Designing fault detection observers for linear systems with mismatched unknown inputs

Algebraic unknown input observers (UIOs) that have been previously reported in the literature can be constructed under the assumption that linear systems with unknown inputs satisfy the so-called observer matching condition. This condition restricts practical applications of UIOs for fault detection and isolation (FDI). We present an algebraic design for fault detection observers (FDOs) for the case in which the observer matching condition is not satisfied. To loosen the restriction imposed by the observer matching condition, the UIO design method combined with the unknown input modeling technique is proposed to design an FDO that decouples the effect of mismatched unknown inputs. To do this, first, unknown inputs that denote the faults of no interest and process disturbances are decomposed into algebraically rejectable unknown inputs and modeled unknown inputs such that the observer matching condition is satisfied. Under the assumption that mismatched unknown inputs are deterministic and can be expressed as the responses of fictitious autonomous dynamical systems, an augmented system is obtained by combining the original system model with the unknown input model. Finally, through the design technique of a UIO for the augmented system, a reduced-order FDO is constructed to estimate an augmented state vector that consists of both the original state variables and the augmentative state variables. The estimated state is then used to generate the residual, which should be designed to be insensitive to unknown inputs while being sensitive to the faults of interest. Two numerical examples are provided to show the usefulness and the feasibility of the presented approach.     

Simultaneous state and input estimation of hybrid systems with unknown inputs

This paper addresses the problem of the simultaneous state and input estimation for hybrid systems when subject to input disturbances. The proposed algorithm is based on the moving horizon estimation (MHE) method and uses mixed logical dynamical (MLD) systems as equivalent representations of piecewise affine (PWA) systems. So far the MHE method has been successfully applied for the state estimation of linear, hybrid, and nonlinear systems. The proposed extension of the MHE algorithm enables the estimation of unknown inputs, or disturbances, acting on the hybrid system. The new algorithm is shown to improve the convergence characteristics of the MHE method by reducing the delay of convergent estimates, while assuring convergence for every possible sequence of input disturbances. To ensure convergence the system is required to be incrementally input observable, which is an extension to the classical incremental observability property.     

Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs

This paper extends the existing results on joint input and state estimation to systems with arbitrary unknown inputs. The objective is to derive an optimal filter in the general case where not only unknown inputs affect both the system state and the output, but also the direct feedthrough matrix has arbitrary rank. The paper extends both the results of Gillijns and De Moor [Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934–937] and Darouach, Zasadzinski, and Boutayeb [Darouach, M., Zasadzinski, M., & Boutayeb, M. (2003). Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39, 867–876]. The resulting filter is an extension of the recursive three-step filter (ERTSF) and serves as a unified solution to the addressed unknown input filtering problem. The relationship between the ERTSF and the existing literature results is also addressed.     

Simultaneous state and unknown input reconstruction using cascaded high-gain observers

This paper presents a systematic routine for jointly reconstruct the state variables and the unknown inputs (UIs) for a large class of nonlinear MIMO systems. After an appropriate change of state coordinates, a set of cascade high-gain observers (CHGO) are designed in such a way that each of them provides an estimation of only one component of the UI vector, except the last one which gives a final adjustment of the whole state variables. Such design achieves a boundary of the state estimation error which can be arbitrarily small by properly specifying the sole synthesis parameter. An illustrative example verifies the effectiveness of the proposed scheme.     

Unbiased minimum-variance state estimation for linear systems with unknown input

The problem of state estimation for a linear system with unknown input, which affects both the system and the output, is discussed in this paper. A recursive optimal filter with global optimality in the sense of unbiased minimum variance over all linear unbiased estimators, is provided. The necessary and sufficient condition for the convergence and stability is also given, which is milder than existing approaches.     

Sliding-mode observers for systems with unknown inputs: A high-gain approach

Sliding-mode observers can be constructed for systems with unknown inputs if the so-called observer matching condition is satisfied. However, most systems do not satisfy this condition. To construct sliding-mode observers for systems that do not satisfy the observer matching condition, auxiliary outputs are generated using high-gain approximate differentiators and then employed in the design of sliding-mode observers. The state estimation error of the proposed high-gain approximate differentiator based sliding-mode observer is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. Finally, the unknown input reconstruction using the proposed observer is analyzed and then illustrated with a numerical example.     

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.     

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.     

Partial state and unknown input estimation for time-delay systems

This article considers the problem of estimating a partial set of the state vector and/or unknown input vector of linear systems driven by unknown inputs and time-varying delay in the state variables. Three types of reduced-order observers, namely, observers with delays, observers without internal delays and delay-free observers are proposed in this article. Existence conditions and design procedures are presented for the determination of parameters for each case of observers. Numerical examples are presented to illustrate the design procedures.     

Low-order state estimators and compensators for dynamical systems with unknown inputs

We propose a new type of state estimator for a class of dynamical systems with unknown inputs. We synthesize the combined low-order estimator-controller compensator and show the stability of the closed-loop system. The results are illustrated with a numerical example.     

On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs

In this paper, a globally optimal filtering framework is developed for unbiased minimum-variance state estimation for systems with unknown inputs that affect both the system state and the output. The resulting optimal filters are globally optimal within the unbiased minimum-variance filtering over all linear unbiased estimators. Globally optimal state estimators with or without output and/or input transformations are derived. Through the global optimality evaluation of this research, the performance degradation of the filter proposed by Darouach, Zasadzinski, and Boutayeb [Darouach, M., Zasadzinski, M., & Boutayeb, M. (2003). Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39, 867-876] is clearly illustrated and the global optimality of the filter proposed by Gillijns and De Moor [Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934-937] is further verified. The relationship with the existing literature results is addressed. A unified approach to design a specific globally optimal state estimator that is based on the desired form of the distribution matrix from the unknown input to the output is also presented. A simulation example is given to illustrate the proposed results.     

Design of unknown input functional observers for nonlinear systems with application to fault diagnosis

This paper considers the design of low-order unknown input functional observers for robust fault detection and isolation of a class of nonlinear Lipschitz systems subject to unknown inputs. The proposed functional observers can be used to generate residual signals to detect and isolate actuator faults. By using the generalized inverse approach, the effect of the unknown inputs can be decoupled completely from the residual signals. Conditions for the existence and stability of reduced-order unknown input functional observer are derived. A design procedure for the generation of residual signals to detect and isolate actuator faults is presented using the proposed unknown-input observer-based approach. A numerical example is given to illustrate the proposed fault diagnosis scheme in nonlinear systems subject to unknown inputs.     

Dynamic observers for linear time-invariant systems

An alternative state estimator structure for linear time-invariant systems named for dynamic observer is presented, which can be considered as an extension of the usual observer in its configuration. As a possible design method for the dynamic observer, an efficient and plausible design algorithm is also provided. The mechanism of the proposed dynamic observer design is the dual of the output feedback controller design. The essential characteristics of the dynamic observer to be qualified as an effective observer are addressed.     

Interval observer design for continuous-time switched systems under known switching and unknown inputs

ABSTRACT

This paper deals with unknown input estimation for switched linear systems in an unknown but bounded error (UBBE) framework. Based on a known switching signal and under the fulfilment of the relative degree property by all the subsystems, a decoupling method is used to make the state partially affected by the unknown input. Assuming that the disturbances and the measurement noises are unknown but bounded with a priori known bounds, lower and upper bounds of the unmeasured state and unknown input are then computed. A numerical example illustrates the efficiency of the proposed methodology.     

Unbiased minimum-variance input and state estimation for linear discrete-time systems

This paper addresses the problem of simultaneously estimating the state and the input of a linear discrete-time system. A recursive filter, optimal in the minimum-variance unbiased sense, is developed where the estimation of the state and the input are interconnected. The input estimate is obtained from the innovation by least-squares estimation and the state estimation problem is transformed into a standard Kalman filtering problem. Necessary and sufficient conditions for the existence of the filter are given and relations to earlier results are discussed.     

Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough

This paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, a recursive filter is derived where the estimation of the state and the input are interconnected. The derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available. The resulting filter has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate.     

Design of linear functional observers for linear systems with unknown inputs

This paper presents a reduced-order linear functional state observer for linear systems with unknown inputs. A simple observer construction procedure is provided. A numerical example is given to illustrate the properties of the observer. The example deals with a linear system comprising of 20 states, 2 inputs, 10 outputs and 5 unknown inputs for which a fourth-order observer is designed to estimate two linear functions of the states.