Communication parameter design for networked control systems with the slotted ALOHA access protocol
Introduction
In the past decade, considerable research results are reported on networked control systems (NCSs) in which a communication network is involved for transmitting measurement data and control signals. For a large-scale NCS, i.e., systems with tens of thousands of nodes (e.g., sensors, actuators and controllers), the network capacity needed can be very large. However, many NCSs have to operate under a limited network capacity, since a large network capacity requires high cost which is generally unaffordable in low-cost implementations. Unfortunately, low-capacity (or low-bandwidth) network channels may cause negative effects on the QoC (quality of control) performances of the NCS (such as stability and contraction rate) due to poor QoS (quality of service) performances, e.g., (low throughput, large delay and large packet dropout rate). To guarantee a certain QoS performance, it is important to resolve the conflicts in channel utilization. This can be achieved, on the one hand, by utilizing a powerful scheduling protocol in the media access control (MAC) layer to guarantee fare and efficient network access of all the nodes in the system [25]. On the other hand, an applicable control algorithm should be designed to deal with the restricted network bandwidth. Since the overall performance of the NCS depends not only on the control algorithm in the control system but also on the scheduling protocol in the communication network, this task is known as the control-communication co-design problem.
There are many papers studying the scheduling issue for networked control purpose [7], [16], [17], [18], [19]. In [17], a sensor scheduling method based on independent and identically distributed stochastic process is presented to guarantee system stability. Optimal scheduling strategies are given in [16] and [7], [18], where the former studies discrete-time systems while the latter two papers are for continuous-time systems. In [19], packet dropouts are considered in studying a stochastic scheduling protocol. However, the main concern of [7], [16], [17], [18], [19] is the system stability and controller design under different scheduling protocols, while little attention is paid to the design of scheduling protocols. In other words, these works didn't consider the co-design of control and communication. The co-design problem is rather complicated due to the strong couplings of the control part and the communication part. Some pacesetting research on this problem can be found in [1], [4], [11], [12], [26], [27], [35]. The paper [11] presents a co-design method by employing a dynamic scheduling rule for sensors and actuators. In [26], different scheduling protocols are studied and the maximum allowable transfer intervals are derived. The paper [27] derives the maximum allowable transfer interval for nonlinear control systems with static and dynamic schedulers. In [1], a rate monotonic scheduling problem is formulated for NCSs under both schedulability constraints and NCS-stability constraints. In [12], a co-design framework is proposed to schedule subsystems using the most regular binary sequences (MRBSs). Paper [4] proposes a co-design method for NCSs using periodical scheduling. In [35], an observer-type feedback controller is designed based on a pair of identified periodical scheduling sequences that preserve reachability and observability.
In communication networks, random protocols are often used in the MAC layer to manage communications among users or nodes. The ALOHA-type protocol is a typical random MAC protocol widely used in many engineering systems [20]. The QoS performance of ALOHA-type protocols has been fully investigated, especially the network throughput [2]. For slotted ALOHA protocol, the throughput is analyzed under different traffic conditions [36]. In [10], the throughput of a slotted ALOHA protocol is investigated by modeling the channel state as a Markov chain. It is shown in [14] that the throughput of a MAC protocol with different exponential backoff algorithms can vary with the initial contention window and the maximum number of retransmission.
In this paper, we are interested in the joint design of both communication parameters and controllers for linear time-invariant systems where all sensors transmit their data to the controller via a single wireless network channel. By dividing the sampling period into a certain number of time slots, we use the slotted ALOHA protocol to schedule the sensors. We aim to design parameters of the slotted ALOHA protocol (such as initial contention window and the maximum number of retransmission) along with the control algorithm to satisfy the QoC requirement of the controlled plant and the QoS requirement of the communication channel. For QoC, we consider mean square stability of the NCS, while for QoS we consider the throughput demanded by the stability of the NCS. It is the first time that the parameters of slotted ALOHA protocol are designed to meet control performance requirements.
The main contributions can be concluded as below:
- (1)
From the perspective of the control part, we obtained the lower bound of throughput satisfying the NCS's stability requirement, and we further provided a control design algorithm to reduce the throughput demanded by the control components.
- (2)
Under the above control design, we derived conditions for the key parameters of the slotted ALOHA protocol, such as the maximum number of retransmission, the initial contention window, and the length of each time slot, so that the mean square stability of the NCS can be guaranteed from both the control part and the communication part.
The remainder of this paper is organized as follows. Section 2 gives the problem formulation and some preliminaries. Section 3 derives the lower bound of throughput satisfying the mean square stability for a NCS where all sensors share a single channel via the slotted ALOHA protocol. Moreover, a smaller throughput is obtained by designing the optimal controller via pole assignment. In Section 4, the design of the parameters of the slotted ALOHA protocol is given. This section is highlighted by a control communication co-design algorithm. In Section 5, the effectiveness of the presented method is illustrated by numerical examples. Section 6 presents the conclusion and some future research topics.
Notation: Rn is the n dimension Euclidian space. Rm × n denotes the set of m × n real matrices. Let be a block diagonal matrix composed by the matrices . I is the identity matrix with compatible dimension. ( · )v denotes the vth power of a value. ‖ · ‖F is the Frobenius norm of a matrix. κF( · ) is the condition number of a matrix. ρ( · ) is the radius of a matrix. ( · )′ denotes the transpose of a matrix. And ⊗ denotes the Kronecker product. spec( · ) denotes the spectrum of a matrix. φ denotes the empty set.
Section snippets
Problem formulations and preliminaries
Consider a networked control system consisting of one controller and n decentralized sensors that communicate with the controller via a single wireless communication channel (see Fig. 1). Assume that sensors are clock driven and they independently measure the system states periodically. The sampled data are transmitted in the form of data packets. The controller computes control values based on the sampled measurement data arriving at the controller. Then, the actuator executes the control
Control design and throughput condition
In this section, we will design the control gain matrix via pole assignment, with which the requirement on the throughput of the slotted ALOHA protocol is derived to guarantee mean square stability of the NCS. We will further provide a controller design method which can minimize the throughput demanded by the NCS.
Note that the NCS in (11) can be viewed as a special case of the jump linear system with 2n subsystems, where n denotes the number of sensors. Then, the following lemma about the mean
Design of parameters for the slotted ALOHA
In this section, the key parameters in the slotted ALOHA protocol will be designed to guarantee the mean square stability of the NCS under the controller designed in Section 3.
Theorem 2 Under the control design in Section 3, the NCS (11) with the slotted ALOHA access protocol is mean square stable if the time slot τ, the initial contention window ω and the maximum retransmission number rmax , satisfy that
Simulations
In this section, several numerical examples are provided to illustrate the theoretical results derived in this paper. The NCS in (7) is assumed to have the following system parameters
Two sensors share a single channel, such that the sensors cannot simultaneously access the network at any time slot. The maximum throughput is for .
First, the control gain design using Algorithm 1 and the lower bound of throughput demanded by the NCS
Conclusions and future work
In this paper, we investigated a control and communication co-design method for discrete-time linear systems where sensors are decentralized over a single wireless network. To solve channel access problems of the sensors, we introduce the slotted ALOHA protocol so that the sensors can share the wireless network. The sampling period is divided into a certain number of time slots. The lower bound of network throughput demanded by the NCS is derived. And the parameters of the slotted ALOHA
Acknowledgments
This work is supported by the National Natural Science Foundation of China (grant no. 61273107, 61573077 and 61703445) and Dalian Leading Talent Project (grant no. 841252).
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