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.     

An optimal adaptive H-infinity tracking control design via wavelet network

In this paper, an optimal adaptive H-infinity tracking control design method via wavelet network for a class of uncertain nonlinear systems with external disturbances is proposed to achieve H-infinity tracking performance. First, an alternate tracking error and a performance index with respect to the tracking error and the control effort are introduced in order to obtain better performance, especially, in reducing the cost of the control effort in the case of small attenuation levels. Next, H-infinity tracking performance, which attenuates the influence of both wavelet network approximation error and external disturbances on the modified tracking error, is formulated. Our results indicate that a small attenuation level does not lead to a large control signal. The proposed method insures an optimal trade-off between the amplitude of control signals and the performance of tracking errors. An example is given to illustrate the design efficiency.     

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.     

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.     

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 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.     

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     

Finite horizon optimal hybrid reference control for improving transient response of feedback control systems

Hybrid reference control (HRC) has been known to improve transient response of a stabilised closed-loop continuous tracking system. In this paper, an optimal HRC strategy, which minimises a finite horizon quadratic performance index in the control and state error, is investigated based on the assumption that the full plant states are available for measurement. A special case of constrained optimisation problems leads to the optimal deadbeat control strategy. Conditions for asymptotic stability to default reference signal are also derived.     

Adaptive control with optimal tracking performance

This paper provides a way to optimise the steady-state tracking performance of the adaptive control system in the presence of unknown external disturbances. A-priori knowledge of the dynamic model of the reference signal to be tracked is not completely known. Especially, the updatable non-empty admissible model set, which is consistent to the a-priori knowledge of the plant parameter and the online measurements, is computed. Treating the overall system performance as the criteria, the nominal model is optimally chosen within the admissible model set. The optimal nominal model is subsequently used to synthesise the optimal closed-loop controller that minimises the steady-state absolute value of the tracking error. Combining the above two aspects, an optimal adaptive control scheme is proposed. Because of the consistency of the identification criteria and control object, the adaptive control scheme proposed in this paper can achieve the overall optimal steady-state tracking performance, and the effect of the interplay between the identification and control of the adaptive system can be handled effectively. In addition, the computable optimal performance is also provided.     

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.     

Data-Based Optimal Tracking of Autonomous Nonlinear Switching Systems

In this paper,a data-based scheme is proposed to solve the optimal tracking problem of autonomous nonlinear switching systems.The system state is forced to track the reference signal by minimizing the performance function.First,the problem is transformed to solve the corresponding Bellman optimality equation in terms of the Q-function(also named as action value function).Then,an iterative algorithm based on adaptive dynamic programming(ADP)is developed to find the optimal solution which is totally based on sampled data.The linear-in-parameter(LIP)neural network is taken as the value function approximator.Considering the presence of approximation error at each iteration step,the generated approximated value function sequence is proved to be boundedness around the exact optimal solution under some verifiable assumptions.Moreover,the effect that the learning process will be terminated after a finite number of iterations is investigated in this paper.A sufficient condition for asymptotically stability of the tracking error is derived.Finally,the effectiveness of the algorithm is demonstrated with three simulation 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.     

基于组合相关函数的CosBOC(10,5)信号无模糊跟踪方法

二进制偏移载波（BOC）调制信号是一种设计精妙的新型卫星导航信号,充分挖掘其性能的前提是解决好其跟踪模糊问题。余弦相位二进制偏移载波（CosBOC）信号的自相关函数相对更复杂且难以简洁统一地表示,其跟踪模糊问题的研究也相对薄弱,为此,以应用于伽利略系统的E6频段上的CosBOC（10,5）信号为研究对象,提出了一种基于组合相关函数的无模糊跟踪方法。该方法首先设计了由数目尽量少的本地复现信号及其移位加权来线性组合构成的本地参考信号;然后通过码跟踪环路该参考信号与接收信号相关后得到无副峰的相关函数,从而消除码鉴别器的跟踪模糊;最后分析了该方法的码跟踪误差。实验结果表明,该方法与类二进制相移键控法（BPSK Like）、峰跳法（Bump Jump）和伪相关函数法（PCF）相比,跟踪环路结构简单,完全抑制了跟踪模糊且整体上具有较好的码跟踪性能。     

Self-tuning servo for stochastic references

In this paper a solution to the problem of tracking an unknown reference signal is proposed. Three-term and error driven mean square optimal controllers are defined. Self-tuning algorithms, providing adaptation to both the process dynamics and the reference signal form are derived from the optimal solution. Their global convergence is proved. Simulation results illustrate their main properties.     

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.     

Trajectory tuning for the tracking problem of the underactuated mechanical systems: application to the PVTOL aircraft

The Planer vertical take-off and landing (PVTOL) aircraft is a typical example of an underactuated mechanical system and has a nonminimum-phase nature. When considering output tracking control, the Input/Output (I/O) linearization method is not appropriate since the stability of the internal dynamics is not guaranteed. Hauser et al. regarded this system as a slightly nonminimum-phase system which approximates to a minimum-phase one. Their control scheme yielded good results when the coupling factor was small, but the results were not acceptable when the coupling factor increased. In this article, we propose two approaches to improve the control performance. First, we consider the approximation error of Hauser's scheme as uncertainty, and apply the Linear Quadratic Regulator (LQR) method, which possesses robustness against uncertainty, to determine the stabilizing feedback coefficients. Second, from the fact that the tracking error is unavoidable, we use the “virtual reference trajectory” to design the tracking control law, and optimize this trajectory to reduce the tracking error between the “actual reference trajectory” and the “resulting trajectory”. This optimization also improves the control performance by choosing a suitable performance index. By using our approach, we achieve better performance even if the coupling factor is increased. We show these results by numerical simulation. This work was presented, in part, at the Seventh International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

基于自适应动态规划的非线性鲁棒近似最优跟踪控制

为克服现有近似最优跟踪控制方法只能跟踪连续可微参考输入的局限,本文针对一类具有未知动态的连续时间非线性时不变仿射系统,提出了一种新的基于自适应动态规划的鲁棒近似最优跟踪控制方法.首先采用递归神经网络建立系统模型,然后建立评价神经网络对最优性能指标进行估计,从而得到最优性能指标偏导数的估计值,进而得到近似最优跟踪控制器,最后利用系统输出与参考输入之间的跟踪误差设计鲁棒项对神经网络建模误差进行补偿.分别针对两个非线性系统进行仿真实验,仿真结果表明了所提方法的有效性和优越性.     

基于子空间跟踪的修正MMSE多用户检测算法

在研究基于子空间跟踪的最小均方误差（MMSE）多用户检测器（MUD）的基础上，为解决原算法因引入特征值估计误差而导致检测性能下降的问题，设计了一种改进的算法——修正的MMSE多用户检测器．采用正交性能优良的OPAST算法跟踪子空间，提出一种基于OPAST的修正MMSE多用户检测算法．仿真结果显示,与基于OPAST的MMSE MUD相比，基于OPAST的修正MMSE MUD算法收敛速度快，输出信干比和误码率性能逼近SVD MUD算法，并且计算复杂度较低．     

新型的卫星导航信号体制设计

为进一步提高导航信号精度、频带效率及增强导航系统可靠性,提出一种基于低密度奇偶校验(LDPC)码的最小频移键控-二进制偏移载波(MSK-BOC)调制信号体制(LDPC-MSK-BOC)。以Compass和GPS信号典型参数为例,采用功率谱密度、码跟踪误差、多径误差包络、误比特率、解调抗窄带/匹配谱干扰品质因数、码跟踪抗窄带/匹配谱品质因数、谱分离系数等参数来评估二进制偏移载波(BOC)和MSK-BOC两种信号的导航性能。理论分析和仿真结果表明:在频带资源受限的条件下,与BOC信号体制相比,所提出的信号体制在码跟踪精度和抗多径方面有更好的性能;同时可进一步提高系统的可靠性,改善频带利用率。     

基于相关函数的改进LMS自适应滤波算法研究

标准LMS算法由于采用了固定的步长因子,致使其算法稳态误差大、实时性差,不适用于非平稳随机过程.提出了一种改进的CLMS变步长自适应滤波算法,该算法采用输入-误差信号的互相关函数控制步长更新,在算法收敛初期步长取较大值,使其有较快的实时跟踪能力,在算法收敛末期步长取较小值,以使其拥有较小的稳态误差.仿真结果表明,相比已有变步长算法,CLMS算法有更好的应用性能.