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.     

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.     

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.     

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.     

Optimal tracking performance of MIMO control systems with communication constraints and a code scheme

This paper investigates the issue of the optimal tracking performance for multiple-input multiple-output linear time-invariant continuous-time systems with power constrained. An H₂ criterion of the error signal and the signal of the input channel are used as a measure for the tracking performance. A code scheme is introduced as a means of integrating controller and channel design to obtain the optimal tracking performance. It is shown that the optimal tracking performance index consists of two parts, one depends on the non-minimum phase zeros and zero direction of the given plant, as well as the reference input signal, while the other depends on the unstable poles and pole direction of the given plant, as well as on the bandwidth and additive white noise of a communication channel. It is also shown that when the communication does not exist, the optimal tracking performance reduces to the existing normal tracking performance of the control system. The results show how the optimal tracking performance is limited by the bandwidth and additive white noise of the communication channel. A typical example is given to illustrate the theoretical results.     

Fundamental performance limitations in tracking sinusoidal signals

This paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies. The tracking performance is measured by an integral square error between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the attainable tracking performance, under any control structure and parameters, in terms of the characteristics and structural parameters of the given plant, as well as those of the reference signal under consideration. It is shown that this fundamental limit depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency-dependent directional information.     

Limitations on maximal tracking accuracy

This paper studies optimal tracking performance issues pertaining to finite-dimensional, linear, time-invariant feedback control systems. The problem under consideration amounts to determining the minimal tracking error between the output and reference signals of a feedback system, attainable by all possible stabilizing compensators. An integral square error criterion is used as a measure for the tracking error, and explicit expressions are derived for this minimal tracking error with respect to step reference signals. It is shown that plant nonminimum phase zeros have a negative effect on a feedback system's ability to reduce the tracking error, and that in a multivariable system this effect results in a way depending on not only the zero locations, but also the zero directions. It is also shown that if unity feedback structure is used for tracking purposes, plant nonminimum phase zeros and unstable poles can together play a particularly detrimental role in the achievable tracking performance, especially when the zeros and poles are nearby and their directions are closely aligned. On the other hand, if a two parameter controller structure is used, the achievable tracking performance depends only on the plant nonminimum phase zeros     

多变量离散系统随机信号跟踪性能极限

研究线性时不变、多变量、离散系统对随机信号的跟踪性能极限问题, 所考虑的随机参考输入信号为布朗运动序列. 研究结果表明此类系统的跟 踪性能极限完全由被控对象的结构特征和参考输入的统计特征决定, 其中, 结构特征指被控对象的非最小相位零点和不稳定极点的位置和方向. 作为特殊情形, 本文给出了参考输入为一致随机信号以及被控对象仅含有单个非最小相位零点和单个不稳定极点时系统跟踪性能极限问题的解. 最后, 给出了两自由度补偿器跟踪系统对随机信号的跟踪性能极限.     

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.     

Performance Limitation of Networked Systems with Network‐Induced Delay and Packet‐Dropout Constraints

In this paper, we deeply investigate the tracking performance limitation of single‐input single‐output (SISO) networked systems with packet dropouts and network‐induced delay constrains, where the power of the tracking error is adopted to measure of the performance and is minimized by searching through all stabilizing one‐parameter controllers, with/without considering the channel input power. Moreover, the explicit expressions of the tracking performance limitation are obtained by applying the spectral factorization technique. Theoretical analysis shows that the tracking performance limitation is closely dependent on non‐minimum phase zeros, unstable poles of a given plant, the characteristics of the reference signals, network‐induced delay and packet dropouts probability of communication channel. Finally, some typical examples and simulations are conducted 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.     

Best tracking and regulation performance under control energy constraint

This paper studies optimal tracking and regulation control problems, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. Our primary objective in this work is to search for fundamental design limitations beyond those known to be imposed by nonminimum phase zeros, unstable poles, and time delays. For this purpose, we solve the problems explicitly by deriving analytical expressions for the best achievable performance. It is found that this performance limit depends not only on the plant nonminimum phase zeros, time delays, and unstable poles-a fact known previously-but also on the plant gain in the entire frequency range. The results thus reveal and quantify another source of fundamental performance limitations beyond those already known, which are nonexistent when only conventional performance objectives such as tracking and regulation are addressed without taking into account the control energy constraint. Among other things, they exhibit how the lightly damped poles, the anti-resonant zeros, as well as the bandwidth of the plant may affect the performance.     

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.     

Fundamental limit of discrete-time systems in tracking multi-tone sinusoidal signals

This paper studies the tracking performance of linear time-invariant multi-variable discrete-time systems. The specific problem under consideration is to track a multi-tone sinusoidal reference signal consisting of linear combinations of a step and several sinusoidal signals, whereas the tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the best attainable performance, under any control structures and parameters, and we seek to determine this limit analytically in terms of the given plant and reference characteristics. Both the full-information and partial-information tracking schemes are formulated and investigated to address these goals, which are concerned with whether or not the reference information is fully available for tracking. Analytical expressions are developed in full generality under full-information tracking, and for a more specialized case under partial-information scheme. In addition, an optimal cheap control design is constructed to show that the performance limit can be attained asymptotically in the full-information case. The results show that in general plant nonminimum phase zeros and reference modes can interact to fundamentally constrain a system's tracking ability. They also show that absence of full reference information can degrade the tracking performance, thus demonstrating an intrinsic trade-off between the tracking objective and the availability of the reference information.     

On performance limitation in tracking a sinusoid

This note studies the performance limitation of a feedback system with a given linear time-invariant (LTI) plant in tracking a sinusoidal signal. It continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal. In this note, we consider the more realistic (and more difficult) situation where the controller only accesses the current and past values of the reference. An explicit formula of the best attainable performance is obtained for a single-input-single-output (SISO) system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid. Compared to the previously studied case when the future of the reference is available, this formula clearly shows the extra effort one has to pay due to the lack of the reference information. A partial result for a multiple-input-multiple-output (MIMO) system is also given.     

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.     

Tracking Performance Bound with Finite Control Energy and Erasure Channel Energy Constraint

This paper studies the tracking performance of the single‐input single‐output (SISO), finite dimensional, linear and time‐invariant (LTI) system under the channel input energy constraint over the Erasure channel. A new performance index is proposed which is minimized over all two‐degree‐of‐freedom stabilizing controllers. The explicit expressions of the lower bound of the performance index and the minimum of the signal‐to‐noise ratio are obtained. The results show that the performance bound is correlated to unstable poles, non‐minimum phase zeros and the packet loss probability. Finally, examples are given to validate the conclusions derived.     

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.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Tracking Performance Achievement for Continuous‐Time Delayed Linear Systems Subject to Actuator Saturation and output Disturbances

Controlling continuous‐time input‐delayed nonminimum‐phase linear systems is addressed in the presence of actuator saturation and output‐disturbances. Focusing on output‐reference tracking, the control design is dealt with in the pseudo‐polynomials ring. A quite appealing L₂ ‐tracking performance is shown to be achievable in the presence of arbitrary inputs i.e. the output reference and the output disturbance. The performance is formulated in terms of a well defined output‐reference mismatch error (ORME), depending on the inputs’ rate and their compatibility with the actuator saturation constraint.