Robust Beamforming Design for IRS-Assisted Downlink Multi-User MISO-URLLC in an IIoT Scenario

Abstract

Intelligent reflecting surfaces (IRS) have recently been regarded as having potential for supporting multi-user (MU) multiple-input single-output (MISO) ultra-reliable and low-latency communication (URLLC) by creating favorable wireless communication links in the industrial IoT (IIoT) scenario. Hence, we studied the joint robust design of a beamformer at a central controller (CC) and phase shifters at the IRS for minimizing the transmit power consumption based on perfect channel state information (CSI) and imperfect CSI, respectively. The design was formulated as a non-convex optimization problem, also taking into account the quality-of-service (QoS) demands of the actuator, unit-modulus constraints of the IRS, and the robustness against the impact of CSI imperfection. Subsequently, we proposed a computationally efficient iterative algorithm to obtain a suboptimal solution by exploiting the penalty method and the successive convex approximation (SCA) for perfect CSI, while the penalty method, S-procedure, and SCA were adopted for imperfect CSI. Finally, our simulation results revealed that (1) the required transmit power consumption for URLLC was significantly reduced by employing the proposed IRS instead of conventional wireless communication without IRS; and (2) the proposed algorithm was robust and effective for the beamforming design.

1. Introduction

With the rapid development of emerging industrial internet of things (IIoT) technology, the conventional industrial wired communication network will be replaced by the industrial wireless communication network, which has the advantage of enhancing deployment flexibility and reducing installation and maintenance costs [1]. Moreover, there are many mission-critical IIoT applications that require ultra-reliable and low-latency communication (URLLC) in order to deliver sensor data and actuation commands at precise instants with designed reliability, such as factory automation (FA), which expects the end-to-end delay to be less than $100 us$, while the block error rate (BER) to be less than ${10}^{-9}$ [2]. To meet the stringent demand of quality-of-service (QoS) in the IIoT applications, finite blocklength (FBL) transmit should be performed in each URLLC packet [3], which will inevitably cause a loss of coding gain and Shannon’s Capacity formula is therefore no longer applicable. Moreover, traditional wireless communication technologies cannot guarantee the crucial requirements of URLLC packets if the channel environment exhibits deep fading. Besides, the traditional wireless communication channel is more challenged because of the uncontrollable channel propagation environment in the IIoT scenario, such as electromagnetic interference, arbitrary movement of objects (robots and vehicles), room dimension, thick structural pillars, etc. These challenges can significantly affect the reliability and performance of wireless communications in IIoT environments. Therefore, it is crucial to investigate new technologies to ensure URLLC in an IIoT scenario.

1.1. Related Works

Intelligent reflecting surfaces (IRS) equipped with a large amount of low-cost and passive reflecting elements comprise a key enabling technology that can significantly improve wireless network performance by creating a programmable communication link to bypass the obstacles between communicating devices [4,5]. Moreover, IRS has attracted extensive research interest from both academics and industry [6,7]. Most of the existing contributions on IRS focus on the transmit design for IRS-aided wireless networks by jointly optimizing the beamformer at the BS and the phase shifters at the IRS for perfect channel state information (CSI), e.g., in [8,9,10,11,12], imperfect CSI, e.g., in [13,14,15], or statistical CSI, e.g., in [15,16,17]. More specifically, the author of [8] minimized the sum transmit power of an IRS-assisted multi-antenna base station (BS) wireless network for single-user systems and multi-user (MU) systems by jointly optimizing the beamforming at the access point (AP) and the continuous phase shifter at the IRS, where the semidefinite relaxation (SDR) technique and the alternate optimization (AO) method was proposed to address this non-convex optimization problem, and the simulation results demonstrated the significant performance gain compared with conventional massive MIMO or multi-antenna amplify-and-forward (AF) relay. Similarly, the sum transmit power minimization problem was further studied in [9] for a single-antenna BS, where a resource allocation problem was proposed by jointly optimizing the transmit power at the BS and the continuous phase shifters at the IRS, and then the dual method and SDR technique was proposed to solve this problem. However, the obtained solution in [8,9] cannot guarantee the convergence to a local optimality solution because of the uncertainty of the Gaussian random technique. The joint beamforming optimization problem was further studied by the author of [10] for an IRS-assisted downlink MU multiple-input single-output (MISO) system, where the weighted sum rate maximized for the continuous and discrete IRS phase shifter was proposed based on the fractional programming technique and AO method, and then the closed-form expression for the passive beamforming at the IRS was derived. The author of [11] proposed a suboptimal low-complexity inexact alternating optimization algorithm to maximize the minimum weighted signal-to-interference-plus-noise ratio (SINR) by jointly optimizing the transmit and reflective beamforming vectors in IRS-aided multi-cell multiple-input single-output (MISO) system, where the SINR performance of cell-edge users can be considerably improved by the deployment of IRS. Besides, the author of [12] minimized the sum transmit power by applying the inner approximation (IA) algorithm to converge to a Karush–Kuhn–Tucker (KKT) solution. However, the accurate acquisition of perfect CSI in [8,9,10,11,12] is a challenging task. Hence, the author of [13] investigated a robust beamforming design for an IRS-aided MISO communication system with an imperfect CSI of the reflection channel, where the minimizing sum transmit power was proposed by jointly optimizing the beamforming at the BS and the continuous phase shifter at the IRS, and then the AO method and S-Procedure were adopted to solve this problem. Moreover, the obtained solution cannot guarantee the convergence to a local optimality solution. A secure IRS-assisted wireless network for maximizing the sum rate in the presence of MU and eavesdroppers was presented in [14], where an AO method, penalty-based approach, SCA, and SDR were developed and a local optimality solution could be obtained. The robust transmit design of an IRS-aided MISO wireless communication system with both imperfect CSI and transceiver hardware impairments was characterized in [15], while the statistical CSI was also considered. The author of [16] formulated a sum transmit power minimization problem subject to the rate outage probability constraints, where the Bernstein-type inequality was used to reformulate the problem. Furthermore, the weighted sum rate was maximized for the constraint of the discrete IRS phase shifter proposed in [17], and the problem was decoupled into two subproblems which were solved by majorization–minimization and the gradient descent method to obtain the optimal passive beamforming at the IRS. A summary of different performance analyses for an IRS-assisted wireless communication system is provided in

Table 1. A summary of different performance analyses for IRS-assisted wireless communication systems.

Ref.	System Setup	CSI	Optimization Problem	Proposed Reflection Design Algorithm	Main Results
[8]	MU MISO	perfect	The sum transmit power minimization constrained by users’ SINR and the continuous phase shifter at the IRS	AO and SDR	Cannot guarantee the convergence to a local optimality solution
[9]	MU SISO			Dual method and SDR
[10]	MU MISO		The weighted sum-rate maximization constrained by the transmit power at the BS and the continuous or discrete phase shifter at the IRS	Fractional programming method and AO	The closed-form expression for the passive beamforming at the IRS was derived
[11]	Multi-cell MU MISO		The minimum weighted SINR maximization constrained by the transmit power at the BS and the continuous phase shifter at the IRS	SCA and AO	Can guarantee the convergence to a local optimality solution
[12]	MU MISO		The sum transmit power minimization constrained by users’ SINR and the continuous phase shifter at the IRS	IA algorithm
[13]		imperfect	The sum transmit power minimization constrained by users’ worst-case achievable rates and the continuous phase shifter at the IRS	AO and S-Procedure
[14]			Users’ worst-case sum rate maximization constrained by the worst-case information leakage to eavesdroppers	AO, SDR, SCA, and penalty-based approaches
[15]		imperfect statistical	The sum transmit power minimization constrained by the worst-case SINR or the maximum outage probability for each user’s SINR combined with the continuous phase shifter at the IRS and the transceiver hardware impairment	SCA, S-Procedure, and BCD
[16]		statistical	The sum transmit power minimization constrained by the maximum outage probability of each user’s rate	Bernstein-type inequality	Cannot guarantee the convergence to a local optimality solution
[17]	MU MIMO		The weighted sum-rate maximization under the statistical CSI error model constrained by the transmit power at the BS and the discrete phase shifter at the IRS	AO, majorization–minimization, and gradient descent methods	The closed-form expression for the active beamforming was derived and the optimal passive beamforming at the IRS was obtained

.

1.2. Motivations and Contributions

All the above-mentioned works only focused on the transitional wireless communication networks with infinite blocklengths (IFBL), where the Shannon’s Capacity formula is an accurate approximation of the achievable rate. Then, in an IIoT scenario with stringent URLLC requirements, because the law of large numbers cannot hold, Shannon’s Capacity formula is no longer applied. Polyanskiy et al. approximated the theoretical boundary of the achievable rate with FBL in [18] and found that it was a non-convex complex function of blocklength, SINR, and BER. Most recently, based on the results in [18], extensive research attention has been devoted to the FBL transmit design [19,20,21,22,23,24] or performance analysis under the FBL [25,26,27,28,29,30,31]. However, all the contributions in [19,20,21,22,23,24,25,26,27,28,29,30,31] did not consider IRS-assisted wireless networks. Only a few scholars have studied the IRS-aided MU MISO-URLLC system in transmit design [32,33,34]. In [32], the authors proposed an iterative algorithm to obtain a suboptimal solution for maximizing the sum rate by joint optimization of the beamformer at the BS and the phase shifters at the IRS under perfect CSI, while taking into account the QoS constraint. In [33], the authors studied a multi-objective optimization problem for maximizing the achievable rate while minimizing the channel blocklength by jointly optimizing the transmit power at the BS, the number of the channel blocklength, and the phase shifters at the IRS under perfect CSI. In [34], the authors investigated the resource allocation design for IRS-assisted MISO orthogonal frequency division multiple access (OFDMA) multi-cell systems under perfect CSI. However, these three works focused on the transmit design under perfect CSI, and the robust design of the system under an IRS-aided MU MISO-URLLC system is not yet available in the existing literature.

Against the abovementioned background, we present a robust beamforming design of the IRS-assisted MU MISO-URLLC system in an IIoT scenario when CSI is perfect or imperfect. Specifically, the main contributions of this paper are summarized as follows:

• To minimize the transmit power consumption of the central controller (CC), both the active beamforming at the CC and the passive beamforming at the IRS were jointly optimized while satisfying the QoS requirement of the actuators. A robust beamforming design was proposed for perfect CSI and imperfect CSI, respectively;
• For perfect CSI, we first transformed the resulting highly coupled non-convex optimization problem into a tractable form with decoupled optimization variables. To address this transformed optimization problem, we proposed an efficient iterative algorithm to obtain a suboptimal solution by exploiting the penalty method and the successive convex approximation (SCA). This algorithm is guaranteed to converge and obtain a suboptimal solution of the original optimization problem. Furthermore, we extended the efficient iterative algorithm by exploiting the S-procedure to address the impact of CSI imperfection;
• Simulation results under various settings of parameters demonstrated the superiority of the proposed algorithm. Particularly, the proposed algorithm in this paper has obvious advantages in reducing the total power consumption of the CC and guaranteeing effectiveness compared with other algorithms.

The main notations and acronyms used in this paper are listed in Appendix A, i.e.,

Table A1. List of main notations.

Notation	Description	Notation	Description
$N$	Antenna number of the CC	$M$	The reflecting number of the IRS
$K$	Number of the actuators	$w_k$	The active beamforming from CC to the $k$-th actuator
$v$	The passive beamforming at the IRS	${\theta }_m$	The phase shift of the $m$-th element’s transmit coefficient at the IRS
$\Theta$	The diagonal reflection-coefficient matrix at the IRS	$h_{bk}$	The channel between the CC and the $k$-th actuator
$H_{br}$	The channel between the CC and the IRS	$h_{rk}$	The channel between the IRS and the $k$-th actuator
${\Phi }_k$	The IRS related cascaded channel to the $k$-th actuator	$R_k$	The channel from BS-IRS to the $k$-th actuator
${\tilde{\Phi }}_k$	The estimated IRS related cascaded to the $k$-th actuator	${\tilde{h}}_{dk}$	The estimated channel between the CC and the $k$-th actuator
${\tilde{R}}_k$	The estimated channel from BS-IRS to the $k$-th actuator	$\Delta h_{bk}$	The direct uncertain channel error
$\Delta {\Phi }_k$	The IRS related cascaded uncertain channel error	$\Delta R_k$	The uncertain channel error from BS-IRS to the $k$-th actuator
${\delta }_{h_{bk}}^2,{\delta }_{{\Phi }_k}^2,{\delta }_k^2$	The radii of the CSI error regions for the $k$-th actuator	$L_k$	The transmit blocklength at the $k$-th actuator
${\epsilon }_k$	The BER at the $k$-th actuator	${\sigma }_k^2$	The power spectral density of the noise at the $k$-th actuator

The main acronyms used in this paper are listed in Table A2.

and

Table A2. List of main acronyms.

Acronym	Description	Acronym	Description
URLLC	ultra-reliable and low-latency communication	SDP	standard semidefinite program
SINR	signal-to-interference-plus-noise-ratio	PSD	positive semidefinite
CSI	channel state information	SCA	successive convex approximation

, respectively.

2. System Model

2.1. Signal Model

As shown in Figure 1, we considered an IRS-assisted downlink MU MISO-URLLC system in an IIoT scenario, where the CC equipped $N$ antennas simultaneously serving $K$ single antenna actuators, which are indexed by $\mathcal{K}=\left\{\begin{array}{ccc}1,& \cdots ,& K\end{array}\right\}$. We assumed the number of IRS phase shifters is $M$, indexed by $\mathcal{M}=\left\{\begin{array}{ccc}1,& \cdots ,& M\end{array}\right\}$. Hence, the passive beamforming vector can be represented as $v={\left[\begin{array}{ccc}{v}_1,& \cdots ,& v_M\end{array}\right]}^H\in {ℂ}^{M\times 1}$, where $v_m=e^{j{\theta }_m}$, $m\in \mathcal{M}$, and ${\theta }_m\in \left[\left.0,2\pi \right)\right.$ represent the phase shifter of the $m$-th reflecting element at the IRS. Then, the reflection coefficient matrix can be modeled as $\Theta =\mathrm{diag}\left(\begin{array}{ccc}{v}_1,& \cdots ,& v_M\end{array}\right)$. We defined the baseband signal $x_k$ of the $k$-th actuator satisfies $\mathbb{E}\left\{x_k\right\}=0$ and $\mathbb{E}\left\{{\left|x_k\right|}^2\right\}=1$. Using linear precoding for all actuators, the transmit signal vector of the CC was generated by superimposing the baseband signals of different actuators with linear precoding. Then, the signal vector of the CC can be formulated as $s={\displaystyle {\displaystyle \sum }_{k=1}^K}{w}_k{x}_k$, where $w_k\in {ℂ}^{N\times 1}$ is the active beamforming vector from the CC to the $k$-th actuator. Thus, the total transmit power consumption can be expressed as ${\displaystyle {\displaystyle \sum }_{k=1}^K}\parallel w_k\parallel {}^2$.

The channel between the CC and the $k$-th actuator between the CC and the IRS, and between the IRS and the $k$-th actuator can be denoted by $h_{bk}\in {ℂ}^{N\times 1}$, $H_{br}\in {ℂ}^{M\times N}$, and $h_{rk}\in {ℂ}^{M\times 1}$, respectively. We assumed all channels satisfy the quasi-static flat fading channel model, while the channel coefficients remain constant over a blocklength and the transmit time consumed is strictly less than the coherence time [35].

Thus, the received signal $y_k$ of the $k$-th actuator can be expressed as

$$y_k=\left(h_{bk}^H+h_{rk}^H\Theta H_{br}\right)w_k{x}_k+{\displaystyle \sum }_{i=1,i\ne k}^K\left(h_{bk}^H+h_{rk}^H\Theta H_{br}\right)w_i{x}_k+n_k,$$

(1)

where $n_k\sim N_{ℂ}\left(0,{\sigma }_k^2\right)$ is the additive white Gaussian noise (AWGN) of the $k$-th actuator. The cascaded channel to the $k$-th actuator in the second term of the right-hand side of (1) can be simplified as $h_{rk}^H\Theta H_{br}=v^H{\Phi }_k$, where ${\Phi }_k=\mathrm{diag}\left(h_{rk}^H\right)H_{br}$. Then, the SINR of the $k$-th actuator can be expressed as

$${\gamma }_k=\frac{{\left|\left(h_{bk}^H+v^H{\Phi }_k\right)w_k\right|}^2}{{\displaystyle {\displaystyle \sum }_{i=1,i\ne k}^K}{\left|\left(h_{bk}^H+v^H{\Phi }_k\right)w_i\right|}^2+{\sigma }_k^2}.$$

(2)

According to [18,36], the transmit rate formula for URLLC can be approximated as

$$r_k\approx {\log }_2\left(1+{\gamma }_k\right)-\frac{Q^{-1}\left({\epsilon }_k\right)}{\sqrt{L_k}}\sqrt{V\left({\gamma }_k\right)}{\log }_{2}e,\left(bits/s/Hz\right),$$

(3)

where $L_k$ is the blocklength of the transmit signal $x_k$, ${\epsilon }_k$ is the BER of the $k$-th actuator, $V\left({\gamma }_k\right)=1-{\left(1+{\gamma }_k\right)}^{-2}$ is the channel dispersion, and $Q^{-1}\left(x\right)$ is the inverse of $Q\left(\cdot \right)$, where $Q\left(x\right)={{\displaystyle \int }}_x^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-\frac{t^2}{2}}dt$. For simple analysis, we defined $D_k=\frac{Q^{-1}\left({\epsilon }_k\right)}{\sqrt{L_k}}{\log }_{2}e$.

2.2. Optimization Problem Formula

In this subsection, we consider the QoS requirement of the actuators, which satisfied $r_k⩾{\tilde{r}}_k,\forall k\in \mathcal{K}$, where ${\tilde{r}}_k$ is the worst QoS requirement of the $k$-th actuator. Then, we propose the following minimizing optimization problem.

$$(\mathrm{P}1)\underset{w_k,v}{\min }{\displaystyle \sum }_{k=1}^K\parallel w_k\parallel {}^2$$

(4)

$$s.t.r_k⩾{\tilde{r}}_k,\forall k\in \mathcal{K},$$

(5)

$$\left|v_m\right|=1,\forall m\in \mathcal{M},$$

(6)

where (4) is the total power consumption at the CC, (5) is the constraint of the $k$-th actuator satisfying the QoS threshold transmit rate, and (6) is the phase shifters constraint at the IRS. Because of the non-convex constraint of (5) and (6) in (P1), although the objective function (4) is convex, (P1) is an NP problem and hard to directly solve.

We introduced Lemma 1 to address the complex expression in (5), where (5) can be further simplified.

Lemma 1.

The transmit rate expression satisfying the QoS constraint for (5), $r_k⩾{\tilde{r}}_k,\forall k\in \mathcal{K}$, is equivalent to ${\gamma }_k⩾{\tilde{\gamma }}_k⩾{\chi }_0,\forall k\in \mathcal{K}$, where ${\tilde{\gamma }}_k=e^{{\tilde{r}}_k+\frac{{\kappa }^{\ast }}{2}-1}$, and ${\chi }_0=\sqrt{\frac{1+\sqrt{1+4D_k^2}}{2}}-1$. The parameter ${\kappa }^{\ast }$ in ${\tilde{\gamma }}_k$ is ${\kappa }^{\ast }=\mathcal{W}\left({}^{2D_k,-2D_k};-4e^{-2{\tilde{r}}_k{D}_k^2}\right)$, which is given by

$$\mathcal{W}\left({}^{{\tau }_1,{\tau }_2};\mu \right)={\tau }_1-{\displaystyle \sum }_{m=1}^{\infty }\frac{1}{m\cdot m!}{\left(\frac{\mu m{e}^{-{\tau }_1}}{{\tau }_2-{\tau }_1}\right)}^m\cdot {\displaystyle \sum }_{n=1}^{m-1}\frac{\left(m-1+n\right)!}{n!\cdot \left(m-1-n\right)!}{\left(\frac{-2}{2m\left({\tau }_2-{\tau }_1\right)}\right)}^n.$$

(7)

Proof. 

According to the characteristic of $r_k$ in URLLC [19], when $0⩽{\gamma }_k<{\chi }_0$, $r_k$ is a monotonically decreasing function about ${\gamma }_k$. When ${\gamma }_k⩾{\chi }_0$, $r_k$ is a monotonically increasing function about ${\gamma }_k$. Therefore, when ${\gamma }_k⩾{\tilde{\gamma }}_k⩾{\chi }_0$ is satisfied, where $r_k\left({\tilde{\gamma }}_k\right)={\tilde{r}}_k$, $r_k$ is a monotonically increasing function. Thus, the transmit rate expression satisfying the QoS threshold constraint for (5), $r_k⩾{\tilde{r}}_k,\forall k\in \mathcal{K}$, is equivalent to ${\gamma }_k⩾{\tilde{\gamma }}_k⩾{\chi }_0,\forall k\in \mathcal{K}$. □

Using Lemma 1, the original optimization problem (P1) can be reformulated as

$$(\mathrm{P}2)\underset{w_k,v}{\min }{\displaystyle \sum }_{k=1}^K\parallel w_k\parallel {}^2$$

(8)

$$s.t.{\gamma }_k⩾{\tilde{\gamma }}_k⩾{\chi }_0,\forall k\in \mathcal{K},$$

(9)

$$\left|v_m\right|=1,\forall m\in \mathcal{M}.$$

(10)

There are two inequalities in (9): ${\gamma }_k⩾{\tilde{\gamma }}_k$ and ${\gamma }_k⩾{\chi }_0$. Since ${\chi }_0$ is a function about $D_k$, the lower bound of ${\tilde{\gamma }}_k$ can be obtained by parameter setting. Therefore, constraint (9) can be further described as ${\gamma }_k⩾{\tilde{\gamma }}_k$, and problem (P2) can be simplified as

$$(\mathrm{P}3)\underset{w_k,v}{\min }{\displaystyle \sum }_{k=1}^K\parallel w_k\parallel {}^2$$

(11)

$$s.t.{\gamma }_k⩾{\tilde{\gamma }}_k,\forall k\in \mathcal{K},$$

(12)

$$\left|v_m\right|=1,\forall m\in \mathcal{M}.$$

(13)

The coupling product square term in constraint (12) with $w_k$ and $v$ is hard to solve. Therefore, the method based on the matrix ascending dimension is proposed to eliminate the square term in (12). Then, the desired signal power of the $k$-th actuator in the numerator term of (2) can be rewritten as

$$\begin{array}{cc}{\left|\left(h_{bk}^H+v^H{\Phi }_k\right)w_k\right|}^2& =2\mathrm{Re}\left\{v^H{\Phi }_k{W}_k{h}_{bk}\right\}+v^H{\Phi }_k{W}_k{\Phi }_k^{H}v+h_{bk}^H{W}_k{h}_{bk}\\ & =Tr\left({\tilde{v}}^H{R}_k^H{W}_k{R}_k\tilde{v}\right)\\ & =Tr\left(W_k{R}_{k}V{R}_k^H\right),\end{array}$$

(14)

where $W_k=w_k{w}_k^H\in {ℂ}^{N\times N}$, $\tilde{v}={\left[\begin{array}{cc}{v}^T& 1\end{array}\right]}^H\in {ℂ}^{\left(M+1\right)\times 1}$, $V=\tilde{v}{\tilde{v}}^H\in {ℂ}^{\left(M+1\right)\times \left(M+1\right)}$, and $R_k=\left[\begin{array}{cc}{\Phi }_k^H& h_{bk}\end{array}\right]\in {ℂ}^{N\times \left(M+1\right)}$.

Substituting (14) into (12), the problem (P3) can be reformulated as

$$(\mathrm{P}4)\underset{W_k,V}{\min }{\displaystyle \sum }_{k=1}^{K}tr\left(W_k\right)$$

(15)

$$s.t.\frac{Tr\left(W_k{R}_{k}V{R}_k^H\right)}{{\displaystyle {\displaystyle \sum }_{i=1,i\ne k}^K}Tr\left(W_i{R}_{k}V{R}_k^H\right)+{\sigma }_k^2}⩾{\tilde{\gamma }}_k,\forall k\in \mathcal{K},$$

(16)

$$V\succcurlyeq 0,W_k\succcurlyeq 0,\forall k\in \mathcal{K},$$

(17)

$$rank\left(V\right)=1,rank\left(W_k\right)=1,\forall k\in \mathcal{K},$$

(18)

$$V_{m,m}=1,1⩽m⩽M+1.$$

(19)

For the optimization problem (P4), the non-convexity arises from constraints (16) because of severely coupled variables and (18) because of the non-convex rank-one constraint. In particular, there are two types of variable coupling: the fractional form of two variables in SINR and the multiplication between $W_k$ and $V$, $Tr\left(W_k{R}_{k}V{R}_k^H\right)$. Hence, this optimization problem is hard to tackle.

3. Joint Beamforming Design under Perfect CSI

We assumed the CSI of all the channel links is perfectly known at the CC in this section. To minimize the total power consumption, the joint beamforming design was proposed to solve the problem (P4).

3.1. Optimization Problem Transformation

We introduced the parameter as $\beta ={\left[\begin{array}{ccc}{\beta }_1,& \cdots ,& {\beta }_K\end{array}\right]}^T$ to address the non-convex constraint in (16), where ${\beta }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K}$, and (16) can be rewritten as

$$Tr\left(W_k{R}_{k}V{R}_k^H\right)⩾{\tilde{\gamma }}_k{\beta }_k,\forall k\in \mathcal{K},$$

(20)

$${\displaystyle \sum }_{i=1,i\ne k}^{K}Tr\left(W_i{R}_{k}V{R}_k^H\right)⩽{\beta }_k-{\sigma }_k^2,\forall k\in \mathcal{K}.$$

(21)

Note that $W_k$ and $V$ are still coupled in the form of multiplication in constraint (20) and (21). Therefore, we used the following equation to decouple the variables.

$$\langle {A,B\rangle }_F=\frac{1}{2}\left({\parallel A+B\parallel }_F^2-{\parallel A\parallel }_F^2-{\parallel B\parallel }_F^2\right),$$

(22)

where ${\langle \cdot \rangle }_F$ represents Frobenius’s inner product, and $\langle {A,B\rangle }_F=Tr\left(A^{H}B\right)$. Thus, substituting (22) into (20) and (21), constraint can be retransformed as

$$(20)\Rightarrow \frac{1}{2}\left({\parallel W_k+R_{k}V{R}_k^H\parallel }_F^2-{\parallel W_k\parallel }_F^2-\left.{\parallel R_{k}V{R}_k^H\parallel }_F^2\right)\right.⩾{\tilde{r}}_k{\beta }_k,\forall k\in \mathcal{K},$$

(23)

$$(21)\Rightarrow {\displaystyle \sum }_{i=1,i\ne k}^K\frac{1}{2}\left({\parallel W_i+R_{k}V{R}_k^H\parallel }_F^2-{\parallel W_i\parallel }_F^2-\left.{\parallel R_{k}V{R}_k^H\parallel }_F^2\right)\right.⩽{\beta }_k-{\sigma }_k^2,\forall k\in \mathcal{K},$$

(24)

where the left-side-term of (23) and (24) is the form of difference of convex function. Thus, for a given point $W_k^n$ and $V^n$ in the $n$-th iteration of the SCA method, the first-order Taylor approximate expansion was adopted in the left-side term of (23) and (24), and the following equation can be obtained.

$$\begin{array}{cc}\mathcal{F}1\left(W_k,V\right)& \triangleq \frac{1}{2}\left\{{\parallel Z_k\parallel }_F^2+2Tr\left(\mathrm{Re}\left[Z_k\left(W_k-W_k^n\right)\right]\right)\right.\hfill \\ & +2Tr\left(\mathrm{Re}\left[\left(R_k^H{Z}_k{R}_k\right)\left(V-V^n\right)\right]\right)\left.-{\parallel W_i\parallel }_F^2-{\parallel R_{k}V{R}_k^H\parallel }_F^2\right\},\end{array}$$

(25)

$$\begin{array}{cc}\mathcal{F}2\left(W_k,V\right)& \triangleq {\displaystyle {\displaystyle \sum }_{i=1,i\ne k}^K}\frac{1}{2}\left\{{\parallel W_i+R_{k}V{R}_k^H\parallel }_F^2-{\parallel W_i^n\parallel }_F^2-{\parallel X_k\parallel }_F^2\right.\hfill \\ & -2Tr\left(\mathrm{Re}\left[W_i^n\left(W_i-W_i^n\right)\right]\right)\left.-2Tr\left(\mathrm{Re}\left[R_k^H{X}_k{R}_k\left(V-V^n\right)\right]\right)\right\},\end{array}$$

(26)

where $Z_k=W_k^n+X_k$, $X_k=R_k{V}^n{R}_k^H$. Then, the non-convex constraint (20) and (21) can be further converted to a convex constraint as follows:

$$\left(20\right)\Rightarrow \mathcal{F}1\left(W_k,V\right)⩾{\tilde{r}}_k{\beta }_k,\forall k\in \mathcal{K},$$

(27)

$$\left(21\right)\Rightarrow \mathcal{F}2\left(W_k,V\right)⩽{\beta }_k-{\sigma }_k^2,\forall k\in \mathcal{K}.$$

(28)

Note that the rank-one constraint (18) in (P4) is the only non-convex constraint. We used the following Lemma 2 to address this non-convex rank-one constraint.

Lemma 2.

For arbitrary $A\in {ℍ}^{N\times N}$,$rank\left(A\right)=1$is equivalent to$tr\left(A\right)-\varrho \left(A\right)⩽0$.

Proof. 

For arbitrary $A\in {ℍ}^{N\times N}$, we have the following inequality:

$$tr\left(A\right)={\displaystyle \sum }_{i=1}^N{\lambda }_i⩾\varrho \left(A\right)=\underset{i=1,\cdots ,N}{\max }\left\{{\lambda }_i\right\},$$

(29)

where ${\lambda }_i$ is the $i$-th eigenvalue of $A$, and the inequality of (29) holds true only if $rank\left(A\right)=1$ holds. Moreover, as $A$ is a Hermitian matrix, the implicit constraint is $tr\left(A\right)-\varrho \left(A\right)⩾0$, which simultaneously with constraint (29) requires that $tr\left(A\right)-\varrho \left(A\right)=0$, where $rank\left(A\right)=1$. □

According to Lemma 2, the rank-one constraint of $W_k$ and $V$ in (18) can be rewritten as

$$tr\left(W_k\right)-\varrho \left(W_k\right)⩽0,\forall k\in \mathcal{K},$$

(30)

$$tr\left(V\right)-\varrho \left(V\right)⩽0,$$

(31)

where the left-side term of (30) and (31) is the form of difference of convex function. Then, using first-order Taylor approximate expansion in the left side term of (30) and (31), the following equations can be obtained:

$${\mathcal{G}}_{W_k}\triangleq tr\left(W_k\right)-\varrho \left(W_k^n\right)-tr\left\{\mathrm{Re}\left[\left({\lambda }_{\max }\left(W_k^n\right){\lambda }_{\max }^H\left(W_k^n\right)\right)\left(W_k-W_k^n\right)\right]\right\},$$

(32)

$${\mathcal{G}}_V\triangleq tr\left(V\right)-\varrho \left(V^n\right)-tr\left\{\mathrm{Re}\left[\left({\lambda }_{\max }\left(V^n\right){\lambda }_{\max }^H\left(V^n\right)\right)\left(V-V^n\right)\right]\right\},$$

(33)

where ${\lambda }_{\max }\left(W_k^n\right)$ and ${\lambda }_{\max }\left(V^n\right)$ denote the eigenvectors of the maximum eigenvalues of $W_k^n$ and $V^n$, respectively. Substituting (32) and (33) into (30) and (31), constraint (30) and (31) can be retransformed as

$$\left(30\right)\Rightarrow {\mathcal{G}}_{W_k}⩽0,\forall k\in \mathcal{K},$$

(34)

$$\left(31\right)\Rightarrow {\mathcal{G}}_V⩽0$$

(35)

.

Then, the penalty function method [37] is used to move constraint (34) and (35) into the objective function. Thus, (P4) can be reformulated as

$$(\mathrm{P}5)\underset{W_k,V,\beta }{\min }Obj$$

(36)

$$s.t.\left(17\right),\left(19\right),\left(27\right),\left(28\right),$$

(37)

$${\beta }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K},$$

(38)

where $Obj\triangleq \left[{\displaystyle {\displaystyle \sum }_{k=1}^K}\left(tr\left(W_k\right)+{\rho }_1{\mathcal{G}}_{W_k}\right)\right]+{\rho }_2{\mathcal{G}}_V$, and ${\rho }_1,{\rho }_2>0$ denotes the penalty factor [37]. Thus, (P5) is a standard semidefinite program (SDP) that can be solved by the CVX tool [38]. To better analyze the problem (P4), we summarize the detailed process for solving (P4) in Algorithm 1.

3.2. Optimization Algorithm Description

In this subsection, we propose an Altmin-1 algorithm for solving (P4) under perfect CSI. Then, Algorithm 1 is summarized as follows:

Algorithm 1: Altmin-1 Algorithm for Solving (P4)

Initialize: Initialize $\left\{W_k^n,V^n\right\}$ to a feasible value and set penalty factor ${\rho }_1>0$ and ${\rho }_2>0$.

1:   Repeat: outer loop;

2:     set iteration index $n=0$;

3:     Repeat: inner loop;

4:    obtain $\left\{W_k^{n+1},V^{n+1}\right\}$ by solving the optimization problem (P5);

5:    set $n\triangleq n+1$, $W_k^n=W_k^{n+1}$, and $V^n=V^{n+1}$;

6:     until the objective function in (P5) converges;

7:   update $\left\{{\lambda }_{\max }\left(W_k^0\right),{\lambda }_{\max }\left(V^0\right)\right\}$ with the current solution $\left\{W_k^n,V^n\right\}$;

8:   set ${\rho }_1\triangleq 2{\rho }_1$, ${\rho }_2\triangleq 2{\rho }_2$;

9:   until the rank-one constraint converges.

Then, there are two loops in Algorithm 1 for solving the optimization problem (P4). In the outer loop, the penalty factor is gradually increased from one iteration to the next as $\left\{{\rho }_1=2{\rho }_1,{\rho }_2=2{\rho }_2\right\}$. Subsequently, the algorithm terminates when the rank-one constraint of $\left\{W_k^n,V^n\right\}$ satisfied. In the inner loop, $\left\{W_k^n,V^n\right\}$ are jointly optimized by iteratively solving the problem (P5). The objective function of (P5) is non-increasing in each iteration of the inner loop.

There are ${\left(KN\right)}^2+{\left(M+1\right)}^2+K$ optimization variables and $5K+M+3$ convex constraints in the problem (P5). Thus, the complexity of Algorithm 1 after each iteration is $\mathcal{O}\left\{{\left(K^2{N}^2+M^2+K\right)}^2\left(5K+M\right)\right\}$ [39].

4. Robust Beamforming Design under Imperfect CSI

We assumed that the CSI of all channel links is not perfectly known at the CC in this section. The well-known norm-bound channel error model was used to describe the channel link error [40]. Then, the channel model between the CC and the $k$-th actuator channel and the IRS-related cascaded channel to the $k$-th actuator could be formulated as

$$h_{bk}\triangleq {\tilde{h}}_{bk}+\Delta h_{bk},{\parallel \Delta h_{bk}\parallel }^2⩽{\delta }_{h_{bk}}^2,\forall k\in \mathcal{K},$$

(39)

$${\Phi }_k\triangleq {\tilde{\Phi }}_k+\Delta {\Phi }_k,{\parallel \Delta {\Phi }_k\parallel }_F^2⩽{\delta }_{{\Phi }_k}^2,\forall k\in \mathcal{K},$$

(40)

where ${\tilde{h}}_{bk}$ and ${\tilde{\Phi }}_k$ denote the estimated direct channel link and the estimated cascaded channel link, respectively. Meanwhile, $\Delta {\tilde{h}}_{dk}$ and $\Delta {\tilde{\Phi }}_k$ denote the uncertain channel error of the direct links and the cascaded links, respectively.

The norm-bound channel error model [40] of $R_k$ can be expressed as

$$\begin{array}{cc}{R}_k& \triangleq \left[{\Phi }_k^H,h_{bk}\right]=\left[{\tilde{\Phi }}_k^H,{\tilde{h}}_{bk}\right]+\left[\begin{array}{cc}\Delta {\Phi }_k^H& \Delta h_{bk}\end{array}\right]\hfill \\ & ={\tilde{R}}_k+\Delta R_k,\hfill \end{array}$$

(41)

where ${\parallel \Delta R_k\parallel }_F^2={\parallel \Delta {\Phi }_k\parallel }_F^2+{\parallel \Delta h_{bk}\parallel }^2={\delta }_{h_{bk}}^2+{\delta }_{{\Phi }_k}^2\triangleq {\delta }_k^2,\forall k\in \mathcal{K}$.

Substituting the imperfect CSI channel model expression (41) into constraint (16) of the problem (P4), this optimization problem can be retransformed as

$$(\mathrm{P}6)\underset{W_k,V}{\min }{\displaystyle \sum }_{k=1}^{K}tr\left(W_k\right)$$

(42)

$$s.t.\frac{Tr\left(W_k{R}_{k}V{R}_k^H\right)}{{\displaystyle {\displaystyle \sum }_{i=1,i\ne k}^K}Tr\left(W_i{R}_{k}V{R}_k^H\right)+{\sigma }_k^2}⩾{\tilde{\gamma }}_k,{\parallel \Delta R_k\parallel }_F^2⩽{\delta }_k^2,\forall k\in \mathcal{K},$$

(43)

$$V\succcurlyeq \mathbf{0},W_k\succcurlyeq \mathbf{0},\forall k\in \mathcal{K},$$

(44)

$$rank\left(V\right)=1,rank\left(W_k\right)=1,\forall k\in \mathcal{K},$$

(45)

$$V_{m,m}=1,1⩽m⩽M+1.$$

(46)

Note that (P6) is a non-convex optimization problem because of the non-convex constraint of (43) and (45).

4.1. Optimization Problem Transformation

We introduced the parameter as $\alpha ={\left[\begin{array}{ccc}{\alpha }_1,& \cdots ,& {\alpha }_K\end{array}\right]}^T$ to tackle the non-convex constraint in (43), where ${\alpha }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K}$, and (43) can be rewritten as

$$Tr\left(W_k{R}_{k}V{R}_k^H\right)⩾{\tilde{\gamma }}_k{\alpha }_k,{\parallel \Delta R_k\parallel }_F^2⩽{\delta }_k^2,\forall k\in \mathcal{K},$$

(47)

$${\displaystyle \sum }_{i=1,i\ne k}^{K}Tr\left(W_i{R}_{k}V{R}_k^H\right)⩽{\alpha }_k-{\sigma }_k^2,{\parallel \Delta R_k\parallel }_F^2⩽{\delta }_k^2,\forall k\in \mathcal{K}.$$

(48)

There is an infinite number of inequality constraints in (47) and (48) because of the impact of CSI imperfection. Thus, Lemma 3 was proposed to transform it into a positive semidefinite (PSD) constraint. Note that $W_k$ and $V$ are still coupled in the form of multiplication in (47) and (48); therefore, we reformulated the expressions by employing the equation as follows.

$$Tr\left(ABC\right)=Tr\left(BCA\right)=Tr\left(CAB\right),$$

(49)

$$Tr\left(A^{H}B\right)=ve{c}^H\left(A\right)vec\left(B\right),$$

(50)

$$vec\left(AXB\right)=\left(B^T\otimes A\right)vec\left(X\right),$$

(51)

where $vec$ represents the vectorization operator of a matrix, and $\otimes$ represents Kronecker’s inner product.

Substituting (49)–(51) into (47) and (48), constraint (47) and (48) can be rewritten as

$$\left(47\right)\Rightarrow ve{c}^H\left(R_k\right)\left(V^T\otimes W_k\right)vec\left(R_k\right)⩾{\tilde{r}}_k{\alpha }_k,{\parallel \Delta R_k\parallel }_F^2⩽{\delta }_k^2,\forall k\in \mathcal{K},$$

(52)

$$(48)\Rightarrow {\displaystyle \sum }_{i=1,i\ne k}^{K}ve{c}^H\left(R_k\right)\left(V^T\otimes W_i\right)vec\left(R_k\right)⩽{\alpha }_k-{\sigma }_k^2,{\parallel \Delta R_k\parallel }_F^2⩽{\delta }_k^2,\forall k\in \mathcal{K}.$$

(53)

We introduced Lemma 3 to tackle the infinite number of inequality constraints in (52) and (53).

Lemma 3.

(S-procedure for complex case [41]) For $F_1,F_2\in {ℍ}^{N\times N}$, $g_1,g_2\in {ℂ}^{N\times 1}$ and $h_1,h_2\in ℝ$, the necessary and sufficient condition for $x^H{F}_{1}x+2\mathrm{Re}\left\{g_1^{H}x\right\}+h_1⩽0\Rightarrow x^H{F}_{2}x+2\mathrm{Re}\left\{g_2^{H}x\right\}+h_2⩽0$ holds if there exists a $\varsigma ⩾0$, which satisfies that

$$\left[\begin{array}{cc}{F}_2& g_2\\ g_2^H& h_2\end{array}\right]\preccurlyeq \varsigma \left[\begin{array}{cc}{F}_1& g_1\\ g_1^H& h_1\end{array}\right],$$

(54)

provided that there exists a point $\widehat{x}$ with $\widehat{x}{F}_1\widehat{x}+2\mathrm{Re}\left\{g_1^H\widehat{x}\right\}+h_1⩽0$.

Using Lemma 3 and introducing the parameter as $\omega ={\left[\begin{array}{ccc}{\omega }_1,& \cdots ,& w_K\end{array}\right]}^T$, where ${\omega }_k⩾0,\forall k\in \mathcal{K}$, constraint (52) can be transformed into a PSD constraint as follows:

$$(52)\Rightarrow \left[\begin{array}{cc}{\omega }_k{I}_{\left(M+1\right)N}& {\mathbf{0}}_{\left(M+1\right)N\times 1}\\ {\mathbf{0}}_{1\times \left(M+1\right)N}& -{\omega }_k{\delta }_k^2-{\tilde{r}}_k{\alpha }_k\end{array}\right]+\left[\begin{array}{cc}J& J{\tilde{r}}_k\\ {\tilde{r}}_k^{H}J& {\tilde{r}}_k^{H}J{\tilde{r}}_k\end{array}\right]\succcurlyeq \mathbf{0},\forall k\in \mathcal{K},$$

(55)

where $J=V^T\otimes W_k$, and ${\tilde{r}}_k=vec\left({\tilde{R}}_k\right)$. To facilitate calculation, the equivalent form of (55) can be obtained by matrix decomposition as

$$(55)\Rightarrow \left[\begin{array}{cc}{\omega }_k{I}_{\left(M+1\right)N}& {\mathbf{0}}_{\left(M+1\right)N\times 1}\\ {\mathbf{0}}_{1\times \left(M+1\right)N}& -{\omega }_k{\delta }_k^2-{\tilde{r}}_k{\alpha }_k\end{array}\right]+E_k^{H}J{E}_k\succcurlyeq \mathbf{0},\forall k\in \mathcal{K},$$

(56)

where $E_k=\left[\begin{array}{cc}{I}_{\left(M+1\right)N}& {\tilde{r}}_k\end{array}\right]$.

Using Lemma 3 and introducing the parameter as $\nu ={\left[\begin{array}{ccc}{\upsilon }_1,& \cdots ,& {\upsilon }_K\end{array}\right]}^T$, where ${\upsilon }_k⩾0,\forall k\in \mathcal{K}$, constraint (53) can be transformed into a PSD constraint as follows:

$$(53)\Rightarrow \left[\begin{array}{cc}{\nu }_k{I}_{\left(M+1\right)N}& {\mathbf{0}}_{\left(M+1\right)N\times 1}\\ {\mathbf{0}}_{1\times \left(M+1\right)N}& {\alpha }_k-{\sigma }_k^2-{\nu }_k{\delta }_k^2\end{array}\right]-\left[\begin{array}{cc}G& G{\tilde{r}}_k\\ {\tilde{r}}_k^{H}G& {\tilde{r}}_k^{H}G{\tilde{r}}_k\end{array}\right]\succcurlyeq \mathbf{0},\forall k\in \mathcal{K},$$

(57)

where $G=V^T\otimes \left(W_{sum}-W_k\right)$, and $W_{sum}={\displaystyle \sum }_{k=1}^K{W}_k$. To facilitate calculation, the equivalent form of (57) can be obtained as follows:

$$(57)\Rightarrow \left[\begin{array}{cc}{\nu }_k{I}_{\left(M+1\right)N}& {\mathbf{0}}_{\left(M+1\right)N\times 1}\\ {\mathbf{0}}_{1\times \left(M+1\right)N}& {\alpha }_k-{\sigma }_k^2-{\nu }_k{\delta }_k^2\end{array}\right]-E_k^{H}G{E}_k\succcurlyeq \mathbf{0},\forall k\in \mathcal{K}.$$

(58)

Now, constraint (45) in (P6) is the only non-convex constraint. We used Lemma 2 to address this non-convex rank-one constraint, and (P6) can be reformulated as

$$(\mathrm{P}7)\underset{W_k,V,\alpha ,\omega ,\nu }{\min }Obj$$

(59)

$$s.t.\left(44\right),\left(46\right),\left(56\right),\left(58\right),$$

(60)

$$w_k⩾0,{\nu }_k⩾0,{\alpha }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K},$$

(61)

where $Obj\triangleq \left[{\displaystyle {\displaystyle \sum }_{k=1}^K}\left(tr\left(W_k\right)+{\rho }_1{\mathcal{G}}_{W_k}\right)\right]+{\rho }_2{\mathcal{G}}_V$, and ${\rho }_1,{\rho }_2>0$ denotes the penalty factor [37]. Note that optimization variables $W_k$ and $V$ are still coupled in the form of Kronecker’s inner product, which is hard to solve. Therefore, the alternate optimization (AO) method was adopted to tackle this non-convex problem (P7), and then the optimization problem was divided into two subproblems.

When an optimization variable $V$ is given, the subproblem of optimization $W_k$ can be written as

$$(\mathrm{P}8.1\mathrm{a})\underset{W_k,\alpha ,\omega ,\nu }{\min }Ob{j}_1$$

(62)

$$s.t.\left(56\right),\left(58\right),$$

(63)

$$W_k\succcurlyeq \mathbf{0},{\omega }_k⩾0,{\nu }_k⩾0,{\alpha }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K},$$

(64)

where $Ob{j}_1\triangleq \left[{\displaystyle {\displaystyle \sum }_{k=1}^K}\left(tr\left(W_k\right)+{\rho }_1{\mathcal{G}}_{W_k}\right)\right]$.

When an optimization variable $W_k$ is given, the subproblem of optimization $V$ can be expressed as

$$\left(\mathrm{P}8.1\mathrm{b}\right)\underset{V,\alpha ,\omega ,\nu }{\min }Ob{j}_2$$

(65)

$$s.t.\left(56\right),\left(58\right),$$

(66)

$$V\succcurlyeq \mathbf{0},V_{m,m}=1,1⩽m⩽M+1,$$

(67)

$${\omega }_k⩾0,{\nu }_k⩾0,{\alpha }_k⩾{\sigma }_k^2,\forall k\in \mathcal{K},$$

(68)

where $Ob{j}_2\triangleq \left({\displaystyle {\displaystyle \sum }_{k=1}^K}tr\left(W_k\right)\right)+{\rho }_2{\mathcal{G}}_V$.

Thus, optimization sub-problems (P8.1a) and (P8.1b) represent a standard SDP, which can be solved by the CVX tool [38]. To better analyze the problem (P7), we summarize the detailed process for solving (P7) in Algorithm 2.

4.2. Optimization Algorithm Description

We propose an Altmin-2 algorithm for solving (P7) in this subsection. Algorithm 2 is summarized as follows:

Algorithm 2. Altmin-2 Algorithm for Solving (P7)

Initialize: Initialize $\left\{W_k^n,V_n\right\}$ to a feasible value, set penalty factor ${\rho }_1>0$ and ${\rho }_2>0$.

1:   Repeat: outer loop;

2:     set iteration index $n=0$;

3:     Repeat: inner loop 1;

4:    Given $V^n$, solve the problem (P8.1a) and obtain $W_k^{n+1}$;

5:    set $n\triangleq n+1$, $W_k^n=W_k^{n+1}$;

6:     until the objective function in (P8.1a) converges;

7:     Repeat: inner loop 1;

8:    Given $W_n^k$, solve the problem (P8.1a) and obtain $V^{n+1}$;

9:    set $n\triangleq n+1$, $V_k^n=V_k^{n+1}$;

10:   until the objective function in (P8.1b) converges;

11:   update $\left\{{\lambda }_{\max }\left(W_k^0\right),{\lambda }_{\max }\left(V^0\right)\right\}$ with the current solution $\left\{W_k^n,V^n\right\}$;

12:   set ${\rho }_1\triangleq 2{\rho }_1$, ${\rho }_2\triangleq 2{\rho }_2$;

13:   until the rank-one constraint converges.

Then, there are two loops in Algorithm 2 for solving the optimization problem (P7). In the outer loop, the penalty factor was gradually increased from one iteration to the next as $\left\{{\rho }_1=2{\rho }_1,{\rho }_2=2{\rho }_2\right\}$. Subsequently, the algorithm terminated when the rank-one constraint of $\left\{W_k^n,V^n\right\}$ was satisfied. In the inner loop 1, when $V^n$ was given, $W_k^n$ was optimized by iteratively solving the problem (P8.1a). The objective function of (P8.1a) was non-increasing in each iteration of the inner loop 1. Similarly, in inner loop 2, when $W_k^n$ was given, $V^n$ was optimized by iteratively solving the problem (P8.1b). The objective function of (P8.1b) was non-increasing in each iteration of the inner loop.

After each iteration, the objective function of the problem (P7) is a monotone non-increasing function, which follows the inequality as follows:

$$Obj\left(W_k^n,V^n\right)\stackrel{\left(a\right)}{⩾}Obj\left(W_k^{n+1},V^n\right)\stackrel{\left(b\right)}{⩾}Obj\left(W_k^{n+1},V^{n+1}\right),$$

(69)

where (a) holds because of the obtained optimal solution $W_k^{m+1}$ by solving (P8.1a), and (b) holds because of the obtained optimal solution $V^{n+1}$ by solving (P8.1b). Therefore, Algorithm 2 is guaranteed to converge.

There are $m=3K$ SDP constraints in problem (8.1a), which contains $n\times n$ PSD matrix, where $n=N$. There are $m=2K+1$ SDP constraints in problem (8.1b), which contains $n\times n$ PSD matrix, where $n=M+1$. According to [39], the algorithm complexity of the SDP problem with $m$ SDP constraints, where the internal elements include $n\times n$ PSD matrix, is $\mathcal{O}\left\{\sqrt{n}\log \frac{1}{ϵ}\left(m{n}^3+m^2{n}^2+m^3\right)\right\}$, where $ϵ$ is the convergence tolerance in Algorithm 2. Thus, the complexity of Algorithm 2 after each iteration is $\mathcal{O}\left\{\log \frac{1}{ϵ}\left[\left(3N^{3.5}+2M^{3.5}\right)K+\left(9N^{2.5}+4M^{2.5}\right)K^2+\left(27N^{0.5}+8M^{0.5}\right)K^3\right]\right\}$ [39].

5. Simulation Results and Analysis

In this section, we evaluate the algorithm performance of an IRS-assisted downlink MU MISO-URLLC system under perfect CSI and imperfect CSI through simulation in an IIoT scenario. Unless specially stated, we denoted the number of antennas at the CC as $N=8$, the number of actuators as $K=4$, the number of phase shifters at the IRS as $M=8$, and the power spectral density of the noise as ${\sigma }_k^2=-174 dBm/Hz$. Moreover, we denoted the blocklength as $L_k=2000$, and the BER as ${\epsilon }_k={10}^{-6}$. Assuming the required transmit rate of the $k$-th actuator holds the same, we defined ${\tilde{r}}_k=0.2 bits/s/Hz$. For simple analysis, we assumed that this IIoT system is located in a two-dimensional coordinate plane. The coordinate position of CC and IRS was $\left(0 m,0 m\right)$ and $\left(50 m,10 m\right)$, respectively. In addition, the actuators were uniformly distributed in a circle with center $\left(70 m,0 m\right)$ and the radius was $10 m$. We denoted the large-scale fading model for the channel sets $G_k\in \left\{h_{dk},H_{br},h_{rk}\right\},\forall k\in \mathcal{K}$ as [10]

$${\Upsilon }_k=-30-10\alpha {\log }_{10}{d}_k,$$

(70)

where $\alpha$ is the path loss exponent, and $d_k$ is the distance between transmitter and receiver in meters. Then, the small-scale fading model for $G_k$ can be expressed as

$$G_k=\sqrt{\frac{\zeta }{1+\zeta }}{G}_k^{LoS}+\sqrt{\frac{1}{1+\zeta }}{G}_k^{NLoS},$$

(71)

where $G_k^{LoS}$ is defined by the product of the steering vector of the transmitter and receiver, and $G_k^{NLoS}$ represents the Rayleigh fading, which satisfies the independent circularly symmetric complex Gaussian (CSCG) distribution with zero means and unit variance. We denoted the Rician fading factor as $\zeta =5$ in this paper. We assumed the path loss exponent between the CC and the $k$-th actuator, between the CC and the IRS, and between the IRS to the $k$-th actuator were ${\alpha }_{br}=2.2$ [42], ${\alpha }_{bk}=4$, and ${\alpha }_{rk}=2$ [13], respectively.

We propose the following schemes to evaluate the effectiveness and robustness of the proposed algorithm:

• Penalty Function Algorithm proposed in [20]: the IRS is not considered in the paper;
• Algorithm 1 in this paper served as the Lower Bound: assuming all channel links are perfectly known at the CC;
• Algorithm 2 in this paper with random IRS: only $W_k$ optimization is considered, and the IRS is generated with a random method;
• Algorithm 2 in this paper with optimized IRS: both $W_k$ and $V$ optimization was considered;
• PCCP algorithm proposed in [13]: both active beamforming and passive beamforming under Shannon’s rate constraint were considered.

Figure 2 plots the convergence behavior of the proposed algorithm under perfect CSI and imperfect CSI in this paper, where the bound CSI error value is ${\delta }_k^2={10}^{-3}$, and the transmit rate is ${\tilde{r}}_k=0.2bits/s/Hz$. It is obvious that both Algorithm 1 and Algorithm 2 were monotonically convergent, where the number of iterations required by Algorithm 1 was significantly higher than Algorithm 2. Actually, Algorithm 1 reached convergence within $6$ iterations, while Algorithm 2 reached convergence within $3$ iterations. The main reason is that each iteration complexity of Algorithm 1 was much higher than Algorithm 2, which holds the same as the previous complexity analysis of the algorithm in this paper. However, the transmit power consumption of the CC under the required transmit rate in Algorithm 1 was lower than that in Algorithm 2, saving about $3 mW$.

Figure 3 illustrates the effectiveness of the proposed algorithm under the impact of the transmit rate and the bound CSI error, where the bound CSI error value was ${\delta }_k^2=\left[\begin{array}{ccc}{10}^{-3}& 3\times {10}^{-4}& 6\times {10}^{-4}\end{array}\right]$, and the transmit rate value was ${\tilde{r}}_k=0.2-2bits/s/Hz$. An interesting observation is that the total power consumption increased with the transmit rate value and the bound CSI error value, where Algorithm 1 served as a lower bound of the URLLC system, and the Algorithm in [13] served as a Shannon upper bound. As the transmit rate value increased, the corresponding SINR in Lemma 1 increased as well, and the channel dispersion in (3) can be seen as a constant. Thus, the performance gap between the Algorithm in [13] and the Algorithm 2 with optimized IRS did not increase with the increase in the transmit rate value. Moreover, the perfect CSI cannot be obtained in practice because of the uncertainty of the channel. Therefore, the performance of Algorithm 1 can only be utilized as the ideal evaluation index of the system. Subsequently, because of the impact of the bound CSI error value and the transmit rate value, the preference gap between Algorithm in [20] and Algorithm 2 with random IRS increased with the transmit rate value’s increase, such as the gap is about $5 mW$ in $0.8 bits/s/Hz$, and the gap is about $18 mW$ in $1.7 bits/s/Hz$. In addition, the performance gap between Algorithm 1 and Algorithm 2 with optimized IRS increased with the transmit rate value’s increase. Furthermore, the power consumption of Algorithm 2 with random IRS was much higher than that of other algorithms, which means that the random IRS in wireless communication cannot improve the performance of the system with the increase in transmit rate value.

Figure 4 and Figure 5 plot the effectiveness of the proposed algorithm under the transmit antennas at the CC and the actuators, where the bound CSI error was ${\delta }_k^2={10}^{-3}$, and the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$. It is promising to see that the performance gap between the Algorithm in [13] and Algorithm 2 with optimized IRS did not increase with the increase in the number of transmit antennas and in the number of actuators. Moreover, the performance achieved by the Algorithm in [20] and Algorithm 2 with random IRS had a negligible performance difference. In Figure 4, the total power consumption decreased with the number of antennas increasing, which means that more antennas can provide a higher spatial degree of freedom, and the fixed actuators can compete for more resources. For instance, in Algorithm 2 with optimized IRS, the increase in the number of the antennas from $6$ to $10$ when the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$ could effectively reduce the transmit power by about $50\%$. On the contrary, in Figure 5, the transmit power consumption increased as the number of actuators increased, which indicates that the fixed resources need to provide more actuators, and the interference term in SINR increased with the number of actuators increasing, which will inevitably increase the transmit power consumption at the CC.

Figure 6 plots the effectiveness of the proposed algorithm under the phase shifters at the IRS, where the bound CSI error was ${\delta }_k^2={10}^{-3}$, and the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$. It is shows that the performance gap between the Algorithm in [13] and Algorithm 2 with optimized IRS was almost fixed with the increase in the number of phase shifters of the IRS. Moreover, the transmit power consumption achieved by Algorithm 2 with optimized IRS, Algorithm 1, and the Algorithm in [13] decreased with the increase in the number of the phase shifters at the IRS. For instance, in Algorithm 2 with optimized IRS, the increase in the phase shifters at the IRS from $10$ to $40$ when the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$ could effectively reduce the transmit power by about $38\%$. In addition, the transmit power consumption in Algorithm 2 with random IRS was almost unchanged, which shows that the random IRS in wireless communication cannot improve the performance of the system with the increase in transmit rate value.

Figure 7 and Figure 8 plots the effectiveness of the proposed algorithm under the blocklength $L_k$ and the BER ${\epsilon }_k$, where the bound CSI error was ${\delta }_k^2={10}^{-3}$, and the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$. As expected, in Figure 7, with the blocklength $L_k$ increasing, the transmit power consumption with finite blocklength (FBL) was close to that of infinite blocklength (IFBL). Furthermore, the transmit power consumption decreased with the increase in the blocklength $L_k$ under FBL. When the transmit rate value was ${\tilde{r}}_k=0.2 bits/s/Hz$, the transmit power consumption for the Algorithm in [13] was higher than that of Algorithm 2 with IFBL for optimized IRS by about $3 mW$. In Figure 8, the transmit power consumption increased with the decrease of the BER ${\epsilon }_k$, where the value of on the X-axis actually was ${\epsilon }_k={10}^{-9}$ for simple expression. In addition, the absolute growth rate decreased with the increase in the BER. For system design purposes, depending on the QoS requirement in an IIoT scenario, an appropriate value of the blocklength $L_k$ and the BER ${\epsilon }_k$ can be chosen in Figure 7 and Figure 8.

6. Conclusions

In this paper, we considered an IRS-assisted downlink MU MISO-URLLC system in an IIoT scenario. We formulated an optimization problem aiming at minimizing the transmit power under perfect CSI and imperfect CSI while taking into account the QoS requirement of the actuator, unit modulus constraints of the IRS, and the robustness against the impact of CSI imperfection. We first transformed the QoS demand into the SINR constraint of the actuator by Lemma 1. Then, for these two cases, we proposed an effective algorithm with algorithm complexity analysis and convergence analysis. Finally, simulation results under various settings of parameters demonstrated the superiority of the proposed algorithm. Particularly, the performance analysis of Algorithm 1 can serve as an ideal lower bound to this system because of the uncertainty of the practice channel. Moreover, in an IIoT scenario with stringent URLLC requirements, Algorithm 2 has obvious advantages in reducing the transmit power consumption and guaranteeing effectiveness compared with other algorithms.