Control of discrete‐time periodic linear systems with input saturation via multi‐step periodic invariant sets

This paper studies local control of discrete‐time periodic linear systems subject to input saturation by using the multi‐step periodic invariant set approach. A multi‐step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. The problems of (robust) estimation of the domain of attraction, (robust) local stabilization (with bounded uncertainties), and disturbance rejection are considered. Compared with the conventional periodic invariant set approach, which has been used in the literature for local stability analysis and stabilization of discrete‐time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing additional auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller, in the presence of norm bounded uncertainties, whose period is the same as the open‐loop system and is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open‐loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Theoretical and numerical comparisons of looped functionals and clock‐dependent Lyapunov functions—The case of periodic and pseudo‐periodic systems with impulses

The stability of uncertain periodic and pseudo‐periodic systems with impulses is analyzed in the looped‐functional and clock‐dependent Lyapunov function frameworks. These alternative and equivalent ways for characterizing discrete‐time stability have the benefit of leading to stability conditions that are convex in the system matrices, hence suitable for robust stability analysis. These approaches, therefore, circumvent the problem of computing the monodromy matrix associated with the system, which is known to be a major difficulty when the system is uncertain. Convex stabilization conditions using a non‐restrictive class of state‐feedback controllers are also provided. The obtained results readily extend to uncertain impulsive periodic and pseudo‐periodic systems, a generalization of periodic systems that admit changes in the ‘period’ from one pseudo‐period to another. The obtained conditions are expressed as infinite‐dimensional semidefinite programs, which can be solved using recent polynomial programming techniques. Several examples illustrate the approach, and comparative discussions between the different approaches are provided. A major result obtained in the paper is that despite being equivalent, the approach based on looped functional reduces to the one based on clock‐dependent Lyapunov functions when a particular structure for the looped functional is considered. The conclusion is that the approach based on clock‐dependent Lyapunov functions is preferable because of its lower computational complexity and its convenient structure enabling control design. Copyright © 2015 John Wiley & Sons, Ltd.     

针对生物振荡网络分析的周期系统目标敏感性方法研究

生物学实验和模型计算结果表明振荡是一种常见的生物学现象, 越来越多的研究人员关注生物系统内部的振荡现象是如何产生的、引起振荡的关键因素是什么等问题. 敏感性分析定量分析系统行为在模型参数、系统输入或者初始条件发生变化时受影响的程度. 对周期系统进行传统的状态敏感性计算时, 得到的灵敏度指标随着时间的增加而发散, 因而对其进行敏感性分析是一项具有挑战性的工作. 本文针对这类系统, 首先提出基本状态敏感性的概念, 由此进一步推导出一种相敏感性分析方法. 在计算周期灵敏度过程中, 提出了一种基于奇异值分解的的改进算法, 简化了基本状态灵敏度的计算. 本文中的目标敏感性分析方法克服了因累积效应引起的发散问题. 通过对一个生物节律模型和一个复杂的信号转导网络系统模型进行敏感性分析, 可以看到改进的周期灵敏度计算方法得到的结果与已有方法一致, 并且新提出的目标敏感性分析方法及其计算在处理存在反应守恒的复杂生物振荡系统分析时是有效的.     

Mathematical modeling of endocrine regulation subject to circadian rhythm

The 2017 Nobel Prize in Physiology or Medicine awarded for discoveries of molecular mechanisms controlling the circadian rhythm has called attention to the challenging area of nonlinear dynamics that deals with synchronization and entrainment of oscillations. Biological circadian clocks keep time in living organisms, orchestrating hormonal cycles and other periodic rhythms. The periodic oscillations of circadian pacemakers are self-sustained; at the same time, they are entrainable by external periodic signals that adjust characteristics of autonomous oscillations. Whereas modeling of biological oscillators is a well-established research topic, mathematical analysis of entrainment, i.e. the nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. Along with sustained periodic rhythms, periodically forced oscillators can exhibit various “irregular” behaviors, such as quasiperiodic or chaotic trajectories.This paper presents an overview of the mathematical models of circadian rhythm with respect to endocrine regulation, as well as biological background. Dynamics of the human endocrine system, comprising numerous glands and hormones operating under neural control, are highly complex. Therefore, only endocrine subsystems (or axes) supporting certain biological functions are usually studied. Low-order dynamical models that capture the essential characteristics and interactions between a few hormones can than be derived. Goodwin’s oscillator often serves as such a model and is widely regarded as a prototypical biological oscillator. A comparative analysis of forced dynamics arising in two versions of Goodwin’s oscillator is provided in the present paper: the classical continuous oscillator and a more recent impulsive one, capturing e.g. pulsatile secretion of hormones due to neural regulation. The main finding of this study is that, while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior due to non-smooth dynamics in entrainment.     

Robust compensation of time delayed plant by continuous‐time high frequency periodic controller with negligible output ripples

The present work addresses the problem of ensuring robust stability to time delayed plants, compensated with continuous‐time high frequency periodic controller. An efficient design methodology is proposed to synthesize the periodic controller for robust compensation of time delayed linear time‐invariant plants. The periodic controller, by virtue of its loop zero‐placement capability, is shown to achieve superior gain as well as phase/delay margin compensation, especially for non‐minimum phase plants having right half plane poles and zeros in close vicinity to each other. The periodic controller is considered in the observable canonical form which results in realizable bounded control input as well as ensuring insignificant periodic oscillations in the plant output. As a consequence, this paper, furthermore, establishes the fact that the periodic controller designed and synthesized with the proposed methodology can be implemented in real‐time with an assurance of model matching and robust zero‐error tracking. Simulation and experimental results are illustrated to establish the veracity of the claims. The closed‐loop system comprising of time‐delayed linear time‐invariant plant with the periodic controller is analyzed employing the averaging principle and presented here explicitly in a meticulous approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust economic model predictive control based on a periodicity constraint

This paper proposes robust economic model predictive control based on a periodicity constraint for linear systems subject to unknown‐but‐bounded additive disturbances. In this economic MPC design, a periodic steady‐state trajectory is not required and thus assumed unknown, which precludes the use of enforcing terminal state constraints as in other standard economic formulations. Instead, based on the desired periodicity of system operation, we optimize the economic performance over a set of periodic trajectories that include the current state. To achieve robust constraint satisfaction, we use a tube‐based technique in the economic MPC formulation. The mismatches between the nominal model and the closed‐loop system with perturbations are limited using a local control law. With the proposed robust tube‐based strategy, recursive feasibility is guaranteed. Moreover, under a convexity assumption, the closed‐loop convergence of the closed‐loop system is analyzed, and an optimality certificate is provided to check if the closed‐loop trajectory reaches a neighborhood of the optimal nominal periodic steady trajectory using Karush‐Kuhn‐Tucker optimality conditions. Finally, through numerical examples, we show the effectiveness of the proposed approach.     

IQC‐based robustness analysis of discrete‐time linear time‐varying systems

The paper investigates the robust stability and performance of uncertain linear time‐varying (LTV) systems using an integral quadratic constraint (IQC) based analysis approach. Specifically, previous theoretical work on IQC‐based robustness analysis of linear time‐invariant (LTI) systems is extended to discrete‐time LTV systems. In the case of a general LTV nominal system, the analysis solution is provided in terms of an infinite‐dimensional convex optimization problem. This optimization problem reduces into a finite‐dimensional semidefinite program when the nominal system in question is finite horizon, periodic, or, more generally, eventually periodic. Finally, the results are applied to an unmanned aircraft control system executing an aggressive maneuver, where the developed techniques are used to find the region in which the aircraft is guaranteed to reside at the end of its planned trajectory. Copyright © 2017 John Wiley & Sons, Ltd.     

果蝇昼夜节律调控网络的一个修正模型

PER和TIM是果蝇两个重要的生物钟蛋白.以往的研究一直认为PER和TIM是在细胞质中结合为二聚体并以二聚体的形式进入细胞核.但2006年Pablo Meyer等人的实验研究表明,PER/TIM复合物在细胞质中分离,然后PER和TIM在很短的时间内独立进入细胞核.根据该项实验结果,我们对果蝇昼夜节律调控模型进行了修正,修正模型反映了per和tim基因的转录翻译及蛋白质的翻译后修饰过程,二次磷酸化的蛋白质PER(P2)、TIM(T2)分别独立进入细胞核并参与后续的调控过程.计算了修正模型的振荡周期并由此确定了新模型所引入的参数值.对修正模型的振荡节律进行数值分析,发现修正模型振荡节律在DD、LD条件下均产生了近于24h的持续周期振荡而在LL条件下呈现出振荡衰减,这些结果与原模型相似,反映出所建模型的合理性.但修正模型对参数对称性的依赖性则更加强烈,具体解释还有待于进一步的工作.     

Periodic event‐triggered control for distributed networked multiagents with asynchronous communication: A predictive control approach

This paper investigates the periodic event‐triggered control problem for distributed networked multiagent systems with interconnected nonlinear dynamics subject to asynchronous communication. A method of state trajectory estimation for the interconnected neighboring agents over each prediction horizon with guaranteed error bounds is addressed to handle the asynchronous communication. Based on it, a distributed robust model predictive control (MPC) is proposed with a distributed periodic event‐triggered scheme for each agent. According to this algorithm, each subsystem generates presumed state trajectories for all its upstream neighbors and computes its own control locally. By checking the designed triggering condition periodically, the optimization problem of MPC will be implemented and solved when the local error of the subsystem exceeds a specified threshold. Then, the optimized control input will be determined and applied until the next time instant when the triggering condition is invoked. Moreover, sufficient condition for ensuring feasibility of the designed algorithm is conducted, along with the analysis of asymptotic stabilization of the closed‐loop system. The illustrative example for a set of coupled Van der Pol oscillators is reported to verify the effectiveness of the proposed approach.     

Robust Actuator‐Fault‐Tolerant Control System Based on Sliding‐Mode Observer for Thrust‐Vectoring Aircrafts

In this paper, a robust actuator‐fault‐tolerant control (FTC) system is proposed for thrust‐vectoring aircraft (TVA) control. To this end, a TVA model with actuator fault dynamics, disturbances, and uncertain aerodynamic parameters is described, and a local fault detection and identification (FDI) mechanism is proposed to locate and identify faults, which utilizes an adaptive sliding‐mode observer (SMO) to detect actuator faults and two SMOs to identify and estimate their parameters. Finally, a fault‐tolerant controller is designed to compensate for these actuator faults, disturbances, and uncertain aerodynamic parameters; the approach combines back‐stepping control with fault parameters and a high‐order SMO. Furthermore, the stability of the entire control system is validated, and simulation results are given to demonstrate the effectiveness and potential for this robust FTC system.     

Adaptive Lipschitz Observer Design for a Mammalian Model

This paper deals with observer designs for a proposed mathematical model of circadian rhythms which exist in almost every living organism. A 7th order model for mammalian circadian rhythms which captures the main dynamic features is considered in this paper. A recent result of one‐sided Lipschitz observer design in the literature is applied to this mammalian model to show a possibility of reducing measurements for circadian models in system biology. The mammalian model presented may contain an uncertainty parameter. An adaptive design of the Lipschitz observer is then applied to deal with this case. Besides detailed designs of both observers, detailed analysis is also performed for nonlinear functions in the mammalian model to show that the Lipschitz observers can indeed be applied. Several simulation studies of the proposed observers are carried out with the results shown in this paper.     

Uncertainty modeling and robust minimax LQR control of multivariable nonlinear systems with application to hypersonic flight

For a class of multi‐input and multi‐output nonlinear uncertainty systems, a novel approach to design a nonlinear controller using minimax linear quadratic regulator (LQR) control is proposed. The proposed method combines a feedback linearization method with the robust minimax LQR approach in the presence of time‐varying uncertain parameters. The uncertainties, which are assumed to satisfy a certain integral quadratic constraint condition, do not necessarily satisfy a generalized matching condition. The procedure consists of feedback linearization of the nominal model and linearization of the remaining nonlinear uncertain terms with respect to each individual uncertainty at a local operating point. This two‐stage linearization process, followed by a robust minimax LQR control design, provides a robustly stable closed loop system. To demonstrate the effectiveness of the proposed approach, an application study is provided for a flight control problem of an air‐breathing hypersonic flight vehicle (AHFV), where the outputs to be controlled are the longitudinal velocity and altitude, and the control variables are the throttle setting and elevator deflection. The proposed method is used to derive a linearized uncertainty model for the longitudinal motion dynamics of the AHFV first, and then a robust minimax LQR controller is designed, which is based on this uncertainty model. The controller is synthesized considering seven uncertain aerodynamic and inertial parameters. The stability and performance of the synthesized controller is evaluated numerically via single scenario simulations for particular cruise conditions as well as a Monte‐Carlo type simulation based on numerous cases. It is observed that the control scheme proposed in this paper performs better, especially from the aspect of robustness to large ranges of uncertainties, than some controller design schemes previously published in the literature. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust fusion steady‐state filtering for multisensor networked systems with one‐step random delay,missing measurements,and uncertain‐variance multiplicative and additive white noises

The robust fusion steady‐state filtering problem is investigated for a class of multisensor networked systems with mixed uncertainties including multiplicative noises, one‐step random delay, missing measurements, and uncertain noise variances, the phenomena of one‐step random delay and missing measurements occur in a random way, and are described by two Bernoulli distributed random variables with known conditional probabilities. Using a model transformation approach, which consists of augmented approach, derandomization approach, and fictitious noise approach, the original multisensor system under study is converted into a multimodel multisensor system with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst‐case subsystems with conservative upper bounds of uncertain noise variances, the robust local steady‐state Kalman estimators (predictor, filter, and smoother) are presented in a unified framework. Applying the optimal fusion algorithm weighted by matrices, the robust distributed weighted state fusion steady‐state Kalman estimators are derived for the considered system. In addition, by using the proposed model transformation approach, the centralized fusion system is obtained, furthermore the robust centralized fusion steady‐state Kalman estimators are proposed. The robustness of the proposed estimators is proved by using a combination method consisting of augmented noise approach, decomposition approach of nonnegative definite matrix, matrix representation approach of quadratic form, and Lyapunov equation approach, such that for all admissible uncertainties, the actual steady‐state estimation error variances of the estimators are guaranteed to have the corresponding minimal upper bounds. The accuracy relations among the robust local and fused steady‐state Kalman estimators are proved. An example with application to autoregressive signal processing is proposed, which shows that the robust local and fusion signal estimation problems can be solved by the state estimation problems. Simulation example verifies the effectiveness and correctness of the proposed results.     

Robust Isolation Of Sensor Failures

Sensor self‐validity check is a critical step in system control and fault diagnostics. In this paper, a robust approach to isolate sensor failures is proposed. First, a residual model for a given system is built off‐line and directly based on input‐output measurement data. The residual model outputs are called “primary residuals” and are zero when there is no fault. Most conventional approaches to residual model generation are indirect, as they first require the determination of state‐space or other models using standard system identification algorithms. Second, a new max‐min design of structured residuals, which can maximize the sensitivity of structured residuals with respect to sensor failures, is proposed. Based on the structured residuals, one can then isolate the sensor failures. This design can also be done in an off‐line manner. It is an optimization procedure that avoids local optimal solutions. Simulation and experimental results demonstrated the effectiveness of the proposed method.     

Using a circadian rhythms model to evaluate shift schedules

As long as the need for shiftwork exists, so will the demand for an optimal scheduling strategy that balances the needs of both industry and the shiftworker. One aim is for schedules that require workers to be on duty during the times that they are most naturally alert and awake. Czeisler et al. (1982) developed a set of circadian rhythm based guidelines intended to aid in designing such shiftwork schedules. This paper takes research one step further by testing such empirical criteria in a mathematical setting. The two-oscillator model of free-run human circadian rhythms developed by Kronauer et al. (1982) was modified to represent the circadian rhythms of a shiftworker on a pre-selected shiftwork schedule. Numerical simulations were used to compare the circadian rhythms produced from a variety of shiftwork schedules to the free-run rhythms. Shift schedules that resulted in circadian rhythms closest to the free-run rhythms were identified as preferred schedules. The numerical results supported Czeisler's findings (1982), indicating the best shift schedules adopt a slow, forward-shifting rotation pattern, rotate shifts after 2-week periods and allow an average of 2 days off per week.     

Using a circadian rhythms model to evaluate shift schedules

As long as the need for shiftwork exists, so will the demand for an optimal scheduling strategy that balances the needs of both industry and the shiftworker. One aim is for schedules that require workers to be on duty during the times that they are most naturally alert and awake. Czeisler et al. (1982) developed a set of circadian rhythm based guidelines intended to aid in designing such shiftwork schedules. This paper takes research one step further by testing such empirical criteria in a mathematical setting. The two-oscillator model of free-run human circadian rhythms developed by Kronauer et al. (1982) was modified to represent the circadian rhythms of a shiftworker on a pre-selected shiftwork schedule. Numerical simulations were used to compare the circadian rhythms produced from a variety of shiftwork schedules to the free-run rhythms. Shift schedules that resulted in circadian rhythms closest to the free-run rhythms were identified as preferred schedules. The numerical results supported Czeisler's findings (1982), indicating the best shift schedules adopt a slow, forward-shifting rotation pattern, rotate shifts after 2-week periods and allow an average of 2 days o^ff per week.     

Semi‐global stabilization via linear sampled‐data output feedback for a class of uncertain nonlinear systems

This paper develops a systematic design scheme to construct a linear sampled‐data output feedback controller that semi‐globally asymptotically stabilizes a class of uncertain systems with both higher‐order and linear growth nonlinearities. To deal with the uncertain coefficients in the systems, a robust state feedback stabilizer and a reduced‐order sampled‐data observer, both in the linear form, are constructed and then integrated together. The semi‐global attractivity and local stability are delicately proved by carefully selecting a scaling gain using the output feedback domination approach and a sampling period sufficiently small to restrain the state growth under a zero‐order‐holder input. Copyright © 2014 John Wiley & Sons, Ltd.     

The Nyquist robust sensitivity margin for uncertain closed‐loop systems

The Nyquist robust sensitivity margin is proposed as a new scalar indicator of robust stability that also provides a meaningful quantitative assessment of the worst sensitivity realized by the uncertain closed loop. After formulating and discussing in detail the underlying optimization problem required for the calculation of the margin, the approach is applied to the characterization of the robust stability of a closed‐loop featuring a linear system with an affine uncertainty structure and a parametric uncertainty set described by a real rectangular polytope. The capabilities of the methodology are illustrated through examples, which include an approach for quantifying alternative robustness margins, such as a parametric stability margin. The computational algorithm is systematic and can be carried out with high numerical precision. Copyright © 2005 John Wiley & Sons, Ltd.     

Features of Circadian Rhythms Relevant for the Design of Shift Schedules

The first part of this paper reviews properties of circadian rhythms when free-running in constant conditions, or in situations where zeitgebers are insufficient for entrainment. The second part deals with problems of entrainment by natural and artificial zeitgebers, with phase controlling effects of sleep and the timing of meals, and with the need to differentiate between entraining and masking effects. The third part discusses phase shifts of circadian rhythms, especially the dependence of the rate of re-entrainment on the direction of the shift, and the splitting of the circadian system into parts which are shifted in opposite directions (re-entrainment by partition). The relevance of these features of circadian rhythms for the design of shift-work schedules is discussed.     

Robust stability analysis of networked systems with varying delays

This article is concerned with networked controlled systems (NCS) with uncertain, varying, bounded transmission delays and asynchronous discrete-time static control laws. It is first shown that the delay variation gives rise to a discrete-time uncertain NCS model; robust stability analysis is carried out via a linear matrix inequality approach which, when combined with a directed parameter search, yields an estimate of robust stability bounds against any variations of the maximum allowable delay (constrained within one sampling period) that the closed-loop system can tolerate. The derived bounds are compared with other techniques relying on the singular values of the perturbed NCS-model. The presented simulation results prove the efficacy of the proposed control scheme.