robust energy efficient control for wireless sensor …jestec.taylors.edu.my/vol 16 issue 1 february...

Journal of Engineering Science and Technology Vol. 16, No. 1 (2021) 065 - 084 © School of Engineering, Taylor’s University

65

ROBUST ENERGY EFFICIENT CONTROL FOR WIRELESS SENSOR NETWORKS VIA UNITY SLIDING MODE

SAFANAH M. RAAFAT*, ALI M. MAHMOOD

Control and Systems Engineering Department, University of Technology, Al-Sina’a St., P.O. Box: 19006, Postal Code: 10066, Baghdad, Iraq

*Corresponding Author: [email protected]

Abstract

Energy management in Wireless Sensor Network (WSN) is one of the key interests in academia due to the finite power resources in sensor nodes. In fact, running out of power could lead finally to node failure, which affects negatively on network lifetime. In this paper, a robust system that contains a Unity Sliding Mode Control (USMC) combined with a stabilizing State Feedback Controller is proposed as a control methodology. The aim is to minimize the power consumption while maintaining the essential data rates in WSN in existence of uncertainties in addition to external disturbances accordingly. For the purpose of ensuring asymptotically stable system, the USMC is combined with optimal Linear Quadratic Regulator (LQR) controller. Moreover, the proposed approach is extended for robust tracking conditions. Simulation results reveal the robust performance of the proposed control strategy.

Keywords: Energy saving, LQR, Pole placement, Power control, Sliding mode control Wireless sensor network.

66 S. M. Raafat and A. M. Mahmood

Journal of Engineering Science and Technology February 2021, Vol. 16(1)

1. Introduction The Wireless Sensor Networks (WSNs) have been used broadly in multiple applications and will be one of the most important pillars in future Internet of Things (IoT) technology [1, 2]. WSNs include number of nodes that are placed in area of interest for collecting and transmitting the information to the main node or commonly known as the sink node for different purposes such as detection and monitoring [3]. Since the nodes are independent and have limited energy resources which communication wirelessly; the energy consumption of data transmission and reception has to be preserved at minimum level. Therefore, in WSN the energy efficiency is one of the key issues in network design [4, 5]. In other words, the sensor node must be able to regulate its transmission rate and the power level accordingly.

Several literature studies have been conducted to focus on the problem of energy efficiency and power consumption in WSNs. The main research trends that have been followed to save energy in WSN can be classified into six main categories [6, 7], which are: i) data reduction represented by minimizing the processing and transmitting of data such as data compression techniques; ii) topology control focuses on adjusting the transmission power for energy saving while keeping network connectivity; iii) energy efficient routing via minimizing the energy consumption of end-to-end transmission for instance, avoiding the low power nodes and sending data only to interested nodes; iv) duty cycling, which denotes the amount of time that the node stays active during their lifetime; v) protocol overhead reduction, which focuses on decreasing the overhead such as the transmission periods of messages and optimization of network flooding; vi) energy harvesting technology by extracting power from the environment via different sources including solar power, wind, magnetic fields, mechanical vibrations and temperature variations.

The work in this paper is mainly lies within the second category, which focuses on the adaptive power transmission among sensor nodes. In fact, when the sensor node needs to provide the desired data rates reliably, a certain level of Signal-to- Interference-plus-Noise-Ratio (SINR) or equivalently a certain thresholds level of transmission power is required. Hence, without a fair management between the data rate and the power, the performance in terms in energy consumption will be degraded particularly with varying of desired rates in the network. In this context, numerous power control techniques have been proposed using different types of objective measures. The authors in [7-9] have designed a distributed control strategy, based on the concepts of game theory for power minimization in WSNs. While the authors in [10-12] introduce power control approaches, which depends on the value QoS requirements.

The authors in [13-15] have used Kalman filter in SINR prediction and estimating the link quality for energy minimization in WSN. The studies of [16-18] have focused on application of machine learning in WSNs for reducing the consumed energy. It worth stating that a large number of power control studies have assumed a perfect SNR measurement. Hence, the authors [19, 20] have propose stochastic models for dynamic power control with noisy measurements in WSNs. Regarding the joint power and rate control in WSN, there are different strategies that have been proposed to describe the relation between the power and the data rate.

Robust Energy Efficient Control for Wireless Sensor Networks via Unity . . . . 67


One of common strategies can be represented by a look up table which involves the power / SINR levels with equivalent rate values However, examining a lookup table leads to increase the complexity which is one of challenges of this strategy. Another technique that can be used is to send a pilot frame with the transmitted data and waiting for feedback, which is denotes as an indicator for desired power of the node. This procedure is also suffering from overhead time or the latency [21].

In this paper, the focus is on a mathematical model, which can be referred to as the joint model. This model has been developed by the authors in [22, 23], which will be explained in detail later in the next section of the paper. The model introduces jointly a mathematical correlation between the SINR, the data rate and the level of congestion in the WSN. The interesting point regarding the joint model is that the power management in WSN can be studied and controlled based on the concepts of control theory. In this model, a combination of data rate, power and are modelled in an interconnected manner for supporting the variation of multiple data rates, saving the transmission power, and minimizing the packet loss. Hence, many researchers have utilized the joint model for application various control strategies as follows.

In [24], the authors used a convex optimization with the Linear Matrix Inequality (LMI) method as a power controller in WSN. The study has focused on uncertainties in the dynamics of the network. In addition, in [25], the power control of WSN has been presented based joint model using Nash game approach which is stochastic linear-quadratic Gaussian.

Furthermore, a group of research studies have investigated the joint model extensively using different controllers. In [26], the adaptive H∞ based on Linear Matrix Inequality (LMI) approach is used, in [27], Model Predictive Control (MPC) has been proposed and similarly the optimization problem has been derived by the LMI. While in [28], an optimal control based on Riccati algebraic equation is developed.

In [29], the power control in WSN is maintained based on of QoS Priority. In [30], the authors apply an adaptive controller that is based on Lyapunov-Krasovskii analysis and LMI. The variable structure control [31] can effectively deals with uncertainties and disturbances. Sliding mode control approach has been applied for many cases [32]. The unit control design procedure provides the sliding mode robustness against any possible uncertainty in the systems’ parameters along with disturbances. Lyapunov function is used to design the control law [33, 34]. In the present work the Unity Sliding Mode Control (USMC) method [35] is utilized to construct a robust WSN control system while considering the uncertainties due to the model of the system and disturbances.

The main contribution in this paper is the development of a robust power transmission control to save energy in WSN using USMC. In order to further optimize the consumed energy, optimal selection of asymptotic eigenvalues has been proposed.

The remains of the paper are structured as follows. In Section 2, the theoretical model of the WSN is presented. In Section 3, a unity sliding mode control is developed with two control techniques, which are the pole placement control and LQR, then the development of tracking controller is described as well. Simulation



and results discussion are shown in Section 4. Conclusions and potential future work will be provided in Section 5.

2. WSN Network Model In WSN, there is a trade-off between the consumed power and transmission data rates along with the level of congestion [24]. Due this fact, these three concepts are required to be mutually addressed for the purpose of enhancing the transmitted rate and minimizing the consumed power simultaneously. This management can be achieved by maintaining a minimum varying between the actual SINR levels with its desired value, where the SNR reflects the status of channel conditions. Therefore, it is highly needed to design an efficient power and rate control methodology, which is more energy-efficient than the existing approaches. In [22-24, 29], the mathematical model is formulated as shown in the following:

Consider a wireless sensor network topology illustrated in Fig. 1. The network involves three types of nodes namely the master node, which is responsible on receiving and processing the data from all nodes; the active node or the transmitting node i that is located within the cell of the master node and the interfering nodes j and l, which are located in other cells and their communication represent interference to node i which is measured in term of SINR for node i at time k as shown in Eq.(1).

𝑠𝑠𝑖𝑖(𝑘𝑘) = 𝐺𝐺𝑖𝑖𝑖𝑖(𝑘𝑘)𝑃𝑃𝑖𝑖(𝑘𝑘)

∑ 𝐺𝐺𝑖𝑖𝑖𝑖(𝑘𝑘)𝑃𝑃𝑖𝑖(𝑘𝑘)+𝜎𝜎𝑖𝑖2(𝑘𝑘)𝑖𝑖∈𝐹𝐹

(1)

where: 𝑠𝑠𝑖𝑖: the actual SINR for node I, 𝐺𝐺𝑖𝑖𝑖𝑖: the gain of the channel of jth to ith nodes, 𝑃𝑃𝑖𝑖 : the transmission power of ith node, 𝜎𝜎𝑖𝑖2: white Gaussian noise power. F: the set of interfering nodes with node i.

Fig. 1. A schematic representation of wireless sensor network [23].

For the purpose of Modeling of power and rate in WSN, firstly, it is assumed that the flow-rate control at each node is expressed in the following equation:

𝑓𝑓𝑖𝑖(𝑘𝑘 + 1) = 𝑓𝑓𝑖𝑖(𝑘𝑘) + 𝜇𝜇[𝑑𝑑𝑟𝑟(𝑘𝑘) − 𝑐𝑐(𝑘𝑘)𝑓𝑓𝑖𝑖(𝑘𝑘)] (2)

where, 𝑓𝑓𝑖𝑖(𝑘𝑘): denotes the flow rate at node i at time k, µ: is a positive step-size,

𝑐𝑐(𝑘𝑘): represents a measurement of congestion in network, 𝑑𝑑𝑟𝑟(𝑘𝑘): controls the value of rate increases for each iteration.



Eq. (2) illustrates the rate control mechanism in computer networks, where the source starts to increase the transmission rate via increasing the frame size at no congestion situations. This procedure is modelled by the additive term µ𝑑𝑑𝑟𝑟(k) and vice versa with the presence congestion, the sender node decreases its rate by the term µ𝑐𝑐(𝑘𝑘)𝑓𝑓𝑖𝑖(𝑘𝑘).

Now, the SINR level at the receiver with the desired data rate can be determined according to Shannon’s capacity formula Eq.(3). This formula shows the correlation between the flow rate 𝑓𝑓𝑖𝑖(𝑘𝑘) and the desired SINR level 𝑠𝑠𝑖𝑖′(𝑘𝑘):

𝑓𝑓𝑖𝑖(𝑘𝑘) = 12 log2�1+𝑠𝑠𝑖𝑖′(𝑘𝑘)� (3)

Eq. (3) is used to select 𝑠𝑠𝑖𝑖(𝑘𝑘) from the given value of 𝑓𝑓𝑖𝑖(𝑘𝑘) Now, it is needed to find a power sequence such that the actual SINR level 𝑠𝑠𝑖𝑖(𝑘𝑘), as calculated by Eq. (1), must approach the desired SINR level si′(𝑘𝑘) , as defined in Eq. (3). Using this fact and the update Eq. (2). Hence, the desired SINR, in dB scale, varies along with the rule in next equation.

�̅�𝑠𝑖𝑖′(𝑘𝑘 + 1) = [1 − µ𝑐𝑐(𝑘𝑘)]�̅�𝑠𝑖𝑖′(𝑘𝑘) + µ′𝑑𝑑𝑟𝑟(𝑘𝑘) (4)

where: µ′ = 20μ/ log2(10) and �̅�𝑠𝑖𝑖(𝑘𝑘) = 10 log 𝑠𝑠𝑖𝑖(𝑘𝑘). The objective is to choose the power control 𝑃𝑃𝑖𝑖(𝑘𝑘) such that the actual SINR levels 𝑠𝑠𝑖𝑖(𝑘𝑘) will follow the desired levels si′(𝑘𝑘).The scheme of achieving this objective derives from the ideas of power control of wireless communications system as shown in Fig. 2.

Fig. 2. Structure of adaptive power control system in WSN node.

According to Fig. 2, the received power is measured at the receiver and then compares this value with the desired reference power. Then the receiver conveys a one-bit signal, to the sender to either increase or decrease its power as follows:

𝑃𝑃�𝑖𝑖(𝑘𝑘 + 1) = 𝑃𝑃�𝑖𝑖(𝑘𝑘) + 𝛼𝛼𝑖𝑖[�̅�𝑠𝑖𝑖′(𝑘𝑘) − �̅�𝑠𝑖𝑖(𝑘𝑘)] (5)

where, the value of 𝛼𝛼𝑖𝑖 represents a step-size parameter and 𝑠𝑠𝑖𝑖(𝑘𝑘) is the actual SINR that is accomplished by 𝑃𝑃�𝑖𝑖(𝑘𝑘). Next let:

𝛽𝛽𝑖𝑖(𝑘𝑘) = 𝐺𝐺𝑖𝑖𝑖𝑖(𝑘𝑘)

∑ 𝐺𝐺𝑖𝑖𝑖𝑖(𝑘𝑘)𝑃𝑃𝑖𝑖(𝑘𝑘)+𝜎𝜎𝑖𝑖2(𝑘𝑘)𝑖𝑖𝑗𝑗𝑗𝑗

which refers to scaling factor that controls how 𝑃𝑃𝑖𝑖(𝑘𝑘) affects 𝑠𝑠𝑖𝑖(𝑘𝑘) in Eq. (5), 𝑠𝑠𝑖𝑖(𝑘𝑘) = 𝛽𝛽𝑖𝑖(𝑘𝑘)𝑃𝑃𝑖𝑖(𝑘𝑘) or, equivalently, in dB scale,

�̅�𝑠𝑖𝑖(𝑘𝑘) = �̅�𝛽𝑖𝑖(𝑘𝑘) + 𝑃𝑃�𝑖𝑖(𝑘𝑘) (6)



�̅�𝛽𝑖𝑖(𝑘𝑘) represents the effective channel gains. Then the value �̅�𝛽𝑖𝑖(𝑘𝑘) is assumed to vary according to the following expression: �̅�𝛽𝑖𝑖(𝑘𝑘 + 1) = �̅�𝛽𝑖𝑖(𝑘𝑘) + 𝑛𝑛𝑖𝑖(𝑘𝑘) , where, 𝑛𝑛𝑖𝑖(𝑘𝑘) refers to zero-mean disturbance of variance 𝜎𝜎𝑛𝑛2 which is independent of 𝑃𝑃�𝑖𝑖(𝑘𝑘). Then substitute model of �̅�𝛽𝑖𝑖(𝑘𝑘) in Eq. (6), then the achieved 𝑠𝑠𝑖𝑖(𝑘𝑘) will be varied as follows:

�̅�𝑠𝑖𝑖(𝑘𝑘 + 1) = (1 − 𝛼𝛼𝑖𝑖) �̅�𝑠𝑖𝑖(𝑘𝑘) + 𝛼𝛼𝑖𝑖�̅�𝑠𝑖𝑖′(𝑘𝑘) + 𝑛𝑛𝑖𝑖(𝑘𝑘) (7)

The key objective is to formulate a power control sequence 𝑃𝑃𝑖𝑖(𝑘𝑘) where the actual SINR level 𝑠𝑠𝑖𝑖(𝑘𝑘) illustrated in Eq. (7) will follow the desired SINR level 𝑠𝑠𝑖𝑖′(𝑘𝑘) in Eq.(4). To address this requirement, the control problem is formulated as follows. First, for the simplicity of notation the node index i is dropped. Second, two-dimensional state vector is introduced:

𝑥𝑥𝑑𝑑 ≜ ��̅�𝑠(𝑘𝑘)�̅�𝑠′(𝑘𝑘)� (8)

Then, by combining Eq. (4) and Eq. (7), the following state-space model is achieved:

𝑥𝑥𝑑𝑑(𝑘𝑘 + 1) = 𝐴𝐴𝑥𝑥𝑑𝑑(𝑘𝑘) + 𝐵𝐵𝑢𝑢𝑑𝑑(𝑘𝑘) + 𝑤𝑤𝑑𝑑(𝑘𝑘)𝑦𝑦𝑑𝑑 = 𝐶𝐶𝑥𝑥𝑑𝑑

� (9)

where,

A = �1−∝ ∝0 1 − 𝜇𝜇𝑐𝑐(𝑘𝑘)� , 𝐵𝐵 = �𝑏𝑏𝑝𝑝𝑏𝑏𝑓𝑓� , 𝐶𝐶 = �1 00 1� ,𝑤𝑤𝑑𝑑 = �

𝑛𝑛(𝑘𝑘)𝜇𝜇′𝑑𝑑(𝑘𝑘)� ,

In this paper, it is necessary to convert the discrete-time model Eq. (9) to an equivalent continuous-time model in order to apply the robust USMC method. The corresponding continuous state equation is:

�̇�𝑥𝑐𝑐 = 𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐(𝑡𝑡) + 𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐(𝑡𝑡) + 𝑤𝑤(𝑡𝑡)𝑦𝑦𝑐𝑐 = 𝐶𝐶𝑐𝑐𝑥𝑥𝑐𝑐

𝑥𝑥𝑐𝑐(0) = 𝑥𝑥𝑑𝑑(0) � (10)

where 𝑥𝑥𝑑𝑑(𝑘𝑘𝑘𝑘) ≅ 𝑥𝑥𝑐𝑐(𝑡𝑡) at 𝑡𝑡 = 𝑘𝑘𝑘𝑘 𝑎𝑎𝑛𝑛𝑑𝑑 𝑢𝑢𝑑𝑑(𝑘𝑘𝑘𝑘) = 𝑢𝑢𝑐𝑐(𝑡𝑡) 𝑎𝑎𝑡𝑡 𝑡𝑡 = 𝑘𝑘𝑘𝑘 , T is the sampling interval.

In addition, the control signal [36]

𝑢𝑢𝑐𝑐(𝑡𝑡) ≅ 𝑢𝑢𝑑𝑑(𝑘𝑘𝑘𝑘) +𝑢𝑢𝑑𝑑(𝑘𝑘𝑘𝑘+𝑘𝑘)−𝑢𝑢𝑑𝑑(𝑘𝑘)

𝑘𝑘(𝑡𝑡 − 𝑘𝑘𝑘𝑘) (11)

𝑘𝑘 = 0,1, 2, … , 𝑘𝑘𝑘𝑘 ≤ 𝑡𝑡 ≤ (𝑘𝑘 + 1)𝑘𝑘 where 𝐴𝐴𝑐𝑐 ≅

2𝑘𝑘

(𝐴𝐴 − 𝐼𝐼𝑛𝑛)(𝐴𝐴 + 𝐼𝐼𝑛𝑛)−1 and 𝐵𝐵𝑐𝑐 ≅12𝑘𝑘�𝐼𝐼𝑛𝑛 −

12𝐴𝐴𝑐𝑐𝑘𝑘�𝐵𝐵

3. The Proposed Control Methodology Sliding Mode Control (SMC) is used as a powerful control algorithm combined with either of the following two control techniques: pole placement or basic optimal control (as LQR). The SMC procedure is formulated in the presence of matched uncertainties that can be associated with the WSN model as well as external disturbance. Figure 3 illustrates the proposed control approach. In the followings, detailed description of each part of the controller will be described.



Fig. 3. Block diagram of the applied control method.

3.1. Basic stabilizing controller design. Let the pair A, B are controllable. Accordingly, a simple state feedback control, as pole placement can be applied. On the other hand, stable closed loop eigenvalues can also be obtained by optimal control approach, where the linear WSN model can be asymptotically stable. Consequently, the zero-input response of the WSN tends to zero (xc(t) → 0) immediately. The less significant the energy related to the state variables the more quickly they tend to zero. Therefore, rapid transitory can be realized by minimizing the energy. For the following index for LQR problem:

𝐽𝐽 = ∫ (𝑥𝑥𝑐𝑐𝑘𝑘𝑄𝑄𝑥𝑥𝑐𝑐 + 𝑢𝑢𝑐𝑐𝑘𝑘𝑅𝑅𝑢𝑢𝑐𝑐)𝑑𝑑𝑡𝑡∞0 (12)

Note that Eq. (12) has two terms of energy: ∫ 𝑥𝑥𝑐𝑐𝑘𝑘𝑄𝑄𝑥𝑥𝑐𝑐 𝑑𝑑𝑡𝑡 ∞0 is associated to the

system states, whereas ∫ 𝑢𝑢𝑐𝑐𝑘𝑘𝑅𝑅𝑢𝑢𝑐𝑐 𝑑𝑑𝑡𝑡 ∞0 is associated to the system input. The

weighting matrix Q is a positive semi-definite symmetric matrix, while R is a positive definite symmetric matrix. The related Riccati equation with this problem is formulated as [37]:

𝐴𝐴𝑐𝑐𝑘𝑘𝑃𝑃1 + 𝑃𝑃1𝐴𝐴𝑐𝑐 − 𝑃𝑃1𝐵𝐵𝑐𝑐𝐵𝐵𝑐𝑐𝑘𝑘𝑃𝑃1 + 𝐶𝐶𝑐𝑐𝑘𝑘𝐶𝐶𝑐𝑐 = 0 (13)

where 𝑃𝑃1 = 𝑃𝑃1𝑘𝑘 ≥ 0

The associated Hamiltonian matrix is:

𝐻𝐻 = � 𝐴𝐴𝑐𝑐 −𝐵𝐵𝑐𝑐𝐵𝐵𝑐𝑐𝑘𝑘

−𝐶𝐶𝑐𝑐𝑘𝑘𝐶𝐶𝑐𝑐 −𝐴𝐴𝑐𝑐𝑘𝑘� (14)

The resulted control law is:

uo =−K𝑥𝑥𝑐𝑐 (15)

where, 𝐾𝐾 ∈ ℛ𝑚𝑚×𝑛𝑛 is chosen such that the matrix

𝐴𝐴𝑜𝑜 = 𝐴𝐴𝑐𝑐 − 𝐵𝐵𝑐𝑐𝐾𝐾 (16)



is Hurwitz with the desired characteristic roots. It is clear that the closed-loop matrix 𝐴𝐴𝑐𝑐 − 𝐵𝐵𝑐𝑐𝐾𝐾 can have any set of eigenvalues through pole assignment or optimal methods to attain the desired dynamics. Accordingly, the tracking error settling rates can be controlled.

3.2. Sliding mode controller design. In order to include parametric uncertainties and disturbance influences, Eq. (10) is rewritten as:

𝑥𝑥�̇�𝑐 = 𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐 + ∆𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐 + 𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐 + ∆𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐 + 𝑑𝑑 (17)

where 𝑥𝑥𝑐𝑐 ∈ ℛ𝑛𝑛 , 𝐴𝐴𝑐𝑐 & ∆𝐴𝐴𝑐𝑐 ∈ ℛ𝑛𝑛×𝑛𝑛 , 𝑢𝑢𝑐𝑐 ∈ ℛ𝑚𝑚 and 𝐵𝐵𝑐𝑐 & ∆𝐵𝐵𝑐𝑐 ∈ ℛ𝑛𝑛×𝑚𝑚 . Here ∆𝐴𝐴𝑐𝑐 and ∆𝐵𝐵𝑐𝑐 denote the matched uncertainties in 𝐴𝐴𝑐𝑐 and 𝐵𝐵𝑐𝑐 respectively. In addition, 𝑑𝑑 represents an external disturbance that satisfies the matching condition:

𝑑𝑑 = 𝐵𝐵𝑐𝑐𝛿𝛿𝑐𝑐 (18)

Then, by considering the previously mentioned uo in Eq.(15), the following control signal is proposed here:

𝑢𝑢𝑐𝑐 = 𝑢𝑢∆ + 𝑢𝑢𝑜𝑜 (19)

Consequently Eq. (17) becomes:

𝑥𝑥�̇�𝑐 = 𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 + ∆𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐 + 𝐵𝐵𝑐𝑐𝑢𝑢∆ + ∆𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐 + 𝑑𝑑 (20)

Let ∆𝐴𝐴𝑐𝑐 = 𝐵𝐵𝑐𝑐𝐴𝐴𝛿𝛿 𝑎𝑎𝑛𝑛𝑑𝑑 ∆𝐵𝐵𝑐𝑐 = 𝐵𝐵𝑐𝑐𝐵𝐵𝛿𝛿 (21)

Then,

∆𝐴𝐴𝑥𝑥𝑐𝑐 + ∆𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐 + 𝑑𝑑 = 𝐵𝐵𝑐𝑐{𝐴𝐴𝛿𝛿𝑥𝑥𝑐𝑐 + 𝐵𝐵𝛿𝛿𝑢𝑢𝑐𝑐 + 𝛿𝛿𝑐𝑐}

= 𝐵𝐵𝑐𝑐{(𝐴𝐴𝛿𝛿 − 𝐵𝐵𝛿𝛿𝐾𝐾)𝑥𝑥𝑐𝑐 + 𝐵𝐵𝛿𝛿𝑢𝑢∆ + 𝛿𝛿𝑐𝑐} (22)

Assumption [35]: The matched uncertainty 𝐴𝐴𝛿𝛿 and 𝐵𝐵𝛿𝛿 and external disturbance 𝛿𝛿(𝑡𝑡) are bounded. Hence, we have

‖(𝐴𝐴𝛿𝛿 − 𝐵𝐵𝛿𝛿𝐾𝐾)𝑥𝑥𝑐𝑐 + 𝐵𝐵𝛿𝛿𝑢𝑢∆ + 𝛿𝛿𝑐𝑐‖ ≤ 𝛼𝛼‖𝑥𝑥𝑐𝑐‖ + 𝛽𝛽‖𝑢𝑢∆‖ + 𝜀𝜀 (23)

where ‖𝐴𝐴𝛿𝛿 − 𝐵𝐵𝛿𝛿𝐾𝐾‖ < 𝛼𝛼, ‖𝐵𝐵𝛿𝛿‖ < 𝛽𝛽, ‖𝛿𝛿𝑐𝑐‖ < 𝜀𝜀, where α, 𝛽𝛽, and 𝜀𝜀 are > 0.

The construction of the uncertainties is described in Appendix A.

3.3. Unit control design The continuous unit control signal u∆ is denoted as [35]

𝑢𝑢∆(𝑡𝑡) = −𝛾𝛾(‖𝑥𝑥𝑐𝑐‖)𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�

)24 (

where ‖. ‖ is the Euclidean norm. Accordingly, the control law in Eq. (19) will be:

𝑢𝑢𝑐𝑐 = −𝛾𝛾(‖𝑥𝑥𝑐𝑐‖)𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�

− 𝐾𝐾𝑥𝑥𝑐𝑐 (25)

Since the system 𝑥𝑥�̇�𝑐 = 𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 is asymptotically stable, there is a Lyapunov function with a positive definite matrix 𝑃𝑃 [38]



𝑉𝑉 = 𝑥𝑥𝑐𝑐𝑘𝑘𝑃𝑃𝑥𝑥𝑐𝑐 (26)

such that �̇�𝑉 < 0 ,∀𝑥𝑥𝑐𝑐 ≠ 0. Accordingly, differentiate 𝑉𝑉 with respect to time to become:

�̇�𝑉 = 𝛻𝛻𝑉𝑉𝑘𝑘𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 + 𝛻𝛻𝑉𝑉𝑘𝑘𝐵𝐵𝑐𝑐 𝑢𝑢∆ + 𝛻𝛻𝑉𝑉𝑘𝑘{∆𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐 + ∆𝐵𝐵𝑐𝑐 𝑢𝑢𝑐𝑐 + 𝑑𝑑 } (27)

�̇�𝑉 can be maintained as negative definite by proper solution of the unite vector gain 𝛾𝛾(‖𝑥𝑥𝑐𝑐‖). Then, Eq. (27) is redefined as follows:

�̇�𝑉 = 𝛻𝛻𝑉𝑉𝑘𝑘𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 + 𝛻𝛻𝑉𝑉𝑘𝑘𝐵𝐵𝑐𝑐 𝑢𝑢∆ + 𝛻𝛻𝑉𝑉𝑘𝑘𝐵𝐵𝑐𝑐{∆𝐴𝐴𝑐𝑐𝑥𝑥𝑐𝑐 + ∆𝐵𝐵𝑐𝑐𝑢𝑢𝑐𝑐 + 𝑑𝑑 } (28)

Meanwhile, �𝛻𝛻𝛻𝛻𝑇𝑇𝐵𝐵𝑐𝑐��𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻��𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�

= �𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�

2

�𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�= �𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉�,

𝛻𝛻𝑉𝑉𝑘𝑘𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 = 2𝑥𝑥𝑐𝑐𝑘𝑘𝑃𝑃𝐴𝐴𝑜𝑜𝑥𝑥𝑐𝑐 = −𝑥𝑥𝑐𝑐𝑘𝑘𝑄𝑄𝑥𝑥𝑐𝑐 ≤ 0

and,

𝛻𝛻𝑉𝑉𝑘𝑘𝐵𝐵𝑐𝑐{(𝐴𝐴𝛿𝛿 − 𝐵𝐵𝛿𝛿𝐾𝐾)𝑥𝑥𝑐𝑐 + 𝐵𝐵𝛿𝛿𝑢𝑢∆ + 𝛿𝛿𝑐𝑐} ≤ ‖𝛻𝛻𝑉𝑉𝑘𝑘𝐵𝐵𝑐𝑐‖‖(𝐴𝐴𝛿𝛿 − 𝐵𝐵𝛿𝛿𝐾𝐾)𝑥𝑥𝑐𝑐 + 𝐵𝐵𝛿𝛿𝑢𝑢∆ + 𝛿𝛿𝑐𝑐‖ ≤ ‖𝐵𝐵𝑘𝑘 𝛻𝛻𝑉𝑉‖(𝛼𝛼‖𝑥𝑥𝑐𝑐‖ + 𝛽𝛽𝛾𝛾 + 𝜀𝜀) Then, Eq. (28) becomes:

�̇�𝑉 ≤ −𝛾𝛾�𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉� + �𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉�(𝛼𝛼‖𝑥𝑥𝑐𝑐‖ + 𝛽𝛽𝛾𝛾 + 𝜀𝜀) = −�𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉�{𝛾𝛾 − 𝛼𝛼‖𝑥𝑥𝑐𝑐‖ − 𝛽𝛽𝛾𝛾 − 𝜀𝜀} = −�𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉�{𝛾𝛾(1 − 𝛽𝛽) − 𝛼𝛼‖𝑥𝑥𝑐𝑐‖ − 𝜀𝜀}

(29)

Since for 𝛾𝛾(‖𝑥𝑥𝑐𝑐‖) =1

(1−𝛽𝛽){𝛼𝛼‖𝑥𝑥𝑐𝑐‖ + 𝜀𝜀 + 𝑘𝑘𝑙𝑙}, 𝑘𝑘𝑙𝑙 > 0 (30)

we have �̇�𝑉 ≤ −𝑘𝑘𝑙𝑙�𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉� < 0 , ∀‖𝑥𝑥𝑐𝑐‖ ≠ 0,

Accordingly, the sliding surface is reached in a finite time since it is a consequence of the unit vector term 𝐵𝐵𝑐𝑐

𝑇𝑇 𝛻𝛻𝛻𝛻�𝐵𝐵𝑐𝑐𝑇𝑇 𝛻𝛻𝛻𝛻�

in Eq. (25). Furthermore, the reaching

time is openly related to the value of the gain 𝛾𝛾(‖𝑥𝑥𝑐𝑐‖). Now, the control law will be given by

𝑢𝑢𝑐𝑐 = −𝐾𝐾𝑥𝑥𝑐𝑐 − 𝛾𝛾(‖𝑥𝑥𝑐𝑐‖)𝑆𝑆‖𝑆𝑆‖

(31)

where 𝑆𝑆 is the vector of the switching function denoted by

𝑆𝑆 = [𝑠𝑠1, 𝑠𝑠2 , … , 𝑠𝑠𝑚𝑚 ]𝑘𝑘 = 𝐵𝐵𝑐𝑐𝑘𝑘 𝛻𝛻𝑉𝑉 = 𝐵𝐵𝑐𝑐𝑘𝑘2𝑃𝑃𝑥𝑥𝑐𝑐 = 𝐺𝐺𝑥𝑥𝑐𝑐 (32)

𝐺𝐺 = 2𝐵𝐵𝑐𝑐𝑘𝑘𝑃𝑃 ∈ ℛ𝑚𝑚×𝑛𝑛, and 𝛾𝛾(‖𝑥𝑥𝑐𝑐‖) as in Eq. (30). And,

‖𝑆𝑆‖ = √𝑆𝑆𝑘𝑘𝑆𝑆2 (33)

The chattering will be induced in system response due to the discontinuous term in (S/‖S‖ in Eq. (31)). The chattering effect can be reduced by resembling the discontinuous term and that will only adjust the state to an area adjacent to the origin. By considering the switching term γ(‖x_c ‖) as a function of the state rather than a constant value, the chattering around the origin will be ((ε+k_l ))/((1-β) ). Finally, the discrete control signal can be easily found by [36, 39]:

𝑢𝑢𝑑𝑑(𝑡𝑡) = 𝑢𝑢𝑐𝑐(𝑡𝑡) (34)

𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘𝑘𝑘 ≤ 𝑡𝑡 ≤ (𝑘𝑘 + 1)𝑘𝑘 , 𝑘𝑘 = 0,1,2, …, N



3.4. Optimal tracking control algorithm The optimal Linear Quadratic Regulator (LQR) controller as described in Section 3.1 will be extended here to LQR - servo controller which is a state feedback optimal control that can track a dynamic reference and reduce the error due to the presence of integral term.

Let the error be (xE), then the following equation can be developed to include the error signal:

�̇�𝒙𝑬𝑬 = 𝒓𝒓𝒌𝒌 − 𝑦𝑦𝑘𝑘 = 𝒓𝒓𝒌𝒌 − 𝐶𝐶𝑥𝑥𝑐𝑐 (35)

where rk is the reference signal. Accordingly, Eq. (10) can be rewritten as follows:

�̇�𝒙𝒌𝒌.𝑬𝑬 =𝑨𝑨𝑬𝑬 𝒙𝒙𝒌𝒌.𝑬𝑬 + 𝑩𝑩𝑬𝑬 𝑢𝑢𝑜𝑜 + 𝒓𝒓𝒌𝒌 + 𝒘𝒘

��̇�𝑥𝑐𝑐�̇�𝒙𝑬𝑬� = �

𝑨𝑨𝒄𝒄 𝟎𝟎𝟐𝟐,𝟐𝟐−𝑪𝑪 𝟎𝟎𝟐𝟐,𝟐𝟐

� �𝒙𝒙𝒄𝒄𝒙𝒙𝑬𝑬� + �

𝑩𝑩𝒄𝒄𝟎𝟎𝟐𝟐,𝟏𝟏

� ⨯ 𝒖𝒖0 + �𝟎𝟎𝟐𝟐,𝟏𝟏𝟏𝟏𝟐𝟐,𝟏𝟏

� ⨯ 𝒓𝒓 + �𝟏𝟏𝟐𝟐,𝟏𝟏𝟎𝟎𝟐𝟐,𝟏𝟏

�𝒘𝒘 (36)

where 𝒙𝒙𝒌𝒌.𝑬𝑬 denotes the augmented state vector [𝒙𝒙𝒄𝒄 𝒙𝒙𝑬𝑬]𝑘𝑘 , 𝑨𝑨𝑬𝑬 is the augmented

system matrix �𝑨𝑨𝒄𝒄 𝟎𝟎𝟐𝟐,𝟐𝟐−𝑪𝑪 𝟎𝟎𝟐𝟐,𝟐𝟐

�, 𝑩𝑩𝑬𝑬 is the input matrix �𝑩𝑩𝒄𝒄𝟎𝟎𝟐𝟐,𝟏𝟏

�, respectively. 𝑨𝑨𝑬𝑬 and 𝑩𝑩𝑬𝑬

need to be controllable [40].

Now, from the augmented system illustration, the control input matrix 𝑢𝑢𝑜𝑜 is obtained from:

𝒙𝒙𝑰𝑰 = [∫(𝑥𝑥𝑟𝑟1 − 𝑥𝑥c1) ∫(𝑥𝑥𝑟𝑟2 − 𝑥𝑥c2)]𝑘𝑘 (37)

𝒙𝒙𝒌𝒌.𝑬𝑬 = [𝑥𝑥c1 𝑥𝑥c2 𝒙𝒙𝑰𝑰 ]𝑘𝑘 (38)

𝒖𝒖𝒐𝒐 = −𝑲𝑲𝑬𝑬 ⨯ 𝒙𝒙𝒌𝒌.𝑬𝑬 (39) where 𝑲𝑲𝑬𝑬 is the control gains.

Similar to Section 3.1, the total control signal uc will be applied according to Eq.19 and as illustrated in Fig. 4, where the tracking algorithm is added to the USMC. Thus, the optimal stabilizing controller combined with the USMC is applied to design a robust nonlinear controller for the WSN system while considering the uncertainty in WSN model and external disturbances, as will be presented in the next Section.

Fig. 4. Block diagram of the optimal tracking-USMC applied method.



4. Simulation Results and Discussion In order to simulate the WSN, Eq. (10) in Section 2 is implemented. Simulation is developed using MATLAB. The performance and the obtained results are based on the parameter settings demonstrated in Table 1.

The result in Fig. 5 shows the bode plot of the WSN, while Fig. 6. shows the root locus plot. It can be clearly seen that the system is unstable. Consequently, experiments were carried out to select a suitable linear state feedback controller K to satisfy that Eq. (16) is Hurwitz. As explained in Section 3, two approaches have been tested for this purpose: Pole Placement (PP) and LQR control. The control equation is applied as in Eq.(15).

Table 1. System simulation parameters [6, 7]. Parameters Description/values desired SINR 20dB ∝ zero mean Gaussian random variable with variance 𝜎𝜎∝2 / 0.2 c the amount of congestion / 0.5 𝑏𝑏𝑝𝑝, 𝑏𝑏𝑓𝑓 1 Desired rate 10M/s 𝜇𝜇′ 0.8 𝑛𝑛(𝑘𝑘) and 𝑑𝑑(𝑘𝑘) zero-mean with variance / 0.01

Fig. 5. The bode plot of the WSN system.

Fig. 6. The root locus of the system.



Using PP, the selected desired characteristic roots are 𝑝𝑝 = [−3 −6]. Fig. 7(a) illustrates the states and output response of the controlled WSN, while Fig. 7(b) shows the control signal. Similarly, the control algorithm of LQR is applied for the same purpose, using suitable values for Q and R. Fig. 8(a) shows the resulted states and output response of the controlled WSN, while Fig. 8(b) illustrates the control signal. Note that in both cases only the effects of uncertainty are considered here.

Figures 7 and 8 illustrate that both controllers can stabilize the system, nevertheless, using LQR can provide less overshoot in the states, output, and control signal than that when using PP. Therefore, the LQR controller will be used for the rest of the simulation experiments.

(a) The States (b) The Control Signal

Fig. 7. Transient response of the WSN system using Pole Placement (PP) Controller, Initial Conditions: x(0)=[2 -3].

(a) The states (b) The control signal

Fig. 8. The transient responses of the WNS system using LQR controller: x(0)=[ 2 -3].

Then, a disturbance effect on the control signal is comprised in the simulation using the same earlier designed state feedback controllers. The disturbance effect is provided by a uniformly distributed signal. The results are shown in Fig. 9. It is clear that despite the highly disturbed control signal the WSN maintained its stable performance. Nevertheless, further improvement can be attained by the inclusion of the USMC, as will be presented next. The parameters α , 𝛽𝛽, and 𝜀𝜀 are selected as 2, 1.4, and 0.3, respectively.

Figure 10. shows the response of the WSN using the combined unity sliding mode and LQR controller, while Fig. 11. presents the resulted sliding surface and phase plane plot of the controlled WSN. The performance is clearly improved as

0 2 4 6 8 10

Time (sec.)

-8

-6

-4

-2

0

2

4

6

8

10

Stat

es a

nd O

utpu

t y

x1,PP

x2,PP

y

0 2 4 6 8 10

Time (sec.)

-50

-40

-30

-20

-10

0

10

Contr

ol Si

gnal

u

0 2 4 6 8 10

Time (sec.)

-4

-3

-2

-1

0

1

2

3

4

Stat

es a

nd O

utpu

t y

x1,LQR

x2,LQR

y

0 2 4 6 8 10

Time (sec.)

-50

-40

-30

-20

-10

0

10

The

Cont

rol S

igna

l



compared with Fig. 8. In Fig. 12, a comparison is illustrated, where faster response and reduced control signal is obtained in presence of uncertainties.

Similarly, Fig. 13. illustrates the performance of the controlled WSN under disturbance, using the USMC combined with LQR, while Fig. 14. shows the resulted sliding surface and phase-plane plot. Although the effect of the noise can be noticed in the control signal, where the disturbance is mainly affecting, the controlled system can still maintain the robustness of stability and performance.

Then in order to obtain tracking of a specific required reference for the states, the arrangement that described in Section 3.3 is applied, as can be seen in Fig. 15. The same previously used initial conditions are used to reach the desired values of x1 and x2, while reference values of 20 dB and 10 M/s are used for the states x1 and x2. Similarly, Fig. 16. shows the robust tracking performance of the controlled WSN in presence of disturbance.


Fig. 9. The transient responses of the WNS system using LQR with disturbance effects on the control signal, x(0)=[ 2 -3 ].

a) The States (b) The Control Signal

Fig. 10. The transient responses of the WSN system using unity control and LQR, x(0)=[ 2 -3 ].

0 2 4 6 8 10

Time (sec.)

-4

-3

-2

-1

0

1

2

3

4

Sta

tes

and

Out

put y

x1,LQR

x2,LQR

y, LQR

0 2 4 6 8 10

Time (sec.)

-50

-40

-30

-20

-10

0

10

The

Con

trol

Sig

nal

0 2 4 6 8 10

Time (sec.)

-4

-3

-2

-1

0

1

2

3

4

Sta

tes

and

outp

ut y

x1,SM+LQR

x2,SM+LQR

y,SM+LQR

0 2 4 6 8 10

Time (sec.)

-35

-30

-25

-20

-15

-10

-5

0

5

Con

trol

Sig

nal u



(a) The sliding surface (b) Phase-plane plot

Fig. 11. The behaviour of the controlled WSN system using unity control and LQR, x(0)=[ 2 -3 ].


Fig. 12. Comparison between the transient responses of WSN system using unity control and LQR, x(0)=[ 2 -3 ].


Fig. 13. The transient responses of the WSN system using unity control and LQR, x(0)=[ 2 -3 ], with disturbance.

0 2 4 6 8 10

Time (sec.)

0

0.5

1

1.5

2

2.5

|S|

104

0 0.5 1 1.5 2 2.5 3 3.5

x1

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

x2

start point 2,-3

0 2 4 6 8 10

k

-4

-3

-2

-1

0

1

2

3

4

Sta

tes

and

outp

ut y

x1,SM+LQR

x2,SM+LQR

y,SM+LQR

x1,LQR

x2,LQR

y,LQR

0 2 4 6 8 10

k

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Con

trol

Sig

nal u

u,SM+LQR

u,LQR

0 2 4 6 8

Time (sec.)

-4

-3

-2

-1

0

1

2

3

4

Sta

tes

and

outp

ut y

x1,SM+LQR

x2,SM+LQR

y,SM+LQR

0 2 4 6 8 10

Time (sec.)

-35

-30

-25

-20

-15

-10

-5

0

5

10

Con

trol S

igna

l u



(a) The sliding service (b) The phase-plane plot

Fig. 14. The behaviour of the controlled WSN System using unity control and LQR, x(0)= [ 2 -3 ], with disturbance.

(a) The state x1 (b) The state x2

(c) The control signal u (d) The phase-plane plot

Fig. 15. Tracking response of the WSN system using unity sliding mode and LQR-servo, x(0)=[ 2 -3 ].

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(a) The state x1 (b) The state x2

(c) The control signal u (d) The phase-plane plot

Fig. 16. Tracking response of the WSN System using unity sliding mode and LQR-servo, x(0)= [ 2 -3 ], with disturbance

5. Conclusions and Potential Future Work One of the interesting correlations in WSN is the trade-off between the data transmission rate, power consumption, and the level of congestion in the network which can be fairly coordinated to enhance energy efficiency.

This paper presents a detailed development of a USMC combined with LQR controller for a linear uncertain WSN system subjected to modelling uncertainties and external disturbances. The obtained controller is robust with respect to the uncertainty in the model of WSN system and to the coordinated external disturbances. A variable switching gain γ(‖x‖) has been utilized to eliminate the effect of the chattering problem in sliding mode control. The developed USMC combined with LQR has effectively derive the uncertain WSN system to robust stability and performance conditions despite the presence of matched disturbances. Optimal power and rate control law have been obtained by solving the matrix algebraic Riccati equation. Then, the optimal tracking LQR-servo algorithm is combined with the USMC. Precise tracking of specified SINR and transmission rate control prove the validity of the applied approach. The simulation results illustrate the effectiveness of the proposed approach.

The next step will be to consider the situation of time-varying delays since their effects have a great impact on the actual network. In addition, real time tuning method as extremum seeking algorithm can be applied to update the parameters of the controller during unexpected disturbances or variations.

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Nomenclatures F The set of interfering nodes with node i 𝐺𝐺𝑖𝑖𝑖𝑖 Gain of the channel of jth to ith nodes 𝑐𝑐(𝑘𝑘) d

Measurement of congestion in network External disturbance

𝑑𝑑𝑟𝑟(𝑘𝑘) Controls the value of rate increase for each iteration 𝑓𝑓𝑖𝑖(𝑘𝑘) Flow rate at node i at time k 𝑛𝑛𝑖𝑖(𝑘𝑘) 𝑃𝑃𝑖𝑖 rk

Zero-mean disturbance of variance 𝜎𝜎𝑛𝑛2 Transmission power of ith node Reference signal

𝑠𝑠𝑖𝑖 Actual SINR for node i T 𝒙𝒙𝒌𝒌.𝑬𝑬

Sampling interval Augmented state vector

Greek Symbols �̅�𝛽𝑖𝑖(𝑘𝑘) 𝜎𝜎𝑖𝑖2

Effective channel gains White Gaussian noise power.

µ Positive step-size Abbreviations

IoT LMI

Internet of Things Linear Matrix Inequality

LQR Linear Quadratic Regulator MPC PP SINR USMC WSN

Model Predictive Control Pole Placement Signal-to- Interference-plus-Noise-Ratio Unity Sliding Mode Control Wireless Sensor Network

References 1. Sobral, J.V.; Rodrigues, J. J.; Rabelo, R.A.; Lima Filho, J.C.; Sousa, N.;

Araujo, H.S.; and Holanda, F.R. (2018). A framework for enhancing the performance of Internet of Things applications based on RFID and WSNs. Journal of Network and Computer Applications, 107, 56-68.

2. Matta, P.; and Pant, B. (2019). Internet-of-things: Genesis, challenges, and applications. Journal of Engineering Science and Technology (JESTEC), 14(3), 1717-1750.

3. Zhang, D.; Li, G.; Zheng, K.; Ming, X.; and Pan, Z.H. (2013). An energy-balanced routing method based on forward-aware factor for wireless sensor networks. IEEE transactions on industrial informatics, 10(1), 766-773.

4. Liu, Y.; Liu, A.; Zhang, N.; Liu, X.; Ma, M.; and Hu, Y. (2019). DDC: Dynamic duty cycle for improving delay and energy efficiency in wireless sensor networks. Journal of Network and Computer Applications, 131, 16-27.

5. Sharma, N.; Singh, K.; and Singh, B.M. (2018). Sectorial stable election protocol for wireless sensor network. Journal of Engineering Science and Technology (JESTEC), 13(5), 1222-1236.



6. Soua, R.; and Minet, P. (2011). A survey on energy efficient techniques in wireless sensor networks. In Proceedings of the 2011 4th Joint IFIP Wireless and Mobile Networking Conference (WMNC 2011). Toulouse, France, 1-9.

7. Liu, Y.; Wu, Q.; Zhao, T.; Tie, Y.; Bai, F.; and Jin, M. (2019). An improved energy-efficient routing protocol for wireless sensor networks. Sensors, 19(20), 4579.

8. Zhu, J.; Jiang, D.; Ba, S.; and Zhang, Y. (2017). A game-theoretic power control mechanism based on hidden Markov model in cognitive wireless sensor network with imperfect information. Neurocomputing, 220, 76-83.

9. Gang, S.; WU, W.Q.; Lei, Y.; DING, C.Y.; and ZHANG, X.Y. (2019). Distributed dynamic power allocation strategy based on energy balance in wireless sensor networks for smart grid. DEStech Transactions on Computer Science and Engineering (cscme).

10. Diaz-Ibarra, M.A.; Campos-Delgado, D.U.; Gutiérrez, C.A.; and Luna-Rivera, J.M. (2019). Distributed power control in mobile wireless sensor networks. Ad Hoc Networks, 85, 110-119.

11. Chincoli, M.; and Liotta, A. (2018). Transmission power control in WSNs: from deterministic to cognitive methods Integration, Interconnection, and Interoperability of IoT Systems (39-57): Springer.

12. Feng, Z.; and Wassell, I. (2016). Dynamic power control and optimization scheme for QoS-constrained cooperative wireless sensor networks. In Proceedings of the 2016 IEEE International Conference on Communications (ICC). Kuala Lumpur, Malaysia, 1-6.

13. Qin, F.; Dai, X.; and Mitchell, J.E. (2013). Effective-SNR estimation for wireless sensor network using Kalman filter. Ad Hoc Networks, 11(3), 944-958.

14. Jayasri, T.; and Hemalatha, M. (2017). Link quality estimation for adaptive data streaming in WSN. Wireless Personal Communications, 94(3), 1543-1562.

15. Sabitha, R.; Bhuma, K.; and Thyagarajan, T. (2017). Transmission power control using state estimation-based received signal strength prediction for energy efficiency in wireless sensor networks. Turkish Journal of Electrical Engineering & Computer Sciences, 25(1), 591-604.

16. Chincoli, M.; and Liotta, A. (2018). Self-learning power control in wireless sensor networks. Sensors, 18(2), 375.

17. Akbas, A.; Yildiz, H.U.; Ozbayoglu, A.M.; and Tavli, B. (2019). Neural network based instant parameter prediction for wireless sensor network optimization models. Wireless Networks, 25(6), 3405-3418.

18. Cheng, H.; Xie, Z.; Wu, L.; Yu, Z.; and Li, R. (2019). Data prediction model in wireless sensor networks based on bidirectional LSTM. EURASIP Journal on Wireless Communications and Networking, 2019(1), 203.

19. Zhu, Y.; Xu, Y.; Guan, S.; and He, J. (2010). Connectivity analysis of random power control in wireless sensor networks. In Proceedings of the 2010 IEEE 12th International Conference on Communication Technology. Nanjing, China, 1248-1251.

20. Wang, Y.; Vuran, M.C.; and Goddard, S. (2010). Stochastic analysis of energy consumption in wireless sensor networks. In Proceedings of the 7th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON). Boston, USA, 1-9.



21. Kapoor, R.; Gupta, R.; Jha, S.; and Kumar, R. (2019). Adaptive technique with cross correlation for lowering signal-to-noise ratio wall in sensor networks. Wireless Personal Communications, 105(3), 787-802.

22. Subramanian, A.; and Sayed, A.H. (2005). Joint rate and power control algorithms for wireless networks. IEEE Transactions on Signal Processing, 53(11), 4204-4214.

23. Subramanian, A.; Khajehnouri, N.; and Sayed, A.H. (2003). A minimum variance rate and power update algorithm for wireless networks. In Proceedings of the 2003 IEEE 58th Vehicular Technology Conference. Orlando, USA, 1468-1472.

24. Zhang, D.; Wang, X.; and Liu, J. (2012). Robust power control method for wireless sensor networks with static output feedback. Journal of Control Theory and Applications, 10(1), 44-49.

25. Kong, S.; and Chen, W. (2016). Joint control of power and rates of wireless networks based on Nash game. International Journal of Automation and Control, 10(1), 1-11.

26. Han, C.; Sun, D.; Dong, Z.; Li, Z.; and Lei, Z. (2011). Adaptive fault-tolerant H∞ power and rate control for wireless networks via dynamic output feedback. In Proceedings of the 2011 9th World Congress on Intelligent Control and Automation. Taipei, Taiwan 141-146.

27. Han, C.; Sun, D.; Liu, L.; Bi, S.; and Li, Z. (2014). Robust model predictive power and rate control for uncertain wireless networks with time-varying delays and input constraints. In Proceedings of the 33rd Chinese Control Conference. Nanjing, China, 5447-5452.

28. Han, C.; Chang, S.; Diao, Q.; Liu, L.; Bi, S.; and Sun, D. (2016). Optimal power and rate control for wireless communication networks with external disturbance. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC). Yinchuan, China, 5592-5595.

29. Han, C.; Chang, S.; Liu, L.; Li, Z.; and Sun, D. (2017). Power and rate control for wireless communication networks based on priority of quality-of-service. In Proceedings of the 2017 36th Chinese Control Conference (CCC). Dalian, China, 2675-2679.

30. Han, C.; Li, M.; Li, Y.; Dong, Z.; and Li, Z. (2017). Adaptive power and rate control for wireless communication networks with nonlinear channel fading and time-varying delay. IFAC-PapersOnLine, 50(1), 6552-6557.

31. Utkin, V. (1977). Variable structure systems with sliding modes. IEEE Transactions on Automatic control, 22(2), 212-222.

32. Vaidyanathan, S.; and Lien, C.H. (2017). Applications of sliding mode control in science and engineering (1st ed.). Springer.

33. Gutman, S.; and Leitmann, G. (1976). Stabilizing feedback control for dynamical systems with bounded uncertainty. In Proceedings of the IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes. Clearwater, FL, USA, 94-99.

34. Gutman, S. (1979). Uncertain dynamical systems - A Lyapunov min-max approach. IEEE Transactions on Automatic control, 24(3), 437-443.



35. Raafat, S.M.; Al-Samarraie, S.A.; and Mahmood, A.M. (2015). Unity sliding mode controller design for active magnetic bearings system. Journal of Engineering, 21(6), 90-107.

36. Shieh, L.S.; Wang, H.; and Yates, R.E. (1980). Discrete-continuous model conversion. Applied Mathematical Modelling, 4(6), 449-455.

37. Fortuna, L.; and Frasca, M. (2012). Optimal and Robust Control: Advanced Topics with MATLAB®. Boca Raton, Florida: CRC press.

38. Khalil, H.K. (2002). Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice hall.

39. Seuret, A.; Hetel, L.; Daafouz, J.; and Johansson, K.H. (2016). Delays and Networked Control Systems. Switzerland: Springer International Publishing.

40. Kim, J.; Yoo, J.; Bae, J.; Kim, C.; and Choi, J. (2015). Two-wheeled self-balancing robot based on LQ-Servo with reduced order observer. In Proceedings of the 4th International Conference on Information Science and Industrial Applications. Busan, South Korea, 6-9

Appendix A Computations of the uncertainty matrices

1) ∆𝐴𝐴 = 𝐵𝐵𝑐𝑐𝐴𝐴𝛿𝛿 ,∆𝐴𝐴 ∈ ℛ𝑛𝑛×𝑛𝑛,𝐵𝐵𝑐𝑐 ∈ ℛ𝑛𝑛×𝑚𝑚 𝑎𝑎𝑛𝑛𝑑𝑑 𝐴𝐴𝛿𝛿 ∈ ℛ𝑚𝑚×𝑛𝑛, where n=2, m=1 for the WSN system. Let ∆𝐴𝐴 be written as:

∆𝐴𝐴 = � 𝑂𝑂 𝑂𝑂𝛿𝛿𝐴𝐴21 𝛿𝛿𝐴𝐴22� (A.1)

where 𝛿𝛿𝐴𝐴21 and 𝛿𝛿𝐴𝐴22 ∈ ℛ𝑚𝑚×𝑚𝑚,𝑚𝑚 = 𝑛𝑛/2 . Also, let 𝐵𝐵 and 𝐴𝐴𝛿𝛿 be written as follows:

𝐵𝐵𝑐𝑐 = �0𝐵𝐵21

� , 0 𝑎𝑎𝑛𝑛𝑑𝑑 𝐵𝐵21 ∈ ℛ𝑚𝑚×𝑚𝑚 (A.2)

𝐴𝐴𝛿𝛿 = [𝐴𝐴𝛿𝛿11 𝐴𝐴𝛿𝛿12], 𝐴𝐴𝛿𝛿11 and 𝐴𝐴𝛿𝛿12 ∈ ℛ𝑚𝑚×𝑚𝑚 (A.3) Now ∆𝐴𝐴 = 𝐵𝐵𝑐𝑐𝐴𝐴𝛿𝛿 becomes:

� 0 0𝛿𝛿𝐴𝐴21 𝛿𝛿𝐴𝐴22� = � 0𝐵𝐵21

� [𝐴𝐴𝛿𝛿11 𝐴𝐴𝛿𝛿12] = �0 0

𝐵𝐵21𝐴𝐴𝛿𝛿11 𝐵𝐵21𝐴𝐴𝛿𝛿12� (A.4)

where the element 𝐵𝐵21 ≠ 0. Therefore 𝐴𝐴𝛿𝛿11 = 𝐵𝐵21−1𝛿𝛿𝐴𝐴21𝐴𝐴𝛿𝛿12 = 𝐵𝐵21−1𝛿𝛿𝐴𝐴22

� (A.5)

2) ∆𝐵𝐵𝑐𝑐 = 𝐵𝐵𝑐𝑐𝐵𝐵𝛿𝛿 ,∆𝐵𝐵𝑐𝑐 ∈ ℛ𝑛𝑛×𝑚𝑚, and 𝐵𝐵𝛿𝛿 ∈ ℛ𝑚𝑚×𝑚𝑚 Let ∆𝐵𝐵𝑐𝑐 be written as:

∆𝐵𝐵𝑐𝑐 = �0

𝛿𝛿𝐵𝐵21� , 0 and 𝛿𝛿𝐵𝐵21 ∈ ℛ𝑚𝑚×𝑚𝑚 (A.6)

and with the aid of Eq. (A.2), we can write ∆𝐵𝐵 = 𝐵𝐵𝐵𝐵𝛿𝛿 as:

� 0𝛿𝛿𝐵𝐵21� = � 0𝐵𝐵21

� 𝐵𝐵𝛿𝛿 = �0

𝐵𝐵21𝐵𝐵𝛿𝛿� (A.7)

Solving for 𝐵𝐵𝛿𝛿 𝐵𝐵𝛿𝛿 = 𝐵𝐵21

−1𝛿𝛿𝐵𝐵21 (A.8)

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