Admissibility analysis of linear singular systems via a delta operator method

In this paper, the problem of admissibility analysis is investigated for linear singular systems through a delta operator method. The delta operator model of a linear singular continuous system is set up first for a special case and then generalised to the common case. It is shown that when the sampling period tends to zero, the delta operator model tends to the corresponding singular continuous system. In order to obtain the above delta operator model, we adopt a linear singular discrete model which contains the sampling period instead of the standard discrete one which contains no sampling period in the modeling process. The relation between the admissibility of the two singular discrete models is analysed and it is proved that the admissibility of one singular discrete model is equivalent to that of the other discrete model. Moreover, based on the relation between the discrete and delta operator models, some admissibility conditions are derived for linear singular delta operator systems. A numerical example is also provided to demonstrate the theoretical results in this paper.     

A bound of conservativeness in sampled-data robust stabilization and its dependence on sampling periods

This paper investigates conservativeness of the ²-induced norm in a sampled-data robust stabilization problem. Namely, it is shown that possible conservativeness of the ²-induced norm can be bounded by difference between the best achievable performance of time-invariant continuous-time controllers and that of sampled-data controllers. Moreover, this difference is proven to approach zero as a sampling period tends to zero, which means the conservativeness above loses importance when a sampling period is small. These results are important because robust stabilization based on the ²-induced norm is computationally inexpensive compared with robust stabilization based on the non-conservative robust stability index. One key lemma plays an essential role in this paper and it should be interesting independently.     

Improving the Stability Behavior of Limiting Zeros for Multivariable Systems Based on Multirate Sampling

It is well-known that existence of unstable sampling zeros is recognized as a major barrier in many control problems, and stability of sampling zeros, in general, depends on the type of hold circuit used to generate the continuous-time system input. This paper is concerned with stability of limiting zeros, as the sampling period tends to zero, of multivariable sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time plant with the relative degrees being two and three, and a sampler in cascade. In particular, the main focus of the paper is how to preserve the stability of limiting zeros when at least one of the relative degrees of a multivariable system is more than two. In this case, the asymptotic properties of the limiting zeros on the basis of normal form representation of continuous-time systems are analyzed and approximate expressions for their stability are discussed as power series expansions with respect to a sufficiently small sampling period. More importantly, unstable sampling zeros of the sampled-data models mentioned above can be avoided successfully through the contribution of this paper, whereas a zero-order hold (ZOH) or a fractional-order hold (FROH) fails to do so. It is a further extension of previous results for single-input single-output cases to multivariable systems.     

Comparing digital implementation via the bilinear and step-invariant transformations

From an analog system K, digital systems K_d and K_bt are often obtained via the step-invariant and bilinear transformations, respectively. For the case when K is stable, it is shown that under certain conditions, K_bt converges to K_d in ?_p-induced norms as the sampling period tends to zero for 1?p?∞. Furthermore, K_bt converges to K_d in the graph metric as the sampling period tends to zero no matter whether K is stable or not.     

线性切换系统的采样数据输出调节问题

本文基于采样控制,解决了线性切换系统的输出调节问题.首先考虑了外部输入信号是常数的情况,给出了采样周期与平均驻留时间的关系.通过构造一类Lyapunov-Krasovskii泛函,使得闭环系统内部稳定且调节输出趋于零.其次考虑了外部输入信号是导数有界且时变的情况,给出了采样数据状态反馈控制器,并依据采样周期和平均驻留时间关系给出的切换条件得到了切换系统实用输出调节问题可解的充分条件.最后通过两个数值例子验证了方法的有效性.     

Necessary and sufficient conditions for solving consensus problems of double‐integrator dynamics via sampled control

In this paper, consensus problems of double‐integrator dynamics via sampled control are investigated. The sampled control protocol is induced from continuous‐time linear consensus protocol by using periodic sampling technology and zero‐order hold circuit. With the obtained sampled control protocol, the continuous‐time multi‐agent system is equivalently transformed into a linear discrete‐time system. Necessary and sufficient conditions are given to guarantee that all the agents asymptotically travel with zero relative positions and common velocities. Furthermore, consensus problem with continuous‐time consensus protocol is re‐analyzed. A necessary and sufficient condition is also obtained which is consistent with the special case when the sampling period tends to zero. The effectiveness of these algorithms is demonstrated through simulations. Copyright © 2009 John Wiley & Sons, Ltd.     

Indirect sampled-data control with sampling period adaptation

It is known that if a continuous-time feedback system is exponentially stable, then the corresponding sampled-data system obtained by sample-hold discretisation with constant sampling period is also exponentially stable, provided that the sampling period τ?>?0 is sufficiently small. In general, it is difficult to estimate how small the sampling period has to be in order to achieve the stability of the sampled-data system. In this article, we present an adaptive mechanism for adjusting the sampling period. This mechanism has the properties that, for every initial state, (i) the adaptation of the sampling period terminates after finitely many time steps and (ii) the state of the adaptive sampled-data system is integrable and converges to zero as time goes to infinity.     

Sampling approximations for non-band-limited harmonizable random signals

As the sampling rate tends to infinity, the classical sampling series of a non-band-limited harmonizable second order random signal is shown to converge to the signal in quadratic mean and in certain cases with probability one, and rates of convergence are derived. Similar results are also shown for certain harmonizable signals with infinite variance.     

The relation between the sampling time and stochastic error for the impulsive response of linear time-independent systems

In this paper the relation between the variance of the impulsive response samples and the sampling time is analyzed with respect to the identification of a linear time-invariant system with one input and one output. The study is made under the hypothesis that only the output data are corrupted by measurement errors. The main result of this paper is a proof that the variance of the impulsive response samples diverges when the sampling time tends toward zero.     

Precompensated Second Order Repetitive Control of an Active Filter Under Varying Network Frequency

Network frequency variations cause a dramatic performance decay in repetitive controller‐based shunt active power filters. This problem may be solved by adapting the sampling period in order to keep the ratio between the network period and the sampling period at a constant value. However, these changes may yield closed‐loop instability. The introduction of a precompensator that forces the plant to remain invariant despite sampling rate changes allows the use of standard LTI methods in control design and stability analysis as well. Moreover, in order to improve robustness in the face of network frequency estimation uncertainty and sampling time quantization, the regular repetitive controller is replaced by a high order one. Experimental results show the validity of the proposal.     

Control of constrained linear systems using fast sampling rates

This paper addresses the problem of optimal control of constrained linear systems when fast sampling rates are utilised. We show that there exists a well-defined limit as the sampling rate increases. An immediate consequence of this result is the existence of a finite sampling period such that the achieved performance is arbitrarily close to the limiting performance.     

Adaptive compensation strategy for the tracking/rejection of signals with time-varying frequency in digital repetitive control systems

Digital repetitive control is a technique which allows to track periodic references and/or reject periodic disturbances. Repetitive controllers are usually designed assuming a fixed frequency for the signals to be tracked/rejected, its main drawback being a dramatic performance decay when this frequency varies. A common approach to overcome this problem consists of an adaptive change of the sampling time according to reference/disturbance period variations. However, the sampling period adaptation implies structural changes that may degrade the closed-loop time response. This article presents a design strategy which allows to compensate for the effect of non-uniform sampling and forces the closed-loop behavior to coincide with that of the system operating under an a priori selected nominal sampling period. The stability of the overall closed-loop system is also analyzed. Experimental results obtained from a mechatronic plant model validate the proposal.     

Digital redesign of continuous systems by matching of states at multiple sampling periods

This paper presents a method of digital approximation of a continuous-data system by the matching of states at multiple sampling periods. Two alternate methods are presented; one method samples the states every sampling period, and alters the forward and feedback gains, so as to match the states of the two systems at the end of N sampling periods. The second method requires the sampling of the states every N sampling periods, with the change of gains at each sampling period. An illustrative example using the one-axis dynamics of the Skylab Satellite is presented.     

Stochastic Bounded Consensus Tracking of Second-Order Multi-Agent Systems with Measurement Noises and Sampled-Data

This paper investigates the stochastic bounded consensus tracking problems of second-order multi-agent systems, where the control input of an agent can only use the information measured at the sampling instants from its neighbors or the virtual leader with a time-varying reference state, and the measurements are corrupted by random noises. The probability limit theory, the algebra graph theory, and some other techniques are employed to derive the necessary and sufficient condition guaranteeing the mean square bounded consensus tracking. It turns out that the maximum allowable sampling period depends on not only the network topology but also the constant feedback gains. Furthermore, the effects of the sampling period on tracking performance, including the tracking speed and the static tracking error, are also analyzed. The results show that reducing the sampling period can accelerate the tracking speed and decrease the static tracking error. Simulations are provided to demonstrate the effectiveness of the theoretical results.     

An eigenvalue condition for sampled weak controllability of bilinear systems

Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case.     

基于小波的DDoS入侵流分析

采用NS2模拟分布式拒绝服务攻击,利用小波方法对受攻击的不同业务流采样样本进行分析,通过理论分析和模拟实验,证明采样方法与Hurst指数的估计关系密切：如果只选择攻击时段内的业务流样本,H值减小;当样本既包含正常流时段又包含攻击时段时,H值增大。     

Comparative study of discretization zero dynamics behaviors in two multirate cases

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems. When the usual digital control with zero-order hold (ZOH) or fractional-order hold (FROH) input is used, unstable zero dynamics inevitably appear in the discrete-time model even though the continuous-time system with relative degree more than two is of minimum phase. This paper investigates the zero dynamics, as the sampling period tends to zero, of sampled-data models composed of a generalized sample hold function (GSHF), a continuous-time nonlinear plant and a sampler in cascade. More precisely, we show how an approximate sampled-data model can be obtained for nonlinear systems with two special GSHF cases such that sampled zero dynamics of the resulting model can be arbitrarily placed. Further, two GSHFs with appropriate parameters provide nonlinear zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time plant. It is also shown that the intersample behavior arising from the multirate input can be localised by appropriately selecting the design parameters based on the stability condition of the zero dynamics. The results presented here generalize well-known ideas from the linear to nonlinear cases.     

采样周期对最优采样值调节器稳定性的影响

本文研究了最优采样值调节器的渐近稳定性,在一般最优采样值调节器和稳定最优采样值调节器的两种情形下,给出了最优采样值调节器为渐近稳定的条件,从而提供了采样周期的选择方法。     

Allowable sampling period for consensus control of multiple general linear dynamical agents in random networks

This article studies the consensus problem for a group of sampled-data general linear dynamical agents over random communication networks. Dynamic output feedback protocols are applied to solve the consensus problem. When the sampling period is sufficiently small, it is shown that as long as the mean topology has globally reachable nodes, the mean square consensus can be achieved by selecting protocol parameters so that n???1 specified subsystems are simultaneously stabilised. However, when the sampling period is comparatively large, it is revealed that differing from low-order integrator multi-agent systems the consensus problem may be unsolvable. By using the hybrid dynamical system theory, an allowable upper bound of sampling period is further proposed. Two approaches to designing protocols are also provided. Simulations are given to illustrate the validity of the proposed approaches.