Modeling and Model Predictive Power and Rate Control of Wireless Communication Networks

Abstract

A novel power and rate control system model for wireless communication networks is presented, which includes uncertainties, input constraints, and time-varying delays in both state and control input. A robust delay-dependent model predictive power and rate control method is proposed, and the state feedback control law is obtained by solving an optimization problem that is derived by using linear matrix inequality (LMI) techniques. Simulation results are given to illustrate the effectiveness of the proposed method.

1. Introduction

Power and rate control for wireless communication networks has attracted a great deal of interest during recent decades. The objective of power and rate control is to guarantee the quality of service (QoS) with the lowest transmit power and the maximum transmission data rate in order to maximize the capacity and throughput of the network. However, low power may lead to the degradation of communication quality or even unreliable connection, while high data rates may cause data bursts in the network. Therefore, transmit power and transmission rate must be effectively controlled for wireless communication networks.

There have been a lot of effective power and rate control methods reported in the literature. Generally speaking, the methods can be broadly classified into three categories: (i) resource allocation under perfect power control [1–4]; that is, the received power is well controlled at the target level, and then the transmission rate and power are optimised by maximizing the throughput, (ii) a snap-shot evaluation of the power and rate allocations pertaining to a given realisation of channel gains for each of the users [5, 6], and (iii) power control based on resource allocation [7–14]; that is, the optimal transmission rate can be obtained according to the resource allocation, and the optimal target signal-to-inference ratio (SIR) can be relatively determined, and then the power control is used to determine an optimal transmit power for each user to drive its actual SIR towards the optimal target SIR. The third method will be used in this paper.

System modeling is an important requirement for power and rate control of wireless communication networks. A simplified linear system model was presented in [7], which considered round-trip delay and treated the effects of channel fading and interference as an additive disturbance. A frame/slot structure was presented in [8], which was used to implement practically a power and rate control system. A high order model as a cascade system was presented in [9, 10], which was used to cancel the effect of time delay. However, parameter uncertainties were not considered in these models. State space models with and without uncertain dynamics were presented in [11], but the time delay was not considered. An extended version of state space model in [11] with state delay was presented in [12], but it only considered the time delay in state and in rate control, while the time delay in input and in power control was not considered. In addition, all the models did not consider the time-varying delays and input constraints. Therefore, a new system model is needed for power and rate control to increase the QoS, which should include uncertainties, input constraints, and time-varying state and input delays in both rate control and power control. This motivates the work of this paper.

A challenge issue in power and rate control is to maintain robustness to time delay [9, 10, 12]. To compensate for the effect of time delay, power and rate control algorithms have been presented by using a multiple-mode Smith prediction filter in [7] and by using a high order model in [9, 10]. From a system-theoretic perspective, a power and rate control algorithm for wireless networks with state delay was presented in [12], which minimized the bound on the error variance between the desired and actual SIR. However, almost all existing power and rate control methods are limited to consider known and constant time delay. Unfortunately, the time delay is often unknown and even time-varying in real network environments.

In the previous work, we presented power and rate control algorithms for wireless networks with time-varying state delay via adaptive control technique [13] and via robust 𝐻_∞ control approach [14]. However, as in [12], the time delay only in rate control was considered, while the time delay in power control was not considered. In fact, the power control is more sensitive to time delay than the rate control. Additionally, only the state time delay was considered in [12–14], but the input delay was not considered. To the best knowledge of the authors, research on power and rate control for wireless communication networks with time-varying delays in both state and control input has not been investigated, which is another motivation of this paper.

Another challenge issue in power and rate control is to maintain robustness to uncertainties and input constraints. It is well known that robust control with linear matrix inequality (LMI) is a powerful technique to cope with uncertainties, and model predictive control (MPC) has strong ability to handle input constraints [15, 16]. Therefore, a robust MPC with LMI can be extended to power and rate control for wireless communication networks with uncertainties and input constraints, which is the third motivation of this paper.

The main contributions of this paper are as follows. (1) In the control theoretic framework, a novel power and rate control system model is presented, which includes uncertainties, input constraints, and time-varying state and input delays in both rate control and power control. (2) A robust model predictive power and rate control method is presented for wireless communication networks with uncertainties, input constraints, and time-varying state and input delays. The state feedback control law is obtained by solving an optimization problem that is derived by using LMI.

The rest of this paper is organized as follows. The power and rate control system model and problem formulation are presented in Section 2. A robust model predictive power and rate control algorithm is presented in Section 3. Simulation results are given in Section 4. And Section 5 concludes the paper.

Notation. In this paper, 𝐀^𝑇 and 𝐀⁻¹ denote the transpose and the inverse of a matrix 𝐀, respectively; 𝑅^𝑛×𝑚 denotes the set of all 𝑛×𝑚 real matrices; 𝐀>0 (𝐀<0) means that 𝐀 is symmetric positive definite (negative definite); 𝐈 is an appropriately dimensioned identity matrix; 𝐝𝐢𝐚𝐠{⋯} denotes a block-diagonal matrix; ‖𝐱‖²=𝐱^𝑇𝐱, ‖𝐱‖²_𝐏=𝐱^𝑇𝐏𝐱; the symmetric terms in a symmetric matrix are denoted by *; for example, 𝐗𝐘^𝑇=.𝐘𝐙𝐗∗𝐘𝐙 (1)

2. System Model and Problem Formulation

In this section, we briefly describe the time delay effects, power control, and rate control in Sections 2.1, 2.2, and 2.3, respectively, and then, in the control theoretic framework, derive the power and rate control system model for uncertain wireless communication networks with input constraints and time-varying delays in both state and control input in Section 2.4 and finally give the problem formulation in Section 2.5.

2.1. Effects of Time Delays

In wireless communication networks, time delays are mainly due to the signal processing and signal transmission [9, 15]. For convenience of analysis, as depicted in Figure 1 for the power control loop, let 𝜏₁ denote the time delay in uplink channel (including the signal processing delay at base station (BS) and the signal transmission delay from mobile station (MS) to BS), and let 𝜏₂ denote the time delay in downlink channel (including the signal processing delay at MS and the signal transmission delay from BS to MS). The total round-trip delay in the network is 𝜏=𝜏₁+𝜏₂. Practical value of the round-trip delay is between 2 and 4 at the sampling rate of 800 Hz [10, 17]. To illustrate the effects of time delays in power control design, let us examine how a power control command 𝑢_𝑝(𝑘+1) is produced. The effect of time delay in rate control design can be straightforward.

Figure 1 

Time delays in power control.

In the uplink power controller, with respect to the operation at MS, there are three main components related to the power control [18]: a quantization reconstructor, a power multiplier (PM), and a power limiter. At time instant 𝑘, the quantization reconstructor is employed to recover the power control update command 𝑢_𝑝(𝑘) sent from BS, which is then fed into the power multiplier, resulting in the transmit power 𝑝(𝑘), which will be limited to the interval [𝑃_min,𝑃_max] due to the physical limitation of power amplifier. Because of the downlink time delay 𝜏₂, the transmit power at MS should be 𝑝(𝑘−𝜏₂). It is then transmitted to BS through the uplink channel with the uplink time delay 𝜏₁. That is, the transmitted power at MS is 𝑝(𝑘−𝜏₂), and the received power at BS is 𝑝(𝑘−𝜏₂−𝜏₁)=𝑝(𝑘−𝜏).

At BS, there are four major components related to the power control [17, 18]: an SIR measurement circuit, an SIR comparator, a power control circuit (PC), and a quantizer. The received SIR 𝛾(𝑘), which is related to 𝑝(𝑘−𝜏), is measured by the SIR measurement circuit. The tracking error can be obtained by comparison between the received SIR 𝛾(𝑘) and the target SIR 𝛾^∗(𝑘) (which can be obtained by the rate controller as shown later). The error is fed into the power controller. The power controller generates a power control update command 𝑢^′_𝑝(𝑘+1) according to an appropriate power control algorithm to make the received SIR 𝛾(𝑘) track the target SIR 𝛾^∗(𝑘) efficiently. The quantizer transfers the update signal from analog 𝑢^′_𝑝(𝑘+1) to digital bits 𝑢_𝑝(𝑘+1) based on a quantization scalar. 𝑢_𝑝(𝑘+1) is then transmitted to MS through the downlink channel.

2.2. Power Control

According to the analysis in Section 2.1, the received SIR at BS for MS 𝑖 at time 𝑘 can be expressed as 𝛾_𝑖(𝑘)=𝑔_𝑖𝑖(𝑘)𝑝_𝑖𝑘−𝜏_𝑖∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑔_𝑖𝑗(𝑘)𝑝_𝑗𝑘−𝜏_𝑗+𝜎²_𝑖, (2) where 𝛾_𝑖(𝑘) is a measurement of the actual SIR, 𝑔_𝑖𝑗(𝑘) represents the channel gain from the 𝑗th MS to the intended BS of the 𝑖th cell, 𝑝_𝑗(𝑘) is the transmit power from the 𝑗th MS, 𝑛 is the number of MSs using the same channel including the intracell and intercell users, 𝜎²_𝑖 is the power of the white Gaussian noise at the receiver of BS, and 𝜏_𝑗 is the round-trip delay. Now let 𝛽_𝑖(𝑘)=𝑔_𝑖𝑖(𝑘)∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑔_𝑖𝑗(𝑘)𝑝_𝑗𝑘−𝜏_𝑗+𝜎²_𝑖 (3) denote the scaling factor that determines how 𝑝_𝑖(𝑘−𝜏_𝑖) affects the achieved 𝛾_𝑖(𝑘) in (2); that is, 𝛾_𝑖(𝑘)=𝛽_𝑖(𝑘)𝑝_𝑖𝑘−𝜏_𝑖. (4)

By introducing logarithmic variables, with 𝛾_𝑖(𝑘)=ln[𝛾_𝑖(𝑘)],  𝛽_𝑖(𝑘)=ln𝛽_𝑖(𝑘), and 𝑝_𝑖(𝑘)=ln𝑝_𝑖(𝑘), we can rewrite (4) as, in dB scale, 𝛾_𝑖(𝑘)=𝛽_𝑖(𝑘)+𝑝_𝑖𝑘−𝜏_𝑖, (5) where 𝛽_𝑖(𝑘) is referred to as the effective channel gain.

Introduce random walk model for 𝛽_𝑖(𝑘) defined by [10] 𝛽_𝑖(𝑘+1)=𝛽_𝑖(𝑘)+𝑛_𝑖(𝑘), (6) where 𝑛_𝑖(𝑘) is a zero mean disturbance of variance 𝜎²_𝑛 and is independent of 𝑝_𝑖(𝑘). This model is based on the assumption that MSs in the network do not jointly optimize their power levels in any centralized manner and only do so independently in a distributed sense.

The distributed power control (DPC) algorithm (in dB scale) for each MS is given by 𝑝_𝑖(𝑘+1)=𝑝_𝑖(𝑘)+𝛼_𝑖𝛾^∗_𝑖(𝑘)−𝛾_𝑖(𝑘), (7) where 𝛼_𝑖 is a step-size parameter that is allowed to vary from one MS to another, 𝛾^∗_𝑖(𝑘) is a desired value of SIR, and 𝛾^∗_𝑖(𝑘)=ln𝛾^∗_𝑖(𝑘).

In order to make the actual SIR 𝛾_𝑖(𝑘) track the desired SIR 𝛾^∗_𝑖(𝑘), as in [12], we employ a power control sequence 𝑢_𝑖𝑝(𝑘) in (7) as follows: 𝑝_𝑖(𝑘+1)=𝑝_𝑖(𝑘)+𝛼_𝑖𝛾^∗_𝑖(𝑘)−𝛾_𝑖(𝑘)+𝑏_𝑖𝑝𝑢_𝑖𝑝(𝑘), (8) for a given factor 𝑏_𝑖𝑝 and control sequence 𝑢_𝑖𝑝(𝑘) to be determined.

Then from (5), (6), and (8), we find that the achieved 𝛾_𝑖(𝑘) varies according to the following rule: 𝛾_𝑖()=𝑘+1𝛽_𝑖(𝑘+1)+𝑝_𝑖𝑘+1−𝜏_𝑖=𝛽_𝑖(𝑘)+𝑛_𝑖(𝑘+)𝑝_𝑖𝑘−𝜏_𝑖+𝛼_𝑖𝛾^∗_𝑖𝑘−𝜏_𝑖−𝛾_𝑖𝑘−𝜏_𝑖+𝑏_𝑖𝑝𝑢𝑘−𝜏_𝑖=𝛽_𝑖(𝑘)+𝑛_𝑖(𝑘)+𝛾_𝑖(𝑘)−𝛽_𝑖(𝑘)+𝛼_𝑖𝛾^∗_𝑖𝑘−𝜏_𝑖−𝛼_𝑖𝛾_𝑖𝑘−𝜏_𝑖+𝑏_𝑖𝑝𝑢𝑘−𝜏_𝑖=𝛾_𝑖(𝑘)−𝛼_𝑖𝛾_𝑖𝑘−𝜏_𝑖+𝛼_𝑖𝛾^∗_𝑖𝑘−𝜏_𝑖+𝑏_𝑖𝑝𝑢𝑘−𝜏_𝑖+𝑛_𝑖(𝑘). (9)

Remark 1. Almost all existing power and rate control methods did not consider the time delay from the transmitted power to the received SIR, that is, assuming 𝜏_𝑖=0 in (2). From the analysis of the dynamics as mentioned in Section 2.1, the effect of this time delay must be considered. This paper considers such a time delay which is mainly different from the existing power and rate control methods.

2.3. Rate Control

Let 𝑓_𝑖(𝑘) denote the flow rate at MS 𝑖 at time 𝑘; for any MS in the network, the following flow-rate control algorithm is given by [12] 𝑓_𝑖(𝑘+1)=𝑓_𝑖(𝑘)+𝜇𝑑(𝑘)−𝑐₁(𝑘)𝑓_𝑖(𝑘)−𝑐₂(𝑘)𝑓_𝑖𝑘−𝜏_𝑖, (10) where 𝜇>0 is a step-size parameter and 𝑐₁(𝑘) and 𝑐₂(𝑘) are measures of the amount of congestion in the network at time 𝑘. 𝑑(𝑘) controls the amount of rate increase per iteration, which is a zero mean random variable with variance 𝜎²_𝑑. 𝜏_𝑖 is nonzero for any controller that incorporates round-trip delay time. Equation (10) is a typical rate control strategy incorporated in wireless communication networks and in computer networks.

According to Shannon’s capacity formula, there is a plausible choice for SIR level in order to achieve a rate value 𝑓_𝑖(𝑘); namely, the SIR level should be at least at a value 𝛾^∗_𝑖(𝑘), that is, related to 𝑓_𝑖(𝑘) given by 𝑓_𝑖(𝑘)=12log₂1+𝛾^∗_𝑖(𝑘). (11)

Usually, during normal network operation, 𝛾^∗_𝑖(𝑘)≫1. Thus, 𝑓_𝑖(𝑘) is proportional to log₂𝛾^∗_𝑖(𝑘). Using the fact and (10), the desired SIR varies, in dB scale, according to the following rule: 𝛾^∗_𝑖(𝑘+1)=1−𝜇𝑐₁(𝑘)𝛾^∗_𝑖(𝑘)−𝜇𝑐₂(𝑘)𝛾^∗_𝑖𝑘−𝜏_𝑖+𝜇𝑑(𝑘), (12) where 𝜇=20𝜇/log₂(10).

Like that mentioned in Section 2.2, as in [12], and considering the round trip time delay, we add a rate control sequence 𝑢_𝑖𝑓(𝑘) in (12) as follows: 𝛾^∗_𝑖(𝑘+1)=1−𝜇𝑐₁(𝑘)𝛾^∗_𝑖(𝑘)−𝜇𝑐₂(𝑘)𝛾^∗_𝑖𝑘−𝜏_𝑖+𝑏_𝑖𝑓𝑢_𝑖𝑓𝑘−𝜏_𝑖+𝜇𝑑(𝑘). (13)

Remark 2. The power and rate control algorithms, presented in [10–12], just considered the time delay in rate control, while the time delay in power control was not considered, that is, in the case 𝜏_𝑖=0 in (2). In fact, the power control is more sensitive to time delay than the rate control. So the time delay in power control must be considered. Furthermore, the time delay in control input was not considered in those algorithms, that is, in the case 𝜏_𝑖=0 in 𝑢_𝑖𝑝(𝑘−𝜏_𝑖) in (9) and in 𝑢_𝑖𝑓(𝑘−𝜏_𝑖) in (13).

2.4. Joint Power and Rate Control

For simplicity of notation, in the following, the MS index 𝑖 is dropped. And we introduce the two-dimensional state vector: 𝐱(𝑘)=𝛾(𝑘)𝛾^∗(𝑘). (14)

Then from (9) and (13), we can obtain 𝐱(𝑘+1)=1001−𝜇𝑐₁(𝑘)𝐱(𝑘)+−𝛼𝛼0−𝜇𝑐₂(𝑘)𝐱()+𝑏𝑘−𝜏_𝑝00𝑏_𝑓𝐮()+𝑛(𝑘)𝑘−𝜏(𝑘),𝜇𝑑 (15) or, more compactly, 𝐱(𝑘+1)=𝐀(𝑘)𝐱(𝑘)+𝐀_𝑑(𝑘)𝐱()()(𝑘),𝑘−𝜏+𝐁𝐮𝑘−𝜏+𝝎 (16) where 𝐀(𝑘)=1001−𝜇𝑐₁(𝑘),𝐀_𝑑(𝑘)=−𝛼𝛼0−𝜇𝑐₂(𝑘),𝑏𝐁=^𝑝00𝑏_𝑓(𝑘)=𝑛(𝑘),𝝎(𝑘),𝜇𝑑 (17) where 𝜏 is a nonzero integer that incorporates round-trip time delay and 𝝎(𝑘) is a zero mean random vector with covariance matrix: 𝐸𝝎(𝑘)𝝎^𝑇(𝑘=𝜎)^2𝑛00𝜇²𝜎²_𝑑. (18)

We now consider uncertainties that arise due to the lack of perfect knowledge about the network dynamics. For example, the congestion control parameters 𝑐₁(𝑘) and 𝑐₂(𝑘) are usually not known exactly and have to be estimated; the estimation process introduces errors in the assumed state-space model. The uncertainty in 𝑐₁(𝑘) can be modelled as [12] 𝑐₁(𝑘)=𝑐₁+𝐷𝐅(𝑘)𝑑, (19) where 𝐅(𝑘) is a zero mean random noise with variance 𝜎²_𝐹, 𝐷 and 𝑑 are known scalars, and 𝑐₁ is unknown but bounded as 𝑐_1,𝑙≤𝑐₁≤𝑐_1,𝑢 (20) for some known positive scalars {𝑐_1,𝑙,𝑐_1,𝑢}. In other words, we allow for both deterministic and stochastic uncertainties in 𝑐₁(𝑘). In this way, the matrix 𝐀(𝑘) in (16) is not known exactly, but it can be modelled as 𝐀(𝑘)=𝐀+Δ𝐀(𝑘), where 𝐀=1001−𝜇𝑐₁,Δ𝐀(𝑘)=𝐷𝐅(𝑘)𝐄_𝑎, (21) where 𝐄_𝑎=000−𝜇𝑑. (22)

Likewise, let 𝑐₂ be bounded as 𝑐_2,𝑙≤𝑐₂≤𝑐_2,𝑢. In this way, the matrix 𝐀_𝑑(𝑘) in (16) is also not known exactly but can be now modelled as 𝐀_𝑑(𝑘)=𝐀_𝑑+Δ𝐀_𝑑(𝑘), where 𝐀_𝑑=−𝛼𝛼0−𝜇𝑐₂,Δ𝐀_𝑑(𝑘)=𝐷𝐅(𝑘)𝐄_𝑑, (23) where 𝐄_𝑑=000−𝜇𝑑_𝑑. (24)

Then the power and rate control system with time-varying state and input delays and uncertainties can be written as 𝐱(𝑘+1)=𝐀(𝑘)𝐱(𝑘)+𝐀_𝑑(𝑘)𝐱𝑘−𝜏_𝑘+𝐁𝐮𝑘−𝜏_𝑘(𝑘),𝐱(𝑘)+𝝎=0,𝑘∈−𝜏_𝑀,0, (25) where Δ𝐀(𝑘) and Δ𝐀_𝑑(𝑘) are time-varying uncertain matrices of the form Δ𝐀(𝑘),Δ𝐀_𝑑(𝑘)=𝐷𝐅(𝑘)𝐄_𝑎,𝐄_𝑑, (26) with 𝐅(𝑘)^𝑇𝐅(𝑘)≤𝐈. Such parameter uncertainties Δ𝐀(𝑘) and Δ𝐀_𝑑(𝑘) are said to be admissible. And it is assumed that, for the set of uncertainties Ω, [𝐀(𝑘)∣𝐀_𝑑(𝑘)]∈Ω. 𝜏_𝑘 is a round-trip time-varying delay and satisfies 0≤𝜏_𝑚≤𝜏_𝑘≤𝜏_𝑀, (27) where 𝜏_𝑚 and 𝜏_𝑀 are known lower and upper delay bounds. The control input is constrained by −𝐮≤𝐮(𝑘+𝑖)≤𝐮,𝑖≥0. (28)

2.5. Problem Formulation

Define the following robust performance index at each time 𝑘: min_{𝐮𝑘+𝑗∣𝑘,𝑗≥0}max_{𝐀𝑘+𝑗∣𝐀𝑑(𝑘+𝑗)∈Ω}𝐽(𝑘), (29) subject to 𝐽(𝑘)=^∞_𝑗=0𝐱(𝑘+𝑗∣𝑘)²_𝐐1+𝐮𝑘+𝑗∣𝑘²_𝐑,𝐱𝐱𝑘+𝑗+1∣𝑘=𝐀𝑘+𝑗𝑘+𝑗∣𝑘+𝐀_𝑑𝑘+𝑗×𝐱𝑘+𝑗−𝜏_𝑘,−∣𝑘+𝐁𝐮𝑘+𝑗∣𝑘+𝝎𝑘+𝑗𝐮≤𝑢𝑘+𝑗=𝐊𝐱𝑘+𝑗∣𝑘≤𝐮,𝑗∈[0,∞), (30) where 𝐽(𝑘) is the performance index function with respect to 𝐱(𝑘), 𝐐₁>0 and 𝐑>0 are weighting matrices, 𝜙(𝐱,𝐮,𝑗)=‖𝐱(𝑘+𝑗∣𝑘)‖²_𝑄1+‖𝐮(𝑘+𝑗∣𝑘)‖²_𝑅 is the stage cost, 𝐱(𝑘+𝑗∣𝑘) is the value of vector 𝐱 at future time 𝑘+𝑗(𝑗>0) predicted at time 𝑘, 𝑥(𝑘∣𝑘)=𝑥(𝑘), and 𝐱(𝑘−𝑖∣𝑘)=𝐱(𝑘−𝑖) for 𝑖>1.

Equations (29)-(30) are a constrained min-max optimization problem corresponding to a worst-case infinite-horizon MPC with a quadratic objective.

The goal of this paper is to find a state feedback control law 𝐮(𝑘)=𝐊𝐱(𝑘) for system (25) via the MPC strategy by solving the optimization problem (29)-(30).

Before ending this section, we introduce the following lemma, which is useful to prove our results.

Lemma 3 (see [19]). Let 𝐃, 𝐄, and 𝐅 be real matrices of appropriate dimensions, with 𝐅^𝐓𝐅≤𝐈, and then there exists a scalar 𝜀≥0, such that the following inequality holds: 𝐃𝐅𝐄+𝐄^𝐓𝐅^𝐓𝐃^𝑇≤𝜀𝐃𝐃^𝑇+𝜀⁻¹𝐄^𝐓𝐄. (31)

3. Model Predictive Power and Rate Control

In this section, we will design the robust state feedback controller for system (25) via the MPC strategy by solving the optimization problem (29)-(30). However, the exact solution to the min-max optimization problem (29)-(30) in general is not tractable. To obtain a practical optimization problem, following a similar approach as in [15], we will relax the min-max problem into another optimization problem which minimizes an upper bound on the worst value of the original cost function 𝐽(𝑘).

Consider a Lyapunov-Krasovskii functional at time 𝑘𝑉(𝐱(𝑘))=𝑉₁(𝐱(𝑘))+𝑉₂(𝐱(𝑘))+𝑉₃(𝐱(𝑘)), (32) where 𝑉₁(𝐱(𝑘))=𝐱^𝑇(𝑘)𝐏𝐱(𝑘),𝑉₂(𝐱(𝑘))=^𝑘−1_{𝑖=𝑘−𝜏𝑘}𝐱^𝑇(𝑖)𝐏_𝐝𝐱(𝑖),𝑉₃(𝐱(𝑘))=^−𝜏𝑚_{𝑗=−𝜏𝑀+1}^𝑘−1_{𝑖=𝑘+𝑗}𝐱^𝑇(𝑖)𝐏_𝐝𝐱(𝑖), (33)(34)(35) where 𝐏=𝐏^𝑇>0, 𝐏_𝑑=𝐏^𝑇_𝑑>0.

An upper bound on the worst value of the cost function 𝐽(𝑘) is obtained whenever the following inequality is satisfied for any [𝐀(𝑘+𝑗∣𝑘)∣𝐀_𝑑(𝑘+𝑗∣𝑘)]∈Ω,𝑗≥0: 𝑉𝐱𝑘+𝑗+1∣𝑘−𝑉𝐱𝑘+𝑗∣𝑘≤−𝜙(𝐱,𝐮,𝑘). (36)

For 𝐽(𝑘) to be finite, we must have 𝐱(∞∣𝑘)=0, and hence 𝑉(𝐱(∞∣𝑘))=0. Hence summing both sides of (36) from 𝑗=0 to 𝑗=∞ yields −𝑉(𝐱(𝑘∣𝑘))≤−𝐽(𝑘), and it follows that 𝐽(𝑘)≤𝑉(𝐱(𝑘∣𝑘))≤𝜐(𝑘), (37) where 𝜐(𝑘)>0 is defined to be the upper bound of the cost function 𝐽(𝑘). This implies max_{𝐀𝑘+𝑗∣𝐀𝑑(𝑘+𝑗)∈Ω,𝑗≥0}𝐽(𝑘)≤𝜐(𝑘). (38)

Then, from (38), the original min-max optimization problem (29)-(30) is turned into the following optimization problem that minimizes this upper bound 𝜐(𝑘): min_{𝜐(𝑘),𝐾,𝑃,𝑃𝑑}𝜐(𝑘), (39) subject to (30) and (38).

Theorem 4. Consider the power and rate control system (25) subject to (26)–(28). If there exist matrices 𝐐=𝐐^𝑇>0, 𝐐_𝑑=𝐐^𝑇_𝑑>0, and 𝐘, 𝐙>0 with appropriate dimension, and scalars 𝜀>0 and 𝜐(𝑘)>0, such that the following optimization problem is solvable: min_{𝜐(𝑘),𝑄,𝑄𝑑,𝑌,𝑍,𝜀}𝜐(𝑘), (40) subject to𝜏−𝐐∗∗∗∗∗∗∗_𝑠+1^1/2𝐐−𝐐_𝑑𝐐∗∗∗∗∗∗_11/2(𝑘)𝐑𝐐𝟎−𝜐𝐈∗∗∗∗∗_1/2𝟎𝟎−𝜐(𝑘)𝐈∗∗∗∗𝟎𝟎𝟎𝟎−𝐐_𝑑(𝑘)∗∗∗𝟎𝟎𝟎𝟎0−𝜐𝐈∗∗𝐀𝐐𝟎𝟎𝟎𝐀_𝑑𝐐_𝑑+𝐁𝐘𝐈𝜀𝐃^𝑇𝐄𝐃−𝐐∗_𝑎𝐐𝟎𝟎𝟎𝐄_𝑑𝐐_𝑑𝟎𝟎−𝜀𝐈≤0,𝝃1∗₃(𝑘)Λ≥0,𝐘𝐙∗_𝑇𝐐≥0,𝐙_𝑗𝑗≤𝐮²_𝑗,𝑗=1,…,𝑚, (41)(42)(43)where 𝜏_𝑠=𝜏_𝑀−𝜏_𝑚, and 𝐐=𝜐𝐏⁻¹,𝐐_𝑑=𝜐𝐏_𝑑⁻¹,𝐘=𝐊𝐐_𝑑,𝝃₃(𝑘)=𝐱^𝑇(𝑘),𝐱^𝑇(𝑘−1),…,𝐱^𝑇𝑘−𝜏_𝑚,𝐱^𝑇(𝑘−𝜏_𝑚−1),…,𝐱^𝑇(𝑘−𝜏_𝑀)^𝑇,𝐐Λ=𝐝𝐢𝐚𝐠𝐐,^𝑑𝜏_𝑠𝐐+1,…,^𝑑𝜏_𝑠,𝐐+1𝜏^𝑑_𝑠,…,𝐐_𝑑, (44) then the state feedback controller 𝐮(𝑘)=𝐊𝐱(𝑘) minimizes the upper bound of the cost function, where the feedback matrix 𝐊 is obtained from the solution of (40)–(43) given by 𝐊=𝐘𝐐_𝑑⁻¹. (45)

The proof of Theorem 4 is presented in the Appendix.

Theorem 4 has given a sufficient condition for the existence of the robust MPC controller at time 𝑘. In order to prove the stability of the closed loop, a lemma is given as follows.

Lemma 5 (see [15]). If the optimization problem in Theorem 4 is feasible at time 𝑘, then it is feasible for all times 𝑘+𝑗, 𝑗>0.

Next, we can give the following theorem to stabilize the closed-loop system.

Theorem 6. If the optimization problem (40)–(43) is feasible at time 𝑘, then the state feedback controller 𝐮(𝑘)=𝐊𝐱(𝑘) from Theorem 4 robustly asymptotically stabilizes the closed-loop system.

Proof. It is known that the optimization problem (40)–(43) is feasible from Lemma 5. So we assume that 𝐏^∗(𝑘), 𝐏^∗_𝑑(𝑘) and 𝐏^∗(𝑘+1), 𝐏^∗_𝑑(𝑘+1) denote the optimal values of problem (40)–(43) at times 𝑘 and 𝑘+1, respectively, and 𝐱^∗(𝑘∣𝑘) and 𝐱^∗(𝑘+1∣𝑘+1) are optimal state correspondingly.
Since 𝐏^∗(𝑘+1) and 𝐏^∗_𝑑(𝑘+1) are optimal, whereas 𝐏^∗(𝑘), 𝐏^∗_𝑑(𝑘) are feasible at time 𝑘+1, we have 𝐱^∗𝑇(𝑘+1∣𝑘+1)𝐏^∗(𝑘+1)𝐱^∗(𝑘+1∣𝑘+1)≤𝐱^∗𝑇(𝑘+1∣𝑘+1)𝐏^∗(𝑘)𝐱^∗(),𝐱𝑘+1∣𝑘+1_𝑇(𝑘+1−𝑖∣𝑘+1)𝐏^∗_𝑑(𝑘+1)𝐱(𝑘+1−𝑖∣𝑘+1)≤𝐱^𝑇(𝑘+1−𝑖∣𝑘+1)𝐏^∗_𝑑(𝑘)𝐱(𝑘+1−𝑖∣𝑘+1). (46)
From Theorem 4, we have an invariant set Γ for the predicted states of system (25), Γ=𝐱(𝑘+𝑖∣𝑘)∈𝑅^𝑛∣𝝃^𝑇₃(𝑘+𝑖∣𝑘)Λ₃𝝃₃()=𝐱()𝑘+𝑖∣𝑘≤𝜐𝑘+𝑖∣𝑘∈𝑅^𝑛∣𝝃^𝑇₃(𝑘+𝑖∣𝑘)Λ𝝃₃(𝑘+𝑖∣𝑘)≤1,𝑖≥0 (47) and 𝐮(𝑘+𝑗∣𝑘)=𝐊𝐱(𝑘+𝑗∣𝑘), 𝑗≥0. Then for any [𝐀(𝑘)∣𝐀_𝑑(𝑘)]∈Ω, we have 𝐱^𝑇(𝑘+1∣𝑘)𝐏(𝑘)𝐱(𝑘+1∣𝑘)<𝐱^𝑇(𝑘∣𝑘)𝐏(𝑘)𝐱(𝑘∣𝑘),𝐱(𝑘∣𝑘)≠0. (48)
Noting 𝐱(𝑘+1−𝑖∣𝑘+1)<𝐱(𝑘+1−𝑖∣𝑘), 𝑖=1,2,…,𝜏_𝑘, that implies 𝐱^𝑇(𝑘+1−𝑖∣𝑘+1)𝐏^∗_𝑑𝐱(𝑘+1−𝑖∣𝑘+1)≤𝐱^𝑇(𝑘+1−𝑖∣𝑘)𝐏^∗_𝑑𝐱(𝑘+1−𝑖∣𝑘),𝑖=1,2,…,𝜏_𝑘. (49)
Since the measured state 𝐱()()(𝑘)𝐱()+𝐀𝑘+1∣𝑘+1=𝐱𝑘+1=𝐀𝑘∣𝑘_𝑑(𝑘)+𝐁𝐊×𝐱𝑘−𝜏_𝑘∣𝑘+𝝎(𝑘) (50) for some [𝐀(𝑘)∣𝐀_𝑑(𝑘)]∈Ω, it must also satisfy the inequality (48). Combining this with (46) and (49) and according to (32)–(35), we can obtain that 𝑉𝐱^∗(𝑘+1∣𝑘+1)≤𝑉𝐱^∗(𝑘∣𝑘). (51)
Therefore, 𝑉(𝐱(𝑘)) is a monotonically nonincreasing and bounded Lyapunov function. Hence, the closed-loop system is robustly asymptotically stable. This completes the proof.

4. Simulation Results

To illustrate the performance of the proposed algorithm, we simulate a network using the model proposed in [12] for the channel gain from the𝑖th MS to its BS. In this model, the channel gain is given by 𝐺_𝑖𝑖(𝑘)=𝑆₀𝑑^−𝛿𝑖_𝑖𝑖(𝑘)⋅10^{0.1𝜁𝑖(𝑘)}, (52) where 𝑆₀ is a function of the carrier frequency and 𝑑^−𝛿𝑖_𝑖𝑖(𝑘) is the exponential path loss, which depends on the distance 𝑑_𝑖𝑖(𝑘) from the MS 𝑖 to its BS and the path-loss exponent, 𝛿_𝑖∈𝑅, which typically takes values between 2 and 6 (we take 4 in the simulation). The term 10^{0.1𝜁𝑖(𝑘)} corresponds to the effect of shadowing (from building, terrain, or foliage), where 𝜁_𝑖(𝑘)∈𝑅 is a zero mean Gaussian random variable with variance 𝜎²_𝜁𝑖, which usually ranges between 6 and 12. The values 𝑐₁(𝑘) and 𝑐₂(𝑘) are chosen as random variables between 0 and 0.5. 𝑛(𝑘) and 𝑑(𝑘) are zero mean with variance 0.01. Moreover, take 𝜇=0.8, 𝛼=0.2.

As indicated in [9, 17], the practical value of the round-trip delay is between 2 and 4 at the sampling rate of 800 Hz and is indicated in [18], and the typical value of the round-trip time delay (sum of the downlink and uplink delay) is between 0 and 4 for IS-95 wireless networks and is 2 for wideband code-division multiple access (WCDMA) wireless networks. In the simulations, we assume the time delay to be changed from 0 to 4, that is, 0=𝜏_𝑚≤𝜏_𝑘≤𝜏_𝑀=4.

The target SIR is set to be 10 dB, and the input constraints are 0<𝐮(𝑘+𝑖)≤1, 𝑖≥0. The parameters of MPC controller are set as follows: the prediction horizon is 𝑁=5 and the control horizon is 𝑁_𝑢=2.

Figure 2 shows the tracking error between the actual SIR and the target SIR by using the proposed model predictive power and rate control method. It is confirmed that the proposed method has good tracking performance for the wireless communication network with time-varying delays, uncertainties, and input constraints. As comparison, the traditional robust power and rate control method presented in [12] is also used for the network with the same parameters; however, only the time delay in state and in rate control is considered, that is, in the case 𝜏_𝑖=0 in (9) and in 𝑢_𝑖𝑓(𝑘−𝜏_𝑖) of (13), and the constant time delay 𝜏=3 is used in the design procedure. The tracking error is also shown in Figure 2. It is shown that the proposed method has less tracking error than the traditional robust power and rate control method.

Figure 2 

Tracking error between the actual SIR and the target SIR.

5. Conclusion

Most existing power and rate control methods are limited to consider known and constant time delay, especially some methods only considering the time delay in state and in power control. This paper has presented a novel power and rate control system model, which includes uncertainties, input constraints, and time-varying delays in both state and input, not only in the rate control but also in the power control. A robust model predictive power and rate control method is proposed, and the state feedback control law is obtained by solving an optimization problem that is derived by using linear matrix inequality (LMI). Simulation result shows that the proposed method has better performance than the traditional power and rate control methods.

Appendix

Proof of Theorem 4. Under the state feedback control 𝐮(𝑘)=𝐊𝐱(𝑘), the closed-loop system can be written as 𝐱(𝑘+1)=𝐀(𝑘)𝐱(𝑘)+𝐀_𝑑𝐱𝑘−𝜏_𝑘+𝝎(𝑘), (A.1) where 𝐀(𝑘)=𝐀+𝐷𝐅(𝑘)𝐄_𝑎,𝐀_𝑑(𝑘)=𝐀_𝑑+𝐁𝐊+𝐷𝐅(𝑘)𝐄_𝑑. (A.2)
Define Δ𝑉(𝐱(𝑘))=𝑉(𝐱(𝑘+1∣𝑘))−𝑉(𝐱(𝑘∣𝑘)); then along the solution of (25), we have Δ𝑉(𝐱(𝑘))=Δ𝑉₁(𝐱(𝑘))+Δ𝑉₂(𝐱(𝑘))+Δ𝑉₃(𝐱(𝑘)),Δ𝑉₁(𝐱(𝑘))=𝐱^𝑇(𝑘+1∣𝑘)𝐏𝐱(𝑘+1∣𝑘)−𝐱^𝑇(𝑘)𝐏𝐱(𝑘)=𝝃^𝑇(𝑘)𝚷^𝑇₁𝐏𝚷₁𝝃(𝑘)−𝐱^𝑇(𝑘)𝐏𝐱(𝑘), (A.3)(A.4) where 𝝃(𝑘)=𝐱^𝑇(𝑘),𝐱^𝑇𝑘−𝜏_𝑘,𝝎(𝑘)^𝑇𝑇,𝚷₁=𝐀,𝐀_𝑑,𝐈,Δ𝑉₂(𝐱(𝑘))=^𝑘_{𝑖=𝑘+1−𝜏𝑘+1}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)−^𝑘−1_{𝑖=𝑘−𝜏𝑘}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)=𝐱^𝑇(𝑘)𝐏_𝑑𝐱(𝑘)−𝐱^𝑇𝑘−𝜏_𝑘𝐏_𝑑𝐱𝑘−𝜏_𝑘+^𝑘−1_{𝑖=𝑘+1−𝜏𝑘+1}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)−^𝑘−1_{𝑖=𝑘−𝜏𝑘+1}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)≤𝐱^𝑇(𝑘)𝐏_𝑑𝐱(𝑘)−𝐱^𝑇𝑘−𝜏_𝑘𝐏_𝑑𝐱𝑘−𝜏_𝑘+^𝑘−1_{𝑖=𝑘+1−𝜏𝑀}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)−^𝑘−1_{𝑖=𝑘−𝜏𝑚+1}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)=𝐱^𝑇(𝑘)𝐏_𝑑𝐱(𝑘)−𝐱^𝑇𝑘−𝜏_𝑘𝐏_𝑑𝐱𝑘−𝜏_𝑘+^{𝑘−𝜏𝑚}_{𝑖=𝑘+1−𝜏𝑀}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖),Δ𝑉₃(𝐱(𝑘=))_{−𝜏𝑚𝑗=−𝜏𝑀+1𝑘𝑖=𝑘+𝑗+1}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)−^𝑘−1_{𝑖=𝑘+𝑗}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)=^−𝜏𝑚_{𝑗=−𝜏𝑀+1}𝐱^𝑇(𝑘)𝐏_𝑑𝐱(𝑘)−𝐱^𝑇𝑘+𝑗𝐏_𝑑𝐱𝑘+𝑗=𝜏_𝑠𝐱^𝑇(𝑘)𝐏_𝑑𝐱(𝑘)−^{𝑘−𝜏𝑚}_{𝑖=𝑘+1−𝜏𝑀}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖). (A.5)(A.6)
Combining (A.4) and (A.6), we have Δ𝑉(𝐱(𝑘))≤𝝃^𝑇(𝑘)Θ₁+𝚷^𝑇₁𝐏𝚷₁𝝃(𝑘), (A.7) where Θ₁=𝐝𝐢𝐚𝐠((𝜏_𝑠+1)𝐏_𝑑−𝐏,−𝐏_𝑑,𝟎).
Considering (36) with 𝐮(𝑘)=𝐊𝐱(𝑘), we obtain Δ𝑉(𝐱(𝑘))+𝜙(𝐱,𝐮,𝑘)=𝝃^𝑇(𝑘)Θ₂+𝚷^𝑇₁𝐏𝚷₁𝝃(𝑘)≤0, (A.8) where Θ₂=𝐝𝐢𝐚𝐠((𝜏_𝑠+1)𝐏_𝑑−𝐏+𝐐₁+𝐊^𝐓𝐑𝐊,−𝐏_𝑑,𝟎). Then Θ₂+𝚷^𝑇₁𝐏𝚷₁≤0. (A.9)
In the following, we will prove that (A.9) is equivalent to (41).
Substituting 𝐐=𝜐(𝑘)𝐏⁻¹ and 𝐐_𝑑=𝜐(𝑘)𝐏_𝑑⁻¹ into (A.9), noting 𝐀 and 𝐀_𝑑 in (A.1), and using Schur complement yield Ξ+Ψ_𝐷𝐅Ψ_𝐸+Ψ^𝑇_𝐸𝐅^𝑇Ψ^𝑇_𝐷≤0, (A.10) where ΦΞ=¹¹∗∗𝟎Φ₂₂∗𝐀𝐀_𝑑Φ+𝐁𝐊𝐐≤0,₁₁=𝜏_𝑠+1𝐐_𝑑⁻¹−𝐐⁻¹+𝜐⁻¹(𝑘)𝐐₁+𝐊^𝐓,Φ𝐑𝐊₂₂=−𝐐_𝑑⁻¹,Ψ_𝐷=𝟎𝟎𝐃^𝑇𝑇,Ψ_𝐸=𝐄_𝑎𝐄_𝑑𝐃^𝑇𝑇. (A.11)
According to Lemma 3, there exists a scalar 𝜀>0, such that the following inequality holds: Ψ_𝐷𝐅Ψ_𝐸+Ψ^𝑇_𝐸𝐅^𝐓Ψ^𝑇_𝐷≤𝜀Ψ_𝐃Ψ^𝑇_𝐷+𝜀⁻¹Ψ^𝑇_𝐸Ψ_𝐸. (A.12)
Using Schur complement, it follows that Φ₁₁∗∗∗𝟎−𝐐_𝑑⁻¹∗∗𝐀𝐀_𝑑+𝐁𝐊𝜀𝐃^𝐓𝐄𝐃−𝐐∗_𝑎𝐄_𝑑𝟎−𝜀𝐈≤0. (A.13)
In order to design the controller and obtain the state feedback gain 𝐊, pre- and postmultiplying (A.13) by  𝐝𝐢𝐚𝐠(𝐐,𝐐_𝑑,𝐈,𝐈), and considering 𝐘=𝐊𝐐_𝑑, we have Φ₁₁∗∗∗𝟎−𝐐_𝑑∗∗𝐀𝐐𝐀_𝑑𝐐_𝑑+𝐁𝐘𝜀𝐃^𝑇𝐄𝐃−𝐐∗_𝑎𝐐𝐄_𝑑𝐐_𝑑𝟎−𝜀𝐈≤0, (A.14) where Φ₁₁=𝜏_𝑠+1𝐐𝐐^𝑇_𝑑𝐐−𝐐+𝜐⁻¹(𝑘)𝐐𝐐_𝟏𝐐+𝜐⁻¹(𝑘)𝐘^𝑇𝐑𝐘. (A.15)
Then using Schur complement again, (41) is obtained. Furthermore, the feedback matrix 𝐊 is given by 𝐊=𝐘𝐐_𝑑⁻¹.
The next step of the proof involves the LMI (42). Equations (34) and (35) can be rewritten as 𝑉₂(𝐱(𝑘))=^𝑘−1_{𝑖=𝑘−𝜏𝑘}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)≤^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)=𝝃^𝑇₁(𝑘−1)Λ₁𝝃₁(𝑘−1), (A.16) where 𝝃₁(𝑘−1)=𝐱^𝑇(𝑘−1),…,𝐱^𝑇𝑘−𝜏_𝑀^𝑇,Λ₁=𝐝𝐢𝐚𝐠𝐏_𝑑,…,𝐏_𝑑,𝑉₃(𝐱(𝑘))=^−𝜏𝑚_{𝑗=−𝜏𝑀+1}^𝑘−1_{𝑖=𝑘+𝑗}𝐱^𝑇(𝑖)𝐏_𝑑𝐱(𝑖)=𝝃^𝑇₂(𝑘−1)Λ₂𝝃₂(𝑘−1), (A.17)(A.18) where 𝝃₂(𝑘−1)=𝐱^𝑇(𝑘−1),𝐱^𝑇(𝑘−2),…,𝐱^𝑇𝑘−𝜏_𝑚,𝐱^𝑇(𝑘−𝜏_𝑚−1),…,𝐱^𝑇(𝑘−𝜏_𝑀+1)^𝑇,Λ₂=𝐝𝐢𝐚𝐠𝜏_𝑠𝐏_𝑑,…,𝜏_𝑠𝐏_𝑑,𝜏_𝑠−1𝐏_𝑑,…,𝐏_𝑑. (A.19)
Summing (33), (A.16), and (A.18), we can conclude that 𝑉(𝐱(𝑘))=𝝃^𝑇₃(𝑘)Λ₃𝝃₃(𝑘), (A.20) where 𝝃₃(𝑘)=𝐱^𝑇(𝑘),𝐱^𝑇(𝑘−1),…,𝐱^𝑇𝑘−𝜏_𝑚,𝐱^𝑇(𝑘−𝜏_𝑚−1),…,𝐱^𝑇(𝑘−𝜏_𝑀)^𝑇,Λ₃=𝐝𝐢𝐚𝐠𝐏,𝜏_𝑠+1𝐏_𝑑,…,𝜏_𝑠+1𝐏_𝑑,𝜏_𝑠𝐏_𝑑,…,𝐏_𝑑. (A.21)
Combining (36) with (A.20), noting 𝐐=𝜐(𝑘)𝐏⁻¹ and 𝐐_𝑑=𝜐(𝑘)𝐏_𝑑⁻¹, and then using Schur complement, the LMIs (42) can be obtained.
Following [15], the input constraint (28) is equivalent to (43). This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61174116), the Beijing Natural Science Foundation (4142014), and the Scientific Research Foundation of North China University of Technology (20130017). The authors would like to thank the reviewers for their helpful suggestions and comments.