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Distributed fixed-time control under directed graph using input shaping
Accepted Manuscript
Distributed fixed-time control under directed graph using input shaping Ti Chen, Jinjun Shan PII: DOI: Reference:
S0016-0032(19)30152-8 https://doi.org/10.1016/j.jfranklin.2018.11.039 FI 3820
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 April 2018 18 September 2018 10 November 2018
Please cite this article as: Ti Chen, Jinjun Shan, Distributed fixed-time control under directed graph using input shaping, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2018.11.039
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Ti Chena , Jinjun Shana,∗ a
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Distributed fixed-time control under directed graph using input shaping
Department of Earth and Space Science and Engineering, York University 4700 Keele St., Toronto, Canada, M3J 1P3
Abstract
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This paper presents novel fixed-time controllers for the distributed tracking of multi-agent systems with double-integrator dynamics based on the input shaping technique under directed graphs. It is assumed that there is no cycle in the directed graph with a globally reachable leader. Distributed fixed-time controllers are designed for cases with various initial conditions by placing input shapers in all communication edges in the graph. Numerical simulations and experimental studies are conducted to verify the effectiveness of the proposed controllers.
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Keywords: Distributed control; fixed-time control; input shaping; multi-agent systems; directed graphs. 1. Introduction
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Cooperative control of multi-agent systems has attracted significant attention due to its wide applications, such as formation flying, attitude alignment, flocking, payload transport, air traffic control, rendezvous and docking [1, 2, 3, 4, 5, 6]. One critical problem in the field of multi-agent systems is consensus, in which a team of agents are expected to reach an agreement by negotiating with their neighbors [7, 8, 9]. The key of consensus problem is to design an appropriate distributed protocol to actuate a group of agents in a cooperative fashion based on a communication graph. Since the second-order dynamics can describe many real physical systems, many researchers have investigated the consensus of second-order multi-agent systems from different perspectives [9, 10, 11, 12, 13, 14]. In ∗
Corresponding author. Email addresses: [email protected] (Ti Chen), [email protected] (Jinjun Shan)
Preprint submitted to Journal of The Franklin Institute
February 21, 2019
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previous studies [7, 11, 15, 9, 14], the convergence time of the consensus controller may be infinite because the controllers are asymptotically or exponentially convergent. Hence, the distributed finite-time or fixed-time control are proposed to drive multi-agent systems converging within a finite time. The main difference between these two control methods is that the finite-time convergence rate depends on the initial conditions and the fixed-time convergence rate does not [16, 17, 18]. Recently, some investigators have focused on distributed finite-time and fixed-time control problems [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. However, the controllers in these work are usually proven to converge to the desired state within a bounded time and contain several control parameters which have a significant effect on the performances of the controllers [27]. Furthermore, the controllers are strongly nonlinear, which will increase the computational burden of the individuals in the multi-agent systems with limited computing capabilities. As stated in [29], it is important and challenging to obtain a distributed controller with a specified settling time. Consequently, Liu and Zhao used Pontryagin’s maximum principle to solve the distributed specified time consensus control problem for a group of harmonic oscillators [29]. However, their controller has complex forms and there exist long-time oscillations before the convergence of the closed-loop systems is achieved. It should be noted that closed-loop second-order multi-agent systems with the feedback of relative positions and local velocities may act like an underdamped vibrational system of high dimension. A feedforward method, known as input shaping, can achieve zero residual vibration in principle. The main advantage of input shaping is that it only alters the shape of the reference command and no additional actuators and sensors are required [30, 31]. Hence, this study tries to solve the consensus problem of multiagent systems with double-integrator dynamics based on an input shaping technique. The main contribution of this study is the development of fixed-time consensus controllers based on an input shaping technique under directed graphs, which contain a spanning tree and have no cycle. Theoretical analyses, numerical simulations and experimental studies are presented to verify the distributed controllers. The main advantages of the proposed controller design method are (i) The designed controllers are very simple; (ii) The closed-loop multi-agent systems will converge at a fixed time, rather than within a finite time; (iii) The convergence time can be calculated analytically and has no relationship with the initial conditions. The remainder of this paper is organized as follows. Section 2 presents some basic knowledge of graph theory and input shaping technique. Fixed2
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time consensus control algorithms are designed in Section 3. Numerical simulations and experimental studies are given in Section 4 and Section 5, respectively. Finally, the conclusions are drawn in Section 6.
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2. Preliminaries
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2.1. Graph theory The information exchanges among the agents can be modeled by a graph G = (V, E), where V = {1, 2, · · · , N } is a nonempty, finite node set and E ⊂ V × V is a set of ordered pairs of nodes. Each element in E is called an edge. The edge (i, j) implies that the node j has access to the information of node i. For the edge (i, j), i is a neighbor of j, agent i is named as the parent node, and j is the child node. The set of the neighbors of node i is represented by Ni . A directed path from node k1 to kl is defined as a sequence of ordered communication edges (kr , kr+1 ) ∈ E, for r = 1, · · · , l −1. A cycle is defined as a directed path that starts and ends at the same node. The (rooted) directed tree is a directed graph where each node has exactly one parent node except for one node, known as the root, which has no parent node and has directed paths to all other nodes. A directed tree is called spanning when it connects all the nodes in the graph. A graph is said to have or contain a directed spanning tree if a subset of the edges forms a directed spanning tree, i.e., there exists at least one node with directed paths to all other nodes in the graph. The adjacency matrix of the directed graph G is represented by A = [aij ] ∈ RN ×N , where aii = 0, aij is a positive value if (j, i) ∈ E and aij = 0 otherwise. The weight aij in this study is set equal to 1Pif (j, i) ∈ E unless otherwise stated. The in-degree of the i-th node is di = N j=1 aij . To study the leader-follower tracking problem, it is assumed that the graph of concern consists of one static leader (node 0) and N follower agents. The leader adjacency matrix is represented by B = diag{b1 , · · · , bN }, where bi = 1 if the leader is the neighbor of the i-th node; otherwise bi = 0. The leader is said to be globally reachable if there exists at least one path from the leader to every follower in the graph. 2.2. Input shaping Input shaping can lead to zero or small vibration of flexible structures by shaping the reference command [31]. As shown in Fig. 1, it is realized by convolving the original command with a sequence of impulses, i.e., input shaper. In this work, a Zero Vibration and Derivative (ZVD) shaper will be employed due to its simplicity and robustness to system uncertainties. 3
Figure 1: Principle of input shaping
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Consider the underdamped oscillatory system governed by
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x ¨ + 2ζω x˙ + ω 2 x = ω 2 f
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where ω, ζ and f are the frequency, the damping ratio and the reference command. The amplitudes and the acting times of the three impulses of ZVD shaper are 1 A0 = , t0 = 0 1 + 2K + K 2 2K π A1 = , t1 = (2) 1 + 2K + K 2 ωd 2 K 2π A2 = , t2 = 2 1 + 2K + K ωd ζπ p √ − where K = e 1−ζ 2 and ωd = ω 1 − ζ 2 . For an arbitrary signal g(t), the signal shaped by the ZVD shaper (2) is represented by IS g(t) . The ω,ζ detailed expression of ISω,ζ g(t) is ISω,ζ g(t) = A0 g(t) + A1 g(t − t1 ) + A2 g(t − t2 ) (3)
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A step signal from 0 to a constant c is defined as ( 0 t<τ Sτ,c (t) = c t≥τ
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Similarly, for an arbitrary signal g(t), the function Sτ,g (t) is defined as ( 0 t<τ Sτ,g (t) = (5) g(t) t ≥ τ Lemma 1. The underdamped oscillatory system (1) with zero initial states will stay at c after t = t2 under the shaped command ISω,ζ S0,c (t) . 4
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−ζω(t−τ )
where zτ,c (t) = c− ce√
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Proof. The response of the underdamped oscillatory system with zero initial states under the step reference command Sτ,c (t) is 0 t<τ Zs (t) = (6) zτ,c (t) t ≥ τ
p sin[ωd (t−τ )+β], in which β = arctan( 1 − ζ 2 /ζ).
For the underdamped oscillatory system (1) under the shaped command ISω,ζ S0,c (t) , the response with zero initial state is
where z¯1 = zt0 ,A0 c (t), z¯2 = detailed expression of z¯3 is z¯3 = c −
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− A1 eζωt1 + A2 eζωt2 )
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z¯1 z¯2 Zss (t) = z¯3
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Hence, the system (1) will stay at c after t = t2 . Consider the reference signal defined as follows ( g(t) t < T G(t) = c t≥T
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where g(t) is an arbitrary function of time and c is a constant.
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Lemma 2. The underdamped oscillatory system (1) with zero initial states will rest at c after t = T +t2 under the shaped reference command ISω,ζ G(t) .
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Proof. The response of the underdamped oscillatory system after t = T under G(t) can be expressed as Rt −ζω(t−s) sin[ω (t − s)]ds ZG (t) = ω 2 0 G(s) d ωd e = ZGT + zT,c (t)
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RT −ζω(t−s) sin[ω (t − s)]ds → 0 as time goes to inwhere ZGT = ω 2 0 G(s) d ωd e finity. Under the shaped signal ISω,ζ G(t) , the response after t = T + t2 5
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G(s) −ζω(t−s) sin[ωd (t − s)]ds 0 ωd e R t−t1 G(s) −ζω(t−t1 −s) 2 +A1 ω 0 sin[ωd (t − t1 − s)]ds ωd e R t−t2 G(s) 2 −ζω(t−t −s) 2 +A2 ω 0 sin[ωd (t − t2 − s)]ds ωd e ζωt ζωt 1 2 = (A0 − A1 e + A2 e )ZGT + z¯3 =c
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Hence, it can be concluded that the underdamped oscillatory system will rest on c after t = T + t2 . However, the initial states of the agents in a network may be different, i.e., there may exist agents starting from non-zero initial conditions in the group. Hence, the input shaping technique will be extended to solve the oscillation control with non-zero initial position and velocity. Due to the linearity of the oscillatory system, the control with non-zero initial position and velocity will be considered separately in this section.
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Lemma 3. The underdamped oscillatory system (1) with initial position x0 and zero initial velocity will converge to x = 0 at t = t2 under the following reference command f = x0 − ISω,ζ S0,x0 (t) (12) Proof. The oscillatory system in such a situation can be expressed as
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x ¨ + 2ζω x˙ + ω 2 x = ω 2 x0 − ω 2 ISω,ζ (S0,x0 (t))
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According to Lemma 1, the above system will converge at t = t2 and the equilibrium position is y = −x0 , i.e., x = 0. For the underdamped oscillatory system (1) with initial velocity x˙ 0 , its response is p x˙ 0 −ζωt Znv (t) = e sin ω 1 − ζ 2 t (15) ωd p It is obvious that one solution to Z˙ nv (t) = 0 is t = tv = arctan( 1 − ζ 2 /ζ)/ωd . Denote xv = Znv (tv ). Lemma 4. The underdamped oscillatory system (1) with zero initial position and non-zero initial velocity x˙ 0 will stay at 0 after t = tv + t2 under the following reference command f = Stv ,xv (t) − ISω,ζ (Stv ,xv (t)) 6
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Proof. The oscillation control with non-zero velocity can be considered as the oscillation control with non-zero position by treating the tv as the new initial time. Hence, x will stay at 0 after t = tv + t2 with the reference command (16) based on Lemma 3. 3. Consensus algorithms
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The aim of this section is to drive a group of agents with double-integrator dynamics from various initial states to track a static leader. It is assumed that all communication edges are directed and there is no cycle in the graph. The desired position xd is determined by a static leader, which is globally reachable. Hence, the information of the leader can be propagated over the whole network. 3.1. Fundamental consensus algorithm
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This section presents a fundamental distributed controller for the leaderfollower consensus of multiple agents with double-integrator dynamics. If the dynamic model of the i-th agent is represented by x ¨i = ui , a fundamental consensus controller can be designed as P aij (xj − xi ) − bi (xi − xd ) ui = −αx˙ i + j∈Ni (17) = −αx˙ i − (di + bi )(xi − xid )
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√ P where 2 di + bi > α > 0 and xid = aij xj /(di + bi ) + bi xd /(di + bi ) is j∈Ni
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called the normalized sum of the positions of the adjacent nodes and the leader (if bi = 1). xid can be considered as the reference command of the i-th agent. Because the directed graph of concern contains a spanning tree with the leader as root and has no cycle, the closed-loop multi-agent systems under the controller (17) are asymptotically stable.
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Remark 1. As shown in the survey [32], there are many applications of consensus algorithm of multi-agent systems with double-integrator dynamics. For example, Ref. [33] studies the formation maneuvers for a group of mobile robots, each of which is described as a double-integrator plant. Hence, this work focuses on the multi-agent systems with double-integrator dynamics.
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3.2. Fixed-time consensus algorithm with input shaping
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The parameter α in the controller (17) will greatly influence the responses of the closed-loop multi-agent systems. For engineering applications, α should be well tuned to meet various requirements to the best of its capabilities. In this section, based on input shaping technique, fixed-time consensus controllers of high performance are proposed to avoid the tuning of α. Three theorems in this section consider the following three situations: (i) consensus control of second-order multi-agent systems with identical initial position and zero initial velocity; (ii) consensus control of second-order multi-agent systems with different initial positions and zero initial velocity; and (iii) consensus control of second-order multi-agent systems with different initial velocities and zero initial positions. Without loss of generality, the identical initial position is set as zero. It should be noted that the situation with non-zero initial position can be converted to that with zero initial position by coordinate translation.
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Theorem 1. Under the following controller ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (18) √ where ωi = di + bi and ζi = α/(2ωi ), the distributed tracking for the second-order multi-agent system with zero initial conditions can be achieved at a fixed time. The convergence time is equal to the maximal value of the weighted length of all paths from node 0 to the nodes with no child node, where ¯ij = 2π/ωdi with q the weight for the edge (j, i) ∈ E is set as a
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Proof. The communication graph of concern contains a spanning tree with the leader as root and has no cycle, i.e., the information of each child node has no effect on its parent node. Hence, as shown in Fig. 2, the nodes in the graph can be divided into multiple layers. The sequence number of layer of each node is determined according to the maximal length of the path from the leader to the node. For example, the i2−1 -th node in Fig. 2 has two paths from the leader. The length of the two paths are 1 and 2, respectively. Hence, the i2−1 -th node is located in the second layer. It should be noted that the reference commands of the nodes in the latter layer are determined by those in the previous layers. For the agents in the Layer 1, their dynamic equations can be expressed as x ¨i1−k + αx˙ i1−k + xi1−k = ISωi1−k ,ζi1−k S0,xd (t) (19) 8
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Figure 2: Layers in the communication graph
where ωi1−k = 1, ζi1−k = α/2 and k = 1, 2, · · · , n1. According to Lemma 1, the system (19) will converge to xd at t = 2π/ωdi1−k , where ωdi1−k = q ωi1−k 1 − ζi21−k . That is, all nodes in Layer 1 will converge to xd at T1 =
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2π/ωdi1−k . For the nodes in Layer 2 which can only receive information from the leader and the i1−k -th node, its original reference command can be classed as a specific function with the format of G(t) in Eq. (9). Such a command will be equal to xd after t = T1 , hence, the node i2−k will q stay at the desired position xd after t = T1 +2π/ωdi2−k , where ωdi2−k = ωi2−k 1 − ζi22−k p with ωi2−k = di2−k + bi2−k and ζlk = α/(2ωi2−k ). There exists a time instant T2 = T1 + maxk∈{1,2,··· ,n2} 2π/ωdi2−k at which all nodes in the second layer will reach the desired position. Similarly, if the maximal convergence time of the nodes in Layer p is denoted by Tp , the nodes in the p + 1 layer will arrive at xd at Tp+1 = Tp + maxk∈{1,2,··· ,n(p+1)} 2π/ωdi(p+1)−k . In this way, all nodes in the graph will converge at a fixed time. Next, a method to calculate the convergence time of the multi-agent system q will be presented. The natural frequency of the i-th agent is ωdi =
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ωi 1 − ζi2 . According to the definition of the controller (18), the settling time of the controller (18) will be added by 2π/ωdi due to the existence of the i-th agent. In the consensus controller design, the edge weights are not relevant, hence, aij for the edge (j, i) ∈ E is set as 1. To analyze the convergence time of the multi-agent systems under the controller in Eq. (18), a weighted directed graph is defined by setting the weight for the edge (j, i) ∈ E as a ¯ij = 2π/ωdi . The weighted length of a path from node 0 to a node in the graph is defined as the summation of the weights of the
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edges in the path. Because the graph of concern has no cycle and contains at least one spanning tree, there must exist at least one node without child node. Such nodes are located at Layer m. Hence, the convergence time of the closed-loop multi-agent systems is equal to the maximal value of the weighted length of all the paths from node 0 to the nodes with no child node.
Remark 2. The controller for the i-th agent depends on the information of itself, its neighbors and the leader in the case of bi = 1, hence, the control algorithm (18) is distributed.
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Remark 3. ωdi can be expressed as q p ωdi = ωi 1 − ζi2 = di + bi − α2 /4
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Figure 3: Delayed convergence times by the nodes with different numbers of neighbors
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Remark 4. The controller in Eq. (18) is designed based on the shaped signal, which can be obtained by shaping all signals flowing through the communication edges. That is, the essence of the proposed controller design method is to place input shapers in all communication edges. For example, Fig. 4(a) indicates a directed graph in which the node 0 has directed paths to all other nodes and there is no cycle. Under the controller in Eq. (18), the communication graph is changed to the topology in Fig. 4(b), i.e., the ZVD shapers are added to all communication edges. Hence, in this case, the 10
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problem can be treated as the multi-agent systems with special communication delays. But the time-delays from the ZVD shaper are too long and the methods in the studies on the consensus control with time-delays [34, 35] are not applicable any more. Consequently, Theorem 1 presents a method to assess the performance of the controller (18) based on the analytical responses of the agents in the group.
Figure 4: Communication graph
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Remark 5. Based on the proposed controller (18), the closed-loop system is a linear multi-agent system with communication delays. However, as shown in the survey paper on distributed fixed-time control methods [27], all existing fixed-time consensus algorithms are developed based on sign function, absolute value function, involution and evolution, i.e., the closed-loop multi-agent systems with the existing fixed-time control methods are strongly nonlinear. Hence, the proposed controller is simpler than the existing ones. Furthermore, as shown in Theorem 1, the proposed controller can drive the multi-agent systems to the desired position at a fixed time for a specified communication graph, rather than within a fixed time.
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Theorem 2. If the agents with double-integrator dynamics are resting at different positions at the initial time, the following control law can drive the i-th agent to the desired position at a fixed time ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (21) +(di + bi )xi0 − (di + bi )ISωi ,ζi S0,xi0 (t) where xi0 is the initial position of the i-th agent. Furthermore, the convergence time of the multi-agent system is the maximal weighted length of all paths from node 0 to the nodes with no child node. 11
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Proof. If xi − xi0 is denoted as yi , the multi-agent systems under controller (21) can be rewritten as y¨i = −αy˙ i − (di + bi ) yi − ISωi ,ζi S0,xid (t) (22) −(di + bi )ISωi ,ζi S0,xi0 (t) Based on the result in Theorem 1, yi will stay at xd − xi0 , i.e., x = xd , after a certain time, which is the maximal weighted length of all paths from node 0 to the nodes with no child node.
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Remark 6. Compared with the controller (18), only additional feedforward loops to compensate for the initial position are added based on local information in the controller (21). Hence, Eq. (21) is a distributed consensus controller.
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Theorem 3. The distributed tracking for the multi-agent system with zero initial position and non-zero initial velocities can be achieved under the following controller ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (23) +(di + bi ) Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t)
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where tvi is the time when the velocity response of the i-th agent with initial velocity x˙ i0 is equal to zero for the first time and xvi is the value of the position response at t = tvi . The convergence time is the maximal value among the convergence time in Theorem 1, and tvi + 2π/ωdi , i = 1, · · · , N .
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Proof. Since all agents of concern are governed by linear dynamics, the command ISωi ,ζi S0,xid (t) from the neighbors and the leader in the case bi = 1 and the command Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t) used to compensate for the initial velocity can be discussed separately. The convergence time under the command ISωi ,ζi S0,xid (t) is the same as the result in Theorem 1. As shown in Lemma 4, the value of tvi is only determined by ωi and ζi , hence, the convergence time under the command Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t) is tvi + 2π/ωdi for the i-th agent. It can be concluded that the convergence time is the maximal value among the convergence time of the situation in Theorem 1, and tvi + 2π/ωdi , i = 1, · · · , N . Remark 7. The consensus controller (23) is distributed because the last two terms in Eq. (23) are determined by the initial velocity and the number of neighbors of the i-th agent and other terms are the same as those in Eq. (18). 12
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Remark 8. The distributed tracking problem with identical non-zero initial velocity, a special case of that in Theorem 3, can also be solved by controller (23). Furthermore, the leader-follower consensus problem with different initial positions and velocities can be solved by combining the results in Theorems 2 and 3 because the multi-agent systems in this work are linear. Remark 9. The input shapers are used to adjust the shape of the reference command for every node. The introduction of the input shapers has no influence on the stability of the closed-loop multi-agent systems. Moreover, it also transforms an asymptotically stable system into one with fixed-time stability.
4. Numerical simulations
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Remark 10. The settling times of the consensus controllers (18), (21) and (23) depend on the communication graph structure and local velocity feedback gain, but not the initial conditions. So the edge weight aij and the local velocity feedback gain α can be adjusted to drive the multi-agent systems to the desired position at the time specified in advance.
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In this section, only the controllers in Eqs. (21) and (23) are verified because the situation in Theorem 1 can be treated as a special case of Theorem 2 or Theorem 3. Consider six agents with double-integrator dynamics, the graph among which is shown in Fig. 4(a). It is obvious that the directed graph has spanning trees and does not contain any cycles. The control parameter α is set as 0.1. In order to verify the effectiveness of the controller (21), at the initial time, the six nodes are assumed to be static at −0.5, 0.5, 1.0, 1.5, 2.0 and 2.5, respectively. The responses of the closed-loop multi-agent systems are shown in Fig. 5. It can be observed that the controller with input shapers can complete the consensus task very well and the convergence time is about 25.2 s, which is determined by the path node 0→node 1→node 2→node 3→node 5 with four edges. It should be noted that both the path node 0→node 1→node 2→node 3→node 6 and the path node 0→node 1→node 2→node 4→node 6 contain four edges. However, node 6 has two neighbors, hence, the convergence time of node 6 is 23.3 s, which is less than the convergence time of node 5. From Fig. 5, it can be found that there exists nonmonotonic maneuvering in the responses of nodes 2∼6 in the first 10 s. The main reason can be explained by taking node 2 as an example. Since node 2 can only drive itself to the desired position by taking the position of node 1 as 13
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Figure 5: Responses of six agents with different initial positions with input shapers
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Next, to test the performance of the controller (23), all the positions of the nodes are chosen as zero at the initial time. The initial velocities of the six nodes are −0.5, 0.5, 1.0, 1.5, 2.0 and 2.5, respectively. As indicated in Fig. 6, the multi-agent systems driven by controller (23) can track the leader quickly. The settling time of the controller (23) is the same as that in Fig. 5 since for ∀i = 1, · · · , 6, tvi + ω2πdi is less than the weighted length of the path node 0→node 1→node 2→node 3→node 5. It is clear that the responses of the nodes 1∼6 are nonmonotonic. This is because the commands used to compensate for the initial velocity of node i do not start working before tvi for i = 1, · · · , 6.
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5. Experimental studies 5.1. Experimental setup The rotary servo system from Quanser Consulting Inc. is employed to verify the proposed controllers in Section 3. As shown in Fig. 7, the experimental setup mainly consists of a SRV02 Rotary Servo Base Unit with a rotary rigid arm, a Q2-USB Data Acquisition Device, an amplifier and a workstation. The rotation of the rigid arm is driven by a DC motor and is 14
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measured through an encoder on the servomotor. The control algorithms are coded in Matlab Simulink and implemented by the Quarc real-time software with the sampling rate of 1 kHz. Under a PD controller, the rigid arm may oscillate around the desired angular position. The oscillation frequency and damping ratio can be obtained through the response of the rigid arm. For the given gains, the response is shown in Fig. 8. Its frequency and damping ratio are 14.439 rad/s and 0.026, respectively. The dynamics of the rotary rigid arm with PD controller can be described by Eq. (1) with ω = 14.439 rad/s and ζ = 0.026. Hence, it can be used to simulate a node in the closed-loop multi-agent systems.
Figure 7: Experimental setup
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every node is, therefore, ordered to occupy the rotary rigid arm system at different moments to finish the experimental verification of the consensus controllers. This is reasonable because, as shown in Fig. 2, the graph can be divided in to several layers and the desired positions of the nodes in the latter layer are determined by the nodes in the previous layers. The experimental studies can be performed layer-by-layer and the study sequence of the nodes in the same layer is arbitrary. For example, based on Fig. 2, the communication graph shown in Fig. 4(a) can be divided into four layers shown in Fig. 9. The first step is to set the angular position of the leader as the reference command to obtain the response of the rotary rigid arm, i.e., the response of node 1. Second, the response of node 2 can be acquired by treating the trajectory of node 1 as the reference command. The responses of nodes 3, 4 and 5 can be generated in a similar way. The reference command of node 6 is the summation of the responses of nodes 3 and 4. This way, the experimental results of all the nodes can be obtained.
Figure 9: Four layers in experimental verification
5.2. Experimental results The graph shown in Fig. 9 is used in the experimental test. The angular position of the leader, i.e., the desired angle, is 1 rad. Similarly, only the controllers in Eqs. (21) and (23) are tested experimentally. 16
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Next, the effectiveness of the consensus controller in Eq. (23) will be verified. At the initial time, all the nodes are located at zero. The initial velocities of the six nodes are the same as those in Fig. 6. The experimental results are shown in Fig. 11. It is obvious that the distributed controller works well and all the agents converge to the desired angle without longtime oscillations. However, the responses are much different from those in the numerical simulation results shown in Fig. 6, especially at the beginning of the consensus task. The main reason is that the frequencies in the numerical simulation are much smaller than those in the experimental studies. For example, the frequencies of node 5 in the numerical simulation and experimental study are 1 rad/s and 14.439 rad/s, respectively. If the initial velocity is 2 (rad/s), the xvi in numerical simulation and experimental study are, respectively, 1.85 and 0.1331 rad, that is, with the same initial velocities, the value of xvi will be larger with smaller frequencies. Hence, before 17
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Figure 11: Experimental results of the six nodes with different initial velocities
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In this study, novel fixed-time consensus controllers are designed based on input shaping for multi-agent systems with double-integrator dynamics under leader-follower directed graphs. It is assumed that the leader is globally reachable and there is no cycle in the graph. By adding input shapers to the communication edges, a novel consensus control design method is proposed for cases with various initial conditions. The input shaper is introduced to alter the reference command of each node to cancel the long-time oscillations of the closed-loop multi-agent systems. Analyses show that the proposed controllers can converge at a fixed time, which can be calculated analytically. Furthermore, the settling time depends on the communication network structure and has no relationship with the initial conditions. The effectiveness of the distributed fixed-time controllers is verified through both numerical simulations and experimental studies, the results of which agree well with the theoretical ones. This study is a trial to solve the fixed-time consensus of multi-agent systems with double-integrator dynamics under directed graphs using the input shaping technique. Further study will involve
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Distributed fixed-time control under directed graph using input shaping Ti Chen, Jinjun Shan PII: DOI: Reference:
S0016-0032(19)30152-8 https://doi.org/10.1016/j.jfranklin.2018.11.039 FI 3820
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 April 2018 18 September 2018 10 November 2018
Please cite this article as: Ti Chen, Jinjun Shan, Distributed fixed-time control under directed graph using input shaping, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2018.11.039
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Ti Chena , Jinjun Shana,∗ a
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Distributed fixed-time control under directed graph using input shaping
Department of Earth and Space Science and Engineering, York University 4700 Keele St., Toronto, Canada, M3J 1P3
Abstract
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This paper presents novel fixed-time controllers for the distributed tracking of multi-agent systems with double-integrator dynamics based on the input shaping technique under directed graphs. It is assumed that there is no cycle in the directed graph with a globally reachable leader. Distributed fixed-time controllers are designed for cases with various initial conditions by placing input shapers in all communication edges in the graph. Numerical simulations and experimental studies are conducted to verify the effectiveness of the proposed controllers.
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Keywords: Distributed control; fixed-time control; input shaping; multi-agent systems; directed graphs. 1. Introduction
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Cooperative control of multi-agent systems has attracted significant attention due to its wide applications, such as formation flying, attitude alignment, flocking, payload transport, air traffic control, rendezvous and docking [1, 2, 3, 4, 5, 6]. One critical problem in the field of multi-agent systems is consensus, in which a team of agents are expected to reach an agreement by negotiating with their neighbors [7, 8, 9]. The key of consensus problem is to design an appropriate distributed protocol to actuate a group of agents in a cooperative fashion based on a communication graph. Since the second-order dynamics can describe many real physical systems, many researchers have investigated the consensus of second-order multi-agent systems from different perspectives [9, 10, 11, 12, 13, 14]. In ∗
Corresponding author. Email addresses: [email protected] (Ti Chen), [email protected] (Jinjun Shan)
Preprint submitted to Journal of The Franklin Institute
February 21, 2019
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previous studies [7, 11, 15, 9, 14], the convergence time of the consensus controller may be infinite because the controllers are asymptotically or exponentially convergent. Hence, the distributed finite-time or fixed-time control are proposed to drive multi-agent systems converging within a finite time. The main difference between these two control methods is that the finite-time convergence rate depends on the initial conditions and the fixed-time convergence rate does not [16, 17, 18]. Recently, some investigators have focused on distributed finite-time and fixed-time control problems [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. However, the controllers in these work are usually proven to converge to the desired state within a bounded time and contain several control parameters which have a significant effect on the performances of the controllers [27]. Furthermore, the controllers are strongly nonlinear, which will increase the computational burden of the individuals in the multi-agent systems with limited computing capabilities. As stated in [29], it is important and challenging to obtain a distributed controller with a specified settling time. Consequently, Liu and Zhao used Pontryagin’s maximum principle to solve the distributed specified time consensus control problem for a group of harmonic oscillators [29]. However, their controller has complex forms and there exist long-time oscillations before the convergence of the closed-loop systems is achieved. It should be noted that closed-loop second-order multi-agent systems with the feedback of relative positions and local velocities may act like an underdamped vibrational system of high dimension. A feedforward method, known as input shaping, can achieve zero residual vibration in principle. The main advantage of input shaping is that it only alters the shape of the reference command and no additional actuators and sensors are required [30, 31]. Hence, this study tries to solve the consensus problem of multiagent systems with double-integrator dynamics based on an input shaping technique. The main contribution of this study is the development of fixed-time consensus controllers based on an input shaping technique under directed graphs, which contain a spanning tree and have no cycle. Theoretical analyses, numerical simulations and experimental studies are presented to verify the distributed controllers. The main advantages of the proposed controller design method are (i) The designed controllers are very simple; (ii) The closed-loop multi-agent systems will converge at a fixed time, rather than within a finite time; (iii) The convergence time can be calculated analytically and has no relationship with the initial conditions. The remainder of this paper is organized as follows. Section 2 presents some basic knowledge of graph theory and input shaping technique. Fixed2
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time consensus control algorithms are designed in Section 3. Numerical simulations and experimental studies are given in Section 4 and Section 5, respectively. Finally, the conclusions are drawn in Section 6.
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2. Preliminaries
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2.1. Graph theory The information exchanges among the agents can be modeled by a graph G = (V, E), where V = {1, 2, · · · , N } is a nonempty, finite node set and E ⊂ V × V is a set of ordered pairs of nodes. Each element in E is called an edge. The edge (i, j) implies that the node j has access to the information of node i. For the edge (i, j), i is a neighbor of j, agent i is named as the parent node, and j is the child node. The set of the neighbors of node i is represented by Ni . A directed path from node k1 to kl is defined as a sequence of ordered communication edges (kr , kr+1 ) ∈ E, for r = 1, · · · , l −1. A cycle is defined as a directed path that starts and ends at the same node. The (rooted) directed tree is a directed graph where each node has exactly one parent node except for one node, known as the root, which has no parent node and has directed paths to all other nodes. A directed tree is called spanning when it connects all the nodes in the graph. A graph is said to have or contain a directed spanning tree if a subset of the edges forms a directed spanning tree, i.e., there exists at least one node with directed paths to all other nodes in the graph. The adjacency matrix of the directed graph G is represented by A = [aij ] ∈ RN ×N , where aii = 0, aij is a positive value if (j, i) ∈ E and aij = 0 otherwise. The weight aij in this study is set equal to 1Pif (j, i) ∈ E unless otherwise stated. The in-degree of the i-th node is di = N j=1 aij . To study the leader-follower tracking problem, it is assumed that the graph of concern consists of one static leader (node 0) and N follower agents. The leader adjacency matrix is represented by B = diag{b1 , · · · , bN }, where bi = 1 if the leader is the neighbor of the i-th node; otherwise bi = 0. The leader is said to be globally reachable if there exists at least one path from the leader to every follower in the graph. 2.2. Input shaping Input shaping can lead to zero or small vibration of flexible structures by shaping the reference command [31]. As shown in Fig. 1, it is realized by convolving the original command with a sequence of impulses, i.e., input shaper. In this work, a Zero Vibration and Derivative (ZVD) shaper will be employed due to its simplicity and robustness to system uncertainties. 3
Figure 1: Principle of input shaping
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Consider the underdamped oscillatory system governed by
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where ω, ζ and f are the frequency, the damping ratio and the reference command. The amplitudes and the acting times of the three impulses of ZVD shaper are 1 A0 = , t0 = 0 1 + 2K + K 2 2K π A1 = , t1 = (2) 1 + 2K + K 2 ωd 2 K 2π A2 = , t2 = 2 1 + 2K + K ωd ζπ p √ − where K = e 1−ζ 2 and ωd = ω 1 − ζ 2 . For an arbitrary signal g(t), the signal shaped by the ZVD shaper (2) is represented by IS g(t) . The ω,ζ detailed expression of ISω,ζ g(t) is ISω,ζ g(t) = A0 g(t) + A1 g(t − t1 ) + A2 g(t − t2 ) (3)
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Similarly, for an arbitrary signal g(t), the function Sτ,g (t) is defined as ( 0 t<τ Sτ,g (t) = (5) g(t) t ≥ τ Lemma 1. The underdamped oscillatory system (1) with zero initial states will stay at c after t = t2 under the shaped command ISω,ζ S0,c (t) . 4
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For the underdamped oscillatory system (1) under the shaped command ISω,ζ S0,c (t) , the response with zero initial state is
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Lemma 2. The underdamped oscillatory system (1) with zero initial states will rest at c after t = T +t2 under the shaped reference command ISω,ζ G(t) .
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Proof. The response of the underdamped oscillatory system after t = T under G(t) can be expressed as Rt −ζω(t−s) sin[ω (t − s)]ds ZG (t) = ω 2 0 G(s) d ωd e = ZGT + zT,c (t)
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Lemma 3. The underdamped oscillatory system (1) with initial position x0 and zero initial velocity will converge to x = 0 at t = t2 under the following reference command f = x0 − ISω,ζ S0,x0 (t) (12) Proof. The oscillatory system in such a situation can be expressed as
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Proof. The oscillation control with non-zero velocity can be considered as the oscillation control with non-zero position by treating the tv as the new initial time. Hence, x will stay at 0 after t = tv + t2 with the reference command (16) based on Lemma 3. 3. Consensus algorithms
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Remark 1. As shown in the survey [32], there are many applications of consensus algorithm of multi-agent systems with double-integrator dynamics. For example, Ref. [33] studies the formation maneuvers for a group of mobile robots, each of which is described as a double-integrator plant. Hence, this work focuses on the multi-agent systems with double-integrator dynamics.
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3.2. Fixed-time consensus algorithm with input shaping
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The parameter α in the controller (17) will greatly influence the responses of the closed-loop multi-agent systems. For engineering applications, α should be well tuned to meet various requirements to the best of its capabilities. In this section, based on input shaping technique, fixed-time consensus controllers of high performance are proposed to avoid the tuning of α. Three theorems in this section consider the following three situations: (i) consensus control of second-order multi-agent systems with identical initial position and zero initial velocity; (ii) consensus control of second-order multi-agent systems with different initial positions and zero initial velocity; and (iii) consensus control of second-order multi-agent systems with different initial velocities and zero initial positions. Without loss of generality, the identical initial position is set as zero. It should be noted that the situation with non-zero initial position can be converted to that with zero initial position by coordinate translation.
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Theorem 1. Under the following controller ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (18) √ where ωi = di + bi and ζi = α/(2ωi ), the distributed tracking for the second-order multi-agent system with zero initial conditions can be achieved at a fixed time. The convergence time is equal to the maximal value of the weighted length of all paths from node 0 to the nodes with no child node, where ¯ij = 2π/ωdi with q the weight for the edge (j, i) ∈ E is set as a
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Proof. The communication graph of concern contains a spanning tree with the leader as root and has no cycle, i.e., the information of each child node has no effect on its parent node. Hence, as shown in Fig. 2, the nodes in the graph can be divided into multiple layers. The sequence number of layer of each node is determined according to the maximal length of the path from the leader to the node. For example, the i2−1 -th node in Fig. 2 has two paths from the leader. The length of the two paths are 1 and 2, respectively. Hence, the i2−1 -th node is located in the second layer. It should be noted that the reference commands of the nodes in the latter layer are determined by those in the previous layers. For the agents in the Layer 1, their dynamic equations can be expressed as x ¨i1−k + αx˙ i1−k + xi1−k = ISωi1−k ,ζi1−k S0,xd (t) (19) 8
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Figure 2: Layers in the communication graph
where ωi1−k = 1, ζi1−k = α/2 and k = 1, 2, · · · , n1. According to Lemma 1, the system (19) will converge to xd at t = 2π/ωdi1−k , where ωdi1−k = q ωi1−k 1 − ζi21−k . That is, all nodes in Layer 1 will converge to xd at T1 =
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2π/ωdi1−k . For the nodes in Layer 2 which can only receive information from the leader and the i1−k -th node, its original reference command can be classed as a specific function with the format of G(t) in Eq. (9). Such a command will be equal to xd after t = T1 , hence, the node i2−k will q stay at the desired position xd after t = T1 +2π/ωdi2−k , where ωdi2−k = ωi2−k 1 − ζi22−k p with ωi2−k = di2−k + bi2−k and ζlk = α/(2ωi2−k ). There exists a time instant T2 = T1 + maxk∈{1,2,··· ,n2} 2π/ωdi2−k at which all nodes in the second layer will reach the desired position. Similarly, if the maximal convergence time of the nodes in Layer p is denoted by Tp , the nodes in the p + 1 layer will arrive at xd at Tp+1 = Tp + maxk∈{1,2,··· ,n(p+1)} 2π/ωdi(p+1)−k . In this way, all nodes in the graph will converge at a fixed time. Next, a method to calculate the convergence time of the multi-agent system q will be presented. The natural frequency of the i-th agent is ωdi =
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edges in the path. Because the graph of concern has no cycle and contains at least one spanning tree, there must exist at least one node without child node. Such nodes are located at Layer m. Hence, the convergence time of the closed-loop multi-agent systems is equal to the maximal value of the weighted length of all the paths from node 0 to the nodes with no child node.
Remark 2. The controller for the i-th agent depends on the information of itself, its neighbors and the leader in the case of bi = 1, hence, the control algorithm (18) is distributed.
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Remark 3. ωdi can be expressed as q p ωdi = ωi 1 − ζi2 = di + bi − α2 /4
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Remark 4. The controller in Eq. (18) is designed based on the shaped signal, which can be obtained by shaping all signals flowing through the communication edges. That is, the essence of the proposed controller design method is to place input shapers in all communication edges. For example, Fig. 4(a) indicates a directed graph in which the node 0 has directed paths to all other nodes and there is no cycle. Under the controller in Eq. (18), the communication graph is changed to the topology in Fig. 4(b), i.e., the ZVD shapers are added to all communication edges. Hence, in this case, the 10
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problem can be treated as the multi-agent systems with special communication delays. But the time-delays from the ZVD shaper are too long and the methods in the studies on the consensus control with time-delays [34, 35] are not applicable any more. Consequently, Theorem 1 presents a method to assess the performance of the controller (18) based on the analytical responses of the agents in the group.
Figure 4: Communication graph
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Remark 5. Based on the proposed controller (18), the closed-loop system is a linear multi-agent system with communication delays. However, as shown in the survey paper on distributed fixed-time control methods [27], all existing fixed-time consensus algorithms are developed based on sign function, absolute value function, involution and evolution, i.e., the closed-loop multi-agent systems with the existing fixed-time control methods are strongly nonlinear. Hence, the proposed controller is simpler than the existing ones. Furthermore, as shown in Theorem 1, the proposed controller can drive the multi-agent systems to the desired position at a fixed time for a specified communication graph, rather than within a fixed time.
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Theorem 2. If the agents with double-integrator dynamics are resting at different positions at the initial time, the following control law can drive the i-th agent to the desired position at a fixed time ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (21) +(di + bi )xi0 − (di + bi )ISωi ,ζi S0,xi0 (t) where xi0 is the initial position of the i-th agent. Furthermore, the convergence time of the multi-agent system is the maximal weighted length of all paths from node 0 to the nodes with no child node. 11
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Proof. If xi − xi0 is denoted as yi , the multi-agent systems under controller (21) can be rewritten as y¨i = −αy˙ i − (di + bi ) yi − ISωi ,ζi S0,xid (t) (22) −(di + bi )ISωi ,ζi S0,xi0 (t) Based on the result in Theorem 1, yi will stay at xd − xi0 , i.e., x = xd , after a certain time, which is the maximal weighted length of all paths from node 0 to the nodes with no child node.
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Remark 6. Compared with the controller (18), only additional feedforward loops to compensate for the initial position are added based on local information in the controller (21). Hence, Eq. (21) is a distributed consensus controller.
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Theorem 3. The distributed tracking for the multi-agent system with zero initial position and non-zero initial velocities can be achieved under the following controller ui = −αx˙ i − (di + bi ) xi − ISωi ,ζi S0,xid (t) (23) +(di + bi ) Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t)
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Proof. Since all agents of concern are governed by linear dynamics, the command ISωi ,ζi S0,xid (t) from the neighbors and the leader in the case bi = 1 and the command Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t) used to compensate for the initial velocity can be discussed separately. The convergence time under the command ISωi ,ζi S0,xid (t) is the same as the result in Theorem 1. As shown in Lemma 4, the value of tvi is only determined by ωi and ζi , hence, the convergence time under the command Stvi ,xvi (t) − ISωi ,ζi Stvi ,xvi (t) is tvi + 2π/ωdi for the i-th agent. It can be concluded that the convergence time is the maximal value among the convergence time of the situation in Theorem 1, and tvi + 2π/ωdi , i = 1, · · · , N . Remark 7. The consensus controller (23) is distributed because the last two terms in Eq. (23) are determined by the initial velocity and the number of neighbors of the i-th agent and other terms are the same as those in Eq. (18). 12
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Remark 8. The distributed tracking problem with identical non-zero initial velocity, a special case of that in Theorem 3, can also be solved by controller (23). Furthermore, the leader-follower consensus problem with different initial positions and velocities can be solved by combining the results in Theorems 2 and 3 because the multi-agent systems in this work are linear. Remark 9. The input shapers are used to adjust the shape of the reference command for every node. The introduction of the input shapers has no influence on the stability of the closed-loop multi-agent systems. Moreover, it also transforms an asymptotically stable system into one with fixed-time stability.
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Remark 10. The settling times of the consensus controllers (18), (21) and (23) depend on the communication graph structure and local velocity feedback gain, but not the initial conditions. So the edge weight aij and the local velocity feedback gain α can be adjusted to drive the multi-agent systems to the desired position at the time specified in advance.
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In this section, only the controllers in Eqs. (21) and (23) are verified because the situation in Theorem 1 can be treated as a special case of Theorem 2 or Theorem 3. Consider six agents with double-integrator dynamics, the graph among which is shown in Fig. 4(a). It is obvious that the directed graph has spanning trees and does not contain any cycles. The control parameter α is set as 0.1. In order to verify the effectiveness of the controller (21), at the initial time, the six nodes are assumed to be static at −0.5, 0.5, 1.0, 1.5, 2.0 and 2.5, respectively. The responses of the closed-loop multi-agent systems are shown in Fig. 5. It can be observed that the controller with input shapers can complete the consensus task very well and the convergence time is about 25.2 s, which is determined by the path node 0→node 1→node 2→node 3→node 5 with four edges. It should be noted that both the path node 0→node 1→node 2→node 3→node 6 and the path node 0→node 1→node 2→node 4→node 6 contain four edges. However, node 6 has two neighbors, hence, the convergence time of node 6 is 23.3 s, which is less than the convergence time of node 5. From Fig. 5, it can be found that there exists nonmonotonic maneuvering in the responses of nodes 2∼6 in the first 10 s. The main reason can be explained by taking node 2 as an example. Since node 2 can only drive itself to the desired position by taking the position of node 1 as 13
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measured through an encoder on the servomotor. The control algorithms are coded in Matlab Simulink and implemented by the Quarc real-time software with the sampling rate of 1 kHz. Under a PD controller, the rigid arm may oscillate around the desired angular position. The oscillation frequency and damping ratio can be obtained through the response of the rigid arm. For the given gains, the response is shown in Fig. 8. Its frequency and damping ratio are 14.439 rad/s and 0.026, respectively. The dynamics of the rotary rigid arm with PD controller can be described by Eq. (1) with ω = 14.439 rad/s and ζ = 0.026. Hence, it can be used to simulate a node in the closed-loop multi-agent systems.
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every node is, therefore, ordered to occupy the rotary rigid arm system at different moments to finish the experimental verification of the consensus controllers. This is reasonable because, as shown in Fig. 2, the graph can be divided in to several layers and the desired positions of the nodes in the latter layer are determined by the nodes in the previous layers. The experimental studies can be performed layer-by-layer and the study sequence of the nodes in the same layer is arbitrary. For example, based on Fig. 2, the communication graph shown in Fig. 4(a) can be divided into four layers shown in Fig. 9. The first step is to set the angular position of the leader as the reference command to obtain the response of the rotary rigid arm, i.e., the response of node 1. Second, the response of node 2 can be acquired by treating the trajectory of node 1 as the reference command. The responses of nodes 3, 4 and 5 can be generated in a similar way. The reference command of node 6 is the summation of the responses of nodes 3 and 4. This way, the experimental results of all the nodes can be obtained.
Figure 9: Four layers in experimental verification
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In this study, novel fixed-time consensus controllers are designed based on input shaping for multi-agent systems with double-integrator dynamics under leader-follower directed graphs. It is assumed that the leader is globally reachable and there is no cycle in the graph. By adding input shapers to the communication edges, a novel consensus control design method is proposed for cases with various initial conditions. The input shaper is introduced to alter the reference command of each node to cancel the long-time oscillations of the closed-loop multi-agent systems. Analyses show that the proposed controllers can converge at a fixed time, which can be calculated analytically. Furthermore, the settling time depends on the communication network structure and has no relationship with the initial conditions. The effectiveness of the distributed fixed-time controllers is verified through both numerical simulations and experimental studies, the results of which agree well with the theoretical ones. This study is a trial to solve the fixed-time consensus of multi-agent systems with double-integrator dynamics under directed graphs using the input shaping technique. Further study will involve
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