Limit cycle analysis of nonlinear sampled-data systems by gain-phase margin approach

This work analyzes the limit cycle phenomena of nonlinear sampled-data systems by applying the methods of gain-phase margin testing, the M-locus and the parameter plane. First, a sampled-data control system with nonlinear elements is linearized by the classical method of describing functions. The stability of the equivalent linearized system is then analyzed using the stability equations and the parameter plane method, with adjustable parameters. After the gain-phase margin tester has been added to the forward open-loop system, exactly how the gain-phase margin and the characteristics of the limit cycle are related can be elicited by determining the intersections of the M-locus and the constant gain and phase boundaries. A concise method is presented to solve this problem. The minimum gain-phase margin of the nonlinear sampled-data system at which a limit cycle can occur is investigated. This work indicates that the procedure can be easily extended to analyze the limit cycles of a sampled-data system from a continuous-data system cases considered in the literature. Finally, a sampled-data system with multiple nonlinearities is illustrated to verify the validity of the procedure.     

PID pitch attitude control for unstable flight vehicle in the presence of actuator delay: Tuning and analysis

In the realm of flight control, proportional–integral–derivative (PID) control is still widely used in practice due to its simple structure and efficiency. The robustness and dynamic performance of PID controller can be evaluated by stability margins. Based on the empirical knowledge about the unstable flight dynamics, the analytical tuning formulas of the PID pitch attitude control with actuator delay are derived with the help of several proper approximations. These tuning formulas can meet the increasing gain and phase margins (iGPM) requirement and avoid time-consuming trial-and-error tuning process. The feasible iGPM area is established in 2-D plane subject to several conditions, especially taking the decreasing gain margin into account, wherein the numerical polynomial solving approaches are employed. The relationship between an existing PD tuning scheme and the proposed PID tuning method is also revealed. The applicable area of the tuning rule is then investigated on the basis of a crucial assumption. Furthermore, the achievable decreasing gain and phase margins (dGPM) area is obtained when the decreasing gain margin is critical; and another tuning rule is derived according to the dGPM specifications. The effect of the actuator delay on the achievable GPM area is demonstrated in a straightforward manner such that the reasonable criteria can be specified. Finally two numerical paradigms are presented to validate the proposed method; and the robustness and dynamic performance of the PID control are also reexamined for unstable flight dynamics.     

Sliding mode control design using generalized relative degree approach: Aerospace application

Relative degree (RD) approach is a powerful tool for obtaining system's input-output dynamics used for output tracking controller designs of minimum phase systems. Designs using the RD alone can fail due both to insufficient control authority in minimum phase systems, and instability of internal/zero dynamics attributed to nonminimum phase systems. A novel definition and a concept of Practical Generalized RD (PGRD) are proposed in this paper and are used in concert with Sliding Mode Control (SMC) to compensate for system perturbations in minimum phase systems. The use of known Generalized Relative Degree (GRD) in nonminimum phase systems allows for the elimination of internal dynamics. However, instability that emerges in the corresponding control dynamic extension is defeating any output tracking controller design. A novel methodology of using GRD for designing continuous SMC in nonminimum phase systems is presented. An algorithm for generating a bounded solution of the unstable dynamic extension is proposed and used in concert with SMC, allowing robust control design for nonminimum phase systems. The efficacy of the proposed GRD-based approaches is demonstrated on a minimum and nonminimum phase rocket attitude control problem both analytically and via simulation.     

Parameter plane control design for a stirred-tank chemical system with slow-fast multirate sampling

The control of a multirate sampled-data, stirred-tank chemical reactor system using a parameter plane method is considered. Due to wide acceptance of proportional-plus- integral-plus-derivative (PID) control in the chemical process industries, a PID controller with a “slow-fast” multirate scheme is used for the chemical reactor system. Based on two related stability equations and using the PID gains as the adjustable parameters, the set of all possible PID gains to maintain the chemical reactor system's stability, and at the same time, to ensure the system has a specified gain margin, phase margin, damping ratio and damping factor is determined. The effects of changing the integer N (which is the ratio of the sampling rates between a slow-and a fast-sampler) and the basic sampling period T on the set of PID gains are examined and the results for single-rate and multirate cases are also studied.     

Diversity incorporated energy adaptation for A-CDMA systems over slow fading channels

In this paper, a theoretical analysis of diversity incorporated Variable Energy Adaptation (VEA) in an Asynchronous Code Division Multiple Access (A-CDMA) system is discussed for Rayleigh, Rician and Nakagami slow fading channels. The adaptation is accomplished by providing the receiver with the capability of measuring the signal energy-to-noise ratio, and controlling the transmitted signal energy by means of a noise-free feedback loop. System parameters such as fading margin, maximum signal-to-noise ratio, and mean transmitter energy gain are derived and plotted for fading channels as a function of the probability of error specification and the probability of unsatisfactory operation. The mean and median probabilities of error are plotted as a function of energy-to-noise ratios for different fading channels. Error probability distribution and density functions are derived and plotted for various signal-fading distributions.     

A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function

In this study, a robust fractional-order controller design methodology for a type of fractional-order or integer-order model with dead time is proposed using phase and gain margin specifications. The delayed Bode’s ideal transfer function is used as a reference model to design the controller analytically. The delay term in delayed Bode’s ideal transfer function provides the exact determination of these frequency domain specifications when the system owns a dead time. The analytical robust controller design problem is transformed to solving four nonlinear equations with four unknown variables, two of which are the desired specifications; namely, phase and gain margins. The remaining two are the phase and gain cross-over frequencies. Next, some conditions are set based on the desired specifications so that nonlinear equations provide a unique solution. The proposed method is compared with the other existing robust controller methods based on the same frequency domain specifications. The simulation results reveal that the proposed method outperforms the other methods and also gives closer outcomes to the desired specifications.     

Recursive reconstruction of sequences from group-delay

A recursive procedure to reconstruct a given sequence from its group delay or phase derivative is given. The procedure is based on the relationships between minimum, maximum phase sequences and their cepstra, and on the modified least squares (MLS) rational approximation. To avoid unwrapping of the phase, the cepstrum of the sequence is calculated from the group delay function. Using a recursive procedure, we find from the cepstrum values a minimum phase sequence with a phase equal to that of the original sequence. The reconstructed sequence is obtained using the MLS procedure to find recursively a rational approximation of the minimum phase sequence. The constraints under which the phase reconstruction is possible are checked with a root distribution algorithm, and we indicate how to modify the sequence when the constraints are not satisfied. Examples illustrate the efficiency of the proposed procedure.     

Cochannel interference effect on BEP performance of SSC receiver in correlated Rician fading

We study the performance of dual branch switch and stay combining (SSC) diversity receivers operating over correlated Rician fading channels, in the presence of correlated Rayleigh distributed cochannel interference (CCI). Fast convergent infinite series representations for both the joint probability density function (PDF) of the SSC input signal-to-interference ratios (SIRs) and the PDF of the SSC output SIR are derived. The last one is applied to semi-analytically study the average bit error probability (ABEP) of differential binary phase shift keying (DBPSK). Furthermore, the optimum switching threshold in the sense of minimum ABEP is obtained numerically and the effects of fading severity and branch correlation on both ABEP and optimum switching threshold are investigated. Numerical and simulation results are presented to support the mathematical analysis.     

融资融券与高管薪酬契约有效性研究

融资融券制度作为近年来中国资本市场重大改革之一,其对实体经济的影响尚未得到重视。本文使用双重差分模型研究了融资融券对高管激励的影响。实证结果发现,融资融券显著提高了上市公司的高管薪酬-会计业绩及高管薪酬-市场业绩敏感度,说明融资融券提升了企业会计信息及市场价格信息质量,从而有助于提高薪酬契约的有效性,具有显著外部治理效应;进一步的机理检验发现,融资融券效用发挥在治理机制较弱的制度环境中更为显著,表明其作为一种替代性的外部治理机制而发挥作用。     

Observer-based finite-time asynchronous sliding mode control for Markov jump systems with time-varying delay

The issue of finite-time sliding mode control (SMC) is studied for a class of Markov jump systems, in which parameter uncertainties, external disturbances and time-varying delay are considered. Firstly, a suitable observer-based SMC law is devised so that state trajectory of the system can reach the designed sliding mode surface in finite-time, the gain of the controller is asynchronous to the mode of original system. Meanwhile, the sufficient conditions of finite-time boundedness in the sliding phase and reaching phase are derived by the time partition strategy. Moreover, the gains of the observer and the observer-based controller will be acquired by using the linear matrix inequalities tool. In fine, emulation products are used to confirm the merits of the SMC strategy.     

System functions for 2D linear phase frequency sampling filters

In this paper, four frequency sampling filter system functions which are classified as Type 1-1, Type 1-2, Type 2-1 and Type 2-2, are developed. Each type of these frequency sampling filter interpolates a frequency response through a specific set of frequency samples and also uses these frequency samples as coefficients in each of their implementations. Each of these system functions are further developed for 2D linear phase filters that have real impulse responses and for 2D linear phase filters that have real impulse responses and fourfold symmetry. The approximate conditions for which these frequency sampling filters can implement narrowband 2D linear phase filters and narrowband 2D linear phase filters with fourfold symmetry more efficiently than direct convolution filters are also derived.     

Reliability Measures and Optimal Maintenance Policies for a Standby System with Repair Facilities Subject to Breakdown

The problem of determining optimal maintenance policies for a repairable standby system whose repair facilities are subject to random breakdowns is considered. The reliability measures of this system are derived by the discrete-state continuous-time Markov model, as well as by a more general probabilistic approach. The dynamic optimization problem is discussed and treated by a generalized version of Pontryagin's minimum principle based on an integral Hamiltonian functional. This is applied to the Markov model of the system for deriving both time-variable and fixed maintenance policies over the whole mission time. Then an appropriate cost function, involving a maintenance cost term and a down-time cost term, is minimized with respect to the vector repair rate function u(t) subject to the practical constraint 0 ⩽ u(t) ⩽ U, where U is a given upper repair rate limit. A particular nontrivial example is given.     

Large angle maneuver and high accuracy attitude pointing steering law for variable speed control momentum gyroscopes

In the variable speed control momentum gyroscopes (VSCMG), the output torque of the VSCMG can be supported by the gyroscopic torque by a large margin with low resolution. The output torque of the VSCMG can be only supported by the reaction torque of the flywheels with high resolution when the gimbals are locking. Consequently, the torque error that is determined by the low spin rate fluctuation or servo tracking error of the gimbals can be vanished, and the VSCMG can be suitable for large maneuver and high accuracy attitude pointing as the actuator. Unfortunately, the singularity of the flywheel torque co-plane may be encountered. In this paper, the singularity of the flywheel torque co-plane is visualized based on geometry method. The singularity can be effectively avoided by locking gimbals. The locking gimbals positions are optimized with the maximum angular momentum. The steering laws with two phases are proposed to operate the task of the large angle maneuver and high accuracy control for a spacecraft. In the phase 1, a large margin torque can be achieved by the new designed steering law, and the gimbals can be steered to the optimization gimbals positions at the final stage of the phase 1. In the phase 2, the high precision torque can be achieved by only steering the flywheels and the locking the gimbals. Consequently, the torque error that is caused by the spin rate fluctuation or servo tracking error of the gimbals can be effectively eliminated.     

Security control for networked control systems with randomly occurring integrity check protection subject to randomly occurring zero-value attacks

This paper investigates the problem of secure control for networked control systems (NCSs) under randomly occurring zero-value attacks (ROZVAs). Specifically, ROZVAs only offset the true signal without injecting obfuscated information or noises, and possess the minimum energy of the added malicious information. To protect system stability against ROZVA, randomly occurring integrity check protection (ROICP) is introduced which prevents malicious data injection with less energy cost than persistently occurring protection. Besides the random phenomena of ROZVA and ROICP, which are characterized by two mutually independent random variables obeying the Bernoulli distribution, the randomly occurring time delays caused by ROICP are also considered in system modelling. According to the built stochastic linear system model, security analysis of the NCS with ROICP subject to ROZVA is carried out and sufficient condition for stochastic stability is derived via a linear matrix inequality (LMI) approach. Based on the proposed condition, a compensation feedback controller is designed to facilitate system stability. Finally, simulation results show the effectiveness of the proposed method.     

Estimation fusion for networked systems with multiple asynchronous sensors and stochastic packet dropouts

This paper studies the asynchronous state fusion estimation problem for multi-sensor networked systems subject to stochastic data packet dropouts. A set of Bernoulli sequences are adopted to describe the random packet losses with different arriving probabilities for different sensor communication channels. The asynchronous sensors considered in this paper can have arbitrary sampling rates and arbitrary initial sampling instants, and may even sample the system non-uniformly. Asynchronous measurements collected within the fusion interval are transformed to the fusion time instant as a combined equivalent measurement. An optimal asynchronous estimation fusion algorithm is then derived based on the transformed equivalent measurement using the recursive form of linear minimum mean squared error (LMMSE) estimator. Cross-correlations between involved random variables are carefully calculated with the stochastic data packet dropouts taken into account. A numerical target tracking example is provided to illustrate the feasibility and effectiveness of the proposed algorithm.     

Incremental H∞ performance for a class of stochastic switched nonlinear systems

In this paper, we investigate the incremental H_∞ performance problem for a class of stochastic switched nonlinear systems by using a state-dependent switching law and the maximum and minimum dwell time approach. By resorting to the state-dependent switching law, some sufficient conditions are provided to cope with the incremental H_∞ performance problem, which can be applied even if all subsystems are unstable. Then, based on the maximum and minimum dwell time scheme, the incremental H_∞ performance problem to be solvable is derived for two cases: one is all subsystems are incrementally globally asymptotically stable in the mean(IGASiM), another is both IGASiM subsystems and unstable subsystems coexist. When all subsystems are IGASiM, the stochastic switched nonlinear system is IGASiM and possesses a incremental L₂-gain under given conditions. When both IGASiM subsystems and unstable subsystems coexist, if the activation time ratio between IGASiM subsystems and unstable ones is not less than a specified constant, the sufficient conditions for the incremental H_∞ performance of the stochastic switched nonlinear system are given. Two numerical examples are given to illustrate the validity of methods proposed.     

非线性控制系统的一般形式及其稳定性

给出了SISO非线控制系统的点相对阶的概念,以此得出一般Brrnes Isidori标准形式。并对于具有最小相位非线性系统,给出了渐近稳定的充分条件;对于具有非最小相位系统,通过设计控制器来构建中心流形的方法,给出了其渐近稳定性的证明     

A guide to the literature on the history of engineering available in the cooper union library: Cooper Union Bulletin. Engineering and science series,Number 28, 46 pages, 18 × 2S CMS., paper. New York,The Cooper Union for the Advancement of Science and Art, 1946.

The linear logarithmic relations between deformation and temperature and pressure, previously empirical, have been directly derived from the defining equations for compressibility and thermal expansivity and shown to apply to all three stages of every deformation.Gibbs' thermodynamic potential function is shown to lead directly to simple and exact expressions for the energy of change of phase and of deformation within any one phase. The latter, a general equation of state, amounts to a simple relation between the four internal pressures.The derived deformation and energy functions are applied to some experimental data on steel tape which included thermal and relaxation observations.Thermodynamic relations governing both elastic and viscous behavior are developed. Plastic behavior involves both single phase and multiple phase applications of the second law. Precise thermal measurements should give the constants.     

基于多元GARCH—VaR的期货组合保证金模型及其应用研究

提出了确定保证金的非线性风险对冲原理、整体风险覆盖原理和动态预测原理,借助多元GARCH(1,1)模型所预测出的多种期货组合的协方差矩阵,计算该期货组合的波动值,并结合风险价值(Value-at-Risk,VaR)思想,建立基于多元GARCH-VaR的多种期货合约组合市场风险评价模型,并利用该模型计算大连商品交易所多种期货合约组合的保证金.本模型的特点一是通过保证金确定的非线性风险对冲原理,利用多元GARCH(1,1)模型在预测风险时对风险进行非线性对冲,解决了SPAN和TIMS系统对组合风险的直接线性相加减,导致预测值不精确问题.二是通过整体风险覆盖原理应用VaR模型计算整体市场风险,应用VaR模型计算任意期货合约组合的整体市场风险,满足了市场监管对整体风险覆盖的需要.三是通过动态预测原理,利用多元GARCH(1,1)模型对期货组合的风险进行预测,准确反映了期货价格时间序列具有波动聚集效应和时变方差效应,保证了预测的准确性.实证结果表明,本模型在保证较高风险防范能力的基础上可降低保证金的收取水平,为期货交易市场价格波动程度的衡量及浮动式保证金的确定方法提供了新的思路.     

非Kolmogorov湍流情况下低阶校正自适应光学系统的性能分析

系统地研究了非Kolmogorov湍流情况下低阶校正自适应光学系统的性能.从理论上分析了非Kolmogorov湍流情况下相位扰动的空间特性和时间特性,推导了相位扰动的空间结构函数、时间结构函数以及时间功率谱.基于哈特曼-夏克波前传感器,提出了一些测量大气参数的新方法.引入模式时间校正因子和模式非等晕因子,分析了低阶校正自适应光学系统的波前残余误差,并首次对扩展目标情况时低阶校正自适应光学系统的非等晕误差进行了研究.此外还根据实际采集的太阳表面米粒结构图像,采用事后处理的方法研究了倾斜校正自适应光学系统的性能.