Design of minimal order stable observers for linear functions of the state via realization theory

Abstract--The design of a minimal order stable observer and a minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multivariable system is discussed. It is shown that both problems can be solved in a straightforward manner using partial realization theory, and several new results are given. These include a strong bound for the dimension of the minimal order stable observer and a simple necessary condition to design the minimal order observer with arbitrary poles that estimates a vector linear function of the state of a multiple-output system. Necessary and sufficient conditions are given for designing a minimal order observer with arbitrary poles for the case of estimating a vector linear function of the state of a single-output system and the case of estimating a scalar linear function of the state of a multiple-output system. A procedure to carry out the design in each of these cases is described. No restrictions whatsoever (except stability) are placed on the possible values of the observer poles. A significant observation of this paper is that the dynamics of the observer are constrained (in all cases) only by the gain matrix in the feedback law to be estimated and the output structure of the given system.     

基于二维混合模型和状态观测器的重复控制设计

针对一类正则线性系统, 提出一种基于状态观测器和二维混合模型的重复控制系统设计方法. 首先, 通过构造一个状态观测器来重构系统的状态, 建立基于重构状态的线性控制律. 然后, 通过独立地考虑重复控制系统的连续控制过程与离散学习行为, 给出基于状态观测器和重构状态反馈的连续/离散二维混合模型. 针对这个混合模型, 运用二维Lyapunov泛函方法, 以线性矩阵不等式(Linear matrix inequality, LMI)的形式给出重复控制系统存在重复控制器和状态观测器的充分条件, 所给条件可用Matlab工具箱方便地求解. 数值仿真验证了本文所提方法的有效性.     

Computer control of a double inverted pendulum

A double inverted pendulum is successfully stabilized at the upright position by using a computer control. The control system is designed based on the state space approach by using a computer aided design program named CADOS developed for this purpose. CADOS was used not only for the analysis and design, but also for the simulation to evaluate the designed system. The controller designed consists of the state variable feedback and the observer. The state variable feedback is determined based on either pol-locations or as the optimal control for the quadratic criterion function. As the observer, the minimal order state observer or a linear functional observer is employed, and it is theoretically proved that the linear functional observer for a multiple inverted pendulum can always be realized by the first order. The designed controller is implemented in the mini-computer used for CADOS and could work satisfactorily. The controller using a linear functional observer requires less computation time and controls the system in more stable way.     

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.     

LMI approach to robust unknown input observer design for continuous systems with noise and uncertainties

The full order robust unknown input observers for continuous systems are presented. The observers are designed for both linear and nonlinear systems considering both noise and uncertainties. First, an unknown input observer is designed for linear systems. The observer is derived based on linear matrix inequality (LMI) approach. Then the observer design problem is extended for a class of nonlinear systems whose nonlinear function satisfies the Lipschitz condition. The main advantage of these observers over the existing works on UIO design is that these can handle both noise and uncertainties simultaneously. The performance of the observers is demonstrated by applying it to the robust state estimation of single link robot arm.     

线性/非线性切换扩张状态观测器的收敛性和稳定性分析

线性/非线性切换扩张状态观测器(linear/nonlinear switching extended state observer,SESO)是为追求更高的控制精度和抗扰能力,在进行状态估计时根据状态误差大小在线性扩张状态观测器(linear extended state observer,LESO)和非线性扩张状态观测器(nonlinear extended state observer,NLESO)之间进行切换的一种扩张状态观测器.但由于该算法存在非线性结构,目前还没有从频域角度对该方法进行收敛性和控制系统稳定性分析的相关研究.针对此问题,以二阶系统为研究对象,首先对SESO的结构及参数选择进行了介绍,然后,分别采用等效增益法和描述函数法,从频域角度对了其状态估计误差的收敛性和基于SESO的自抗扰控制系统的稳定性进行了研究.结果表明,三阶SESO具有良好的收敛性,能实现对系统状态和"总扰动"的误差估计,并且选择适当的SESO参数,可避免极限环的产生.     

非线性切换系统基于观测器的容错控制器设计

针对一类具有执行器故障的非线性切换系统,研究基于系统状态估计和故障重构的容错控制问题.首先,在具有平均驻留时间(ADT)的切换信号下,设计一种切换PI观测器作为状态-故障估计器,以达到对系统状态和故障同时渐近估计的目的;其次,基于线性矩阵不等式,给出非线性切换系统观测器存在的充分条件;再次,基于PI观测器给出的状态和故障估计,提出一种非线性切换系统的容错控制器设计方法;最后,以一个电子电路为仿真实例进行分析,验证了所提出方法的可行性和适用性.     

含未知输入的广义系统的状态与输入估计

研究了含未知输入的广义系统的输入解耦观测器设计问题. 在系统脉冲能控的条件下通过系统输入-状态对的非奇异变换, 把此问题等价地转化为正常状态空间系统的相应问题. 用大家熟知的方法设计正常状态空间系统的观测器, 从而得到广义系统的输入解耦观测器. 然后用广义系统的观测器状态和系统输出的线性组合渐近估计系统的状态与未知输入.     

A state observer for nonlinear systems

A state observer design methodology for a class of nonlinear systems is proposed. The nonlinear observer design methodology is, in fact, an extension of linear observer design techniques using transformations related to linear observability matrices     

A memoryless state observer for discrete time-delay systems

This paper presents a reduced-order memoryless state observer with a γ-stability margin (0<γ<1) for linear discrete-time systems with multiple delays in both the state and control vectors. The dynamics of the observer are derived from a set of unstable and/or poorly damped eigenvalues of the system. A simple and systematic design method is presented. A numerical example is given to illustrate the properties of the new observer and its design method     

不确定变时滞系统的状态观测器与基于观测是器的鲁棒控制器设计

研究了具有变时滞的不确定系统的状态观测器和基于观测器的鲁棒控制器设计问 题,其中不确定性是时变的,不要求满足匹配条件.通过构造增广系统,利用线性矩阵不等式 (LMI)方法,获得了该不确定系统存在状态观测器和基于观测器的鲁捧控制器的充分条件, 同时给出了相应的状态观测器和基于观测器的鲁棒控制器.     

高阶不确定非线性系统的线性自抗扰控制

针对一类具有内部动态和外部扰动未知的SISO高阶非线性系统,提出一种通用的线性自抗扰控制方案.该方案基于单参数调节的高增益观测器思想,分别设计线性跟踪微分器、线性扩张状态观测器和线性状态误差反馈控制律.利用Lagrange中值定理和Cauchy-Schwarz不等式将系统总扰动的微分值转化为关于系统估计和跟踪误差的函数,可以解决因系统控制增益未知所导致的控制量微分值难以预先确定的问题.在此基础上,基于Lyapunov稳定性定理证明闭环系统误差信号有界,并进一步分析得到系统估计和跟踪误差与控制器参数的定量关系,即都可以随观测器增益的增大而达到无限小.仿真比较结果验证了所提出方案的有效性,与韩式自抗扰控制方案相比,该方案结构简单,调节参数少,易于工程实现.     

Observer theory in optimal control for systems with time delay†

An observer theory for estimating the state variables of deterministic linear time-invariant dynamic forced system is presented. The variation in the cost performance when some of the state variables are estimated by using a linear observer system is evaluated.     

Control design based on a linear state function observer

A control design method based on a linear state function observer is proposed. The method is a semi-inverse design procedure in that the control law is not designed before the observer system, but is a result of the observer design. However, the observer design is not completely independent of the control design, but seeks to yield a feedback signal that is close to a prescribed control law. First, the observer design problem is considered as the reconstruction of a linear function of the state vector. The linear state function to be reconstructed is the given control law. Then, based on the derivation for linear state function observers, the observer design is formulated as a parameter optimization problem. The optimization objective is to generate a matrix that is close to the given feedback gain matrix. Based on that matrix, the form of the observer and a new control law can be determined. The semi-inverse design procedure can yield a reduced-order observer with dimension considerably smaller than that of the system. Two numerical examples are used to demonstrate the proposed design procedure.     

Robust reduced-order sliding mode observer design

In this article, a triple state and output variable transformation-based method combined with linear matrix inequality (LMI) techniques to design a new robust reduced-order sliding mode observer for perturbed linear multiple-input and multiple-output systems is developed. The state and output variables of the original system are triple transformed into suitable canonical form coordinates to facilitate the design of a reduced-order observer. The existing transformations are summarised in this study and presented systematically. A new combined observer configuration is proposed and compared with another type of observers. Global asymptotical stability LMI and sliding mode existence conditions for the coupled observer error system are derived using Lyapunov full quadratic form. Reaching and sliding modes of motion of decoupled observer error system are discussed as well. Two numerical and simulation examples are given to illustrate the usefulness of the proposed design techniques.     

Adaptive nonlinear observer for state and unknown parameter estimation in noisy systems

This paper proposes a novel adaptive observer for Lipschitz nonlinear systems and dissipative nonlinear systems in the presence of disturbances and sensor noise. The observer is based on an H_∞ observer that can estimate both the system states and unknown parameters by minimising a cost function consisting of the sum of the square integrals of the estimation errors in the states and unknown parameters. The paper presents necessary and sufficient conditions for the existence of the observer, and the equations for determining observer gains are formulated as linear matrix inequalities (LMIs) that can be solved offline using commercially available LMI solvers. The observer design has also been extended to the case of time-varying unknown parameters. The use of the observer is demonstrated through illustrative examples and the performance is compared with extended Kalman filtering. Compared to previous results on nonlinear observers, the proposed observer is more computationally efficient, and guarantees state and parameter estimation for two very broad classes of nonlinear systems (Lipschitz and dissipative nonlinear systems) in the presence of input disturbances and sensor noise. In addition, the proposed observer does not require online computation of the observer gain.     

具有状态观测器的鲁棒重复控制系统设计

针对一类时变不确定系统,研究基于状态观测器的鲁棒重复控制系统设计问题.通过状态观测器重构系统状态,利用重构状态进行状态反馈.基丁2维系统稳定性理论,应川线性矩阵不等式(LMI)方法,推导出了重复控制系统存在状态观测器和重复控制器的一个LMI条件,并用LMI的町行解给出了状态观测器和重复控制器参数的具体形式.最后用数值仿真验证了本文所提方法的有效性.     

Nonlinear observer design for GNSS-aided inertial navigation systems with time-delayed GNSS measurements

Global navigation satellite system (GNSS) receivers suffer from an internal time-delay of up to several hundred milliseconds leading to a degeneration of position accuracy in high-dynamic systems. With the increasing interest in GNSS navigation, handling of time-delays will be vital in high accuracy applications with high velocity and fast dynamics. This paper presents a nonlinear observer structure for estimating position, linear velocity, and attitude (PVA) as well as accelerometer and gyro biases, using inertial measurements and time-delayed GNSS measurements. The observer structure consists of four parts; (a) attitude and gyro bias estimation, (b) time-delayed translational motion observer estimating position and linear velocity, (c) input delays for inertial and magnetometer measurements, and (d) a faster than real-time simulator. The delayed PVA and gyro bias estimates are computed using a uniformly semiglobally exponentially stable (USGES) nonlinear observer. The high-rate inertial measurements are delayed and synchronized with the GNSS measurements in the state observer. The fast simulator integrates the inertial measurements from the delayed state estimate to provide a state estimate at current time. The sensor measurements are carefully synchronized and the estimation procedure for the GNSS receiver delay is discussed. Experimental data from a small aircraft are used to validate the results.     

基于观测器的时滞系统鲁棒控制器的设计

研究了一类不确定时滞系统基于观测器的鲁棒镇定问题,系统的不确定性时变未知且范数有界,目的是设计状态观测器和线性无记忆观测状态反馈控制器,使其能够镇定一类状态和控制输入不确定性时滞系统。基于Lyapunov稳定性理论,采用线形矩阵不等式这一有效工具,给出了系统基于观测状态反馈鲁棒镇定的充分条件,并且利用线形矩阵不等式的解构造了使得系统鲁棒稳定的基于观测状态反馈控制器,所得结果与时滞相关,从而相对减弱了控制器设计的保守性。数值算例表明了所提出的设计方法的有效性。     

State estimation and unknown input reconstruction via both reduced-order and high-order sliding mode observers

This paper considers the problems of the simultaneous estimation of the system states and the unknown inputs for linear systems when the so-called observer matching condition is not satisfied. An auxiliary output vector is introduced so that the observer matching condition is satisfied with respect to it. A high-order sliding mode observer is considered to get the exact estimates of both the auxiliary outputs and their derivatives in a finite time based on the system measured outputs. After this, a reduced-order observer is constructed by using the estimated auxiliary outputs as the new system outputs. The reduced-order observer is able to asymptotically estimate the system states without suffering the influence of the unknown inputs. A kind of unknown input reconstruction method based on both the state and the auxiliary output derivative estimates is developed. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed methods.