# Joint Design of Control and Power Efficiency in Wireless Networked Control System

Authors: Yan Wang; Zhicheng Ji
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper presents a joint design method for wireless networked control system (WNCS) to balance both the demands of network service and the control performance. Since the problems of power consumption, communication reliability, and system stability exist simultaneously and interdependently in WNCS, most of the achieved results in the wireless network and wired networked control system cannot be used directly. To coordinate the three problems, sampling period is found to be the linking bridge. An adaptive sampling power efficiency algorithm is proposed to manage the power consumption such that it can meet the demands of network life span. The sampling period is designed to update periodically on the constraints of network schedulability and system stability. The convergence of the power efficiency algorithm is further proved. The sampling period is no longer a fixed value, however; thus, increasing the difficulty in modeling and controlling such a complicated time-varying system remains. In this work, a switched control system scheme is applied to model such a WNCS, and the effect of network-induced delay is considered. Switched feedback controllers are introduced to stabilize the WNCS, and some considerations on stability condition and the bounds of the update circle for renewing sampling period are discussed. A numerical example shows the effectiveness of the proposed method.


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## Body

## 1. Introduction

Wireless networked control systems are composed of distributed fields and plant devices (sensors, actuators, and controllers) interconnected via a wireless network [1]. The sensors, controllers, and actuators exchange information with one another through the wireless network. Sensors collect the status or outputs of plants at every sampling instant, packet the data with time-stamp, and then send the information to each corresponding controller via the wireless network. Controllers then compute the control variables as soon as they receive the newest data from their plants. After that, the control variables are sent to their corresponding actuators. Plants will not update their status until they receive the newest command from controllers. Wireless capabilities clearly provide opportunities to be more inventive in system organization [2]. The use of wireless network offers several advantages compared with a conventional wired networked control system in terms of cost, maintenance, scalability, and implementation flexibility. However, wireless nodes also have obvious weak points such as reliability and availability, particularly the power consumption. Power efficiency technologies are important research areas in wireless network applications.Most studies on WNCS analyze the effects of the wireless medium on overall closed-loop control systems in [3–11]. The first critical analysis on the use of wireless control is performed by Kumar in [3]. Kumar explores the impact of different protocol layers, from routing to physical, on control performance. In [4], Liu and Goldsmith analyzed the performance of a control system in terms of variation in data rate, error correcting codes, and different maximum bounds in the number of retransmissions. They also studied the impact of IEEE 802.11b medium access control. In [5], Colandairaj et al. discuss the impact of data flow on the stability of a closed-loop wireless network control system based on IEEE 802.11. Sampling rate adaption is proposed as a codesign solution to enhance control and wireless network performance. Different from our work in this paper, the purpose of adapting the sampling rate is to optimize bandwidth utilization not to save power. A robust and dynamic cross-layer communication architecture for wireless networked control system is presented in [6] by Israr et al. The protocol stack for WNCS comprises five layers. Each layer contributes to the overall goal of reliable, power-efficient communication. However, the control performance is not taken into account in their work.A number of studies related to power efficiency in wireless sensor networks (WSNs) and wireless networks have also been conducted in [12–18]. Current power efficiency research always falls into two categories: one is reducing the power consumption of each single node in the network and the other is balancing the power consumption of all the nodes in network. In [12], Colandairaj et al. present a dynamic power control strategy to minimize the communication power consumption of nodes by varying the transmission rate. A protocol intending to balance power consumption from the remaining battery power of the node-based routing policy is proposed by Liang and Yang in [13]. The nodes with greater remaining power are allocated with more communication tasks. In [15], Kim et al. propose a lifetime-based routing strategy in which the survival time is estimated according to the residual power and current ratio of power consumption. The path with the longest node survival time is selected for data transmission.Recently, limited studies in [19–21] are conducted on effective power saving strategies that specifically target at WNCS. Fischione et al. [20] propose a trade-off between wireless output power related to reliability and power consumption, where a physical characteristic model revealed quantitative relations with communication outage probability. They also focus on the lower layer optimal protocol design by considering the application layer requirements. Lino [21] discusses the optimal sleep mode control of wireless network nodes and proposes a trade-off method between control performance and power consumption. An optimal control strategy is applied to optimize the control period. In [22], event-predictive control for power saving of wireless networked control system is discussed. The key idea is to save power by maximizing the control interval with constrains of appropriate control performance. The proposed control method is rather complicated and requires online optimization mixed integer programming, which reduces the practicability. Thus, a simpler trade-off method for WNCS is required.Power consumption, communication reliability, and system stability exist simultaneously and react with one another in wireless networked control systems. Supposed that the three factors are interdependent, most results achieved in wireless network power management and wired networked control systems cannot be directly applied to WNCS. Thus, the motivation of this paper is to find a bridge which can link the three factors and make a balance among these factors through the bridge parameter, such that the overall satisfactory performance can be achieved. Fortunately, the sampling period of sensor node is found to be the bridge parameter. From this point, a joint design method of adaptive sampling power efficiency algorithm and coordinated control method are discussed in this paper. An updating rule of sampling period is presented to satisfy the demands of wireless life span under constrains of network schedulability and control system stability. Convergence of the power efficiency algorithm is further proved. Subsequently, the control system is a varying-period system since the sampling periods of sensors are time-varying. It is then modeled as a class of switched control system with two types of behavior in each update period. The switched control law is applied to stabilize the control system and stability conditions are discussed. Also, the choosing rule of update period is given.The remaining sections are organized as follows. Section2 is the problem formulation. Section 3 presents the adaptive power efficiency algorithm. Section 4 discusses the coordinated wireless networked control system modeling and design method. Numerical simulation is given in Section 5. Section 6 is the conclusion of this paper.


## 2. Problem Formulation

### 2.1. Description of WNCSs

Consider the wireless networked control systems shown in Figure1. There are three kinds of node in the system. Power consumption varies for the different kinds of node.Figure 1
Structure of wireless networked control systems.Besides, some necessary assumptions are made in this paper as the follows.Assumption 1.
The power of sensor and actuator nodes is supplied by battery while the power of controller is supplied by base station.Assumption 2.
The sensor and the actuator are clock driven while the controller is event driven. The sampling data is packed in one packet for transmission with time stamp.Assumption 3.
There exists transmission delay in the control loop, and it is assumed to be less than one sampling period.


### 2.2. Analysis of Power Consumption in WNCSs

Sensor power is consumed by three processes: data sampling, sample data reading by the ADC, and data transfer. The power consumption of the controller node is also consumed by three processes: receiving data, calculating control variables, and sending data packet. The power of the actuator is consumed by two processes: receiving data and D/A conversion. The power consumption of different tasks is shown in Table1 (see in [6]).Table 1
Power consumption of different tasks.
Task
Energy consumption (nAh)
Receive data
8
Transfer data
20
Read data
0.011
Sample data
1.08From the table, we get the following two conclusions:(1)
when the sensor transfers the same amount of data as the actuator receives, it will consume 2.5 times more power than the actuator will consume.(2)
data transfer consumes over 90% of the total sensor power consumption.Given that the power required by control nodes can be supplied by the base station in most situations, the power required by the sensor nodes and actuators can be provided by batteries. Thus, sensors utilize the maximum amount of power consumption in WNCS. Managing the power consumption of sensors is the key to prolonging the survival time of the wireless network. A direct and effective method is to reduce as much of the transmission consumption as possible by properly adjusting the amount of sample data. This principle is the basis of our power control algorithm.In wireless networks, the average power consumption of sending a packet can be described as [3]
(1)E-=b×packet_size+c,
where b is the coefficient of power consumption and c is the fixed power consumption of the node sending a data packet. According to Assumptions 1 and 2, a sensor node sends a packet to the corresponding controller at every sampling time. Given the lack of packet retransmission, the survival time of the sensor can be described as
(2)L=∑i=1sum(t(i)-t(i-1)),
where L is the survival time of the sensor node, sum is the maximum number of packets sent by the sensor given an initial power, sum=⌊Einit/E-⌋, ⌊x⌋ is the integral part of x, Einit is the given initial power of the sensor, t(i),  i∈{1,2,…,sum} is the time that the sensor node sends the ith data packet, and t(0) is the initial time.Survival time is dependent on transfer intervals. It can be prolonged by increasing the transfer intervals. Based on this premise as well as on the knowledge of the relationship between sampling period and control performance, we can cooperatively design the control and the network performances by adaptively adjusting the sampling period with a proper rule.


## 2.1. Description of WNCSs

Consider the wireless networked control systems shown in Figure1. There are three kinds of node in the system. Power consumption varies for the different kinds of node.Figure 1
Structure of wireless networked control systems.Besides, some necessary assumptions are made in this paper as the follows.Assumption 1.
The power of sensor and actuator nodes is supplied by battery while the power of controller is supplied by base station.Assumption 2.
The sensor and the actuator are clock driven while the controller is event driven. The sampling data is packed in one packet for transmission with time stamp.Assumption 3.
There exists transmission delay in the control loop, and it is assumed to be less than one sampling period.


## 2.2. Analysis of Power Consumption in WNCSs

Sensor power is consumed by three processes: data sampling, sample data reading by the ADC, and data transfer. The power consumption of the controller node is also consumed by three processes: receiving data, calculating control variables, and sending data packet. The power of the actuator is consumed by two processes: receiving data and D/A conversion. The power consumption of different tasks is shown in Table1 (see in [6]).Table 1
Power consumption of different tasks.
Task
Energy consumption (nAh)
Receive data
8
Transfer data
20
Read data
0.011
Sample data
1.08From the table, we get the following two conclusions:(1)
when the sensor transfers the same amount of data as the actuator receives, it will consume 2.5 times more power than the actuator will consume.(2)
data transfer consumes over 90% of the total sensor power consumption.Given that the power required by control nodes can be supplied by the base station in most situations, the power required by the sensor nodes and actuators can be provided by batteries. Thus, sensors utilize the maximum amount of power consumption in WNCS. Managing the power consumption of sensors is the key to prolonging the survival time of the wireless network. A direct and effective method is to reduce as much of the transmission consumption as possible by properly adjusting the amount of sample data. This principle is the basis of our power control algorithm.In wireless networks, the average power consumption of sending a packet can be described as [3]
(1)E-=b×packet_size+c,
where b is the coefficient of power consumption and c is the fixed power consumption of the node sending a data packet. According to Assumptions 1 and 2, a sensor node sends a packet to the corresponding controller at every sampling time. Given the lack of packet retransmission, the survival time of the sensor can be described as
(2)L=∑i=1sum(t(i)-t(i-1)),
where L is the survival time of the sensor node, sum is the maximum number of packets sent by the sensor given an initial power, sum=⌊Einit/E-⌋, ⌊x⌋ is the integral part of x, Einit is the given initial power of the sensor, t(i),  i∈{1,2,…,sum} is the time that the sensor node sends the ith data packet, and t(0) is the initial time.Survival time is dependent on transfer intervals. It can be prolonged by increasing the transfer intervals. Based on this premise as well as on the knowledge of the relationship between sampling period and control performance, we can cooperatively design the control and the network performances by adaptively adjusting the sampling period with a proper rule.


## 3. Adaptive Sampling Power Efficiency Algorithm

### 3.1. Update Rule of Sampling Period

For the consideration of simplicity and generality, we choose one of the control loops in the WNCS to describe the power control algorithm. LetTmin⁡ be the lower bound of the sampling period to guarantee the schedulability of the network and let Tmax⁡ be the upper bound of the sampling period to ensure system stability. Supposing that there are M candidate sampling periods for choosing in the allowable range between the maximum and minimum bounds, Tl∈[Tmin⁡,Tmax⁡],  l=1,2,…,M. Furthermore, an update period TM,TM>Tmax⁡ is designed for the sampling period renewal. The sampling period is renewed at each update instant as follows:
(3)T(j+1)=T(j)+ΔT(j+1)ΔT(j+1)=-sgn⁡(L^(j)-Le)·S(Tmax⁡-Tmin⁡)M-1,
where
(4)S=min⁡{⌊2(M-1)|L^(j)-Le|((EinitE--∑l=1Mkl(j)TMTl-TMTj)(Tmax⁡-Tmin⁡))-1⌋hhhhhhh×((EinitE--∑l=1Mkl(j)TMTl-TMTj)hhhhhhhlhh(EinitE--∑l=1Mkl(j)TMTl-TMTj)×(Tmax⁡-Tmin⁡))-1⌋,M-1};Le is the demand of sensor survival time; T(j) is the value of the sampling period in the jth updating interval; sgn⁡(x) is the signal of scalar x; ⌊x⌋ is the integral part of scalar x; kl(j) is the number of updating intervals in which the sampling period is Tl during the previous j updates; L^(j) is the current predicted survival time of sensor node calculated by the following formula:
(5)L^(j)=jTM+Erem(j)E-×T(j);Erem(j) is the current remaining power at the updating instant jTM.


### 3.2. Convergence of Power Efficiency Algorithm

Theorem 1.
For the WNCS described in Figure1, considering the update rule of adaptive sampling period (3), if the minimum sampling period satisfies
(6)Tmin⁡>Lesum,
then the actual survival time of sensor node will reach its expected value Le through the proposed rule of sampling period update.Proof.
According to Formula (1), we have
(7)L^(j+1)=(j+1)TM+Erem(j+1)E-×T(j+1).
The remaining power relationship at the two adjacent updating instants is given by(8)Erem(j+1)=Erem(j)-TMT(j)×E-.
We assume that the sensor node has sampled the plant with the sampling periodTl for kl(j) times from the initial to the current time instant. The remaining power can be calculated based on the initial power and consumed power:
(9)Erem(j)=Einit-∑l=1Mkl(j)TME-Tl.
Power control error is defined ase(j)=L^(j)-Le. According to formulas (7), (8), and (9), the dynamics of the error can be described as
(10)e(j+1)=e(j)+K(j)ΔT(j+1),
where K(j) is defined as K(j)=sum-∑l=1Mkl(j)(TM/Tl)-(TM/T(j)). At the current time instant, K(j) is a known variable.
The following Lyapunov function is introduced to prove convergence of the adaptive sampling power efficiency algorithm:(11)V(j)=12e2(j).
Considering Formula (10), it follows that
(12)ΔV(j)=V(j+1)-V(j)=K(j)ΔT(j+1)e(j)+12K2(j)ΔT2(j+1).
With formula (3), we obtain
(13)e(j)>0,ΔT(j+1)<0e(j)<0,ΔT(j+1)>0}⟹ΔT(j+1)e(j)<0,|ΔT(j+1)||e(j)|>12K(j)ΔT2(j+1).
From inequalities (13), it can be concluded that ΔV(j)<0. Furthermore, to guarantee that the minimum survival time can reach the expected value, the minimum sampling period is bounded by Tmin⁡≥Le/sum. Consequently, the error system is stable, and the survival time can converge to the expected value if the conditions in Theorem 1 are satisfied.Remark 2 (prediction of survival time).
The actual survival time is unavailable at the current instant because the power consumption is time varying. However, it can be predicted by the known information of the remaining power and sampling period at the current instant. Formula (5) provides the prediction and indicates that the survival time of the node will be L^(j) if the sensor node maintains the sampling period T(j) as unchanged from the current instant t=jTM. The prediction of the survival time serves as a substitute for real survival time and is used to calculate the new sampling period.Remark 3 (lower bound of sampling period).
Taking IEEE 802.11b as an example, the lower bound of the sampling period of sensorTmin⁡ can be determined by the following formula:
(14)Tmin⁡=max⁡{Lesum,Smc×2×NQ},
where (Smc×2×N)/Q, is the allowable minimum sampling period when the wireless network can be schedulable (see in [6]), and
(15)Q=SmcTDIFS+(CWmin⁡×TSIFS/2)+Tframe+TSIFS+TACK(16)Tframe=SPHYRl+SMAC+SmcRt,Tframe=SPHYRl+SMAC+SmcRt.Rt is the transmission rate, Rl is the legacy transmission rate, Smc is the measurement-control data size, SPHY is the size of control frame in physical layer, and SMAC is the data size of ACK and is a confirmed sign in the header of TCP data packet that confirms the received TCP message. TSIFS is the shortest time period of the 802.11b protocol for the interval of frames requiring immediate response. TDIFS is the time segment for the interval of the time frame of the distributed coordination function for sending in IEEE 802.11b. TPIFS is the time segment for the interval of the time frame of the centralized coordination function for sending, which satisfies
(17)TPIFS=TDIFS-Tslot,Tslot=TPIFS-TSIFS.
CW is the contention window. Wireless network parameters under the 802.11b direct sequence spread spectrum are shown in Table 2.Table 2
Wireless network parameters under 802.11b direct sequence spread spectrum.
Parameter
Value
R
t
11 Mbps
R
l
1 Mbps
T
SIFS, TDIFS, Tslot
10, 50, 20 us
S
MAC, SPHY
34, 24 bytes
S
mc
80 bytes
S
ACK
14 bytes + PHY header
CWmin, CWmax
32, 1024Remark 4 (upper bound of sampling period).
For a SISO system, the maximum sampling period can be obtained using Shannon sampling theorem. For a MIMO system, the following method can be used to obtain the upper bound. If the system feedback control law is given ahead, thenTmax⁡ can be obtained by solving the following optimal problem:
(18)max⁡Ts.thh∥eig(eAT+∫0TeATdtBK0)∥<1,
where A is the system matrix and B is the control input matrix. K0 is the Kalman gain, which satisfies
(19)Re(eig(A+BK0))<0,
where Re(υ) denotes the real part of υ and eig(X) denotes the eigenvalues of matrix X. Gain K0 can be determined by the pole assignment in the continuous time domain.However, the above optimal problem is difficult to solve directly. The following iteration method can be used to obtain the approximate optimal value ofTmax⁡:Step 1.
Letq=1, and the initial value of Tmax⁡ is Tmax⁡(q)=2Tmin⁡. If the condition can satisfy the constraints of the optimal problem, go to Step 2; or else, go to Step 3.Step 2.
Letq=q+1, and let Tmax⁡(q)=2Tmax⁡(q-1). If the condition can fulfill the constraints of the optimal problem, cycle Step 2; or else, end the iteration and then let Tmax⁡=Tmax⁡(q-1).Step 3.
LetTmax⁡(q)=(1+(3/4)q)Tmin⁡. If the condition still does not satisfy the constraints of the optimal problem, let q=q+1 and cycle Step 3; or else end the iteration and let Tmax⁡=Tmax⁡(q).


## 3.1. Update Rule of Sampling Period

For the consideration of simplicity and generality, we choose one of the control loops in the WNCS to describe the power control algorithm. LetTmin⁡ be the lower bound of the sampling period to guarantee the schedulability of the network and let Tmax⁡ be the upper bound of the sampling period to ensure system stability. Supposing that there are M candidate sampling periods for choosing in the allowable range between the maximum and minimum bounds, Tl∈[Tmin⁡,Tmax⁡],  l=1,2,…,M. Furthermore, an update period TM,TM>Tmax⁡ is designed for the sampling period renewal. The sampling period is renewed at each update instant as follows:
(3)T(j+1)=T(j)+ΔT(j+1)ΔT(j+1)=-sgn⁡(L^(j)-Le)·S(Tmax⁡-Tmin⁡)M-1,
where
(4)S=min⁡{⌊2(M-1)|L^(j)-Le|((EinitE--∑l=1Mkl(j)TMTl-TMTj)(Tmax⁡-Tmin⁡))-1⌋hhhhhhh×((EinitE--∑l=1Mkl(j)TMTl-TMTj)hhhhhhhlhh(EinitE--∑l=1Mkl(j)TMTl-TMTj)×(Tmax⁡-Tmin⁡))-1⌋,M-1};Le is the demand of sensor survival time; T(j) is the value of the sampling period in the jth updating interval; sgn⁡(x) is the signal of scalar x; ⌊x⌋ is the integral part of scalar x; kl(j) is the number of updating intervals in which the sampling period is Tl during the previous j updates; L^(j) is the current predicted survival time of sensor node calculated by the following formula:
(5)L^(j)=jTM+Erem(j)E-×T(j);Erem(j) is the current remaining power at the updating instant jTM.


## 3.2. Convergence of Power Efficiency Algorithm

Theorem 1.
For the WNCS described in Figure1, considering the update rule of adaptive sampling period (3), if the minimum sampling period satisfies
(6)Tmin⁡>Lesum,
then the actual survival time of sensor node will reach its expected value Le through the proposed rule of sampling period update.Proof.
According to Formula (1), we have
(7)L^(j+1)=(j+1)TM+Erem(j+1)E-×T(j+1).
The remaining power relationship at the two adjacent updating instants is given by(8)Erem(j+1)=Erem(j)-TMT(j)×E-.
We assume that the sensor node has sampled the plant with the sampling periodTl for kl(j) times from the initial to the current time instant. The remaining power can be calculated based on the initial power and consumed power:
(9)Erem(j)=Einit-∑l=1Mkl(j)TME-Tl.
Power control error is defined ase(j)=L^(j)-Le. According to formulas (7), (8), and (9), the dynamics of the error can be described as
(10)e(j+1)=e(j)+K(j)ΔT(j+1),
where K(j) is defined as K(j)=sum-∑l=1Mkl(j)(TM/Tl)-(TM/T(j)). At the current time instant, K(j) is a known variable.
The following Lyapunov function is introduced to prove convergence of the adaptive sampling power efficiency algorithm:(11)V(j)=12e2(j).
Considering Formula (10), it follows that
(12)ΔV(j)=V(j+1)-V(j)=K(j)ΔT(j+1)e(j)+12K2(j)ΔT2(j+1).
With formula (3), we obtain
(13)e(j)>0,ΔT(j+1)<0e(j)<0,ΔT(j+1)>0}⟹ΔT(j+1)e(j)<0,|ΔT(j+1)||e(j)|>12K(j)ΔT2(j+1).
From inequalities (13), it can be concluded that ΔV(j)<0. Furthermore, to guarantee that the minimum survival time can reach the expected value, the minimum sampling period is bounded by Tmin⁡≥Le/sum. Consequently, the error system is stable, and the survival time can converge to the expected value if the conditions in Theorem 1 are satisfied.Remark 2 (prediction of survival time).
The actual survival time is unavailable at the current instant because the power consumption is time varying. However, it can be predicted by the known information of the remaining power and sampling period at the current instant. Formula (5) provides the prediction and indicates that the survival time of the node will be L^(j) if the sensor node maintains the sampling period T(j) as unchanged from the current instant t=jTM. The prediction of the survival time serves as a substitute for real survival time and is used to calculate the new sampling period.Remark 3 (lower bound of sampling period).
Taking IEEE 802.11b as an example, the lower bound of the sampling period of sensorTmin⁡ can be determined by the following formula:
(14)Tmin⁡=max⁡{Lesum,Smc×2×NQ},
where (Smc×2×N)/Q, is the allowable minimum sampling period when the wireless network can be schedulable (see in [6]), and
(15)Q=SmcTDIFS+(CWmin⁡×TSIFS/2)+Tframe+TSIFS+TACK(16)Tframe=SPHYRl+SMAC+SmcRt,Tframe=SPHYRl+SMAC+SmcRt.Rt is the transmission rate, Rl is the legacy transmission rate, Smc is the measurement-control data size, SPHY is the size of control frame in physical layer, and SMAC is the data size of ACK and is a confirmed sign in the header of TCP data packet that confirms the received TCP message. TSIFS is the shortest time period of the 802.11b protocol for the interval of frames requiring immediate response. TDIFS is the time segment for the interval of the time frame of the distributed coordination function for sending in IEEE 802.11b. TPIFS is the time segment for the interval of the time frame of the centralized coordination function for sending, which satisfies
(17)TPIFS=TDIFS-Tslot,Tslot=TPIFS-TSIFS.
CW is the contention window. Wireless network parameters under the 802.11b direct sequence spread spectrum are shown in Table 2.Table 2
Wireless network parameters under 802.11b direct sequence spread spectrum.
Parameter
Value
R
t
11 Mbps
R
l
1 Mbps
T
SIFS, TDIFS, Tslot
10, 50, 20 us
S
MAC, SPHY
34, 24 bytes
S
mc
80 bytes
S
ACK
14 bytes + PHY header
CWmin, CWmax
32, 1024Remark 4 (upper bound of sampling period).
For a SISO system, the maximum sampling period can be obtained using Shannon sampling theorem. For a MIMO system, the following method can be used to obtain the upper bound. If the system feedback control law is given ahead, thenTmax⁡ can be obtained by solving the following optimal problem:
(18)max⁡Ts.thh∥eig(eAT+∫0TeATdtBK0)∥<1,
where A is the system matrix and B is the control input matrix. K0 is the Kalman gain, which satisfies
(19)Re(eig(A+BK0))<0,
where Re(υ) denotes the real part of υ and eig(X) denotes the eigenvalues of matrix X. Gain K0 can be determined by the pole assignment in the continuous time domain.However, the above optimal problem is difficult to solve directly. The following iteration method can be used to obtain the approximate optimal value ofTmax⁡:Step 1.
Letq=1, and the initial value of Tmax⁡ is Tmax⁡(q)=2Tmin⁡. If the condition can satisfy the constraints of the optimal problem, go to Step 2; or else, go to Step 3.Step 2.
Letq=q+1, and let Tmax⁡(q)=2Tmax⁡(q-1). If the condition can fulfill the constraints of the optimal problem, cycle Step 2; or else, end the iteration and then let Tmax⁡=Tmax⁡(q-1).Step 3.
LetTmax⁡(q)=(1+(3/4)q)Tmin⁡. If the condition still does not satisfy the constraints of the optimal problem, let q=q+1 and cycle Step 3; or else end the iteration and let Tmax⁡=Tmax⁡(q).


## 4. Modeling and Stability Analysis of Adaptive Sampling Period WNCS

### 4.1. Modeling of Adaptive Sampling Period WNCS

Considering the generality, one of the control loops in the WNCS is chosen as an example to illustrate the modeling approach. For the WNCS power consumption managed by algorithm in Section2, the dynamics of the control system is time-varying. Since the sampling period varies among the M candidates, the system can be considered a switched system with M modes from the perspective of the switched control system scheme. Each switching mode is corresponded to one of the candidates. According to Assumption 1, the sensor and actuator nodes are assumed to be clock-driven. It results that the switching occurs at some of the sampling instants. Additionally, according to Assumption 3, the inevitable existence of network-induced delay is taken into account and it is less than one sampling period.We consider a plant of control loopi in the WNCS with the following dynamics:
(20)x˙i(t)=Aixi(t)+Biui(t)yi(t)=Cixi(t),
where xi(t)∈Rn is the plant state, ui(t)∈Rm is the control input, and yi(t)∈Rp is the plant output. Ai∈Rn×n,  Bi∈Rn×m, and Ci∈Rp×n are the matrices of state, control input, and output matrices, respectively. Due to the generality of i, we omit the superscript i in the following model description and deduction.Discretizing system (20) with sampling rate Tl and considering network-induced delay less than one sampling period, the discrete dynamics of the open control loop can be described as
(21)x(k+1)=Φl(k)x(k)+Γl(k)u(k-1)y(k)=Cx(k),
where Φl(k)=eATl(k); Γl=∫0Tl(k)eAtdtB; and l(k) is the identification of sampling period at the kth sampling instant, l(k)∈Z→l={0,1,…,M-1}.For the discrete switched system (21), a switched state feedback controller is introduced in the following form:
(22)u(k)=Kγ(k)x(k),
where γ(k)∈Z→l={0,1,…,M-1} denotes the switching signal used in the control.Consequently, the closed-loop WNCS can be written as(23)[x(k+1)x(k)]=[Φl(k)Γl(k)Kγ(k)I0][x(k)x(k-1)].


### 4.2. Stability Analysis of Adaptive Sampling Period WNCS

The control gainsKγ(k) are assumed to be designed in such a way that the closed-loop system is asymptotically stable when γ(k)=l(k). Ideally, the switching signal used in control γ(k) is the same as the real signal l(k). However, this view is unrealistic in WNCS with network-induced delay, where γ(k)=l(k-1). The evolution over one sampling period can be described by two distinct types of behavior: the nominal and mixed mode sampling periods, as shown in Figure 2.Figure 2
Evolution over one period of switched WNCS with two types of behavior.(
1
)
Nominal sampling period is when the system evolution uses the right switching information:
(24)γ(k)=l(k-1),l(k)=l(k-1).(
2
)
Mixed mode sampling period is when the system command uses a wrong feedback gain
(25)γ(k)=l(k-1),l(k)≠l(k-1).DenotingΔm as the samples spent in the mixed mode, since the delay is less than one sampling period, it follows that Δm has a range of 0≤Δm≤1. Moreover, we assume that the system is controlled using the right gains for at least Δn samples before another switching occurs.The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feedback control gains when the switching signal is temporarily uncertain. Consider the scalarsθn>0,  θm>0 and the symmetric positive definite matrices Pln,  P(l,γ)m with (l,γ)∈l×l, which satisfy the following matrix inequalities:
(26)Φ-(l,γ)T(P(l,γ)m)-1Φ-(l,γ)-θmP(l,γ)m<0,∀(l,γ)∈l×lΦ-(l,l)TPlnΦ-(l,l)-θnPln<0,∀l∈l.Moreover, consider the following two scalars,(27)βn=maxl∈leigmax⁡(Pln)minl∈leigmin⁡(Pln),βm=max(l,γ)∈l×leigmax⁡(P(l,γ)m)min(l,γ)∈l×leigmin⁡(P(l,γ)m),
where eigmax⁡(X) and eigmin⁡(X) denote the maximum and the minimum eigenvalues of matrix X, respectively.Then, the stability of the closed-loop WNCS (23) can be guaranteed by the following theorem.Theorem 5.
Letθn*,  θm* be the solutions of the optimization problems θn*=min⁡θn and θm*=min⁡θm subject to matrix inequalities (26). Closed-loop system (23) is asymptotically stable if
(28)βn·βm·(θn*)Δn·θm*<1.Proof.
We consider the following Lyapunov functions:(29)Vn(k)=x-T(k)Plnx-(k),Vm(k)=x-T(k)P(l,γ)mx-(k).
Inequalities (13) show that
(30)Vn(k1)<(θn*)k1-k0Vn(k0).∀k∈[k0,k1), if the right switching signal is used in the control, l(k)=γ(k)=l(k0), then
(31)Vm(k1)<(θm*)k1-k0Vm(k0).∀k∈[k0,k1), if the switching signal in the control is not necessarily the same as the real signal, the pair l(k),  γ(k) takes an arbitrary value l(k0),  γ(k0) in l×l.
Since the controller gainsKl are designed to make the matrices Φ-(l,l) stable, the scalar θn is smaller than one, θn<1. The scalar θm may be greater than one, θm>1, since the gains Kγ are not designed to stabilize combinations other than Φ-(γ,γ).
Combining inequalities (26) yield
(32)min⁡l∈leigmin⁡(Pln)∥x-(k1)∥2<(θn*)k1-k0max⁡l∈leigmax⁡(Pln)∥x-(k0)∥2∀l∈lmin⁡(l,γ)∈l×leigmin⁡(P(l,γ)m)∥x-(k1)∥2<max⁡(l,γ)∈l×leigmax⁡(P(l,γ)m)∥x-(k0)∥2∀(l,γ)∈l×l.
With the definitions ofβn and βm, we can obtain the state vector norm decay or growth rate in a nominal regime and in an uncertain switching signal regime as follows:
(33)∥x-(k1)∥2<βn·(θn*)k1-k0∥x-(k0)∥2,∀l∈l∥x-(k1)∥2<βm·(θm*)k1-k0∥x-(k0)∥2,∀(l,γ)∈l×l.
Letksm describe the instants when the closed-loop system jumps to a mixed mode with uncertain switching signal, and let ksn be the instance when the system enters into a normal regime. With definitions and bounds of Δm and Δn, it follows that
(34)0≤Δm=ksn-ksm≤1,ks+1m-ksn≤Δn.
Without loss of generality, we assume that the system starts with a mixed-mode behavior,ksm<ksn. The system behavior in time interval k∈[ksm,ks+1m) is then analyzed.
Given thatl(k)≠γ(k),  ∀k∈[ksm,ksn) and l(k)=γ(k),  ∀k∈[ksn,ks+1m), using inequalities (26), the norm of the state at the end of the sequence can be upper bounded asd
(35)∥x-(ks+1m)∥2<βm·βn·(θm*)ksn-ksu·(θn*)ks+1m-ksn∥x-(ksm)∥2<∥x-(ks+1m)∥2<βm·βn·θm*·(θn*)Δn∥x-(ksm)∥2.
It indicates that closed loop (16) will be asymptotically stable if condition (28) in Theorem 5 is satisfied.


### 4.3. Choosing Rule of Update PeriodTM

Theorem 6.
Consider the WNCS with adaptive sampling period rule (3). The WNCS can be stabilized by the switched state feedback controllers (22), whereas the survival time can meet its expected value Le, if the update period TM satisfies the following conditions:
(36)(1)TM≥max⁡(Δin+1)[T1max⁡,T2max⁡,…,Tnmax⁡](2)⌊LeTM⌋·TM≤EinitE-min⁡{T1min⁡,T2min⁡,…,Tnmin⁡},
where Δin is the least nominal sampling period of control loop i and the solution of Theorem 5, [T1max⁡,T2max⁡,…,Tnmax⁡], is the least common multiple of the maximum sampling period for all the n control loops in the WNCS.Proof.
According to Theorem5, control loop i in the WNCS should stay at least Δin in a nominal sampling period in one updating interval for stability. With the consideration of 0≤Δim≤1, if the update period satisfies condition (1), all control loops in the WNCS will meet the demands of their least nominal sampling periods and will be stable. As a complete system, WNCS is composed of n control loops that will be stable when condition (1) is satisfied.
Condition (2) in Theorem 6 provides the upper bound of the updating period. For the power efficiency algorithm in Theorem 1, with the definition of kl(j), there exists
(37)∑l=1Mkl(j)TMTl+TMTj≤⌊LeTM⌋·TMmin⁡{T1min⁡,T2min⁡,…,Tnmin⁡}.
If condition (2) in Theorem 6 is satisfied, the item ((Einit/E-)-∑l=1Mkl(j)(TM/Tl)-(TM/Tj)) in the denominator of formula (3) will be greater than zero, which guarantees that formula (3) has physical meaning and is solvable. The survival time will, thus, reach the expected value by applying the power efficiency algorithm.


## 4.1. Modeling of Adaptive Sampling Period WNCS

Considering the generality, one of the control loops in the WNCS is chosen as an example to illustrate the modeling approach. For the WNCS power consumption managed by algorithm in Section2, the dynamics of the control system is time-varying. Since the sampling period varies among the M candidates, the system can be considered a switched system with M modes from the perspective of the switched control system scheme. Each switching mode is corresponded to one of the candidates. According to Assumption 1, the sensor and actuator nodes are assumed to be clock-driven. It results that the switching occurs at some of the sampling instants. Additionally, according to Assumption 3, the inevitable existence of network-induced delay is taken into account and it is less than one sampling period.We consider a plant of control loopi in the WNCS with the following dynamics:
(20)x˙i(t)=Aixi(t)+Biui(t)yi(t)=Cixi(t),
where xi(t)∈Rn is the plant state, ui(t)∈Rm is the control input, and yi(t)∈Rp is the plant output. Ai∈Rn×n,  Bi∈Rn×m, and Ci∈Rp×n are the matrices of state, control input, and output matrices, respectively. Due to the generality of i, we omit the superscript i in the following model description and deduction.Discretizing system (20) with sampling rate Tl and considering network-induced delay less than one sampling period, the discrete dynamics of the open control loop can be described as
(21)x(k+1)=Φl(k)x(k)+Γl(k)u(k-1)y(k)=Cx(k),
where Φl(k)=eATl(k); Γl=∫0Tl(k)eAtdtB; and l(k) is the identification of sampling period at the kth sampling instant, l(k)∈Z→l={0,1,…,M-1}.For the discrete switched system (21), a switched state feedback controller is introduced in the following form:
(22)u(k)=Kγ(k)x(k),
where γ(k)∈Z→l={0,1,…,M-1} denotes the switching signal used in the control.Consequently, the closed-loop WNCS can be written as(23)[x(k+1)x(k)]=[Φl(k)Γl(k)Kγ(k)I0][x(k)x(k-1)].


## 4.2. Stability Analysis of Adaptive Sampling Period WNCS

The control gainsKγ(k) are assumed to be designed in such a way that the closed-loop system is asymptotically stable when γ(k)=l(k). Ideally, the switching signal used in control γ(k) is the same as the real signal l(k). However, this view is unrealistic in WNCS with network-induced delay, where γ(k)=l(k-1). The evolution over one sampling period can be described by two distinct types of behavior: the nominal and mixed mode sampling periods, as shown in Figure 2.Figure 2
Evolution over one period of switched WNCS with two types of behavior.(
1
)
Nominal sampling period is when the system evolution uses the right switching information:
(24)γ(k)=l(k-1),l(k)=l(k-1).(
2
)
Mixed mode sampling period is when the system command uses a wrong feedback gain
(25)γ(k)=l(k-1),l(k)≠l(k-1).DenotingΔm as the samples spent in the mixed mode, since the delay is less than one sampling period, it follows that Δm has a range of 0≤Δm≤1. Moreover, we assume that the system is controlled using the right gains for at least Δn samples before another switching occurs.The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feedback control gains when the switching signal is temporarily uncertain. Consider the scalarsθn>0,  θm>0 and the symmetric positive definite matrices Pln,  P(l,γ)m with (l,γ)∈l×l, which satisfy the following matrix inequalities:
(26)Φ-(l,γ)T(P(l,γ)m)-1Φ-(l,γ)-θmP(l,γ)m<0,∀(l,γ)∈l×lΦ-(l,l)TPlnΦ-(l,l)-θnPln<0,∀l∈l.Moreover, consider the following two scalars,(27)βn=maxl∈leigmax⁡(Pln)minl∈leigmin⁡(Pln),βm=max(l,γ)∈l×leigmax⁡(P(l,γ)m)min(l,γ)∈l×leigmin⁡(P(l,γ)m),
where eigmax⁡(X) and eigmin⁡(X) denote the maximum and the minimum eigenvalues of matrix X, respectively.Then, the stability of the closed-loop WNCS (23) can be guaranteed by the following theorem.Theorem 5.
Letθn*,  θm* be the solutions of the optimization problems θn*=min⁡θn and θm*=min⁡θm subject to matrix inequalities (26). Closed-loop system (23) is asymptotically stable if
(28)βn·βm·(θn*)Δn·θm*<1.Proof.
We consider the following Lyapunov functions:(29)Vn(k)=x-T(k)Plnx-(k),Vm(k)=x-T(k)P(l,γ)mx-(k).
Inequalities (13) show that
(30)Vn(k1)<(θn*)k1-k0Vn(k0).∀k∈[k0,k1), if the right switching signal is used in the control, l(k)=γ(k)=l(k0), then
(31)Vm(k1)<(θm*)k1-k0Vm(k0).∀k∈[k0,k1), if the switching signal in the control is not necessarily the same as the real signal, the pair l(k),  γ(k) takes an arbitrary value l(k0),  γ(k0) in l×l.
Since the controller gainsKl are designed to make the matrices Φ-(l,l) stable, the scalar θn is smaller than one, θn<1. The scalar θm may be greater than one, θm>1, since the gains Kγ are not designed to stabilize combinations other than Φ-(γ,γ).
Combining inequalities (26) yield
(32)min⁡l∈leigmin⁡(Pln)∥x-(k1)∥2<(θn*)k1-k0max⁡l∈leigmax⁡(Pln)∥x-(k0)∥2∀l∈lmin⁡(l,γ)∈l×leigmin⁡(P(l,γ)m)∥x-(k1)∥2<max⁡(l,γ)∈l×leigmax⁡(P(l,γ)m)∥x-(k0)∥2∀(l,γ)∈l×l.
With the definitions ofβn and βm, we can obtain the state vector norm decay or growth rate in a nominal regime and in an uncertain switching signal regime as follows:
(33)∥x-(k1)∥2<βn·(θn*)k1-k0∥x-(k0)∥2,∀l∈l∥x-(k1)∥2<βm·(θm*)k1-k0∥x-(k0)∥2,∀(l,γ)∈l×l.
Letksm describe the instants when the closed-loop system jumps to a mixed mode with uncertain switching signal, and let ksn be the instance when the system enters into a normal regime. With definitions and bounds of Δm and Δn, it follows that
(34)0≤Δm=ksn-ksm≤1,ks+1m-ksn≤Δn.
Without loss of generality, we assume that the system starts with a mixed-mode behavior,ksm<ksn. The system behavior in time interval k∈[ksm,ks+1m) is then analyzed.
Given thatl(k)≠γ(k),  ∀k∈[ksm,ksn) and l(k)=γ(k),  ∀k∈[ksn,ks+1m), using inequalities (26), the norm of the state at the end of the sequence can be upper bounded asd
(35)∥x-(ks+1m)∥2<βm·βn·(θm*)ksn-ksu·(θn*)ks+1m-ksn∥x-(ksm)∥2<∥x-(ks+1m)∥2<βm·βn·θm*·(θn*)Δn∥x-(ksm)∥2.
It indicates that closed loop (16) will be asymptotically stable if condition (28) in Theorem 5 is satisfied.


## 4.3. Choosing Rule of Update PeriodTM

Theorem 6.
Consider the WNCS with adaptive sampling period rule (3). The WNCS can be stabilized by the switched state feedback controllers (22), whereas the survival time can meet its expected value Le, if the update period TM satisfies the following conditions:
(36)(1)TM≥max⁡(Δin+1)[T1max⁡,T2max⁡,…,Tnmax⁡](2)⌊LeTM⌋·TM≤EinitE-min⁡{T1min⁡,T2min⁡,…,Tnmin⁡},
where Δin is the least nominal sampling period of control loop i and the solution of Theorem 5, [T1max⁡,T2max⁡,…,Tnmax⁡], is the least common multiple of the maximum sampling period for all the n control loops in the WNCS.Proof.
According to Theorem5, control loop i in the WNCS should stay at least Δin in a nominal sampling period in one updating interval for stability. With the consideration of 0≤Δim≤1, if the update period satisfies condition (1), all control loops in the WNCS will meet the demands of their least nominal sampling periods and will be stable. As a complete system, WNCS is composed of n control loops that will be stable when condition (1) is satisfied.
Condition (2) in Theorem 6 provides the upper bound of the updating period. For the power efficiency algorithm in Theorem 1, with the definition of kl(j), there exists
(37)∑l=1Mkl(j)TMTl+TMTj≤⌊LeTM⌋·TMmin⁡{T1min⁡,T2min⁡,…,Tnmin⁡}.
If condition (2) in Theorem 6 is satisfied, the item ((Einit/E-)-∑l=1Mkl(j)(TM/Tl)-(TM/Tj)) in the denominator of formula (3) will be greater than zero, which guarantees that formula (3) has physical meaning and is solvable. The survival time will, thus, reach the expected value by applying the power efficiency algorithm.


## 5. Numerical Example

Simulation studies are performed on a WNCS closed by an IEEE 802.11b wireless network with two control loops sharing the network resources. The two control loops are assumed to have the same dynamics but with different initial conditions:(38)x˙i(t)=[-1-0.100.95]xi(t)+[-0.15-0.43]ui(t)hhhhhhhhhhhhhhhlhyi(t)=[11]xi(t),hhhhhhhhhhhhhhlhhhhhhhhhhhi=1,2x1(0)=[-55]T,x2(0)=[-510]T.The wireless network parameters are set as in Table2. The time delays in both loops are less than one sampling period. Some other necessary parameters are given as shown in Table 2.In Table3, the minimum and maximum sampling periods are computed by the methods in Section 3. Solving inequality (28) in Theorem 5, it yields Δ1n=Δ2n=8,  Δm=1. Also, the update period can be chosen by Theorem 6 as TM=150ms.Table 3
Simulation parameters.
Parameter
Value
Initial energy of both sensorsEinit
0.15 J
Sensor 1 expected survival timeLe1
75 s
Sensor 2 expected survival timeLe2
70 s
Unit transmission energyE-
25 dbm
Number of sampling period candidatesM
10
The minimum sampling period of loop1T1min⁡
1 ms
The maximum sampling period of loop1T1max⁡
256 ms
The minimum sampling period of loop2T2min⁡
1 ms
The maximum sampling period of loop2T2max⁡
256 msSolving matrix inequalities (18) and (19), the controller gains of ten switching modes can be obtained as in Table 4.Table 4
Controller gains of ten sampling modes.
Sampling period (/ms)
Controller gain
T
0
=
1
K
0
=
[
1117.6
1118.8
]
T
1
=
29
K
1
=
[
39.4837
40.5512
]
T
2
=
58
K
2
=
[
19.4678
20.4319
]
T
3
=
86
K
3
=
[
12.5668
13.4279
]
T
4
=
114
K
4
=
[
9.0353
9.7947
]
T
5
=
143
K
5
=
[
6.8663
7.5265
]
T
6
=
171
K
6
=
[
5.383
5.947
]
T
7
=
199
K
7
=
[
4.295
4.769
]
T
8
=
227
K
8
=
[
4.225
3.852
]
T
9
=
256
K
9
=
[
3.707
2.95
]With the above simulation parameters and controller gains, the curves of survival time prediction, power consumption, and control output of both control loops are shown in Figures3 to 8.Figure 3
Sensor 1 survival time prediction.Analyzing the simulation curves, we have the following results.(1)
Figures3, 6, 4, and 7 imply that both sensor 1 and sensor 2 can meet their expected survival time requirements.(2)
The power consumption in three cases of minimum sampling, maximum sampling, and the proposed adaptive sampling is compared in Figures4 and 7. It is obvious that the power is consumed much faster than the other two cases. In the case of the adaptive sampling, the power consumption varies according to both the requirements of the control performance and survival time. At the beginning, the power consumption curves vary quickly and more power are consumed because of the control systems not reaching stable yet. Then, after the control systems are settled, it tends to the minimum consumption rate which is corresponding to the maximum sampling period.(3)
Figures5 and 8 are the control outputs of loop1 and loop2 in the three cases mentioned above. The figures show that the control systems can be stabilized through the proposed joint design methods. In three cases, the adaptive sampling can get the control performances closed to the case of minimum sampling.(4)
Combining Figures4, 5, 7, and 8, it can be concluded that the proposed joint design method achieves a tradeoff between the performances of control and power efficiency.Figure 4
Sensor 1 power consumption comparison in three cases.Figure 5
Control loop1 output comparison in three cases.Figure 6
Sensor 2 survival time prediction.Figure 7
Sensor 2 power consumption comparison in three cases.Figure 8
Control loop2 output comparison in three cases.


## 6. Conclusion

This paper presents a joint design method for wireless networked control systems with limited power constraint. A power efficiency algorithm based on the adaptive sampling period is put forward to satisfy the demands of sensor survival time and system stability. Then, the time-varying control system with transmission delay is modeled as a switched system with uncertain switching signals. A dwell-time-dependent control method is discussed to guarantee the stability of WNCS. Simulation results show the effectiveness of the proposed method and indicate that it can achieve good tradeoff performance. Methods by which to reduce power consumption from the aspect of a single node as well as balancing power consumption from the global network perspective are worthy of further exploration.



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*Source: 257079-2014-08-06.xml*