Shuying Zhang, Xiaohui Zhao＊

College of Communication Engineering, Jilin University, Changchun 130012, China

* The corresponding author, email: xhzhao@jlu. edu.cn

Abstract: We propose a distributed closedloop power control scheme for a cognitive radio network (CRN) based on our developed state space model of the CRN. The whole power control process is separated into outer control loop and inner control loop in order to solve different problem. In outer loop, the interference temperature (IT) constraint is transformed to a performance index minimized by a state feedback controller to obtain an appropriate target signal to interference plus noise ratio (SINR) of secondary user (SU). For ideal channel model and random time-varying channel model, our designed controller is a linear quadratic regulator (LQR) and a linear quadratic Gaussian (LQG) regulator respectively.While in inner loop, SU controls its transmit power to make the instantaneous SINR track the corresponding target and ensure the IT constraint under the limited threshold. The closed-loop stability of the CRN is proved and the performance of proposed control scheme is presented by computer simulations, which shows that this scheme can effectively guarantee both the requirement of SINR and IT constraint for all SUs.

Keywords: cognitive radio; power control;state space model; LQR; LQG

I. INTRODUCTION

Wireless spectrum resource is a key limiting factor in the development of wireless communication for its exorbitant price and scare nature. Cognitive radio (CR) as a technology of improving the efficiency of the spectrum utilization has been proposed for more than 20 years to solve this problem [1]. However,inappropriate reuse of the spectrum licensed to primary users (PUs) by secondary users (SUs)inevitably leads to mutual interference and performance degradation for both PUs and SUs. Therefore, how to well control the transmit power of SU in a cognitive radio network(CRN) is one of the important and challenging task to better realize the mutual share of the valuable spectrum resource. In other words, a reasonable power control scheme can limit the interference from SUs and guarantee communication quality of CRN [2].

Power control for the CRN has been studied for more than ten years and there have been a lots of schemes emerging [3-7]. In these schemes, each SU terminal obtains its power by solving an optimization objective function with available information and certain constraints. With the development of study, other important factors, such as imperfect spectrum sensing, energy harvesting, beam-forming,quality of experience (QoE), etc. have also been considered in some recent works [8-11].Summarizing these studies, we know that they almost all use convex optimization theory to maximize or minimize one or more target functions parameterized by communication performance indicator like transmission rate,throughput, energy efficiency (EE), outage probability, bit error ratio (BER) and so on,subject to transmit power constraint and protection mechanism for PUs, like interference temperature (IT) constraint or outage probability constraint of PU. This kind of solution from the results of these studies is an open loop power control in the perspective of control theory that often considers dynamic control process and closed-loop property. Furthermore, in control theory, the controlled object needs to be described as a dynamic system.Although the structure of open loop control is simple to design, it barely has ability to resist external disturbance and is not better reflect the behavior of a complicated CRN system or harsh wireless communication environment.

On the contrary, closed-loop control based on the description of the controlled system model has obvious advantages that it can make the system stable with a predefined target and well reject different disturbance to guarantee system performance even with model uncertainties [12]. In fact, control theory can provide a dynamic solution for a control problem with robustness and adaptability to the complicated environment. Hence, it is necessary to introduce this theory to the power control of CRNs. There have been some interesting research works using state space model and feedback control methods to obtain effective and flexible low complexity power control algorithms in the particular networks [13-16].Other research results also inspire us to extend control theory to the power allocation problem for CRNs in more practical and robust manners [17, 18].

However, the existence of different constraints produces difficulties in finding control strategy to the power control problem of CRNs. Although the authors in [19] have successfully introduced PID control and model predictive control (MPC) to solve the problem,the treatment to IT constraint makes their control scheme lack of validity and robustness.Coincidentally, in [20], the authors propose an energy-saving adaptive transmit power control based on BER feedback for a cognitive personal area network (CPAN) in the Multiband Orthogonal Frequency Division Multiplexing Ultra-wideband (MB-OFDM UWB) system,where each CPAN device controls its transmit power using a nonlinear approximation of power-time curve to make the total power in an MB-OFDM band and the BER lower than the corresponding acceptable threshold respectively. Unfortunately, the interference from SU to PUs is not discussed instead of just considering the total power constraint in an MB-OFDM band, which may result in serious disturbance to the primary links sometime. In general, it is not direct and easy to manipulate control theory or typical control method to design a power controller for a CR system when modeling the dynamics of the network with the constraints.

In conventional network, the general scheme of transmit power control is divided in two parallel processes, namely outer loop and inner loop [16, 21], where the inner loop power control updates the transmit power at each time slot to keep the instantaneous SINR of each link as close as possible to a target SINR provided by the outer loop. Usually, the outer loop adjusts the target SINR to satisfy various communication requirements. For instance,the scheme proposed in [21] is to maintain the lowest acceptable quality of service (QoS),the control strategy in [13] is to get a tradeoff among power consumption, data transmission rate and congestion levels in a wireless network, and the scheme in [15] is to minimize the queue backlog/delay for a dynamic wireless network environment. These works motivate us to consider interference suppression of CRN as a control requirement of outer loop,and we treat some necessary constrains by transforming them to system performance index. Precisely, we convert the IT constraint to a controlled output indicator for the optimized controller design.

Other important consideration in the power control of CRN is the time-varying and random nature of wireless channels. Tradition-ally, the influence of channel uncertainty has been treated in most power control schemes based on robust optimization theories in recent studies, the most used namely the Bayesian approach and the worst-case approach [22-24]. They still require calculating the optimal value from given objective function with all available information and different constraints. However, these methods do not solve the problem of the channel uncertainty in the sense of a dynamic control process or closedloop control system. Actually, there have been attacking robust control theory, such as linear quadratic Gaussian (LQG) control, H2/ H∞control and so on [13, 15, 16], to design robust controllers for the systems with parameter uncertainties and disturbances. We strongly believe that we can copy with the power control problem of CRNs in a dynamic and robust manner by using the appropriate control theories, since CRN system is a completely random, dynamic, and time-varying with delay.

Based on the above brief discussions, in this study, we propose a distributed power control scheme based on linear quadratic optimal control theory for a CRN. First, we formulate the outer control loop as a linear state space model where both IT constraint and QoS of SU are included in a performance index. In this loop, a dynamic target SINR is adjusted by a local linear quadratic regulator (LQR) or a LQG regulator according to the IT constraint without or with the channelfluctuation respectively. Then, in the inner control loop, each link of SU only uses the standard power control algorithm of [21] to control its own power by local measurement to track the given target SINR.

The major contributions of this paper are as follows:

– The dynamic description of power control for CRN is formulated as a linear state space model using variable transformation from the linear scale to the logarithmic scale to establish the foundation for us to use control theory for the controller design of the CRN;

– Without increasing additional assumptions compared with the previous works, an effective real time power control strategy for nominal CRN model has been proposed on the basis of LQR and LQG regulator to track a predesigned SINR for the satisfaction of communication of SUs under the consideration of either without or with the channelfluctuation. The closed-loop control stability analysis of this CRN is provided.The remainder of the paper is organized as follows. Section II introduces the network model in a state space representation. The power control problem formulation based on state space model and the controller design are given in Section III. The overall process of our proposed scheme is summarized in Section IV.Section V gives the stability analysis of overall system. Simulation results are presented in Section VI and conclusion is drawn in Section VII.

II. CRN MODE

Fig. 1. CRN model.

We consider a CRN structure given in figure 1 consisting of a primary network and a secondary network, in which the relative positions among all users describe logic relationship rather than geography. The primary network is assumed as a cellular network with one base station (PU-BS) and L terminals, namely PU receivers (PU-RXs) with index l, l∈L and set L ={1,…,L}. The form of the secondary network is not restricted as a certain one without loss of generality. We assume that there are M SUs working in the network whose index is i, i∈M and set M={1,…,M}. We use SU-TX and SU-RX to denote the transmitter and the receiver of SU respectively. In this CRN, time is divided into slots whose size is appropriately chosen to support the applications, and the system is assumed wide synchronized to slot level. In this study, we only focus on power control of SU when PUs transmit data in downlinks (the channels from BS to mobile terminals), and the all study results can be easily transferred to the uplink case(the channels from mobile terminals to BS),since the latter is a special case of the former.

According to the requirement of IT, the transmit power of the ithSU p ki() at the time slot k must satisfy that the interference from SUs to the lthPU cannot exceed its IT threshold. This strict constraint can be formulated as

Our purpose is to design a distributed power control strategy only using local information to allocate power to each SU. In order to simplify the subsequent control model, we use decibel scale=10lgx to represent variable x. Substituting the variables of (2) with the decibel values, we can get

Let gij(k) denote the channel power gain of the jthSU-TX to the ithSU-RX,andrepresent the interference plus noise from other SUTX to the ithSU-RX at the time slot k, then the effective channel power gain is given asIt is worth mentioning that the interference from PUs to SU is included in the background noise with varianceFurthermore, the SINR of the ithSU can be expressed asand its decibel value can be written as

In order to deduce the dynamics of CRN,we introduce the following important result.

When the secondary network is stationary,the dynamics of the channel power gain gii(k )can be modeled as

The proof of this result is that the channel power gain can be described as a first-order Markov random model in the decibel scale as formulated in [26]

Observing (8), we can see that (5) can be true when the secondary network is stationary,namely a=1.

Using the same development, we can obtain that both the channel power gain dynamics of PU and the interference plus noise dynamics among SUs can also be formulated as the forms ofandand si(k) are the models of the channel power gain fluctuation and the interference fluctuation respectively [13] [15].

After some mathematical calculation, the variation of the corresponding variables required under the decibel scale can be written as

III. PROBLEM FORMULATION AND POWER CONTROLLER DESIGN

In this section, we will formulate the power allocation problem from the perspective of control theory. The total process of power control is divided into outer loop control and inner loop control, where the role of outer loop is to provide a target SINR according to communication needs and then each SU controls its transmit power to make the received SINR track to the target SINR in the inner loop.

In CRN, the normal communications of SUs need to be satisfied, and the interference to PUs should be maintained below a given threshold as well. Thus, in order to satisfy the above communication requirements for both SU and PU, we introduce an auxiliary variable(k)defined as the target SINR controlled by our proposed controller for SU to adjust its allocated power for the following control goals:

– The instantaneous SINR γi(k) tracks the target SINR(k);

To update the target SINR, we introduce a control variable ui( k ) for the ithSU link formulated as

In this way, the transmit power of SU is controlled in every time slot to make the actual SINR approach the target SINR. In the consideration that the control mode should be as simple as possible for its implementation, we modify the classical standard power control of[21] to track the time-varying target SINR

where 0＜αi＜1, for providing “smoothing”effects to reduce the change of transmission power from one time slot to the next one, is the standard power control gain allowed to vary from one link to another. The initial power(0) is assumed to be different in different links. According to (4), (10) and (12), we obtain the following adjustable instantaneous SINR

From the goal of the control strategy, the target SINR of each SU is achieved under the condition that the IT constraint should be ensured. To reach this purpose, we substitute(12) and (9) into (3) to obtain the interference power

Our aim is to select power control sequence {pi( k )} such that the actual SINR of SU must approach as closely as possible to the target one defined by (11) subject to the given IT constraint. To realize this goal, we control the distance between the IT threshold and the actual valueand the one between the target SINR and the actual valueto be as small as possible during a periodof time T with N samplings. Thus, we introduce two kinds of state variablesandwhose expressions can be derived as

Then we define an element state vectorwith L+1 dimensions for the SU link i. Thus, combining (15) and (16), we build the general state space model for the SU link i as

where the coefficient matrices are as follows

The fluctuation about the corresponding channel power gain wi(k) can be considered as an exogenous disturbance in this state space model defined as

and its property will be discussed later.

From (17), we must verify its controllability, since it is necessary for a control problem described by state space model. We calculate the controllability matixwe change the state space m

odel (17) as the following

In order to achieve control objectives, we define a cost function as

It is easy to know that the rank of the controllability matrix is only 2 instead of L+1,which means this state space model is not controllable. The reason is that there is only one working interference entry which must belong to the most vulnerable PU. Therefore, in power control process, we only need to consider this interference entry of the link from SU to PU in every time slot k and the interference of other links can be well restricted. We assume lminis the index of PU whose transmission is the most easily disturbed, and the selection of lminwill be discussed in Section IV. Only considering the distance of the primary link lminbetween the IT threshold and the actual value without interrupting its communication. This often depends on a fixed IT threshold. Thus we assign(k) is larger than(k) and ri( k) for the better tracking of the IT a priori,which means that SU can better use available spectrum resources. The power control based on the above state space model can be formulated as a state feedback control problem in different forms with or without considering channel power gain fluctuation, which is tofind a control gain to satisfy the communication requirements for both PUs and SUs.

3.1 Controller design without channelfluctuation

If we design a controller using the state space model without channel fluctuation, we can consider that each channel power gain is invariable during a certain packet transmission period, so that(k) in (23) can be eliminated and the stochastic power control can be simplified to a standard LQR problem [27]

And the controller of the discrete-time LQR problem is given as

where Pi(k) is the solution of the following discrete-time Riccati equation

It is worth noting that the assumption of the state space model without channelfluctuation is not reliable from a practical point of view.Neglecting the consideration of the gain uncertainties of all channels first, the dynamic of co-channel interference plus noise among SUs should not be ignored in the process of practical communication. Although we have modeled the dynamic of co-channel interference plus noise as a steadyfirst-order Markov random model, the updated transmit power plays an important role for its variation in every time slots. Certainly, the assumption of invariable channel power gain is valid for the random nature of wireless channels. By contrast, the power control with consideration of the channel fluctuation should be more practical. Thus we will design a more practical method to control the power of SUs in next subsection.

3.2 Controller design with channelfluctuation

We have analyzed the importance of the channel power gain uncertainty and the dynamic of co-channel interference plus noise, thus we will propose our control algorithm with considering these two factors in this subsection.Note that the known information a priori of the channel fluctuation directly determines the control mode and precision [13, 15]. If the stochastic distribution of the exogenous disturbance is assumed as the Gaussian, the controller can be formulated as the LQG solution,otherwise more complex stochastic control theory can be introduced. Here we only study the former case to show the effectiveness of our method.

Based on the assumption provided in [13],we also assume(k) is a 2×1 random vector with covariance matrixIn the LQG problem, an auxiliary variable called output measurement vector yi(k) should be considered and written as a linear function of the state vector(k), that is

In general, the coefficient matrix Ciis assigned as a unit matrix and the measurement noise vi(k) with covariance matrixis independent with(k). The power control based on the LQG formulation is such that we find a controller(k) to minimize the following stochastic quadratic cost function subject to the state space constraint (23), i.e,

where E{?} is a mean for a variable.

As mentioned above, we can adjust the weight of each term of Qi(k) and solve this stochastic LQG problem [28]. The solution of LQG regulator is combination of the LQR and the Kalman filter, and this controller can be written as

where Ki(k) has been given in (26), and the estimated state(k) is the output of the Kalmanfilter

Fig. 2. Block diagram of the ith SU transceiver pair with standard power control algorithm in local loop.

IV. THE OVERALL POWER CONTROL PROCESS

Without considering any transmission delay, the cascade power control block diagram of the ithSU is depicted in figure 2. First, the SU-RX decodes the information from traffic environment,including the received SINRand the interference to all PUs. To explain little more, the PU who most need to be considered to establish the state space should be picked out. To get the index of the most vulnerable PU is to select the index having minimal difference between the IT threshold and the actual interference, which can be formulated as

Based on these parameters, the outer loop updates the target SINR by using the optimal control( k ) from (26) or LQG regulator(31). Then each SU can utilize the standard power control method (12) to adjust its own power to reach the new target SINR as well as to satisfy the IT constraint.

More clearly, we summarize our proposed power control scheme in algorithm 1.

V. STABILITY ANALYSIS

When we put all state space model of SU together, the dynamic of the whole CRN can be expressed as

We briefly discuss the stability of state space model (23). From control theory, we can deduce that (23) is internal stability if the closed-loop system

with zero-input , the pair ofis controllable,is detectable, and Qi(k) is symmetric positive definite [30]. It is obvious that (23) is internal stability since these conditions are all satisfied. External stability is also called input-output stability which means a bounded input always produces a bounded output, also called BIBO stability. Since the solution of the previous Riccati equation (28)can guarantee the external stability in discrete-time linear system [16], we can conclude that system (36) is stable.

VI. SIMULATION RESULTS

We set up a system-level simulation environment with a smaller scale of network, in order to facilitate the analysis of the performance of this CRN. The relative locations among all users are generated randomly in an area of 2000 meters radius where PU-RXs distribute around PU-BS and SU-RXs surround their own transmitters as presented in figure 3. From figure 3, we can see that there are 2 PU links and 3 SU links, and we conduct the simulations on frame with transmit power and interference without the consideration of modulation and coding.

In this simulation scenario, we consider the path loss and shadowing for all links regardless of fast fading, since the period of power updated is large enough for fast fading. We define dijis the distance from transmitter j to receiver i, then the coherent channel power gain

is a lognormal distribution according to [31],whereis a constant depending on antenna characteristics and average channel attenuation, d0is a reference distance for far-field antenna, η is the path loss exponent and ψdBis a Gaussian distributionrandom variable with mean zero and variance. And the correlation of shadowing over distance can be characterized as

Algorithm 1. Distributed power control process.

Fig. 3. Simulation environment.

As discussed in Section II, the secondary system is stationary, which means that the velocity v tends to zero, then

These above parameters can be obtained to approximate either an analytical or empirical model. In our simulation, we set d0=10m,, and the carrier frequencywhereand c=3× 108m/s,and transmission bandwidth is 1MHz. As previously mentioned, the background noise includes the interferences from PUs to SU,which is assumed as=4.2× 10?8W,=2.2× 10?8W and=2.4× 10?8W, respectively. And we give the IT threshold=5.89× 10?10W and=6.88 × 10?10W. On the basic of the environment, each SU-TX adjusts its power by the algorithm given in table 1.

6.1 Power control performance of LQR

Fig. 4. Transmit power and interference with different αi.

First, we analyze a complete power control process of each SU when the channel has nofluctuation. In the control process, the setting parameters play an important role. For the control weights of the cost function described earlier in Section III, according to our experiments, we choose,and ri( k )=1, respectively. We only take SU2 as an example to demonstrate the controlled transmit power and the interference for PU1 in figure 4 since the interference between the two users may be the most obvious in view of their geographical locations in figure 3. From this figure, we can see that the ultimate power arrived are almost same no matter what choice of αiwithin a certain range is, however the convergence rate is significantly different with different αi. The larger αiis, the more quickly the power converges. But, we cannot only pursue the convergence speed without considering the influence that higher αiresults in large power increase in each time slot, which means that the interference from SU to PU would be beyond the IT threshold. Although we do notfind the overshoot or the excessive interference on PU in the second subfigure of interference in figure 4, it will occur when the initial power is slightly larger. Thus, αishould be selected appropriately according to practical communication situation. In the subsequent experiments, we set αi=0.48 for both convergence speed and interference consideration in the assumed communication scenario.

In addition, with the setting of parameters,the power control process of each SU link is presented in figure 5, including the controlled transmit power, the interference to each PU and the track of target SINR. Figure 5 (a) illustrates each link can well control its own

power and complete power control in several time slots. From figure 5 (b), we can see that each PU can be protected by using the proposed power control scheme, since the interference from all SUs to each PU reaches the IT threshold at the most rather than beyond it,where Ave-Ith1 is defined as the average IT threshold of the 1stPU and Ave-Ith2 is the 2ndPU’s. Meanwhile, we also find that the communication performance of each SU can also be ensured from figure 5 (c), since the target SINR can be adapted according to the degree of how much interference of SUs to every PU.When the interference to PU is far away from corresponding IT threshold, the target SINR is increased appropriately by the proposed controller. However, the target SINR does not increase very much if the interference is close to the threshold. These results tell us that the proposed LQR power control algorithm can satisfy the communication requirement for both SUs and PUs at the same time slot.

6.2 Power control performance of LQG regulator

We know that the good performance of the LQR power control scheme displayed in figure 5 depends heavily on the assumption of the channel without fluctuation. This ideal communication condition would not happen in the real situation but only give us the opportunity to develop new control method easily and understand the mechanism of the whole control process. Therefore, we should go even further with our study in considering time-varying wireless channel more for the actual communication environment. We propose an algorithm based on LQG regulator for stochastic channel model in order to illustrate its performance in this problem shown in figure 6. We conduct the experiment on the basis of the previous parameter setting of the nominal model, and assume the uncertain parameters wi(k) and vi(k) generated as Gaussian random vector.Accurately, the change of wi(k) is caused by both the fluctuation of the channel and the power control. Thus we can assume the change of channel power gain follows a zero mean Gaussian random vector with variance 0.01 and vi(k) is defined as a zero mean Gaussian random vector with variance 1.

Fig. 5. Results of LQR based power control for each SU.

Fig. 6. Results of power control using different algorithm in stochastic channel.

In figure 6, we present comparison of the performance of LQR power control with that of power control based on LQG regulator for the stochastic channel. As shown in figure 6(a) and figure 6 (b), the transmit power controlled by LQR varies up and down since the channel state changes over time, in contrast,the transmit power of LQG regulator is much more smoothly because of the robustness of this method. Comparing figure 6 (c) with figure 6 (d), we can also see that the proposed LQR control strategy for nominal model cannot well control the interference produced by SUs to be below the IT threshold all the time when the actual channel power gain changes randomly. But, the designed LQG regulator can keep the interference from each SU to each PU under the corresponding IT threshold at most time, especially in the later stages of the period. In figure 6 (d), despite the interference is seldom little above the IT threshold for this case, the given conservative measure has been considered in the manner of averaging IT constraint to further protect the communication of the primary link. Finally, the superiority of LQG strategy in the sense of the SINR tracking is revealed in figure 6 (f) comparing with that of the SINR tracking of LQR strategy in figure 6 (e), which shows that the LQG regulator can make each SU quickly follow its own the target SINR but the LQR does not.

In order to illustrate the superiority of our proposed power control scheme, we give some numerical simulations of the performance comparison between our algorithm and two other algorithms, the distributed optimal algorithm based on convex optimal technology and its corresponding robust version (called ConOpt and WorstCase algorithm respectively), where the uncertainty of the channel and measurement noise are also considered. The WorstCase algorithm deals with the various uncertainties [22, 24] in which the uncertainty is measured as the worst percentage 0.2.

In figure 7 and figure 8, the performance comparison is presented from two aspects, the total interference from all SUs for each PU and the total data transmission rate of SUs.As shown in figure 7 for the total interference for each PU, ConOpt cannot well protect the communication of PUs when the channel uncertainty and the measurement noise are considered. The protection for PUs by WorstCase depends on the priori knowledge about uncertainty, which is the worst percentage in this simulation. However, in this environment, ouralgorithm still works well. Certainly, this better performance is obtained by the scarification of the data transmission rate. Thus, from figure 8, we can see that the total data transmission rate of our algorithm is little bit lower compared with those in other two algorithms.But in the sense of the protection for PU in the CRN, our algorithm outperforms ConOpt and WorstCase algorithms.

Fig. 8. Total data transmission rate of SU for difference algorithms.

VII. CONCLUSIONS

Fig. 7. Total interference for each PU versus time slot for difference algorithms.

In this paper, we propose a power control scheme for a CRN based on formulating the traditional power control problem into a dy-namic state space model. The objective lies in thefinding optimal controller to adjust the target SINR of SU according to IT constraints.Considering the two scenarios without or with channel fluctuation, the proposed LQR and LQG regulator both achieve our control goal through the analysis of the computer simulations. The simulation results show that all SUs can allocate its power to satisfy the IT constraints which implies that the communication requirement for each SU is guaranteed. More interestingly, our proposed state space model is very simple and useful in the design of power control in CRNs since it provides other powerful mean to solve conventional power allocation problem under the optimization approaches with new perspective.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant No.61571209).