Robust fault detection using Luenberger-type unknown input observers-a parametric approach

A new parametric observer-based approach for robust fault detection in multivariable linear systems with unknown disturbances is proposed. The residual is generated through utilizing a Luenberger function observer. By using a parametric solution to a class of generalized Sylvester matrix equations, a parametrization is proposed for the residual generator on the basis of a Luenberger function observer. By further properly constraining the design parameters provided in the Luenberger observer design, the effect of the unknown disturbance is decoupled from the residual signal. The proposed approach provides all the degrees of freedom and is demonstrated to be simple and effective.     

矩阵方程AX-EXF=BY的通解及其应用

给出矩阵方程AX—EXY=BY的一个完全解析的、具有显式表达式和完全自由度的参数解(X,Y)．这里假设矩阵束(E,A B)为R-能控的,F为任意的方阵．相比于现有结论,求解算法不要求矩阵A和F具有特殊的形式,且对它们的特征值没有任何的限制,此外,本文给出的通解还具有结构简洁的特点,作为一个应用,给出了广义系统正常Luenberger函数观测器的一种参数化的设计方法,算例证明了方法的有效性．     

On the solution of a matrix equation in the theory of Luenberger observers

An explicit solution is presented for a matrix equation involved in the design of a Luenberger observer to realize an acceptable approximation to a desired state-variable feedback lawu = u_{0} + Kxfor the systemdx/dt = Ax + Bu, y = Dx.     

矩阵方程AX-EXF=BY的通解及其应用

给出矩阵方程 AX-EXY=BY的一个完全解析的、具有显式表达式和完全自由度的参数解 (X,Y) .这里假设矩阵束 (E,A,B) 为R-能控的, F为任意的方阵.相比于现有结论,求解算法不要求矩阵A和F具有特殊的形式,且对它们的特征值没有任何的限制.此外,本文给出的通解还具有结构简洁的特点.作为一个应用,给出了广义系统正常Luenberger函数观测器的一种参数化的设计方法.算例证明了方法的有效性.     

鲁棒Luenberger观测器设计

观测器控制系统中的观测器条件是系统的状态观测值渐近收敛于系统真实状态的根本 条件.本文首先提出了Luenberger观测器设计的一种参数方法,然后根据使观测器条件误差 为最小的准则,考虑了具有参数摄动的系统的鲁棒Luenberget观测器设计问题,给出了简单、 有效的算法.仿真结果说明了本文方法的有效性.     

线性系统对偶Luenberger观测器设计

提出了线性系统对偶Luenberger观测器的参数化设计方法.基于一类广义Sylvester矩阵方程的显式参数化解,根据一些自由参数给出了对偶Luenberger观测器所有增益矩阵的参数化表达.该设计方法能提供所有的设计自由度,它们能进一步用来实现系统的其他性能指标.数值例子显示了该方法的设计过程.     

An efficient method for observer design with approximately linear error dynamics

Exact error linearization is a well-known full-order observer design method which yields linear time-invariant, stable error dynamics in normal form coordinates. The difficult procedure associated with exact error linearization has led to the simpler extended Luenberger observer design. However, extended Luenberger observers can only guarantee a first-order approximation to the exact error linearization design. This paper provides an explicit formula involving only derivative (Lie derivative and bracket) and matrix inversion operations for computing an Nth-order approximation of the exact error linearization observer. Hence, the proposed method is a computational simple tool for designing observers for systems admitting exact error linearization observers. Two examples demonstrate the application of the method.     

线性系统的干扰解耦观测器设计

此文提出线性系统的Luenberger函数观测器关于干扰解耦的条件,指出它与环路复现 (LTR)的关系.并结合Luenberger观测器设计的一种参数化方法,给出了线性系统干扰解 耦Luenberger观测器的一种简单、有效的设计方法.     

A robust controller for trajectory tracking of a DC motor pendulum system

This work presents a solution to the output feedback trajectory tracking problem for an uncertain DC motor pendulum system under the effect of an unknown bounded disturbance. The proposed algorithm uses a Proportional Derivative (PD) controller plus a novel on-line estimator of the unknown disturbance. The disturbance estimator is obtained by coupling a standard second-order Luenberger observer with a third-order sliding modes differentiator. The Luenberger observer provides estimates of the motor angular position and velocity. Moreover, an ideal disturbance estimator in terms of the Luenberger observer error and its first and second time derivatives is obtained from the observer error formulae; these time derivatives are not available from measurements. Subsequently, the sliding modes third-order differentiator allows obtaining estimates of these time derivatives in finite time. The estimates replace the real values of the first and second time derivatives in the ideal disturbance estimator thus producing a practical disturbance estimator, and also permit obtaining an estimate of the motor angular velocity. A depart from previous approaches is the fact that the disturbance is not directly estimated by the Luenberger observer or the third-order differentiator. Numerical simulations and real-time experiments validate the effectiveness of the proposed approach.     

Eigenspectrum-based extended Luenberger observers for a class of distributed parameter systems

State estimation is an important problem in distributed parameter system especially with nonlinear dynamics in industrial process. An extended Luenberger observer based on the eigen-spectrum of the system operator is developed in this paper to handle this problem. The distributed parameter system is projected into a finite-dimensional subspace where a low-order ordinary differential equation describing the dominant dynamics of the system is derived. A Luenberger observer extended with a nonlinear part is developed based on that dominant dynamics. A sufficient condition is given in this paper for the convergence of the estimated error. Finally, by applying the developed design method to the temperature estimation of a catalytic rod, the achieved simulation results show the effectiveness of the proposed observer.     

Output feedback second‐order sliding mode control of the cart on a beam system

We propose an output feedback second‐order sliding mode controller to stabilize the cart on a beam system. A second‐order sliding mode controller is designed using a Lyapunov function‐based switching surface and finite‐time controllers, while the state estimator is designed based on the Luenberger‐like observer. The proposed observer extends the applicability of Luenberger‐like observer to nonlinear systems that are not input–output linearizable, but can be approximately input–output linearized. The approximation is based on the physical property of the system, wherein certain terms in the total energy are neglected. Extensive numerical simulations validate the robustness of the proposed controller to parametric uncertainties using estimated states. Copyright © 2009 John Wiley & Sons, Ltd.     

Controller parametric robustification using observer-based formulation and multimodel design technique

This note deals with controller robustification using equivalent observer based structure. The purpose of the method is to improve the parametric robustness of an initial controller, synthesized, for example, by using the LQ,H/sub /spl infin//, or /spl mu/ technique. The method uses equivalent Luenberger observer formulation and multimodel design procedure for the parametric robustification. First, the initial controller is put under equivalent observer structure. Second, from this observer-based formulation, the controller is reshaped to increase the parametric robustness without paying attention to the closed-loop poles coming from the controller dynamics. Note that during this step, changes in the controller initial performances are minimized. Another advantage of the equivalent observer based formulation lies with the fact that it could directly be used to schedule the controller (dynamic and feedback parts). Finally, the global method (equivalent observer plus robustification) is applied on the robust control of the space shuttle described in /spl mu/-analysis and synthesis toolbox.     

Parametric compensator design in the frequency domain

The parametric approach to the design of observer based compensators has hitherto only been formulated in the time domain. It yields an explicit parametric expression for the state feedback matrix (observer gain) given the closed loop eigenvalues and the corresponding sets of invariant parameter vectors. Using the polynomial approach to the design of observer based compensators this contribution presents an equivalent parameterization in the frequency domain. By introducing the closed loop poles and the set of so-called pole directions as new design parameters, one obtains expressions in parametric form for the polynomial matrix D(s) (D(s)), parameterizing the state feedback (state observer) in the frequency domain. It is shown how the pole directions are related to the invariant parameter vectors used in the time domain approach. Another new result is the parametric design of reduced order observers both in the frequency domain, and derived from those results, in the time domain. The proposed design procedure is also used to provide a parametric solution for the optimal LQG control problem in the presence of partially perfect measurements. Simple examples demonstrate the design procedure.     

Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback

We study the problem of global stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither controllable nor observable. A new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers. While the smooth state feedback control law is constructed based on the tool of adding a power integrator, the observer design is new and carried out by developing a machinery, which makes it possible to assign the observer gains one-by-one, in an iterative manner. Such design philosophy is fundamentally different from that of the traditional "Luenberger" observer in which the observer gain is determined by observability. In the case of linear systems, our design method provides not only a new insight but also an alternative solution to the output feedback stabilization problem. For a class of high-order nonhomogeneous systems, we further show how the proposed design method, with an appropriate modification, can still achieve global output feedback stabilization. Examples and simulations are given to demonstrate the main features and effectiveness of the proposed output feedback control schemes.     

采样数据系统最优诊断观测器设计

研究采样数据(sampled-data,SD)系统最优诊断观测器的直接设计方法.首先从传递函数角度分析一般离散时间系统诊断观测器的Luenberger条件,由此构造出SD诊断观测器,并推导出其混杂动态关系.然后将SD故障检测问题定义为一个特定的比值型优化问题,并应用互内外分解法和代数方法构制其最优诊断观测器,使得产生的离散时间残差对连续时间未知输入具有鲁棒性,但对故障却是敏感的.仿真结果验证了所提设计方法的有效性.     

二阶动力学系统高阶PI观测器的参数化设计方法

本文研究二阶动力学系统的高阶比例积分(PI)观测器设计问题．基于Sylvester矩阵方程的解，提出了该观测器设计的参数化方法．该参数化方法给出了该类观测器增益矩阵的参数化表达式，其所含参数除了满足两个约束条件之外是完全自由的．这些参数为控制系统设计提供了全部自由度，可通过优化等手段选择这些参数来满足某些性能要求，如干扰解耦、故障检测和鲁棒性等．此外，该PI观测器的参数化设计方法直接利用二阶动力学系统的原始参数矩阵，只涉及二阶动力学系统n维参数矩阵的运算．给出一个数值例子，以说明所提方法的简单性和有效性．     

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.     

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.     

Nonlinear observer design using Lyapunov’s auxiliary theorem

The present work proposes a new approach to the nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer theory, the proposed approach develops a nonlinear analogue. The formulation of the observer design problem is realized via a system of singular first-order linear PDEs, and a rather general set of necessary and sufficient conditions for solvability is derived by using Lyapunov’s auxiliary theorem. The solution to the above system of PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.