A Bandwidth Allocation Strategy based on Multirate Sampling Method in Networked Control System

A Bandwidth Allocation Strategy based on MultiMate Sampling Method in Networked Control System

Zhiwen Wang, Hongtao Sun2

Abstract. Considering of the resource constraint, a bandwidth allocation strategy with multirate sampling method is proposed to achieve the control and scheduling co-design of networked control system (NCS). The proposed bandwidth allocation strategy integrates static scheduling method with dynamic scheduling method and the final bandwidth of each closed loop is determined by three steps. First, the basic sampling period is selected based on the maximum available bandwidth of network. Then, the initial bandwidth allocation method is given on the basis of the sampling period which is determined by the different characteristics of each closed loop, and the central bandwidth can be obtained in this step. At last, the final bandwidth can also be adjusted dynamically on the basis of the state change of each plant. In order to verify the validity of this bandwidth allocation strategy, the stability of control system and the bandwidth utilization of network system are shown through simulation experiments.

Keywords: networked control system; co-design; bandwidth allocation; multirate sampling

1 Introduction

Recently, the research of networked control system (NCS) which sensors, controllers and actuators are connected through the communication network has attracted increasing attention [1]. There is a common view that the performance of NCS not only correlated with the control algorithms but also with the network resource allocation. In order to resolve the problems that caused by QoP and QoS mutually influence and restricted each other, Steo [2] put forward the thought of control and scheduling co-design for NCS which adopt an effective information scheduling scheme to achieve a minimum utilization of network resources against the background of guaranteeing the control performance.

In the control and scheduling co-design of NCS, control system and network system should be considered at the same time. Fortunately, the sampling period establishes the relationship between the QoP and QoS. A shorter sampling period can supply a better performance for control system that means bigger bandwidth occupancy and a longer one can supply a subprim performance for control system that means smaller bandwidth occupancy. In order to provide an appropriate bandwidth for each control loop on the background of bandwidth limitation, an appropriate sampling period should be determined [3, 4]. About research on co-design of networked control system, most works devote itself to the controller design on the context of the time varying sampling period which mainly focuses on the QoP [5]. But these works would attach more important to the “combination” rather than the “co-design”. Bandwidth management is considered as a feasible way to achieve the co-design of NCS because it takes the different characteristics of each single closed loop into account. An optimal bandwidth allocation strategy which determine the sampling period according to the state feedback of each axis is presented in [6]. A QoS-adaptive is designed in [7] which allocate the bandwidth to each control loop by measuring the network QoS parameters. Ultimately, the bandwidth allocation strategy is still focus on the adjustment of sampling periods. In fact, distribution and vast scale are the invariable property of NCS, it is impossible to adopt a constant sampling period to achieve the expected performance. So the multirate sampling method is the inevitable choice in the control and scheduling co-design of NCS. In addition, the rational arrangement of sampling period provides a feasible way to allocate the network bandwidth.

This paper is aim to elaborate the idea of multirate sampling method we proposed and organized as follows. The NCS model and related description are given in section 2.The bandwidth allocation strategy of initial allocation and redistribution which based on the multirate sampling method is given in section 3. The simulation results is shown to illustrate the above two methods in section 4 and some conclusions are obtained in section 5.

2 The NCS Model and Problem Description

The NCS considered here consisting of a collection of LTI plants. Each subsystem communicates with its controller via a shared communication network, as shown in Fig.1.

The continuous-time state-space model of the plant can be described by the following standard form:

(1)

where is the state of plant, is control input for loop, is the time delay of the closed loop, are real matrix and with appropriate dimensions.

Fig. 1The NCS model

In order to illustrate the basic problems about sampling period in detail, some assumptions are given as follows.

Assumption1: The sensors are time driven; controllers and actuators are event driven. The sample signal and the control signal is hold by the zero-order-holder (ZOH).

Assumption2: the data of each node is single packet transmission and no packet dropout occurred.

Assumption3: [8] In order to guarantee the stability of each closed loop, the maximum allow delay bound (MADB) which denotes as is given at first. If the sampling period is bigger than, the stability of control system will be not guaranteed. The above condition can be represented as (2) if the control system is stable:

(2)

Assumption4: The relationship between the quality of performance and bandwidth of the closed loop can be represented as:

(3)

The time delay which arises from the introduction of network is an important factor which should be considered and it can degrade the performance of control system. The sample data which is defined as the state signal arrives at the controller after some time delays which are called. Then, the controller computes the control signal by the data which is received from the sensors and it would generate anther time delay which is called the computation delay. However, it can be include in the network delay. The control signal is sent by the network from the controller and arises the additional the time delays which called. After, the control signal is arrive at the actuator and hold by the ZOH too. However, the time delay is mainly made up with the delay from the sensor to controller and the delay from controller to actuator for simplicity. The total time delay of each closed loop can be obtained by (4) and the detail is shown in Fig. 2.

(4)

Fig. 2 The time delay of each closed loop

Generally, the above sampling control system is a discrete system and calculates its control valve based on input valve at each sample period. So, Shanon sampling theory is the basic of the analysis. Sampling system (1) with can be represented as follows if the time delay is considered:

(5)

where,， and is the sampling period of the control loop, , is the same as (1).

The control loops are connected via network that the mass data need to be transmitted through it and the bandwidth limitation should be considered. Because of the different characteristic of each closed loop, the amount of data that need to be transmitted is also different. In addition, the bandwidth requirement is also different when the state of plant is changing at different times. Obviously, it would lead to the waste of bandwidth and bad performance of NCS if a constant or average sampling period is adopted. Hence, the main problem which needs to be dealt with in network is to determine the sampling period for each single closed loop. There are two problems to be solved in this paper. One is the method to determine the initial sampling period which based on the characteristic for a single closed loop and another is the method to adjust the sampling period dynamically which based on the state change of plant.

3 Bandwidth Allocation with Multirate Sampling Method

In this section, the method of multirate sampling period is illustrated and the bandwidth allocation strategy which based on multirate sampling method is given. The essence of the multirate sampling is to allocate a rational bandwidth for each closed loop from different perspectives. It makes sense that the stability is guaranteed for control subsystems. In the following, the detail of bandwidth allocation strategy which based on the multirate sampling method is given.

3.1 The Initial Bandwidth Allocation

The main purpose of the initial bandwidth allocation is to determine the rough bandwidth based on the different characteristics of each closed loop. Here, the bandwidth which determined by the initial allocation is called the central bandwidth and it can vary in some extent. However, the condition (2) is met through the initial bandwidth allocation and all closed loop can be stabilized simultaneously. At the same time, the time delay can be reduced to the condition (6).

(6)

From the perspective of network system, the total bandwidth which denotes as is invariable and the definition is given below.

Definition1: Denote the total bandwidth of network is the maximum amount of data can be transmitted in unit time which is represented as:

(7)

where is the minimum the sampling period which called the basic sampling period. Here, the total bandwidth is main consideration in view of the network system and it is tiny different from the bandwidth which combine the control system.

Then, the bandwidth of each closed loop which combines the control system can be defined as below [9].

(8)

where is the time spent on messaging required to perform operation and it is assumed to a constant for each closed loop.

However, the bandwidth summation of each closed loop should not exceed the total bandwidth and it can be represented as (9).

(9)

In order to determine the bandwidth of each closed loop with an appropriate sampling period, the basic sampling period can be used. A suitable positive integer is selected to determine the sampling period for each closed loop which can be represented as (10).

(10)

According to the requirement of each closed loop, the bandwidth can be determined in a different proportion which based on the basic sampling period. By the initial allocation, the bandwidth of each closed loop is given as follows.

(11)

Based on this method, the sampling period can be determined flexible according to the demand of each closed loop and the initial bandwidth allocation is completed. However, the conditions (2), (9) are still need to be satisfied.

3.2 The Dynamic Bandwidth Allocation

The main contribution of initial bandwidth allocation is to determine the central bandwidth. But it is still irrational to adopt a constant sampling period because the state is uncertain when the plants are in different situation. So, the time varying sampling period should be determined dynamically based on the state change of each plant when system running. The idea of this method is illustrated in Fig.3. is monitoring period which much bigger than a sampling period in following figure. If the state change of the plant is smoothly, a bigger sampling period should be adopted while a smaller bandwidth is needed. In turn, if the state change of the plant is severely, a smaller sampling period should be adopted while a bigger bandwidth is needed. In order to make full use of the bandwidth resource, the remaining bandwidth of the smaller requirement of closed loops should allocate to the bigger requirement ones. The detail method is given as follows.

Fig.3 The time varying sampling period

At first, another suitable positive integer is defined. The varying sampling period have a range of around the central bandwidthwhich can be represented as:

(12)

where.

Then, the relation between the control performance and the bandwidth is expressed as the function for each closed loop.

(13)

In this situation, the expected bandwidth of the closed loop should be set as if the expected control performance is achieved. Here, the definition of should be based on (2) and is the actual requirement bandwidth for closed loop.

Subsequently, the closed loops are divided into three sets:, and. For set, it can supply the extra bandwidth when the condition (13) is satisfied. For set, it needs extra bandwidth from the set. The other closed loops is working the central bandwidth and neither the bandwidth is required nor surplus bandwidth can supply.

If the expected control performance is met based on (13) and there is surplus bandwidth, it can be denoted as and represented as:

(14)

where can be calculated by (13).

Suppose there are closed loops achieve the expected control performance and some extra bandwidth is remained, the total surplus bandwidth can be calculated as follows.

(15)

where.

Similarly, if the expected control performance is not met based on (13) and there is extra bandwidth is required, it can be denoted as and represented as:

(16)

where can be obtained from (15).

Suppose there are closed loops do not achieve the expected control performance and some extra bandwidth is remained, the total extra bandwidth requirement can be calculated as follows.

(17)

where.

The relationship of the set, and satisfies:.

In the following, the main problem is how to allocate the bandwidth to the bandwidth. Here, another function is defined which can reflect the state change of the plant. The concept of the average change rate is used and it defined as:

(18)

where the is the first derivative and denote as..

The total average change rate can be given:

(19)

So the bandwidth redistribution can follow the idea of the proportional fairness [10]. The faster the state change, the bigger bandwidth should be allocated. It can be calculated by:

(20)

Through above analysis, the actual bandwidth of each closed loop is given as follows:

(21)

In particular, if, all closed loops can achieve the expected control performance, however, if, only the closed loop in set and set can achieve the expected control performance and the control performance of other loops only can be improved.

4 Simulation Results

For simplicity, two closed loops in NCS are supposed and the simulation result is shown with MALTAB. The controller design which based on augmented state space can be seen in [11].

Loop1: Loop2:

The network parameters are set in the following. The basic sampling period is.Let, the central bandwidth is.If the expected control performance of closed loop 1 is met when the bandwidth is equal to, then, the closed loop2 can obtain another bandwidth and it is equal to. At last, the initial state is the same in loop1 and loop2 and let it equal to.The stability is given by the following simulation results.

Fig.4 (1) The initial response of loop1. (2) The initial response of loop2.

After using this bandwidth width allocation strategy, the bandwidth consumption of loop1, loop2 and its sum is given by Fig.5.

Fig.5 The bandwidth consumption

The bandwidth of loop1 becomes smaller while the bandwidth of loop 2 becomes bigger. The reduce bandwidth of loop 1 can be added to loop 2. However, the total redistribution bandwidth is less than the total initial allocation. So it is feasible for this bandwidth allocation strategy.

5 Conclusions

In this paper, multirate sampling method is applied to bandwidth allocation. The control and network feature are considered at the same time. The multirate sampling method includes two aspects. On the one hand, the different sampling periods is employed to adapt the different characteristics of a single closed loop; on the other hand, the sampling period is also varying when the state of each plant is changing. The main purpose of this bandwidth allocation strategy is to adjust the sampling period and make it more correspond with the control and network feature. At last, a simple simulation experiment is given to demonstrate the validity of this bandwidth allocation strategy.

6 Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (2009JC11 and 2009QN120). Also, it is partially supported by Doctor Foundation (BS03200901) and Excellent Young Teachers Foundation (Q201012) of Lanzhou University of Technology.

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