Functional observability and the design of minimum order linear functional observers

The design of a minimum-order linear functional observer for linear time-invariant systems has been an open problem for over four decades. This technical note provides a solution to this problem. The technical note also introduces the concept of Functional Observability/Detectability and shows that the well-known concept of Observability/Detectability is a special case of Functional Observability/Detectability.

A fault detection scheme for time-delay systems using minimum-order functional observers

This paper presents a method for designing residual generators using minimum-order functional observers to detect actuator and component faults in time-delay systems. Existence conditions of the residual generators and functional observers are first derived, and then based on a parametric approach to the solution of a generalized Sylvester matrix equation, we develop systematic procedures for designing minimum-order functional observers to detect faults in the system. The advantages of having minimum-order observers are obvious from the economical and practical points of view as cost saving and simplicity can be achieved, particularly when dealing with high-order complex systems. Extensive numerical examples are given to illustrate the proposed fault detection scheme. In all the numerical examples, we design minimum-order residual generators and functional observers to detect faults in the system.

Minimum-order filter design for partial state estimation of linear systems with multiple time-varying state delays and unknown input delays

Designing delay-dependent functional observers for LTI systems with multiple known time-varying state delays and unknown time-varying input delays is studied. The input delays are arbitrary, but the state delays should be upper-bounded. In addition, two scenarios of slow-varying and fast-varying state delays are investigated. The results of the paper can also be considered as one of the first contributions considering unknown-input functional observer design for linear systems with multiple time-varying state delays. Based on the Lyapunov Krasovskii approach, delay-dependent sufficient conditions of the exponential stability of the observer in each scenario are established in terms of linear matrix inequalities. Because of using effective techniques, such as the descriptor transformation and an advanced weighted integral inequality, the proposed stability criteria can result in larger stability regions compared with the other papers that study functional observers for time-varying delay systems. Furthermore, to help with the design procedure, a genetic algorithm-based scheme is proposed to adjust a weighting matrix in the established linear matrix inequalities. Two numerical examples illustrate the design procedure and demonstrate the efficacy of the proposed observer in each scenario.

Design of reduced-order observers for neutral time-delay systems

This paper presents a method for the design of reduced-order observers for a class of linear time-delay systems of the neutral-type. Conditions for the existence of reduced-order observers that are capable of asymptotically estimating any given function of the state vector are derived. A step-by-step design procedure is given for the determination of the observer parameters. A numerical example is given to illustrate the design procedure.

Time-delay systems : design of delay-free and low-order observers

This note provides a comprehensive treatment on the design of functional observers for linear systems having a time-varying delay in the state variables. The designed observers possess attractive features of being low-order and delay-free and hence they are cost effective and easy to implement. Existence conditions are derived and a design procedure for finding low-order observers is given.

Design of reduced-order state/unknown input observers: a descriptor system approach

This paper addresses the problem of estimating simultaneously a linear function of both the state and unknown input of linear system with unknown inputs. By adopting the descriptor system approach, the problem can be conveniently solved. Observers proposed in this paper are of low-order and do not include the derivatives of the outputs. New conditions for the existence of reduced-order observers are derived. A design procedure for the determination of the observer parameters can also be easily derived based on the derived existence conditions

A system decomposition approach to the design of functional observers

This paper reports a system decomposition that allows the construction of a minimum-order functional observer using a state observer design approach. The system decomposition translates the functional observer design problem to that of a state observer for a smaller decomposed subsystem. Functional observability indices are introduced, and a closed-form expression for the minimum order required for a functional observer is derived in terms of those functional observability indices. © 2014 Taylor & Francis.

Approach to fault detection of time-delay systems using functional observers

A new methodology is reported for designing functional observers to detect actuator faults of a class of time-delay systems where the matrix pair (A, C) is not observable. First, a generalised state transformation is used to transform the system into new coordinates where the delay term associated with the state vector is injected into the system's output and input. Then, a minimum-order functional observer is designed to construct a residual function that can trigger system faults. The finding is significant as it is now possible to detect faults of time-delay systems where the pair (A, C) is not required to be observable. A numerical example is given to illustrate the effectiveness of the proposed design approach. © The Institution of Engineering and Technology 2014.

Minimal unknown-input functional observers for multi-input multi-output LTI systems

Designing minimum possible order (minimal) disturbance-decoupled proper functional observers for multi-input multi-output (MIMO) linear time-invariant (LTI) systems is studied. It is not necessary that a minimum-order unknown-input functional observer (UIFO) exists in our proposed design procedure. If the minimum-order observer cannot be attained, the observer's order is increased sequentially through a recursive algorithm, so that the minimal order UIFO can be obtained. To the best of our knowledge, this is the first time that this specific problem is addressed. It is assumed that the system is unknown-input functional detectable, which is the least requirement for the existence of a stable UIFO. This condition also is a certificate for the convergence of our observer's order-increase algorithm. Two methodologies are demonstrated to solve the observer design equations. The second presented scheme, is a new design method that based on our observations has a better numerical performance than the first conventional one. Numerical examples and simulation results in the MATLAB/Simulink environment describe the overall observer design procedure, and highlight the efficacy of our new methodology to solve the observer equations in comparison to the conventional one.

Linear functional observers with guaranteed ε-convergence for discrete time-delay systems with input/output disturbances

The problem of designing linear functional observers for discrete time-delay systems with unknown-but-bounded disturbances in both the plant and the output is considered for the first time in this paper. A novel approach to design a minimum-order observer is proposed to guarantee that the observer error is ϵ-convergent, which means that the estimate converges robustly within an ϵ-bound of the true state. Conditions for the existence of this observer are first derived. Then, by utilising an extended Lyapunov-Krasovskii functional and the free-weighting matrix technique, a sufficient condition for ϵ-convergence of the observer error system is given. This condition is presented in terms of linear matrix inequalities with two parameters needed to be tuned, so that it can be efficiently solved by incorporating a two-dimensional search method into convex optimisation algorithms to obtain the smallest possible value for ϵ. Three numerical examples, including the well-known single-link flexible joint robotic system, are given to illustrate the feasibility and effectiveness of our results.

Minimal-order functional observer-based residual generators for fault detection and isolation of dynamical systems

This paper examines the design of minimal-order residual generators for the purpose of detecting and isolating actuator and/or component faults in dynamical systems. We first derive existence conditions and design residual generators using only first-order observers to detect and identify the faults. When the first-order functional observers do not exist, then based on a parametric approach to the solution of a generalized Sylvester matrix equation, we develop systematic procedures for designing residual generators utilizing minimal-order functional observers. Our design approach gives lower-order residual generators than existing results in the literature. The advantages for having such lower-order residual generators are obvious from the economical and practical points of view as cost saving and simplicity in implementation can be achieved, particularly when dealing with high-order complex systems. Numerical examples are given to illustrate the proposed fault detection and isolation schemes. In all of the numerical examples, we design minimum-order residual generators to effectively detect and isolate actuator and/or component faults in the system.

Partial-state observers for nonlinear systems

This note deals with the design of reduced-order observers for a class of nonlinear systems. The order reduction of the observer is achieved by only estimating a required partial set of the state vector. Necessary and sufficient conditions are derived for the existence of reduced-order observers. An observer design procedure based on linear matrix inequalities is given. A numerical example is given to illustrate the design method.

Design of state observers for interconnected time-delay systems via a coordinate transformation approach

This paper considers the design of state observers for interconnected time-delay systems using a coordinate transformation method. Through such a transformation, the system that has interconnection and state delays is metamorphosed into a new system that injects time-delay information into its input and output terms, before reintroducing them back into the latter system, effectively coupling the delay terms into the IO injection terms and eliminating the delay values from the state variables. Next, full-order and reduced-order observers are designed based on the transformed system. Finally, the observed states of the transformed system that correspond to the original system is used to deduce the estimates of the original system. A numerical example is provided of an interconnected time-delay system.

Partial state estimation of LTI systems with multiple constant time-delays

Functional observer design for Multi-Input Multi-Output (MIMO) Linear Time-Invariant (LTI) systems with multiple mixed time delays in the states of the system is addressed. Two structures for the design of a minimum-order observer are considered: 1 - delay-dependent, and 2 - internal-delay independent. The parameters of the delay-dependent observer are designed using the Lyapunov Krasovskii approach. The delay-dependent exponential stability of the observer for a specified convergence rate and delay values is guaranteed upon the feasibility of a set of Linear Matrix Inequalities (LMIs) together with a rank condition. Using the descriptor transformation, a modified Jensen's inequality, and improved Park's inequality, the results can be less conservative than the available functional observer design methods that address LTI systems with single state delay. Furthermore, the necessary and sufficient conditions of the asymptotic stability of the internal-delay independent observer are obtained, which are shown to be independent of delay. Two illustrative numerical examples and simulation studies confirm the validity and highlight the performance of the proposed theoretical achievements.