Existence conditions for unknown input functional observers

This article presents necessary and sufficient conditions for the existence and design of an unknown input Functional observer. The existence of the observer can be verified by computing a nullspace of a known matrix and testing some matrix rank conditions. The existence of the observer does not require the satisfaction of the observer matching condition (i.e. Equation (16) in Hou and Muller 1992, ‘Design of Observers for Linear Systems with Unknown Inputs’, IEEE Transactions on Automatic Control, 37, 871–875), is not limited to estimating scalar functionals and allows for arbitrary pole placement. The proposed observer always exists when a state observer exists for the unknown input system, and furthermore, the proposed observer can exist even in some instances when an unknown input state observer does not exist.

Quasi-decentralized functional observers for the LFC of interconnected power systems

This letter presents a novel approach to the load-frequency control (LFC) of interconnected power systems. Based on functional observers theory, quasi-decentralized functional observers (QDFOs) are designed to implement any given global PI state feedback controller. The designed functional observers are decoupled from each other and also of low-order; thus, they are cost effective and easy to implement. Although the proposed approach is applicable to N- area power systems, an example of a two-area interconnected power system with reheat thermal turbines is considered for simplicity.

Fault detection for dynamical systems using first-order functional observers

This paper reports a new result on the fault detection of dynamical systems by employing only first-order functional observers. Indeed, we show that fault detection can be achieved by utilizing first-order functional observers. The advantages for having such simple structured observers are obvious from the economical and practical points of view as significant cost saving can be achieved. We derive existence conditions and an algorithm for the generation of residual signals to detect faults using firstorder functional observers. Two numerical examples are given to illustrate the proposed fault detection scheme. In one of the examples, a two-area interconnected power system with reheat thermal turbines is considered where only a first-order functional observer is designed to detect faults in the power system.

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.

A new algorithm to design minimal multi-functional observers for linear systems

Designing minimum possible order (minimal) observers for Multi-Input Multi-Output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for Linear Time-Invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

A practical functional observer scheme for interconnected time-delay systems

This paper proposes a partially distributed functional observer scheme for a class of interconnected linear systems with very strong non-instantaneous subsystems interaction and with time delays in the local states and in the transmission of output information from the remote subsystems. A set of easily verifiable existence conditions is established and upon its satisfaction, simple distributed observers are designed using a straightforward design procedure. Simulation results of a numerical example are given to substantiate the feasibility of the approach.

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.

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.

Minimal multi-functional observers for linear systems using a direct approach

The direct approach in designing functional observers was first presented in [1] for estimating a single function of the states of a Linear Time-Invariant (LTI) system. One of the benefits of the direct scheme is that it does not require solving the interconnected Sylvester equations that appear in the other observer design approaches. In the present paper, the direct approach is extended to reconstruct multiple functions of the states in such a way that the minimum possible order of the observer is achieved. The observer is designed so that an asymptotic functional observer can be obtained with arbitrary convergence rate. In the proposed methodology, it is not necessary that a reduced order observer exists for the desired functions to be estimated. To release this limitation, an algorithm is employed to find some auxiliary functions in the minimum required number to be appended to the desired functions. This method assumes that the system is functional observable. This assumption however is less restrictive than the observability and detectability conditions of the system. A numerical example and simulation results explain the efficacy and the benefits of the proposed algorithm.

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.

Design of reduced-order positive linear functional observers for positive time-delay systems

This brief considers a new problem of designing reduced-order positive linear functional observers for positive time-delay systems. The order of the designed functional observers is equal to the dimension of the functional state vector to be estimated. The designed functional observers always nonnegative at any time and they converge asymptotically to the true functional state vector. Moreover, conditions for the existence of such positive linear functional observers are formulated in terms of linear programming (LP). Numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.