A numerical sufficiency test for the asymptotic stability of linear time-varying systems

A numerical sufficiency test for the asymptotic stability of linear time-varying Hurwitz systems is proposed. The algorithmic procedure constructs a bounding tube in which the state is guaranteed to stay. The continuous-time system is evaluated at discrete time instants, for which successive quadratic Lyapunov functions are generated. The tube is constructed based on: (i) a conservative estimate of the state evolution, from a discrete time instant to the next, obtained from the corresponding Lyapunov function, and (ii) re-evaluation of the tube diameter at each discrete time instant to account for variations in the plant matrix. The numerical test is illustrated in simulation via both a stable and an unstable system.     

Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs

In this paper, we propose an adaptive observer for a class of uniformly observable nonlinear systems with nonlinear parametrization and sampled outputs. A high gain adaptive observer is first designed under the assumption that the output is continuously measured and its exponential convergence is investigated, thanks to a well defined persistent excitation condition. Then, we address the case where the output is available only at (non uniformly spaced) sampling instants. To this end, the continuous-time output observer is redesigned leading to an impulsive observer with a corrective term involving instantaneous state impulses corresponding to the measured samples and their estimates. Moreover, it is shown that the proposed impulsive observer can be put under the form of a hybrid system composed of a continuous-time observer coupled with an inter-sample output predictor. Two design features are worth to be emphasized. Firstly, the observer calibration is achieved through the tuning of a scalar design parameter. Secondly, the exponential convergence to zero of the observation and parameter estimation errors is established under a well defined condition on the maximum value of the sampling partition diameter. More specifically, the observer design is firstly carried out in the case of linear parametrization before being extended to the nonlinear one. The theoretical results are corroborated through simulation results involving a typical bioreactor.     

A state observer for continuous oscillating systems under intrinsic pulse-modulated feedback

A static gain observer for linear continuous plants with intrinsic pulse-modulated feedback is analyzed. The purpose of the observer is to asymptotically drive the state estimation error to zero and synchronize the sequence of pulse modulation instants estimated by the observer with that of the plant. Conditions on the observer gain matrix locally stabilizing the observer error along an arbitrary periodic plant solution are derived and illustrated by simulation for the case of pulsatile testosterone regulation.     

An uncertainty-based approach to discrete-time fault estimation observer design for nonuniformly sampled systems

This paper considers the fault estimation problem of nonuniformly sampled system in which sensor sampling is performed at aperiodic interval. After being discretized at sampling instant, the nonuniformly sampled system is modeled as an equivalent polytopic system with norm bounded uncertainties. A discrete-time time-varying fault estimation observer with multiple design freedom is then constructed, and a sufficient condition given in linear matrix inequality (LMI) is provided to obtain the constant filter gain and ensure not only the asymptotic stability of fault estimation error but also the robustness of uncertainties. Compared with the existing observer designed based on continuous-time delay approach, the proposed one has a better estimation accuracy and less conservatism and is easy for digital implementation. A numerical simulation and a quadruple-tank benchmark are used to demonstrate the effectiveness and superiority of the proposed method.     

Predictor-based continuous-discrete nonlinear observer design subject to aperiodic sampled and delayed measurement

In this paper, a predictor-based continuous-discrete nonlinear observer is proposed for a class of nonaffine Lipschitz nonlinear systems with aperiodic sampled delayed output measurements and external disturbance. Firstly, this study introduces a class of delayed sampling hybrid nonlinear systems. By employing Lyapunov techniques and trajectory-based stability theory, sufficient conditions are established for ensuring input-to-state stability of the delayed sampling hybrid nonlinear system with respect to external disturbances. Then the predictor-based continuous-discrete nonlinear observer is designed which consists of a continuous-time observer, a compensating injector and a predictor. The continuous-time observer is utilized to obtain continuous and delay-free state estimation. The compensating injector is designed to compensate for output errors that occur between sampling instants. Furthermore, the predictor is employed to obtain delay-free output error information, which is then utilized by the continuous-time observer for feedback correction. The proposed observer's exponential input-to-state property against the disturbance is proved via the proposed hybrid system stability theory. The effectiveness of the proposed observers has been demonstrated through numerical simulations and performance comparisons with the zero-order holder-based observer and the output predictor-based observer designs in order to highlight the effectiveness and advantages of the proposed observer.     

State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers

This paper proposes a linear parameter varying (LPV) interval observer for state estimation and unknown inputs decoupling in uncertain continuous-time LPV systems. Two different problems are considered and solved: (1) the evaluation of the set of admissible values for the state at each instant of time; and (2) the unknown input observation, i.e. the design of the observer in such a way that some information about the nature of the unknown inputs affecting the system can be obtained. In both cases, analysis and design conditions, which rely on solving linear matrix inequalities (LMIs), are provided. The effectiveness and appeal of the proposed method is demonstrated using an illustrative application to a two-joint planar robotic manipulator.     

Comments on ‘Driving a linear constant system by a piecewise constant control’

Earlier results on controllability by means of piecewise constant controls are reviewed. Implications of the resulting freedom in the choice of the sampling instants are also indicated as well as consequences for the corresponding observability problem, i.e. state estimation by means of output samples.     

State estimation for linear hybrid systems with periodic jumps and unknown inputs

In order to solve the state estimation problem for linear hybrid systems with periodic jumps and unknown inputs, some hybrid observers are proposed. The proposed observers admit a Luenberger‐like structure and the synthesis is given in terms of linear matrix inequalities (LMIs). Therefore, the proposed observer designs are completely constructive and provide some input‐to‐state stability properties with respect to unknown inputs. It is worth mentioning that the structure of the hybrid observers, as well as the structure of the LMIs, depends on some observability properties of the flow and jump dynamics, respectively. Then, in order to compensate the effect of the unknown inputs, a hybrid sliding‐mode observer is added to the Luenberger‐like observer structure, providing exponential convergence to zero of the state estimation error despite certain class of unknown inputs. The existence of the hybrid observers and the unknown input hybrid observer is guaranteed if and only if the hybrid system is observable and strongly observable, respectively. Some numerical examples illustrate the feasibility of the proposed estimation approach.     

Time-varying increasing-gain observers for nonlinear systems

A full-order state observer for a class of nonlinear continuous-time systems is presented as generalization of the high-gain observer for having a time-varying gain that is let to be small in the first time instants, increases over time up to its maximum, and then is kept constant. The global stability of the resulting estimation error is proved by means of a Lyapunov functional via a change of coordinate. The design of such an observer is obtained by solving a nonlinear programming problem and using series expansions to set the time-varying gain.     

Event-triggered transmission for linear control over communication channels

We consider an exponentially stable closed loop interconnection between a continuous-time linear plant and a continuous-time linear controller, and we study the problem of interconnecting the plant output to the controller input through a digital channel. We propose an event-triggered transmission policy whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. Global asymptotic stability is guaranteed when the plant state is available or when an estimate of the state is available (provided by a classical continuous-time linear observer). Under further assumptions, the transmission policy guarantees global exponential stability of the origin.     

Observers for state-affine systems

A system whose discrete-time model is linear in state variables and nonlinear in control variables is considered. Under the assumption of generic observability (under single experiment) an exact observer algorithm is suggested. Also, it is shown that the existence of a general nonlinear asymptotic observer implies the existence of a linear one. Two design procedures for asymptotic observers are presented. Surprisingly, one of them extends to a rather complete solution for continuous-time systems with piecewise constant controls provided discrete output measurements are more frequent than switches of control values.     

Stability of a class of discrete-time nonlinear recursive observers

This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer.     

一类受扰线性采样数据系统的鲁棒执行器故障估计

针对一类含有外部干扰的线性采样数据系统, 本文研究了执行器故障估计问题. 首先, 文章设计了一种用于估计系统执行器故障的鲁棒故障估计观测器, 在连续采样时间间隔内, 观测器增益矩阵指数变化. 之后, 通过联合增广状态估计误差与故障误差, 并利用Lyapunov-Krasovskii泛函和线性矩阵不等式技术, 给出了保证误差系统全局渐近稳定的充分条件. 同时, 建立的线性矩阵不等式条件涉及调谐参数和最大采样间隔, 可以较好改善状态和故障估计性能. 最后, 通过对某型民航飞机模型实例进行仿真, 验证了本文所提方法具有更好的跟踪效果     

OBSERVER-BASED HYBRID CONTROL OF SAMPLED-DATA UNCERTAIN SYSTEM WITH INPUT TIME DELAY

This paper proposes a new digital redesign method for determining the hybrid controller of a continuous-time system with input time delay using an observer-based digital controller. The proposed method together with the genetic algorithms is used to determine: (1) the interval digital model of a continuous-time uncertain system with input time delay, (2) the interval digital redesign control law and (3) the interval digital observer of the original continuous-time uncertain observer with input time delay. Moreover, the result is less conservative than those obtained by the existing interval methods. A discrete-time observer is built by the original continuous-time observer with input-time-delay control law and a predictor such that the estimated states of the redesigned discrete-time observer closely match those of the original continuous-time observer with input time delay not only at sampling instants but also the behavior of the system state during the sampling interval is optimal by minimizing the hybrid performance index. The digitally redesigned observer-based controller can closely match the states of the digitally redesigned uncertain sampled-data system with those of the original continuous-time observer-based controlled uncertain system with input time delay.     

Set-valued observer design for a class of uncertain linear systems with persistent disturbance and measurement noise

In this paper, a class of linear systems affected by parameter variations, additive noise and persistent disturbances is considered. The problem of designing a set-valued state observer, which estimates a region containing the real state for each time instant, is investigated. The techniques for designing the observer are based on positive invariant set theory. By constructing a set-induced Lyapunov function, it is shown that the estimation error converges exponentially to a given compact set with an assigned rate of convergence.     

Using set invariance to design robust interval observers for discrete‐time linear systems

Based on interval and invariant set computation, an interval version of the Luenberger state observer for uncertain discrete‐time linear systems is proposed in this work. This new interval observer provides a punctual estimation of the state vector and guaranteed bounds on the estimation error. An off‐line and an on‐line approach to characterize, in a guaranteed way, the estimation error are introduced. Compared with the existing approaches, the proposed interval observer design method is not restrictive in terms of required assumptions, complexity, and on‐line computation time. Furthermore, the convergence issue of the estimation error is well established and to reduce the conservatism of the estimated state enclosure induced by the bounded additive state disturbance and noise measurement, an H_∞ method to compute the optimal observer gain is proposed. The performance of the proposed state estimation approach are highlighted on different illustrative examples.     

Synthesis of linear digital controllers using dead-beat observers for systems with inaccessible states†

A procedure is given to synthesize linear digital controllers for uniformly observable linear time-varying discrete data systems where some of the state variables are not available for feedback. Dead-beat observer theory is used and the controller generates the exact optimal control law in at most q?1 sampling periods, where q is the observability index of the system.     

Adaptive control of continuous-time overmodeled plants

This paper presents an adaptive controller that will stabilize and asymptotically regulate any continuous-time, linear time-invariant plant which has one input and one output. The only assumptions on the plant are observability, controllability, and the a priori knowledge of an upper bound on the plant order. The adaptive control concept combines continuous-time state observation and sampled-data control with a generalized, plant-dependent hold function. The feedback gains and the hold function are adaptively tuned     

Design of tracking observers for locally jointly observable systems

This paper considers distributed state estimation of continuous-time linear system monitored by a network of multiple sensors. Each sensor can only access locally partial measurement output of the system and effectively communicates with its neighbors to cooperatively achieve the asymptotic estimation of the target full system state. For a constructive design, we shall incorporate the concept of system immersions and propose a class of distributed tracking observers for the problem under a reasonable condition of the locally joint observability. Moreover, as a direct application of the proposed observer design, we further present an interesting leader-following consensus design for multi-agent system.