This paper focuses on the principle for designing reduced-order fuzzy-observer-based actuator fault reconstruction for a class of nonlinear systems. The problem addressed can be indicated as an approach for a kind of reduced-order fuzzy observer design with special gain matrix structure that depends on a given matching condition specification. Using the Lyapunov theory, the stability conditions are obtained and expressed in terms of linear matrix inequalities, and the conditions for asymptotic estimation of actuator faults are derived. Simulation results illustrate the observer design procedure and demonstrate the actuator fault reconstruction effectiveness and performance.

Automated diagnosis has been one of the most fruitful applications in sophisticated control systems, with potential significance for domains in which systems diagnosis must proceed, while the system is operative and testing opportunities are limited by operational considerations. A real problem is usually to fix the system with faults so that it can continue its mission for some time with some limitations in functionality. Consequently, diagnosis is a part of a larger problem known as Fault Detection, Identification and Reconfiguration (FDIR). The classical principles include observer-based methods, parity space methods, and parameter identification based methods, which have been thoroughly studied (see, e.g., [

Observer design is an actual research topic, important in the observer-based fault estimation, and in the fault detection and isolation [

Recently, fault estimation and reconstruction are preferred as an option to fault detection, where, instead of generating residuals, observer-based methods are used to reconstruct sensor and actuator fault signals in nonlinear systems. These practices primarily use adaptive and unknown input observer structures (see e.g., [

An alternative approach is the Takagi-Sugeno (TS) fuzzy approximation of the nonlinear system model equations. Since the TS fuzzy method provides the suitable model for a certain class of nonlinear dynamic systems [

System state observers based on TS fuzzy models are principally realized in the same structures as the linear observers [

Because fault reconstruction provides a direct estimate of the size and severity of a fault, the location of the fault is so known, and the fault isolation step can be deleted. Establishing a general approach for fault reconstruction in systems described by TS models, or finding conditions under which fault reconstruction is well possible, is still an open task [

Considering the author’s previous work [

The remainder of this paper is organized as follows. Sections

Throughout the paper, the following notations are used:

The systems under consideration fall in a class of multi-input and multioutput (MIMO) nonlinear dynamic systems, which in the state-space form are represented as

It is considered that the number of the nonlinear terms in the vector function

Thus, constructing the set of membership functions

Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set [

Note, the model (

It is supposed in the next that the aforementioned TS model does not include parameter uncertainties or external disturbances, and all premise and output variables are measurable.

Let

If

The singular value decomposition (SVD) of

Let the output matrix

Applying SVD to

Using the congruence transform (

Substituting (

Substituting (

The fault input matrix and the output matrix

The matching condition, given in Proposition

Standard applications of TS fuzzy principle in nonlinear system fault diagnosis exploit the fuzzy observers as residual generators. The procedure of fault detection covers the residual generation by the fuzzy observers and their evaluation. Thus, the reconstruction error, or any function of it, is used as fault residual signal that is as a rule zero in the fault free case and nonzero otherwise [

The fuzzy observer to the fault-free system (

The fuzzy observer (

If the above conditions hold, the set of the observer gain matrices is given as

Introducing the estimation error between the fault-free (

Note, to apply for actuator fault reconstruction, an adaptive structure of the full-order state observer can be used [

Problem of the interest is to design the asymptotically stable reduced-order observer based on the TS fuzzy model of the fault-free nonlinear system (

Considering the affine TS fuzzy system (

Since (

It is evident that

The reduced-order TS fuzzy observer (

Using (

If (

The asymptotic stability condition of the autonomous part of (

Substituting (

Since (

Note, the form of the time derivative (

The equalities (

Ones explaining the variable

Evidently, (

Using the equivalency of the stability conditions, an actuator fault estimation structure based on reduced-order TS fuzzy observer can be discussed.

To obtain an actuator fault estimation structure based on reduced-order TS fuzzy observer, the matching condition (

The estimation error dynamics of the reduced-order TS fuzzy observer (

The system with an actuator fault is described as

Evidently,

The estimation error dynamic (

If the above conditions hold, the observer gain matrix is given as

Satisfying (

Using (

Since

If

Considering the fact that the reduced-order TS fuzzy observer does not contain any information about actuator faults, the next reconstruction principle can be used.

Designed with respect to

Since (

Taking the actuator fault reconstructor as given by (

Note, matrix pseudoinverse in (

Referring to [

Using SVD of the output matrix

The scalar LMI variable

Thus, solving (

For simulation purposes only, the equilibrium of the system was stabilized by the fuzzy feedback controller

In simulations was considered the fault which does not cause closed-loop system instability, modeled by a fault starting at any time instant in the system equilibrium state. Applying the above-designed reduced-order observer-based actuator fault estimation, the fault responses for the nonlinear system are given in Figures

The second actuator fault signal.

The reconstruction of the actuator fault signal.

From the simulation results of Figures

Generalized design method of a reduced-order observer-based actuator fault estimation scheme is developed, as augmentation of unknown observers synthesis for one class of nonlinear systems described by TS fuzzy model. This is achieved by manipulation of observer asymptotic stability with respect to the proposed matching conditions. Design conditions for asymptotic estimation of actuator faults are derived in terms of LMI, using standard LMI procedures to manipulate the reducer-order observer stability. Because of the specific observer gain matrix structure, the estimated unmeasurable part of the system state is free of actuators faults. By examining the estimated state vector, it is presented that using a numerical realization of time derivative of the state vector estimate, the actuator fault signals can be faithfully reconstructed.

Proposed scheme is able to simultaneously estimate the time-varying actuator faults, as well as the system state variables with a good accuracy, is easy to implement, and can be applied to a reasonably wide class of systems satisfying the matching condition. Presented simulations have shown that the proposed design task is feasible and effective.

The work, presented in this paper, was supported by VEGA, the Grant Agency of Ministry of Education and Academy of Sciences of Slovak Republic, under Grant no. 1/0256/11. This support is very gratefully acknowledged.