod model, to describe the aggregate motion of a flock of birds, a school of fish, and so on, in the natural world. In 1995, Vicsek et al. introduced a simple model with an innovative heading (directional) formula to investigate selfassemble behaviours of flocks, referred to as the Vicsek model today. Thereafter, Olfati and Murray (Saber and Murray, 2002 , Saber and Murray, 2003a , Saber and Murray, 2003b , Saber and Murray, 2004 and Saber and Murray, 2007 ) studied the coordinated control of multiagent systems by designing a specific control protocol for each agent, with formal formulations given in (Saber and Murray, 2003a and Saber and Murray, 2003b ) for flocking of multiagent systems, which sublimed the Biod model mentioned above.
Briefly speaking, the investigation on coordinated control of multiagent systems demands integrated knowledge from dynamical systems, control theory, graph theory, mechanics, distributed computation, and so on. Following the works mentioned above, coordinated control of multiagent systems, including consensus, flocking and swarming, had become a research forefront in the last two decades on which a large volume of literature can be found (Hong et al., 2006 , Jiang and Wang, 2010 , Li et al., 2010 , Li et al., 2013a , Li et al., 2013b , Saber and Murray, 2002 , Saber and Murray, 2003a , Saber and Murray, 2003b , Saber and Murray, 2004 , Saber and Murray, 2007 , Su et al., 2009 , Su et al., 2011 , Su et al., 2014a , Su et al., 2014b , Tanner et al., 2004 and Yu et al., 2010 ). Among the existing results, coordinated tracking, also called leaderfollowing coordinated control, has received particularly significant attention, which aims at guiding all agents to track virtual/active leaders of the systems.
Roughly, research on coordinated control of multiagent systems can be considered from three perspectives:
In the existing literature on coordinated control of multiagent systems, for example those mentioned above and some references therein, it is assumed that no limitation is imposed to the movement of each agent in the consensus algorithms implementation. However, in reality, it is impractical for agents to move with absolutely freely during the process towards consensus. To tackle this kind of problems, Nedic et al. (2010) considered communication constraints, where every agent is restricted lying in a convex set in the motion space, and a distributed “projected consensus algorithm” is designed to successfully guide all agents to track the intersection of their individual constraint sets. As a direct extension, Lin and Ren (2014) analyzed the constrained consensus problem for multiagent systems with an unbalanced topology and time delays.
Compared with the abovediscussed communication constraints, input saturation (Lin, 1998 , Lin, 1999 and Saberi et al., 1996 ) is more important in practical situations, which can induce deterioration or instability of the underlying systems or networks, such as the windup phenomenon. Recently, a large number of works have been devoted to coordinated control of multiagent systems subject to input saturation. In such a setting, the control input to agent u_{i} will always be limited within bound interval [−ω ,ω ] and this can be described by a saturation function, sat (u_{i} ). One of the intuitive difficulties induced by input saturation is the nonlinearity. To handle this, the socalled parameterized lowgain feedback technique is introduced ( Lin, 1999 and Saberi et al., 1996 ). Roughly speaking, by lowgain feedback, the control input u_{i} for agent i can be turned to arbitrarily small to within the saturation bound interval so that the saturation nonlinearity can be avoided. There are two kinds of approaches to designing lowgain feedback laws, the eigenstructure assignment based design and the Algebraic Riccati equation based design. The current study on coordination control of multiagent systems with input saturation mainly relies on the second approach ( Chen et al., 2015 , Fan, 2015 , Su et al., 2013 , Su et al., 2014a , Su et al., 2014b , Su et al., 2015a , Su et al., 2015b , Su et al., 2015c , Wang et al., 2015a , Wang et al., 2015b , Wang and Wang, 2015 , Yang et al., 2014a , Zhang et al., 2015 , Zhao and Lin, 2014a and Zhao and Lin, 2014b ). On the other hand, as pointed out by (Lin, 1999 ), parameterized highgain feedback laws can lead the systems to achieve higher performances beyond consensus tracking, for example to achieve robust consensus tracking or robust swarm tracking ( Su et al., 2015a , Su et al., 2015c and Wang et al., 2015a ). With the help of the lowgain and highgain feedback techniques, semiglobal and global coordinated control of multiagent systems with input saturation were studied in ( Chen et al., 2015 , Fan, 2015 , Su et al., 2013 , Su et al., 2014a , Su et al., 2014b , Su et al., 2015a , Su et al., 2015b , Su et al., 2015c , Su et al., 2015d , Wang et al., 2015a , Wang et al., 2015b , Wang and Wang, 2015 , Yang et al., 2014a , Zhang et al., 2015 , Zhao and Lin, 2014a and Zhao and Lin, 2014b ) and in ( Meng et al., 2013 , Yang et al., 2014b , Zhang and Chen, 2015 , Zhang et al., 2014 , Zhao and Lin, 2014a and Zhao and Lin, 2014b ), respectively. Taking a panoramic view of the existing investigations on coordinated control of multiagent systems with input saturation, the content of this survey focuses on analyzing the effects of the network topology on the coordinated control performance and finding distributed protocols to achieve consensus effectively, not only for general linear systems ( Chen et al., 2015 , Fan, 2015 , Meng et al., 2013 , Su et al., 2013 , Su et al., 2014a , Su et al., 2014b , Su et al., 2015a , Su et al., 2015b , Su et al., 2015c , Su et al., 2015d , Wang et al., 2015a , Wang et al., 2015b , Wang and Wang, 2015 , Yang et al., 2014a , Yang et al., 2014b , Zhang et al., 2014 , Zhang et al., 2015 , Zhao and Lin, 2014a , Zhao and Lin, 2014b and Zhang and Chen, 2015 ); but also for nonlinear systems (Zhang et al., 2014 ).
The rest of the article is organized as follows. “Preliminaries” section provides some preliminaries. “Semiglobal coordinated control of multiagent systems subject to input saturation” and “Global coordinated control of multiagent systems subject to input saturation” sections introduce the semiglobal and global coordinated control of multiagent systems with input saturation, respectively. “Conclusions” section concludes the presentation.
Throughout, N is the number of agents, R^{+} and R^{n × m} are the sets of real numbers and n × m real matrices, respectively; for a square matrix A , A^{T} is its transpose; ARE means algebraic Riccati equation; sign (·) is the signum function; ∥·∥_{∞} denotes the infinite norm.
For a system consisting of N agents, labelled by 1, 2, …, N , the switching directed interactions among agents can be described by a triplet with node set , edge set if agent j can access the information of agent i at time t } and adjacent matrix having

A directed path is a sequence of directed edges in the form of (v_{1} ,v_{2} ), (v_{2} ,v_{3} ), …, with all . A directed graph has a directed spanning tree if there exists at least one agent that has a directed path to every other agent. A directed graph is called strongly connected if any two distinct nodes of the graph can be connected via a path that follows the directions of the edges in the graph. The Laplacian matrix is defined by , where is a diagonal matrix with the i th diagonal element equal to . Denote and order the eigenvalues of as 0 = λ_{1} ( ) ≤ λ_{2} ( ) ≤ ⋯ ≤ λN ( ). It is well known in reference Godsil et al. (2001) that if and only if is connected.
Let be an augmented graph generated by a leader (labelled by 0) and the above undirected graph . The symmetric Laplacian matrix of is denoted by , where π :[0,∞) → Γ is a switching signal whose value at time t is the index of the graph at time t , and Γ is finite. Moreover, define = diag(h_{1} (t ), h_{2} (t ), …, h_{N} (t )), where if agent i is a neighbour of the leader at time t , then h_{i} (t ) = 1; otherwise h_{i} (t ) = 0.
Note that the notation above will be independent of t if the topology of the graph is nonswitching (i.e., fixed).
The semiglobal coordinated control of multiagent systems with input saturation can be achieved by utilizing the parameterized AREbased lowgain feedback design technique (Lin, 1999 , Su et al., 2013 , Su et al., 2014a , Su et al., 2014b , Zhao and Lin, 2014a and Zhao and Lin, 2014b ). AREbased lowgain feedback means a family of feedback laws in which a parameterized gain matrix, P (ɛ ), approaches zero as the lowgain parameter ɛ ∈ (0,1] approaches zero. By choosing a proper ɛ ∈ (0,1], the control input would not go beyond the saturation domain and thus the nonlinearity induced by the saturation function can be avoided. The current investigations on this topic mainly include lowgain state feedback consensus tracking ( Lin, 1999 , Su et al., 2013 , Zhao and Lin, 2014a and Zhao and Lin, 2014b ) and lowgain output feedback consensus tracking ( Lin, 1999 , Su et al., 2013 and Su et al., 2014b ) problems.
Specifically, consider a multiagent system with N agents, labelled as 1, 2, …, N , where all agents move in the n dimensional Euclidean space according to the following dynamics:

( 1) 
where x_{i} ∈ R^{n} is the state of agent i , u_{i} (t ) is the control input to i , sat_{ω} (·) is the saturation function satisfying
Moreover, A ∈ R^{n × n} and B ∈ R^{n × m} are constant matrices satisfying the following assumption.
The pair (A ,B ) is asymptotically null controllable with bounded control (ANCBC), i.e.,
The dynamics of the leader, labelled as 0, is described by

( 2) 
where x_{0} (t ) is the state of the leader.
The network consisting of N agents and a leader, denoted as , satisfies the following assumption.
The graph contains a spanning tree rooted at the leader.
To emphasize the role of the lowgain parameter ɛ in control input, u_{i} (t ) can be specifically expressed as u_{i} (t ,ɛ ). By choosing a proper ɛ * ∈ (0,1], it can ensure u_{i}(t ,ɛ* )_{∞} ≤ ω .
For a system with agent dynamics (1) and leader dynamics (2), it is said to achieve semiglobal consensus tracking if, for any a priori given bounded set χ ⊂ R^{n} , there is an ɛ * > 0 such that for any ɛ ∈ (0,ɛ *],

provided that x_{i} (0) ⊂ χ for all i = 0, 1, 2, …, N .
Su et al., 2013 , Su et al., 2014a and Su et al., 2014b were the first to investigate the semiglobal consensus tracking problem for multiagent systems with input saturation, wherein both problems were discussed on fixed and switching topologies, respectively. They showed that the problem of consensus tracking for system (1) can be addressed in two steps:

where constant γ > 0 corresponds to the system topology.
The above two steps show that the network topology and the control input both play critical roles in consensus problems for multiagent systems. In Su et al., 2014a and Su et al., 2014b , Su et al. designed a distributed u_{i} as

( 3) 
The authors in reference Su et al. (2013) defined a Lyapunov function and then chose a proper to guarantee that . For a priori given bounded set χ ∈ R^{n} , let c > 0 be a constant such that

Taking the derivative of V (t ) along with (1) , (2) and (3) , it follows that, under Assumption 1 and Assumption 2 , control input (3) can guide all agents in system (1) to track the leader (2) semiglobally.
Furthermore, Wang and Wang (2015) provided a periodically intermittent type of controllers to deal with the semiglobal consensus tracking problem for system (1), with the control input designed as follows:

( 4) 
where T > 0 is a control time period and 0 < k < T is the control width. Then, Chen et al. extended the periodically intermittent consensus tracking ( Wang and Wang, 2015 ) to a more general setting on eventbased synchronization of discretetime linear systems with input saturation. In eventtriggered control, the controllers update information only on the occurrence period of an event. Specifically, the system uses the current information and the sampling information of the last step to determine when to make an update. In so doing, a concerned issue is how to remove the requirement for keeping track of the current information, moment by moment, so as to save control energy? Along this research line, the selftriggered control strategy was proposed (Wang et al., 2015b ). Selftriggering control is a more feasible sampledata control method, where the system determines the next sampling instant according to the last sampling rather than doing sampling at prearranged instants.
In addition, Su et al. (2014b) constructed a state observer for each agent as follows:

( 5) 
where and F is a matrix such that A − FC is Hurwitz. The existence of such a matrix F is ensured by the following assumption.
The pair (C ,A ) is detectable.
The observerbased feedback law for agent i was designed in Su et al. (2014b) as follows:

( 6) 
and semiglobal consensus tracking on fixed and switching topologies, respectively, were analyzed in detail in Su et al., 2014a and Su et al., 2014b . Following Su et al. (2014b) , Fan (2015) studied the semiglobal consensus problem with relative output feedback and input saturation on systems with directed switching topologies, while Zhao and Lin (2014b) focused on semiglobal consensus tracking of general linear multiagent systems with position and rate saturation, with the agent dynamics described by

( 7) 
where x_{i} ∈ R^{n} is the plant state, y_{i} ∈ R^{r} is the output, v_{i} ∈ R^{m} is the actuator state, u_{i} ∈ R^{m} is the control input.
Notice that all the literature mentioned above focused on homogenous linear systems. Yang et al. (2014b) considered heterogeneous linear systems, where agents have different linear dynamics, with analysis on semiglobal regulation of output synchronization for such networks subject to actuator saturation.
Differing from the objective of semiglobal consensus tracking, which is to drive the followers to track the leader, containment control refers to controlling multiagent systems with more than one leader and aims at guiding the followers to move into the convex hull spanned by the multiple leaders in the motion space. Su et al., 2014a , Su et al., 2015b and Su et al., 2015d studied the containment problem for multiagent systems with input saturation. Specifically, for system (1), suppose that there exist M leaders (labelled as N + 1, N + 2, …, N + M ) and each leader adjusts itself by following the linear dynamics

( 8) 
The convex hull of the M virtual leaders is

Containment of multiagent systems with input saturation on a network having a fixed topology was studied in Su et al. (2014a) . Then, as a direct extension, the focus in Su et al. (2015b) is on containment of systems with input saturation subject to switching topologies, while both statecontainment and outputcontainment are discussed in Su et al. (2015d) by constructing intermittent lowgain feedback laws.
Beyond the AREbased lowgain feedback, Zhang et al. (2015) proposed a model predictive flocking control scheme for a discretetime secondorder multiagent system with input constraints based only on neighbouring measurements, which can direct a rigidlattice flock with a nonzero final velocity to achieve the goal.
As a further extension of the lowgain feedback technique, the lowandhigh feedback technique was developed by combining both lowgain and highgain feedback laws together to deal with the coordinated tracking problem for multiagent systems with input saturation and communication noise. Unlike the lowgain feedback, which is characterized by a parameter ɛ ∈ (0,1], the highgain feedback is equipped with a parameter p ∈ (0 + ∞). With the help of the highgain feedback, a system can achieve better performance beyond consensus such as semiglobal robust swarming and semiglobal robust consensus. Su et al. (2015c) took the lead in investigating the robust coordinated tracking problem for multiagent systems with input saturation and communication noise as well as a dead zone constraint. For system

( 9) 
with a leader (2), the input control u_{i} is designed as

( 10) 
where p > 0 refers to the highgain parameter and is the saturation function satisfies the following:

where the constants δ > 0 and b > 0.
Note that if b = 0, the saturation function is the standard saturation function, but if b > 0, it is the standard saturation function with ideal deadzone characteristics.
In (10) , g : R^{n} × R_{+} → R^{m} represents disturbance uncertainties and satisfies the following assumption.
The uncertain element g (v ,t ) = [g_{1} (v ,t ), g_{2} (v ,t ), …, g_{m} (v ,t )] = −g (−v ,t ) ∈ R^{m} is piecewise continuous in t , locally Lipschitz in v , and its norm is bounded by a known function as

where D_{0} is a known nonnegative constant and the known function g_{0} : R_{+} → R_{+} is locally Lipschitz and satisfies g_{0} (0) = 0.
Su et al. (2015c) addressed the semiglobal robust consensus and swarm tracking problems for multiagent systems with input saturation and communication noise. For any a priori given bounded set χ ⊂ R^{n} , with all the initial states of all the agents selected from χ , the robust semiglobal consensus tracking and swarm tracking of system (9) and leader (2) can be achieved, respectively, if lim_{t → ∞ } ∥x_{i} (t ) − x_{0} (t )∥ = 0, i = 1, 2, …, N , holds and ∥x_{i} (t ) − x_{0} (t )∥, i = 1, 2, …, N , remains in a prescribed set χ_{0} ⊂ R^{n} with 0 ∈ χ_{0} after some finite time.
Following Su et al. (2015c) , Wang et al. (2015a) extended state coordinated tracking to output coordinated tracking, where to handle coordinated control of systems with input saturation and communication noise, a new lowandhighgain feedback technique was developed by combining the dulowgain feedback and the quasihighgain feedback together.
One of the outstanding features of semiglobal coordinated control is that the initial value of each agent is required to be in a priori given bounded set. To meet this presumption, one has to determine the values of the lowgain and highgain parameters, ɛ and ρ , respectively, and the control processes are often difficult to carry out. To avoid such difficulties, a novel approach is referred to the scheduling lowandhighgain feedback.
In scheduling lowandhighgain feedback, the lowgain parameter ɛ is a function of the states which can adjust so that the control input u_{i} is always located in the saturation domain. Su et al. (2015a) studied the robust consensus tracking and robust swarm tracking of multiagent systems with input saturation and communication noise via the AREbased scheduling lowandhighgain feedback approach. It requires information of all agents to determine the lowgain parameter ɛ . Zhao and Lin (2014a) suggested avoiding this problem by constructing a lowgain nonlinear feedback control law, in which each agent uses the information of other agents through multihop paths in the communication network. The number of hops each agent uses to obtain such information is not bigger than the largest algebraic multiplicity of the eigenvalues on the unit circle of the system matrix. Without using the lowgain feedback technique, the global consensus tracking problem for multiagent systems subject to input saturation is studied in Meng et al. (2013) and Yang et al. (2014a) by carrying out state transformation to the model. Very recently, Chen et al. (2015) generalized this state transformation to eventbased global synchronization of systems with input saturation.
This article offers a brief overview of the recent research and development in coordinated control of multiagent systems subject to input saturation. To date, the investigations on this topic include semiglobal and global coordinated controls, mainly from the perspectives of network topologies and distributed control algorithms. Lowgain feedback technique is widely used to solve the semiglobal consensus tracking problem, while lowandhighgain feedback is introduced to achieve better performance beyond consensus tracking, e.g., to robust consensus tracking and robust swarm tracking. The extension of converting semiglobal results to global ones is not trivial, because it not only has to follow the distributed rule but also need to guarantee the control input to remain in the saturation domain. Future works are expected to address this important but challenging problem.
The authors declare that there is no conflict of interest.
This work is supported by the National Natural Science Foundation of China (Nos. 61374176 , 61473189 , 61473129 ), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Nos. 61221003), the Program for New Century Excellent Talents in Universities from the Chinese Ministry of Education (No. NCET120215 ), the Fundamental Research Funds for Central Universities (HUST No. 2015TS025 ), and the Program for Changjiang Scholars and Innovative Research Teams in Universities (IRT1245).
Published on 05/10/16
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