Robust Mixed $\text {H}_{2} / \text {H}_{\infty }$ Power Control Algorithm for Cognitive Radio Networks

Abstract

In this paper, a basic dynamical state-space system model with two state feedback controllers and tracking signal-to-interference-plus-noise (SINR) of secondary user (SU) and interference to primary user (PU) generated from the whole secondary user (SUs) is proposed based on the automatic control framework to guarantee the quality of service (QoS) both primary user (PU) and SU in underlay cognitive radio communication network, which considers the constraint of the average interference temperature. On the above foundation, a delay state-space model with the constraint of the weighted interference temperature is proposed under the time delay, multiple access interference (MAI), channel fading, measurement errors, and uncertain noise. Moreover, the robust mixed $\text {H}_{2} / \text {H}_{\infty }$ power tracking algorithm (RPCA) with adaptive step is proposed to obtain a robust optimal SU’s SINR tracking error under the minimization the worst-case and the minimum interference to PU tracking error, which can compensate the effects of the above factors. In comparison with the conventional power control algorithm (CPCA), the proposed mixed $\text {H}_{2} / \text {H}_{\infty }$ RPCA can ensure the communication quality of SUs and reduce the interference to PU from the all SUs.

Appendix A

1.1 Channel Model

The above $g_{i i}(k)$, $g_{j i}(k)$, $g_{l i}(k)$ are time-varying power transmission gains in wireless communication channel, which are collectively called $g_{l i}(k)$. It is composed of long-term fading $g_{l f}(k)$ and short-term fading $g_{s f}(k)$[14] . Thus we have

$$\begin{aligned} g(k)=g_{l f}(k)+g_{s f}(k) \end{aligned}$$

(45)

The short-term fading $g_{s f}(k)$ is fast power fluctuations caused by the motion of PUs and SUs, and follow the Rayleigh distribution.

The long-term fading $g_{l f}(k)$ includes an associated with distance-dependent path loss and a shadow fading caused by barriers of the transmission link, it can be expressed as

$$\begin{aligned} g_{l f}(k)=g_{0}-10 \alpha \log _{10}(D)+S(k) \end{aligned}$$

(46)

where $g_{0}$ represents the path loss at the reference distance and is a constant. D is the distance of transmitter and receiver. $\alpha$ is path loss factor. The shadow fading S(k) is and has logarithmic normal distribution characteristics in signal level. It can be seen as a first order autorgressive model [AR(1)]–filtered gaussian white noise. The models of the shadow fading S(k) and corresponding autocorrelation function $R_{s}(k)$ are given as

$$\begin{aligned} S(k)= & {} \exp \left( \frac{\ln (\beta ) v(k) T_{s}}{D_{e}}\right) S(k-1)+n(k-1) \end{aligned}$$

(47)

$$\begin{aligned} R_{s}(k)= & {} \sigma _{s}^{2}(k) \exp \left( \frac{\ln (\beta ) v(k) T_{s}}{D_{e}}\right) ^{|k|} \end{aligned}$$

(48)

where $\beta$ denotes autocorrelation coefficient between receiver and transmitter at distance $D_{e}$ , v(k) is velocity of the mobile terminal at time slot k, $T_{s}$ is the sampling period, $n(k-1)$ is a white Gaussian process of the zero mean value at time slot $k-1$ and variance

$$\begin{aligned} \sigma _{n}^{2}(k)=\left( 1-\left[ \exp \left( \frac{\ln (\beta ) v(k) T_{s}}{D_{e}}\right) \right] ^{2}\right) \sigma _{s}^{2}(k) \end{aligned}$$

(49)

at time slot k. The $\sigma _{s}(k)$ represents standard deviation of the shadow fading at time slot in the actual communication environment.

Short-term fading is caused by channel fading fast power fluctuations and follow the Rayleigh distribution, short-term fading by famous Jack model simulation in this paper.