Consensus Control of Dual-Rate Multi-Agent Systems With Quantized Communication

The present study discusses the consensus control of dual-rate multi-agent systems, where the sampling/communication interval of quantized data is an integer multiple of the control interval. A conventional multi-agent system uses a dynamic quantizer which is designed in a single-rate system where the intervals are equal, i.e., the control interval length is the same as the communication interval length. However, a dynamic quantizer designed in a dual-rate system is expected to have improved control performance. In the present study, an objective function is divided into a quantization term, which is related to the quantization error, and the remaining term. The proposed dual-rate dynamic quantizer is designed such that the quantization term is minimized. Finally, in numerical examples, the proposed dual-rate method is quantitatively evaluated by comparing with the conventional single-rate method, and the effectiveness of the proposed method is demonstrated.


I. INTRODUCTION
Owing to the advanced development of microprocessors and network technologies, most electronic devices are now able to be connected to networks. A networked control system is a core technology of network-based systems such as the Internet of Things and Cyber-Physical Systems [1].
In these control systems [2], [3], since the data are communicated through networks, completely lossless data transfer is realistically almost impossible because of the intrinsic properties of a network such as communication delay, packet loss, and data rate limits. Communication delay, in particular, is accrued due to physical restrictions and is also inevitable in cases of heavy computation load, such as for data analysis and generation. Furthermore, packet loss can occur, wherein a part of the transferred data is simply lost during the data communication. Data rate limits also affect the data transfer performance, albeit less directly. Since the network used for the data communication is a discrete time system, the transferred data are discretized. The discretization consists of the sampling and quantizing operations, the former is the discretization of the time domain, and the latter is the quantization/discretization of the signal value. A higher The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen . resolution on the sampling and the quantization results in better performance. However, this resolution level is restricted by the specification of data rate limits, to add to this problem, and the control system might be unstable because of low resolution.
The present study investigates the performance deterioration caused by data rate limits, where the sampling/communication interval and the quantization level are restricted, on the consensus control of multi-agent systems [4]. In static quantization, because the continuous signal is discretized by rounding off only the present value, the performance is strongly degraded by low resolution. To compensate for this type of performance degradation, a consensus control method for multi-agent systems using a probability quantizer has been proposed [5], [6]. In this method, however, since the value of the quantization error is divided by the sampling interval, the quantization error increases with long sampling/communication intervals. To address this issue, in a conventional consensus control method of multi-agent systems [7], the data transferred to networks are quantized using a dynamic quantizer [8], which does not arbitrarily decide the sampling interval. Because the conventional methods are designed in the single-rate system where the intervals are equal, i.e., the control interval length is the same as the communication interval length. Therefore, in order to improve the degradation of consensus performance due to the quantization error, frequent communication, i.e., short sampling intervals, is required. In the continuous time domain, a steady-state optimization method [9], the consensus conditions [10], and nonlinear systems [11] have been studied. Furthermore, an adaptive dynamic quantizer has been proposed for continuous-time second-order multi-agent systems with input quantized [12].
The present study investigates a design method for the consensus control of dual-rate multi-agent systems [13], where the sampling interval of the quantized controlled value is an integer multiple of the updating interval of the control input. Following the conventional method for dual-rate multi-agent systems [14], a controller was previously designed for optimizing not only at the sampling instants but also between them, and the implementation was therefore difficult. In the present study, the conventional single-rate design method [7] is extended to the dual-rate system so that the effect of the quantization error at the sampling instant is minimized.
In the present study, N and R denote the spaces of integer and real numbers, respectively, and ⊗ is the Kronecker product. 0 i,j is an i × j matrix of which all the elements are 0 and 1 i,j is an i × j matrix of which all the elements are 1.

II. CONTROLLED SYSTEM
Consider a multi-agent system which consists of N a (∈ N) agents, in which a network of communication is a non-directed graph. A block diagram for agent i (i = 1, · · · , N a ) is illustrated in Fig. 1, where P i and C i denote the controlled plant and the controller in agent i, respectively. Furthermore, the state of agent i is quantized by quantizer Q i , and the quantized value is transferred through the network. The dynamics of P i are described by an integral system as follows: where x i [k] ∈ R is the state, and u i [k] ∈ R is the control input to agent i, which is decided by C i designed as a distributed controller given as where h ∈ R is the controller gain, and x Qi [k] is the quantized value of the state of agent i, Here, a ij is the i, jth element of the adjacency matrix A, and is defined by Eq. (3) decided by the edge set of graph G, ε, as follows: The data used in the controller are under the following assumptions: Assumption 1: • The states of agents are quantized to be transferred in the network as digital signals.
• The state of an agent i, which is itself used in controller C i , is quantized because the sensor performance is assumed to be equal to the quantization level. All the states used in the controllers are therefore quantized data.
Quantizer Q i in Fig. 1 is implemented by the following dynamic quantizer [8]: where ζ i [k] ∈ R is the state of the quantizer, and the initial value is ζ i [0] = 0. Here, q denotes the static quantizer defined as where d ∈ R is the quantization level. Consequently, a is the maximum integer which is less than or equal to α ∈ R. Furthermore, A Q [k] ∈ R, B Q ∈ R and C Q ∈ R are the design parameters of the dynamic quantizer. In the present study, the parameters are designed to compensate for the control degradation caused by both the quantization and the long communication interval.
Regarding the quantization error in agent i as noise w i [k], Eq. (5) is arranged as Substituting the control law (Eq. (2)) into the plant (Eq. (1)), the next relational expression is given as Using the above-mentioned expression, all the agents are summarized as follows: where L is the graph Laplacian of G, defined as 97558 VOLUME 8, 2020 in which degree matrix D is given as follows: Eq. (9) can be rewritten in vector form as Additionally, the dynamic quantizer is also described in vector form as where

III. DUAL-RATE MULTI-AGENT SYSTEM
In the present study, because of the network performance constraints, the communication interval, where the state of agents is transferred through networks, is an integer (N > 1) multiple of the updating interval of the control input.
Since it is assumed that agents communicate with adjacent agents at every step (N = 1), X [k + i] (i = 1, · · · , N ) are listed as . . .
The equations are then summarized as follows: where the step length of K ∈ N is N times that of k, and K + 1 denotes K + N .
Thus, the control input is updated at every step, whereas the state of agents is measured every N steps. The system is therefore a dual-rate system, where the states of agents are not measured at steps K + j (j = 1, · · · , N − 1). The relationship between the interval of control input u i [k] and that of agent state x i [k] is illustrated in Fig. 2.
where D eg and A dj are the degree matrix and the adjacent matrix of the dual-rate network system. Next, the dynamics of the quantizers between the sampling instants are summarized as follows: N a (N −1) . . .
Since the future variables at steps K+j (j = 1, · · · , N − 1) are included in X f [K] and X f Q [K] used in Eq. (16) and VOLUME 8, 2020 Eq. (17), X f [K] and X f Q [K] can be rearranged using accessible variables. Using Eq. (16), X f [K] is written as follows: Expanding the elements which consist of X can then be rewritten as follows: , respectively, and their relational expressions are given as follows: is obtained as follows: Using Eq. (18) and Eq. (20), Eq. (16) can be rewritten as In the same way, Eq. (17) can also be rewritten as Note that W f [K] included in Eq. (21) and Eq. (22) represents unknown future noise. In the present study, the design parameters of the dynamic quantizer are designed so that the effect of W f [K] is minimized.

B. PARAMETER DESIGN OF DYNAMIC QUANTIZER
The objective function δ[K] is defined as Here, X Ideal [K] denotes the ideal state of X[K], in which non-quantized states of agents are transferred with the shortest communication interval (N = 1), and is set as The objective function used in the conventional method [14] evaluates the error between the actual state with the ideal state at non-communication steps from K − N + 1 to K − 1 as well as at communication step K. Consequently, extremely high control performance is demanded, and the solution cannot be easily obtained. In contrast, in the present study, a realistic objective function is used where the error at the communication step is evaluated, and the optimal design is therefore achieved. From Eq. (21), when the initial states are obtained, X[K] is calculated as follows: 97560 VOLUME 8, 2020 Using Eq. (25), the objective function is written as The equation is divided into two terms, δ QE [K], which is related to the quantization error, and the remaining δ ETC [K], as follows: The design objective of the present study is to minimize the effect of the quantization error, and the design parameters of the dynamic quantizer are decided based on the minimization of δ QE [K]. Theorem 1: The design parameters of an optimal dynamic quantizer are given as follows: where H ij denotes block element i, j of H, of which the block size is N a × N a . The optimal value of δ QE [K] is given as follows: δ Opt Proof: From Eq. (27), the supremum of δ QE [K] is described as To minimize the coefficient of the quantization error, Eq. (30) is rewritten as follows: is a lower unitriangular. The second term of the right-hand side in Eq. (31) is rewritten as follows: where As a result, the optimal parameters for minimizing the second term of the right-hand side in Eq. (31) are given by Eq. (28).
Additionally, A Q [k] when k = jN − 1 in Eq. (28) is obtained by using the fminsearch function in MathWorks MATLAB software and is given as   The simulation results are shown in Fig. 4. For comparison, the single-rate control results with a long control interval are also shown in Fig. 5, where both the control and communication intervals are 0.4 s and the time-invariant A Q is 1 [7].
To evaluate the performance quantitatively, 500 trials are simulated under the same conditions except for the initial states  the variances of each agent in the last 100 steps (290 s < t ≤ 300 s) is shown in Table 1. Furthermore, Table 2 shows the probability that the norm of the difference between the consensus value and the state of the agents is less than or equal to 1/10 of the quantization level d. These results demonstrate that the dual-rate system has superior consensus control performance compared to the single-rate system.

VI. CONCLUSION
In the present study, we propose an optimal method for a quantizer used in the consensus control of dual-rate multi-agent systems where the quantized states of agents are transferred through a network for which the sampling/communication interval is an integer multiple of the control interval. To minimize the effect of the quantization error, which deteriorates the consensus control performance of multi-agent systems, a dual-rate multi-agent system is designed using a dynamic quantizer. The design method of the dynamic quantizer in this dual-rate system is proposed such that the state of agent is close to the ideal state. Additionally, the effectiveness of the proposed method is demonstrated through numerical examples.
Since the proposed method is restricted to the dual-rate system, where all communication and control intervals are equal, in future studies we plan to extend the method to non-uniform interval systems. Furthermore, communication delay and packet loss, which occur almost universally in network communication systems, are not taken into account in the present study, and hence, must also be coped with or addressed accordingly. NOZOMU ARAKI received the Ph.D. degree from the University of Hyogo, Japan, in 2007. He is currently working as an Assistant Professor with the Department of Mechanical Systems, University of Hyogo. His research interests include control engineering, signal processing, and medical engineering.
YASUO KONISHI received the Ph.D. degree in mechanical engineering from Keio University, Tokyo, Japan, in 1989. He is currently a Professor with the University of Hyogo and is the Head of the Laboratory of Control Engineering. His research focuses on the game-theory approach to mechanical design, time-optimal control by binary input using MLDS, and positioning control with static and kinetic friction using hybrid controllers. VOLUME 8, 2020