Consensus of Multi-Agent Systems with Input Constraints Based on Distributed Predictive Control Scheme

: Consensus control of multi-agent systems has attracted compelling attentions from various scientific communities for its promising applications. This paper presents a discrete-time consensus protocol for a class of multi-agent systems with switching topologies and input constraints based on distributed predictive control scheme. The consensus protocol is not only distributed but also depends on the errors of states between agent and its neighbors. We focus mainly on dealing with the input constraints and a distributed model predictive control scheme is developed to achieve stable consensus under the condition that both velocity and acceleration constraints are included simultaneously. The acceleration constraint is regarded as the changing rate of velocity based on some reasonable assumptions so as to simplify the analysis. Theoretical analysis shows that the constrained system steered by the proposed protocol achieves consensus asymptotically if the switching interaction graphs always have a spanning tree. Numerical examples are also provided to illustrate the validity of the algorithm.


Introduction
Consensus means that a group of dynamic agents agree upon a certain quantity of interests such as position and orientation, and which is one of the most fundamental problems in multi-agent systems (MASs) [Olfati-Saber, Fax and Murray (2007); Zhan and Li (2013)]. Due to the promising applications both in military and civil areas, especially in fields such as multi-robot systems and sensor networks, consensus control for multi-agent systems has attracted great attention from various domains [Dong and Geng (2015); Olfati-Saber and Murray (2004); Sahin (2005) ;Brambilla, Ferrante, Birattari et al. (2013)]. Consensus-seeking problems should be addressed using distributed protocols based on local information since these systems are very large-scale while the included individuals only have limited situational awareness ; Kia, Cortés and Martínez (2015); Liu, Dou and Sun (2016)]. Many distributed consensus control algorithms have been put forward in robotics and previous results to discrete-time double-integrator consensus problems with directed switching interaction topologies and acceleration constraints. However, owing to the limitation of available power and safety reasons, which is often encountered in practical applications, constraints on velocity should also be included. Therefore, we are trying to develop a discrete-time consensus protocol for a class of MASs with single-integrator dynamics and switching topologies, and both the velocity and acceleration constraints are included simultaneously under the MPC framework. The remainder of this paper is organized as follows. In Section 2, some necessary preliminary results and lemmas are described together with problem description. Section 3 gives an MPC protocol for MASs with single-integrator dynamics and constraints. Thereafter, the corresponding stability analysis is provided in Section 4. Numerical examples are provided in Section 5 to illustrate the validity of the algorithm and Section 6 summarizes this paper. Throughout this paper, n I and n 1 denote identity matrix and the column vector of all ones of dimension n , respectively.
( ) n R M represents square matrices of order n and the operator "⊗" denotes the Kronecker product. Let . The notation  denotes Euclidean norm.

Preliminary and problem description
Before discussing the main problem addressed in this note, we also need some necessary preliminary results and lemmas on graph and matrix theory first. 2). Considering the following constrained optimization problem, 3). Considering the following constrained optimization problem, min ( ), s.t. || || , V is a constant matrix and K is a nonsingular and compatible matrix. Suppose the quadratic function ( ) g U achieves its unique minimal point at * U , then it follows:

Proof:
The proof of part 1) and 2) can be found in Cheng et al. ], part 3) can be obtained analogously as part 2). Hence, only the proof of part 4) is given in this note. According to the definition of ( ) g U , it can be obtained that One can directly calculate the minimal point As we have known that * 0.

Lemma 2.2 [Ren and Beard
If the graph associated with A has a spanning tree, then A is SIA (stochastic, indecomposable and aperiodic). That is , where y is a nonnegative column vector satisfing = Ay y and T = y y 1 . (2015)]: Let 1 , , k A A  be a finite set of SIA matrices with the property for each sequence 1 , ,

Lemma 2.3 [Cheng, Fan and Zhang
is SIA. Then for each infinite sequence 1 2 , ,

Consensus with velocity constraint only
Consider that a system consists of n agents with discrete-time single-integrator dynamics and velocity constraints given as below.
Np and Nc are the prediction and control horizon, respectively, fulfilling Nc Np ≤ .
Based on the nominal model (8) and velocity ( ) i k V to be designed, the position of agents for the instants k t + , 1, , t Np =  can be obtained as follows Then the above iteration can be rewritten in a compact form as: 1 0 0 0 As prepared above, the optimization problem with velocity constraints in the MPC scheme designed for agent i in a finite time horizon Np is described as follows: , and α is a positive weight coefficient.
Note that the definition of , ( only depends on the position of agent i and its neighbors, which indicates that this control scheme is distributed. For simplicity, we rewrite the MPC cost function (10) into a compact form and substituting (9) in, it derives: Eq. (11) is a quadratic function, whose minimum-value point * ( ) i k V can be calculated by whose first m entry will be actually implemented as the control input at sampling instant k, i.e., If we take the velocity constraints || ( 1) || into consideration, the control input can be obtained as (13) based on Lemma 2.1:

Consensus with velocity and acceleration constraints
Due to the limitation of structural strength or available overload, which is often encountered in practical networked systems, constraints on acceleration should also be considered. Therefore, both the velocity and acceleration constraints are considered simultaneously in this subsection. Theorem 1: Assume each agent of the system has discrete-time second-order dynamics with input constraints given as follows: ( . .|| ( ) || ,|| ( ) || , is the acceleration of i .
Suppose the constraint on turning rate is ignored and coordinated turn can always be implemented, that is, the velocity and acceleration are exactly in line. One can derive the actual control input as is the optimal velocity when no input constraint exists, * min{ ,|| ( 1)

Proof:
Based on (14) and ( ) i k u to be designed, the velocity of agent i for the instant k t + , 1, , t Nc =  , can be obtained as follows Then the above iteration can be rewritten in a compact form as:  Thus by virtue of (14) and (17), the position of agent i can be rewritten in a compact form as: Substituting (17) If we take the constraints * || ( ) || i k u ≤ u into consideration and suppose that the velocity and acceleration are exactly in line, we have || ( 1) || || ( ) || || ( 1) || , (20) where ˆ( ) i k v denote the actually implemented control input.
On the other hand, || ( ) || i k v should fulfill the velocity constraint, i.e., || ( ) , on basis of part 2) and 3) of Lemma 2.1, then the actually implemented control input can be rewritten as (16) The proof is thus completed. Remark 1: Note that although the double-integrator dynamics of the system is adopted, is still implemented as the control input, while ( ) i k u is regarded as the changing rate of velocity input. No specific expression of * ( ) i k U or * ( ) i k u is given as it is indeed an intermediate variable used in the derivation process. Remark 2: As a matter of fact, the control input under different constraints can be obtained by changing the value of v and u . When v or u is set as positive infinity, it means that the corresponding constraints for v or u will not be activated.

Stability analysis
In this section, we will present the convergence analysis of the consensus protocol.
Before seeking to analytically solve the problem, we first give the solution with a much simpler form to simplify the analysis. Theorem 2: If the quadratic optimization problem (11) is feasible with constraints described in (15), then the control input * ( ) i k v without constraints has an equivalent expression as below: x r (22) with 0 i σ > .

Proof:
By the definition of x P and , ( ) i x k R , we know that , ( | ) R is an mNpdimensional column vector which can be rewritten as follows: , , Recall (12) and substitute (23) in, it yields that ( ) (24) Consider that only the first m entry of * ( ) i k V will be actually implemented as the control input, it follows that where p φ is the p-th row of ( ) x q P is the q-th column of x P .
Then it derives that * , Thus the proof is completed. Remark 3: One can see that the consensus protocol is not only distributed but also only depends on the errors of states between agent i and its neighbors. In fact, just as described in many of the literature, such errors are usually sufficient for consensus control. Substituting the result of Lemma 4.1 into (16), the actual implemented control input ˆ( ) i k v can be rewritten as x r (28) Theorem 3: Consider a system consists of n agents, whose dynamics and constraints of each agent are described in (14) and (15) with control input given in (28). Then there exists a stochastic matrix ( ) k D such that the discrete time update scheme of the system can be written as

Proof:
Recall the definition of , ( ) i x k r and substitute into the control input (28), it gives where ( ) i k L is the i-th row of the Laplacian matrix associated with the interaction graph.
Hence, it follows from (29) Hence, it follows from (14) satisfies the conditions that all off-diagonal elements are nonnegative and all its row sums are equal to 1, which indicates ( ) k D is a (row) stochastic matrix.
Thus the proof is completed.

Remark 4:
One can see from Lemma 4.2 that the system achieves asymptotic consensus for any initial condition, if and only if there exists an infinite matrices sequence such that lim ( ) lim ( 1) (0) (0) , where SS X denotes the consensus state of the system.
Theorem 4: Suppose that the interaction graph, denoted as ( ) G k , changes at time t kT = , and keeps fixed across uniformly bounded and non-overlapping time interval [ , ) kT kT T + . Let a finite set G denote all the possible interaction topologies of the system, then ( ) G k G ∈ . The constrained system (14) with control input described in (16) achieves consensus asymptotically if the infinite sequence ( ) G k always has a spanning tree.

Proof:
From Lemma 4.2, we know that ( ) k D describes the corresponding interaction topology of ( ) G k and is a stochastic matrix with positive diagonal entries. The assumption that interaction graph ( ) G k always has a spanning tree indicates that the graph associated with ( ) k D has a spanning tree. By virtue of Lemma 2.2, we know that ( ) k D is SIA. Then by applying Lemma 2.3, one gets that Thus the consensus state is obtained.

Simulation study
In this section, numerical examples are presented to illustrate the feasibility of the distributed MPC consensus protocol by a planar problem. G . An arrow from node j heading to node i implies agent i receives information of agent j; thus j is a neighbor of i. Both 1 G and 2 G have a directed spanning tree Consider a system of 5 n = agents moving in two-dimensional plane, that is 2 m = . The interaction topology switches from 1 G to 2 G periodically with period 0.1 T = , which is sufficiently small and satisfies the constraint condition 1 i i T η σ < . The graphs 1 G , 2 G are defined in Fig. 1 and both Fig. 4; one can see that both velocity and acceleration meet the condition of constraints, respectively. It can be observed from Figs. 4(a) and (b) that the velocity changes at a relatively gentle rate since the acceleration is constrained, while the results containing only velocity constraints in (c) and (d) are just the reverse. This implies the efficiency of our algorithm and also highlights the necessity of taking both velocity and acceleration constraints into consideration simultaneously. Thus the effectiveness of Theorem 1 is demonstrated.  (Fig. 3). The first is the frequent graph switching over time. The second but the most is that there is no constraint on the turning rate of velocity vectors, which leads to sharp changes in velocity direction. This conclusion can be further confirmed by Fig.  4(a), in which one can see that the amplitude of velocities has no big fluctuations.

Conclusions
In this paper, a distributed MPC scheme has been developed to achieve consensus for MASs with single-integrator dynamics, input constraints and switching directed interaction topologies. The control inputs in analytic form are obtained under different constraints. Under the condition that the switching interaction graphs always have a spanning tree, we prove that the system containing velocity and acceleration constraints can achieve consensus asymptotically. Further extensions of this work will concern constraints on the turning rate of velocity vectors, which is a key factor to eliminate the large oscillations in control input.