Optimal tracking performance of MIMO discrete‐time systems with communication constraints

The optimal tracking problem for multiple‐input multiple‐output linear‐time‐invariant discrete‐time systems with communication constraints in the feedback path is studied in this paper. The tracking performance is measured by the energy of the error signal between the output of the plant and the reference signal. The objective is to obtain an optimal tracking performance, attainable by all possible stabilizing compensators. It is shown that the optimal tracking performance consists of two parts, one depends on the nonminimum phase zeros and zero direction of the given plant, as well as the reference input signal direction, and the other depends on the nonminimum phase zeros, unstable poles, and pole direction of the given plant, as well as the bandwidth and additive white Gaussian noise of the communication channel. It is also shown that, if the constraint of the communication channel does not exist, the optimal tracking performance reduces to the existing tracking performance of the control system without communication constraints. A typical example is given to illustrate the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

多通道约束下的网络控制系统的最优跟踪性能

本文针对双通道约束下的线性时不变网络控制系统的随机信号跟踪性能极限问题进行了研究.网络通信包含通信噪声和通信带宽两种信道因素.被控系统考虑是非最小相位和不稳定系统,并且系统包含多个不同的非最小相位零点和多个不同的不稳定极点.对上行通道和下行通道都存在通信带宽约束及高斯白噪声影响的情形,从频域角度,通过采用双自由度控制器和尤拉参数化方法,获得了此类网络控制系统的最优可达的跟踪性能.研究结果表明网络控制系统的跟踪性能极限完全由被控对象的结构特征(非最小相位零点、不稳定极点以及被控对象的系统增益),参考输入信号和网络特性(高斯白噪声的统计特征、通信信道带宽)所决定.最后,仿真结果检证了所得结果的正确性.     

An Optimal Tracking Performance of MIMO NCS with Quantization and Bandwidth Constraints

This paper presents an optimal tracking performance of multiple‐input multiple‐output (MIMO) networked control systems (NCSs) with quantization and bandwidth constraints. In this study, we simultaneously consider the encoding‐decoding, quantization and bandwidth of communication channel. The optimal tracking performance of NCSs is obtained by spectral factorization technique and partial fraction. The obtained results demonstrate that the optimal tracking performance is influenced by the nonminimum phase zeros and unstable poles as well as their directions for a given plant. In addition, it is shown that characteristics of reference signal, encoding‐decoding, quantization, and bandwidth of communication channel are also closely related to tracking performance. Finally, the efficiency of proposed tracking performance is verified by typical examples.     

Best Tracking Performance of Networked Control Systems Based on Communication Constraints

The best tracking problem for a single‐input‐single‐output (SISO) networked control system with communication constraints is studied in this paper. The tracking performance is measured by the energy of the error signal between the output of the plant and the reference signal. The communication constraints under consideration are finite bandwidth and networked induced‐delay. Explicit expressions of the minimal tracking error have been obtained for networked control systems with or without communication constraints. It is shown that the best tracking performance dependents on the nonminimum phase zeros, and unstable poles of the given plant, as well as the bandwidth and networked induced‐delay. It is also shown that, if the constraints of the communication channel do not exist, the best tracking performance reduces to the existing tracking performance of the control system without communication constraints. The result shows how the bandwidth and networked induced‐delay of a communication channel may fundamentally constrain a control system's tracking capability. Some typical examples are given to illustrate the theoretical results.     

Optimal Performance in Tracking Stochastic Signal Under Disturbance Rejection

The analysis method of optimal tracking performance is proposed for multiple‐input multiple‐output (MIMO) linear time‐invariant (LTI) systems under disturbance rejection. An H₂ criterion of the error signal between the output of the plant and the reference signal is used as a measure for the tracking performance. Spectral factorization is applied to obtain the optimal solution of the system tracking error. The explicit expressions are derived for this minimal tracking error with respect to random reference signals under disturbance rejection. It is shown that the nonminimum phase zeros, the zero direction, the unstable poles, the pole direction of a given plant, statistical characteristics of the reference input signal, and disturbance signal have a negative effect on a feedback system's ability to reduce the system error with disturbance rejection. The results show that the optimal tracking performance will further be damaged because of disturbance rejection. Some typical examples are given to illustrate the theoretical results.     

编解码和时延约束下的网络化控制系统最优跟踪性能

本文研究了具有丢包、时延、编解码等通信资源受限下多输入多输出离散时间网络控制系统的最优跟踪性能. 基于频域方法, 采用二元随机过程来模拟数据包丢失, 并假设信道噪声是加性高斯白噪声(AWGN), 推导了在丢包、信道噪声、时延和编解码影响下的跟踪性能极限. 采用单参数补偿器(SDOF), 利用互质分解、Youla参数化等工具得到了编解码和时延约束下的网络控制系统最优跟踪性能的显式表达式. 结果表明, 跟踪性能与对象的固有特性(非最小相位零点与不稳定极点的位置和方向)、时延、丢包率和AWGN 功率谱密度密切相关.     

网络化系统周期信号的跟踪

研究线性时不变、单变量、离散网络化系统对周期信号的跟踪问题.与现有文献考虑的参考输入信号大都为常见的能量信号所不同的是,本文参考输入信号是离散时间周期方波功率信号.相应地,研究系统对基于功率谱的参考输入信号功率的响应,系统的跟踪性能通过输入信号与受控对象输出之差的功率来衡量,而最优跟踪性能采用跟踪误差的平均功率来度量.考虑的网络化控制系统仅上行通道存在丢包误差的影响,把丢包过程看作两个信号的合成,一是确定性信号,二是随机过程,进而丢包误差描述为源信号和白噪声之间乘积.根据被控对象和随机过程的性质,采用Parseval等式、维纳–辛钦定理和范数矩阵理论得到该系统跟踪性能极限的下界表达式.仿真结果表明,所设计的控制器能实现对周期信号的有效跟踪,进而验证了结论的正确性.     

Fundamental limitations in control over a communication channel

Fundamental limitations in feedback control is a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a continuous-time, unstable, non minimum phase (NMP) plant with time-delay over an additive Gaussian coloured noise communication channel. By working on a linear setting the infimal SNR for stabilisability is defined as the infimal achievable H₂ norm between the channel noise input and the channel signal input. The result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros, time-delay as well as the channel NMP zeros, bandwidth, and channel noise colouring.     

Optimal regulation performance of discrete time-delay systems based on quantization and input energy constraints

In this paper, we study the optimal regulation problem of networked control systems and propose a new performance index for a given discrete time-delay system. The regulation performance of the controlled plant is investigated by considering the effects of various constraints on the communication channel such as quantization, bandwidth, and packet dropouts using frequency domain methods and two-degree-of-freedom control techniques. The results show that the regulation performance is not only related to the location and direction of the non-minimum phase zeros and unstable poles of a given system but also related to the internal time delay of the controlled plant. Packet dropouts, quantization, and bandwidth limitations can also negatively affect the optimal performance. In addition, the trade-off of the input energy constraint can also make the optimal regulation performance suffer. Finally, the reliability of this innovative result is illustrated by some simulation examples.     

多通道网络化系统跟踪性能极限

本文基于白噪声和编码影响研究多通道网络化系统跟踪性能极限.网络化系统的跟踪性能指标是通过跟踪误差的能量来定义的,采用谱分解技术和范数矩阵理论得到多通道网络化系统跟踪性能极限的下界表达式.研究结果显示了对象的固有特性(非最小相位零点、零点方向、不稳定极点和极点方向)和多通道的编码器及白噪声决定网络化系统跟踪性能极限;同时也说明了多通道网络是如何影响系统跟踪性能极限.仿真结果验证了该结论的正确性.     

基于丢包和带宽限制网络化系统稳定性分析

基于数据包丢失和网络带宽限制研究了网络化系统稳定性问题.通信网络的特征通过数据包丢失、带宽和加性白噪声来体现.采用频域的方法得到了网络化系统稳定所需信噪比的最小(极限)值.该最小(极限)值说明了网络化系统稳定所需信噪比是由通信网络通道的丢包率、带宽、系统的不稳定极点的位置和非最小相位零点的位置决定.研究结果进一步揭示网络化系统稳定所需信噪比的最小(极限)值与系统本质特征(非最小相位零点的位置和不稳定极点的位置)和通信网络参数(数据包丢失和带宽)的关系.仿真结果说明该理论的正确性.     

Lower bound of tracking performance with finite control energy and channel energy constraint

This paper studies the tracking performance of the single-input single-output (SISO), finite dimensional, linear and time-invariant (LTI) system over an additive white Gaussian noise (AWGN) channel with finite control energy and channel input energy constraint. A new performance index is proposed which is minimized over all stabilizing two-degree-of-freedom controllers. The explicit expressions of the lower bound of the tracking performance and the minimum of signal-to-noise required are obtained. The results show that the lower bound is correlated to the unstable pole, nonminimum phase zero and the channel scaling factor. Finally, one example is given to validate the conclusions by adopting the special inner-outer factorization.     

The problem of LQG optimal control via a limited capacity communication channel

The paper addresses a LQG optimal control problem involving bit-rate communication capacity constraints. A discrete-time partially observed system perturbed by white noises is studied. Unlike the classic LQG control theory, the control signal must be first encoded, then transmitted to the actuators over a digital communication channel with a given bandwidth, and finally decoded. Both the control law and the algorithms of encoding and decoding should be designed to archive the best performance. The optimal control strategy is obtained. It is shown that where the estimator-coder separation principle holds, the controller-coder one fails to be true.     

Limitations in tracking systems

In this paper, we obtain information theoretical conditions for tracking in linear time-invariant control systems. We consider the particular case where the closed loop contains a channel in the feedback loop. The mutual information rate between the feedback signal and the reference input signal is used to quantify information about the reference signal that is available for feedback. This mutual information rate must be maximized in order to improve the tracking performance. The mutual information is shown to be upper bounded by a quantity that depends on the unstable eigenvalues of the plant and on the channel capacity. If the channel capacity reaches a lower limit, the feedback signal becomes completely uncorrelated with the reference signal, rendering feedback useless. We also find a lower bound on the expected squared tracking error in terms of the entropy of a random reference signal.We show a misleading case where the mutual information rate does not predict the expected effect of nonminimum phase zeros. However, mutual information rate helps generalize the concept that there is a tradeoff when tracking and disturbance rejection are simultaneous goals, and a constraint communication channel is present in the feedback loop. Examples and simulations are provided to demonstrate some of the results.     

An H∞ suboptimal robust control approach for systems with uncertainties and data dropouts

This paper studies the design of control systems subject to plant uncertainties and data losses in the channel connecting the plant sensor with the controller. The controller design has two main objectives. The first one is to robustify the control law against plant uncertainties. The other one is to achieve good performance by minimising the variance of the error signal. Data losses are modelled as an independent and identically distributed sequence of Bernoulli random variables. For analysis and design, this random variable is replaced by an additive noise plus gain channel model. To cope with structural uncertainties in the model of the plant, an H_∞ control technique is employed. The controller is synthesised in order to make the closed-loop system robust against structural uncertainties of the nominal model, while achieving optimal performance of the system in the presence of dropouts.     

Control of Continuous-Time Linear Gaussian Systems Over Additive Gaussian Wireless Fading Channels: A Separation Principle

This note is concerned with the control of continuous-time linear Gaussian systems over additive white noise wireless fading channels subject to capacity constraints. Necessary and sufficient conditions are derived, for bounded asymptotic and asymptotic observability and stabilizability in the mean square sense, for controlling such systems. For the case of a noiseless time-invariant system controlled over a continuous-time additive white Gaussian noise channel, the sufficient condition for stabilizability and observability states that the capacity of the channel C must satisfy C > Sigma_{{i;Re(lambdai(A))ges0}} Re(lambda_i(A)), where A is the system matrix and lambda_i(A) denotes the eigenvalues of A. The necessary condition states that the channel capacity must satisfy C ges Sigma _{{i;Re(lambdai(A))ges0}} Re(lambda_i(A)). Further, it is shown that a separation principle holds between the design of the communication and the control subsystems, implying that the controller that would be optimal in the absence of the communication channel is also optimal for the problem of controlling the system over the communication channel.     

单通道最优和自校正去卷平滑器及其应用

基于白噪声估计理论，本文提出了单通道ＡＲＭＡ信号的一种新的最优和自校正去卷滤波器和平滑器，可处理非平稳ＡＲＭＡ输入信号及不稳定和非最小相位系统，并给出了在雷达跟踪系统中的仿真应用例子。     

A new information fusion white noise deconvolution estimator

The white noise deconvolution or input white noise estimation problem has important applications in oil seismic exploration, communication and signal processing. By the modern time series analysis method, based on the autoregressive moving average （ARMA） innovation model, a new information fusion white noise deconvolution estimator is presented for the general multisensor systems with different local dynamic models and correlated noises. It can handle the input white noise fused filtering, prediction and smoothing problems, and it is applicable to systems with colored measurement noises. It is locally optimal, and is globally suboptimal. The accuracy of the fuser is higher than that of each local white noise estimator. In order to compute the optimal weights, the formula computing the local estimation error cross-covariances is given. A Monte Carlo simulation example for the system with Bernoulli-Gaussian input white noise shows the effectiveness and performances.     

基于量化控制信号的线性系统的跟踪性能极限

研究基于量化控制信号的线性系统的跟踪性能极限. 所研究的被控对象是稳定的线性时不变(LTI)系统, 参考跟踪信号是阶跃信号. 跟踪性能通过对象输出与参考信号之差的能量来衡量. 为了达到渐近跟踪, 提出了一个新的量化方案. 方案包括两部分: 一部分是在初始时刻将控制信号的稳态值通过网络传输给被控对象并且存储在其输入端; 另一部分则是通过对数量化器, 对控制信号的瞬态部分(即控制信号与其稳态值之差)进行量化, 然后传输给被控对象. 最后该量化信号与稳态控制信号相加后作用于被控对象. 这里, 假设对数量化误差是     

Regulation performance limitation of networked time-delay systems with incomplete information

In this paper, the regulation performance limitation of networked time-delay systems is studied. The communication network is mainly affected by parameters such as packet dropouts, encoding-decoding, interference signal, and channel noises. Non-minimum phase zeros, unstable poles, and time delay are all considered for a given plant. The corresponding regulation performance expression is derived using coprime factorization and spectral decomposition techniques in the frequency domain. The results indicate that the regulation performance of the system is related to the inherent characteristics of the given plant, including non-minimum phase zeros, unstable poles, and time delay. Additionally, network communication parameters such as white Gaussian noise, packet dropouts, encoding-decoding, and external interference signals all affect the regulation performance of networked time-delay systems. Finally, some simulation examples are provided to demonstrate the effectiveness of the theory.