A Reduced-Order TS Fuzzy Observer Scheme with Application to Actuator Faults Reconstruction

Abstract

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.

1. Introduction

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., [1, 2] and the references therein).

Observer design is an actual research topic, important in the observer-based fault estimation, and in the fault detection and isolation [3–5]. The nonlinear system theory, exploiting Lipschitz condition, is emerged as an approach capable of use in the state estimation design for nonlinear systems [6], although Lipschitz condition is a restrictive limitation and many classes of systems may not be satisfied. Application of this principle in state estimator design results only in a sufficient condition for the asymptotic stability of estimation error, and, in fact, there is no straightforward method for selecting the observer gain to satisfy such conditions [7]. Because of strong restrictions, an observer structure with adaptively adjusted parameters is proposed in [8], where Lipchitz constant can be unknown. Concerning fault detection, for example, in [9–11] there are proposed sliding-mode observers. Since they are conditioned by matching conditions, these approaches are not sufficient to ensure safe operation in all applications.

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., [12–14]), ensuring disturbance rejection and robustness properties of fault estimation.

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 [15], the well-known nonlinear observers are based on TS fuzzy system model. Using TS fuzzy model, a nonlinear system is represented by the fuzzy rules. Each rule utilizes the local system dynamics by a linear model, and the nonlinear system is represented by a collection of fuzzy rules. In this sense, the TS fuzzy model can be viewed as an expansion of piecewise linear partition for the nonlinear system. Since such description allows the utilization of system state representation, model order reduction and error approximation problems have to be solved using the projection methods [16], generally given in the form of linear matrix inequality (LMI) constraints.

System state observers based on TS fuzzy models are principally realized in the same structures as the linear observers [17–20], and design principles usually used techniques based on LMIs. Research in TS fuzzy observers application in fault detection and isolation has attracted many investigators and was the subject of widely scattered publications (see, e.g., [21–24]), mainly focused on the LMI-based observer design, to ensure the stability of the residuals and to optimize the quadratic performance of residual transfer matrix with respect to exogenous disturbance.

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 [25]. Sophisticated fault estimation schemes were proposed especially for systems with disturbances and uncertainties, where, for example, Gao et al. [26] propose the fuzzy descriptor observer, potentially applicable to sensor fault estimation. In contrast, Xu et al. [27] present an estimation algorithm, based on the integrated fuzzy observer and the inverse system model, for nonlinear actuator fault estimation. The principle of the inverse system model is combined with sliding mode also in [27], since sliding mode observers can be employed in fault estimation if systems are uncertain owing to their insensitivity to matched uncertainties or disturbances. Certainly, the basic approach to actuator faults estimation is based on TS adaptive observers [17], in spite of high-order observer dynamics. On the other hand, few results have been reported to reduced-order observer-based fault reconstruction [8, 28], despite the importance of relative-order dynamics of reduced-order observers for systems without disturbances.

Considering the author’s previous work [29], the main contribution of the paper is to examine one principle for designing of reduced-order-observer-based actuator fault estimation for a class of continuous-time nonlinear MIMO systems, approximated by TS models. Comparing with the approach given in [14], a new design method is proposed to construct a set of linear reduced-order observers, combined by fuzzy rules, to estimate unmeasurable part of the system state vector. Based on the stable observer set, the actuator fault estimation scheme is developed to guarantee asymptotic estimation of actuator faults. The structure of the design conditions is motivated by the need for feasibility, while under defined matching conditions the stability of the reduced-order observer is assured.

The remainder of this paper is organized as follows. Sections 2 and 3 describe TS fuzzy model properties for given class of nonlinear systems and the design principle of the reduced-order observer based on TS model, respectively. The actuator fault reconstruction, using reduced-order fuzzy observer, is outlined in Section 4, especially with respect to observer design principle, matching condition, and stability. In Section 5, one illustrative example is given, and simulation results are presented to confirm the validity of the proposed fault reconstruction scheme. Finally, Section 6 draws some concluding remarks.

Throughout the paper, the following notations are used: 𝐱^𝑇, 𝐗^𝑇 denotes the transpose of the vector 𝐱 and matrix 𝐗, respectively, diag[⋅] denotes a block diagonal matrix, for a square matrix 𝐗=𝐗^𝑇>0 (resp., 𝐗=𝐗^𝑇<0) means that 𝐗 is a symmetric positive definite matrix (resp., symmetric negative definite matrix), the symbol 𝐈_𝑛 represents the nth order unit matrix, ℝ denotes the set of real numbers, and ℝ^𝑛×𝑟 denotes the set of all 𝑛×𝑟 real matrices.

2. Takagi-Sugeno Fuzzy Models

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 𝐪̇(𝑡)=𝐚(𝐪(𝑡))+𝐁(𝐪(𝑡))𝐮(𝑡)+𝐁_𝑓𝐟(𝑡),𝐲(𝑡)=𝐂𝐪(𝑡), (2.1) where 𝐪(𝑡)∈ℝ^𝑛, 𝐮(𝑡), 𝐮_𝑓(𝑡)∈ℝ^𝑟, and 𝐲(𝑡)∈ℝ^𝑚 are vectors of the state, input, actuator fault, and output variables, respectively, 𝐂∈ℝ^𝑚×𝑛, 𝐁_𝑓∈ℝ^𝑛×𝑟 are real matrices, 𝐚(𝐪(𝑡))∈ℝ^𝑛, 𝐁(𝐪(𝑡))∈ℝ^𝑛×𝑟 are bounded nonlinear functions of 𝐪(𝑡), and 𝐟(𝑡)∈ℝ^𝑟 is an actuator fault. It is assumed that 𝐚(𝐪(𝑡)) is bounded in associated sectors, that is, in the regions within the system will operate, 𝐚(0)=0, only actuator faults can occur, and if no actuator fault occurs then 𝐮_𝑓(𝑡)=𝟎, for all 𝑡≥0.

It is considered that the number of the nonlinear terms in the vector function 𝐚(𝐪(𝑡)) is 𝑝 and there exists the set of the nonlinear sector functions {𝑤_𝑙𝑗(𝜃_𝑗(𝑡)),𝑙=1,2,…,𝑝,𝑗=1,2,…,𝑘} such that 𝑤_𝑙1(𝜽(𝑡))=1−^𝑘_𝑗=2𝑤_𝑙𝑗𝜃_𝑗(𝑡), (2.2) where 𝑘 is the number of sector functions, and 𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)⋯𝜃_𝑞(𝑡) (2.3) is the vector of premise variables. A premise variable represents any measurable variable (in a simple case it can be directly a state variable) and none of the premise variables depend on the inputs 𝐮(𝑡).

Thus, constructing the set of membership functions 𝑤_𝑖(𝜽(𝑡))=∏^𝑝_𝑙=1|𝑗𝑤_𝑙𝑗(𝜃_𝑗(𝑡)), 𝑖=1,2,…𝑠,𝑠=2^𝑘 from all combinations of the sector functions, the states of the system with an actuator fault are inferred as follows: 𝐪̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)+𝐁_𝑖𝐮(𝑡)+𝐁_𝑓𝐟(𝑡),𝐲(𝑡)=𝐂𝐪(𝑡), (2.4)(2.5) where the system output is given by the relation (2.5) and ℎ_𝑖(𝜽(𝑡=𝑤))^𝑖(𝜽(𝑡))∑^𝑠_𝑖=1𝑤_𝑖(𝜽(𝑡)) (2.6) is the averaging weight for the ith rule, representing the normalized grade of membership (membership function). By definition, the membership functions satisfy the following convex sum properties: 0≤ℎ_𝑖(𝜽(𝑡))≤1,^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))=1,∀𝑖∈⟨1,…,𝑠⟩, (2.7)𝐀_𝑖∈ℝ^𝑛×𝑛 is the Jacobian matrix of 𝐚(𝐪(𝑡)) with respect to 𝐪(𝑡)=𝐪_𝑖, 𝐁_𝑖∈ℝ^𝑛×𝑟 is the matrix equal to 𝐁(𝐪_𝑖), and 𝐪_𝑖 is the center of the 𝑖-th fuzzy region, described by the associated sector function. It is evident that the fuzzy model is achieved by fuzzy amalgamation of the linear subsystem models.

Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set [19], but determines a function with state variables as arguments, and the corresponding function is a local function for the fuzzy region that is described by the premise part of the rule. Thus, using linear functions, a system state is described locally (in fuzzy regions) by linear models, and at the boundaries between regions the linear interpolation is used between the corresponding local models.

Note, the model (2.4), (2.5) is mostly considered for analysis, control, and state estimation of nonlinear systems.

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.

3. Basic Preliminaries

Definition 3.1 (null space of the matrix). Let 𝐄∈ℝ^ℎ×ℎ, rank(𝐄)=𝑘<ℎ be a rank deficient matrix. Then the null space 𝑁_𝐄 of 𝐄 is the orthogonal complement of the row space of 𝐄.

Lemma 3.2 (orthogonal complement). If 𝐄∈ℝ^ℎ×ℎ, rank(𝐄)=𝑘<ℎ, is a rank deficient matrix, then an orthogonal complement 𝐄^⟂ of 𝐄 is 𝐄^⟂=𝐄^∘𝐔^𝑇_𝐸2, (3.1) where 𝐔^𝑇_𝐸2 is the null space of 𝐄 and 𝐄^∘ is an arbitrary matrix of appropriate dimension.

Proof (see, e.g., [30]). The singular value decomposition (SVD) of 𝐄 gives 𝐔^𝑇_𝐸𝐄𝐕_𝐸=𝐔^𝑇_𝐸1𝐔^𝑇_𝐸2𝐄𝐕_𝐸1𝐕_𝐸2=𝐒_𝐸𝟎₁₂𝟎₂₁𝟎₂₂, (3.2) where 𝐔^𝑇_𝐸∈ℝ^ℎ×ℎ is the orthogonal matrix of the left singular vectors of 𝐄, 𝐕_𝐸∈ℝ^ℎ×ℎ is the orthogonal matrix of the right singular vectors of 𝐄, and 𝐒_𝐸∈ℝ^𝑘×𝑘 is the diagonal positive definite matrix of the form 𝐒_𝐸=diag𝜎_𝐸1⋯𝜎_𝐸𝑘,𝜎_𝐸1≥⋯≥𝜎_𝐸𝑘>0, (3.3) which diagonal elements are the singular values of 𝐄. Using orthogonal properties of 𝐔_𝐸 and 𝐕_𝐸, that is, 𝐔^𝑇_𝐸𝐔_𝐸=𝐈_ℎ,𝐕^𝑇_𝐸𝐕_𝐸=𝐈_ℎ, and 𝐔^𝑇_𝐸1𝐔^𝑇_𝐸2𝐔_𝐸1𝐔_𝐸2=𝐈₁𝟎𝟎𝐈₂,𝐔^𝑇_𝐸2𝐔_𝐸1=𝟎, (3.4) respectively, where 𝐈_ℎ∈ℝ^ℎ×ℎ is the identity matrix, then 𝐄 can be written as 𝐄=𝐔_𝐸𝐒_𝐸𝐕^𝑇_𝐸=𝐔_𝐸1𝐔_𝐸2𝐒_𝐸𝟎₁₂𝟎₂₁𝟎₂₂𝐕^𝑇_𝐸1𝐕^𝑇_𝐸2=𝐔_𝐸1𝐔_𝐸2𝐒_𝐸1𝟎₂=𝐔_𝐸1𝐒_𝐸1, (3.5) where 𝐒_𝐸1=𝐒_𝐸𝐕^𝑇_𝐸1. Thus, (3.4) and (3.5) imply 𝐔^𝑇_𝐸2𝐄=𝐔^𝑇_𝐸2𝐔_𝐸1𝐔_𝐸2𝐒_𝐸1𝟎₂=𝟎. (3.6) It is evident that for an arbitrary matrix 𝐄^∘ it is 𝐄^∘𝐔^𝑇_𝐸2𝐄=𝐄^⟂𝐄=𝟎, (3.7) which implies (3.1). This concludes the proof.

Lemma 3.3 (congruence transform). Let the output matrix 𝐂 be of full column rank, rank𝐂=𝑚, then there exists a new coordinate system such that 𝐂 takes the structure 𝐂_𝑎=[𝐈_𝑚0].

Proof. Applying SVD to 𝐂 gives 𝐂=𝐔𝐒𝟎𝐕^𝑇=𝐔𝐒𝐈_𝑚𝟎𝐕^𝑇, (3.8) where rows of 𝐔^𝑇∈ℝ^𝑚×𝑚 are left singular vectors of 𝐂, and columns of 𝐕∈ℝ^𝑛×𝑛 are right singular vectors of 𝐂, all ordered in such way to be associated with the singular values of 𝐂, written as diagonal elements of 𝐒∈ℝ^𝑚×𝑚, 𝐒=diag𝜎₁⋯𝜎_𝑚,𝜎₁≥⋯≥𝜎_𝑚>0. (3.9) Using the notations 𝐖⁻¹=𝐔𝐒,𝐓_𝑎=𝐕^𝑇,𝐂_𝑎=𝐈_𝑚𝟎, (3.10) where 𝐓^𝑇_𝑎=𝐓^𝑇_𝑎1𝐓^𝑇_𝑎2,𝐓^𝑇_𝑎1∈ℝ^𝑛×𝑚, (3.11) then (3.8) implies 𝐂=𝐖⁻¹𝐂_𝑎𝐓_𝑎,𝐂_𝑎=𝐖𝐂𝐓_𝑎⁻¹. (3.12) Note if 𝐂 is of rank 𝑚, 𝐖∈ℝ^𝑚×𝑚 is a regular matrix, and 𝐓_𝑎∈ℝ^𝑛×𝑛 is an orthogonal matrix such that 𝐓_𝑎⁻¹=𝐓^𝑇_𝑎=𝐕. This concludes the proof.

Lemma 3.4. Using the congruence transform (3.12), each linear submodel of fault-free TS fuzzy model (2.4), (2.5) can be partitioned such that 𝐪̇_𝑎1𝑖𝐪̇_𝑎2𝑖=𝐀_𝑎11𝑖𝐀_𝑎12𝑖𝐀_𝑎21𝑖𝐀_𝑎22𝑖𝐪_𝑎1(𝑡)𝐪_𝑎2(𝑡)+𝐁_𝑎1𝑖𝐁_𝑎2𝑖𝐮(𝑡)+𝐁_𝑎𝑓1𝐁_𝑎𝑓2𝐟(𝑡)𝐲(𝑡),=𝐖⁻¹𝐯(𝑡),𝐯(𝑡)=𝐈_𝑚𝟎𝐪_𝑎(𝑡)=𝐈_𝑚𝟎𝐪_𝑎1(𝑡)𝐪_𝑎2(𝑡)𝐪_𝑇𝑎,(𝑡)=𝐪^𝑇_𝑎1(𝑡)𝐪^𝑇_𝑎2(𝑡),𝐪_𝑎(𝑡)=𝐓_𝑎𝐪(𝑡), (3.13)(3.14)(3.15)(3.16) where 𝐪_𝑎1(𝑡)∈ℝ^𝑚, 𝐪_𝑎2(𝑡)∈ℝ^𝑛−𝑚, 𝐁_𝑎1𝑖∈ℝ^𝑚×𝑟, 𝐁_𝑎𝑓1∈ℝ^𝑚×𝑟, respectively.

Proof. Substituting (3.11) into (2.5) gives 𝐲(𝑡)=𝐖⁻¹𝐂_𝑎𝐓_𝑎𝐪(𝑡)=𝐖⁻¹𝐂_𝑎𝐪_𝑎(𝑡). (3.17) Thus, using 𝐯(𝑡)=𝐂_𝑎𝐪_𝑎(𝑡), (3.18) (3.17) implies (3.14), and with (3.11), (3.12), then (3.18) implies (3.15).
Substituting (3.16) in (2.4), it can be obtained 𝐪̇_𝑎(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑖𝐪_𝑎(𝑡)+𝐁_𝑎𝑖𝐮(𝑡)+𝐁_𝑎𝑓𝐟(𝑡), (3.19) where 𝐀_𝑎𝑖=𝐓_𝑎𝐀_𝑖𝐓_𝑎⁻¹,𝐁_𝑎𝑖=𝐓_𝑎𝐁_𝑖,𝐁_𝑎𝑓=𝐓_𝑎𝐁_𝑓, (3.20) and partitioning accordingly to (3.15), (3.20) implies (3.13). This concludes the proof.

Proposition 3.5 (matching condition). The fault input matrix and the output matrix 𝐁_𝑎𝑓, 𝐂 satisfies the conditions rank𝐁_af=𝑟, rank𝐂>rank𝐁_𝑓, respectively, that is, 𝑚>𝑟, and the matrix 𝐁_𝑎𝑓 takes the structure 𝐁_𝑎𝑓=𝐂^𝑇_𝑎𝐁_𝑎𝑓1. (3.21)

The matching condition, given in Proposition 3.5 seems to be restrictive theoretically, but fortunately, for many practical control systems it is satisfied. In addition, comparing with the static decoupling control principle [31], the condition reflects inserting at least one redundant output sensor into the sensor structure.

4. Full-Order TS Fuzzy Observer

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 [1, 2].

The fuzzy observer to the fault-free system (2.4), (2.5) is constructed as follows: 𝐪̇_𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀𝐪_𝑒(𝑡)+𝐁𝐮(𝑡)+𝐀_𝑖𝐪_𝑒(𝑡)+𝐉_𝑖𝐲−𝐲_𝑒(𝑡,𝐲_𝑒)(𝑡)=𝐂𝐪_𝑒(𝑡), (4.1)(4.2) where 𝐪_𝑒(𝑡)∈ℝ^𝑛 is the estimation of the system state vector, 𝐉_𝑖∈ℝ^𝑛×𝑚,𝑖=1,2,…,𝑠, is the set of the observer gain matrices. The design conditions are given by the next lemma.

Lemma 4.1. The fuzzy observer (4.1), (4.2) is stable if there exist a positive definite symmetric matrix 𝐏>0, 𝐏∈ℝ^𝑛×𝑛, and matrices 𝐙_𝑖∈ℝ^𝑛×𝑚, 𝑖=1,2,…,𝑠, such that 𝐏=𝐏^𝑇𝐀>0,_𝑖𝐏+𝐏𝐀_𝑖−𝐙^𝑇_𝑖𝐂^𝑇−𝐙_𝑖𝐂<0,∀𝑖. (4.3)(4.4)
If the above conditions hold, the set of the observer gain matrices is given as 𝐉_𝑖=𝐏⁻¹𝐙_𝑖. (4.5)

Proof. Introducing the estimation error between the fault-free (2.4) and (4.1) as follows: 𝐞(𝑡)=𝐪(𝑡)−𝐪_𝑒(𝑡), (4.6) and taking into account the time derivative of 𝐞(𝑡), it can be obtained 𝐞̇(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖−𝐉_𝑖𝐂𝐞(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑒𝑖𝐞(𝑡), (4.7) where the observer system matrices are 𝐀_𝑒𝑖=𝐀_𝑖−𝐉_𝑖𝐂,𝑖=1,2,…,𝑠. (4.8) Defining the quadratic positive definite Lyapunov function of the form 𝑣(𝐞(𝑡))=𝐞^𝑇(𝑡)𝐏𝐞(𝑡), (4.9) where 𝐏>0, then after evaluation of its derivative with respect to 𝑡 it is obtained 𝑣̇(𝐞(𝑡))=𝐞̇^𝑇(𝑡)𝐏𝐞(𝑡)+𝐞^𝑇(𝑡)𝐏𝐞̇(𝑡). (4.10) Substituting (4.7) in (4.10) gives 𝑣̇(𝐞(𝑡))=𝐞^𝑇(𝑡)𝐏^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑒𝑖𝐞(𝑡)+𝐞^𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀^𝑇_𝑒𝑖(𝑡),𝑣̇(𝐞(𝑡𝐏𝐞))=𝐞^𝑇(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐏𝐀_𝑒𝑖+𝐀^𝑇_𝑒𝑖𝐏𝐞(𝑡), (4.11)(4.12) respectively. It is evident that (4.12) is negative if there exist a set of gain matrices 𝐉_𝑖∈ℝ^𝑛×𝑚,𝑖=1,2,…,𝑠, and a symmetric positive definite matrix 𝐏∈ℝ^𝑛×𝑛 such that 𝐀_𝑖−𝐉_𝑖𝐂^𝑇𝐏+𝐏𝐀_𝑖−𝐉_𝑖𝐂<0,∀𝑖. (4.13) Setting 𝐏𝐉_𝑖=𝐙_𝑖, (4.14) (4.13) implies (4.4). This concludes the proof.

Note, to apply for actuator fault reconstruction, an adaptive structure of the full-order state observer can be used [12].

5. Reduced-Order TS Fuzzy Observer

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 (2.4), (2.5).

Theorem 5.1. Considering the affine TS fuzzy system (2.4), (2.5), then the reduced-order TS fuzzy observer takes the form 𝐩̇_2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘_2𝑒𝑖(𝑡),𝐪^∘_2𝑒𝑖(𝑡)=𝐀_𝑎𝑒𝑖𝐩_2𝑒(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡)𝐪_𝑎2𝑒,(𝑡)=𝐩_2𝑒(𝑡)+^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐯(𝑡), (5.1)(5.2)(5.3) where 𝐀_𝑎𝑣𝑖=𝐀_𝑎21𝑖−𝐉_𝑖𝐀_𝑎11𝑖+𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗,𝐀_𝑎𝑒𝑖=𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖, (5.4)(5.5) and 𝐉_𝑖∈ℝ^{(𝑛−𝑚)×𝑚},𝑖=1,2,…,𝑠 is the set of gains.

Proof. Since (3.13) can be partitioned as 𝐀_𝑎12𝑖𝐪_𝑎2(𝑡)=𝐪̇_𝑎1𝑖(𝑡)−𝐀_𝑎11𝑖𝐯(𝑡)−𝐁_𝑎1𝑖𝐮(𝑡),𝐪̇_𝑎2𝑖(𝑡)=𝐀_𝑎21𝑖𝐯(𝑡)+𝐀_𝑎22𝑖𝐪_𝑎2(𝑡)+𝐁_𝑎2𝑖𝐮(𝑡), (5.6)(5.7) then the TS fuzzy observer can be defined as follows: 𝐪̇_𝑎2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^•_{𝑎2𝑒𝑖}(𝑡),𝐪^•_{𝑎2𝑒𝑖}(𝑡)=𝐀_𝑎21𝑖𝐯(𝑡)+𝐀_𝑎22𝑖𝐪_𝑎2𝑒(𝑡)+𝐁_𝑎2𝑖𝐮(𝑡)+𝐉_𝑖𝐪̇_𝑎1𝑖(𝑡)−𝐀_𝑎11𝑖𝐯(𝑡)−𝐁_𝑎1𝑖𝐮(𝑡)−𝐀_𝑎12𝑖𝐪_𝑎2𝑒(𝑡), (5.8)(5.9) where 𝐪_𝑎2𝑒(𝑡)∈ℝ^𝑛−𝑚 is an estimation of the unmeasurable part of system state vector, and 𝐉_𝑖,𝑖=1,2,…,𝑠,𝐉_𝑖∈ℝ^{(𝑛−𝑚)×𝑚}, is the set of the observer gain matrices. Now, (5.8), (5.9) can be rewritten as 𝐪̇_𝑎2𝑒(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐪̇_𝑎1𝑖(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^⋄_𝑎2𝑖(𝑡),𝐪^⋄_𝑎2𝑖(𝑡)=𝐁_𝑎2𝑖𝐮(𝑡)+𝐀_𝑎21𝑖𝐯(𝑡)+𝐀_𝑎22𝑖𝐪_𝑎2𝑒(𝑡)−∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗𝐯(𝑡)+∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗𝐯(𝑡)+𝐉_𝑖−𝐀_𝑎11𝑖𝐯(𝑡)−𝐁_𝑎1𝑖𝐮(𝑡)−𝐀_𝑎12𝑖𝐪_𝑎2𝑒(𝑡)−∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗𝐯(𝑡)+∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗𝐯(𝑡). (5.10)(5.11) Defining the new state variable 𝐩_2𝑒(𝑡)=𝐪_𝑎2𝑒(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐯(𝑡), (5.12) then (5.12) implies (5.3), and defining the left side of (5.10) as 𝐩̇_2𝑒(𝑡)=𝐪̇_𝑎2𝑒(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐪̇_𝑎1𝑖(𝑡), (5.13) it can be obtained 𝐩̇_2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘_2𝑒𝑖𝐪^∘_2𝑒𝑖=𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖𝐩_2𝑒(𝑡)+𝐁_𝑎2𝑖−𝐉_𝑖𝐁_𝑎1𝑖𝐮(𝑡)+𝐀_𝑎21𝑖−𝐉_𝑖𝐀_𝑎11𝑖𝐯(𝑡)+𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐉_𝑗𝐯(𝑡). (5.14)(5.15) Since 𝐁_𝑎2𝑖−𝐉_𝑖𝐁_𝑎1𝑖=−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖 (5.16) with (5.4), (5.5), and (5.16) then (5.14), (5.15) implies (5.1), (5.2).
It is evident that 𝐯(𝑡)=𝐪_𝑎1(𝑡)=𝐩₁(𝑡). (5.17) This concludes the proof.

Theorem 5.2 (reducer-order TS fuzzy observer stability). The reduced-order TS fuzzy observer (5.1), (5.2) is asymptotically stable if there exist a symmetric positive definite matrix 𝐏^∘∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} and matrices 𝐙^∘_𝑖∈ℝ^{(𝑛−𝑚)×𝑚},𝑖=1,2,…,𝑠 such that 𝐏^∘=𝐏^∘𝑇𝐀>0,_{𝑇𝑎22𝑖}𝐏^∘+𝐏^∘𝐀_𝑎22𝑖−𝐀^𝑇_𝑎12𝑖𝐙_𝑖^∘𝑇−𝐙^∘_𝑖𝐀_𝑎12𝑖<0. (5.18)(5.19) If the above conditions hold, the set of the observer gain matrices is given as 𝐉_𝑖=𝐏^∘−1𝐙^∘_𝑖. (5.20)

Proof. Using (5.1), (5.2), it can be rewritten as 𝐩̇_2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐩_2𝑒(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡), (5.21) and, with (5.4), the autonomous part of (5.21) takes the form 𝐩̇_2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖𝐩_2𝑒(𝑡). (5.22) Defining the quadratic positive definite Lyapunov function: 𝑣𝐩_2𝑒(𝑡)=𝐩^𝑇_2𝑒(𝑡)𝐏^∘𝐩_2𝑒(𝑡), (5.23) where 𝐏^∘=𝐏^∘𝑇>0, 𝐏^∘∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} then, after evaluation of derivative with respect to 𝑡, it is obtained 𝑣̇𝐩_2𝑒(𝑡)=𝐩^𝑇_2𝑒(𝑡)𝐏^∘𝐩̇_2𝑒(𝑡)+𝐩̇^𝑇_2𝑒(𝑡)𝐏^∘𝐩_2𝑒(𝑡)<0. (5.24) Substituting (5.22) into (5.24) gives 𝑣̇𝐩_2𝑒(𝑡)=𝐩^𝑇_2𝑒(𝑡)𝐏^∘𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐩_2𝑒(𝑡)+𝐩^𝑇_2𝑒(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀^𝑇_𝑎𝑒𝑖𝐏^∘𝐩_2𝑒(𝑡)𝑣̇𝐩<0,_2𝑒(𝑡)=𝐩^𝑇_2𝑒(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐏^∘𝐀_𝑎𝑒𝑖+𝐀^𝑇_𝑎𝑒𝑖𝐏^∘𝐩_2𝑒(𝑡)<0, (5.25)(5.26) respectively. Thus, (5.26) is negative, if there exist a set of matrices 𝐉_𝑖,𝑖=1,2,…,𝑠, and a matrix 𝐏^∘ such that 𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖^𝑇𝐏^∘+𝐏^∘𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖<0,∀𝑖. (5.27) Setting 𝐏^∘𝐉_𝑖=𝐙^∘_𝑖 (5.28) (5.27) implies (5.19). This concludes the proof.

Remark 5.3. If (5.19) is infeasible, then it can be set 𝐀^𝑇_𝑎22𝑖𝐏+𝐏𝐀_𝑎22𝑖−𝐀^𝑇_𝑎12𝑖𝐙^𝑇_𝑖−𝐙_𝑖𝐀_𝑎12𝑖<−𝐐, (5.29) where 𝐐=𝐐^𝑇>0, 𝐐∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} is a symmetric positive definite matrix. It is obvious, based on the inequality (5.29), such design condition implies inherently the more conservative solution.

Theorem 5.4 (stability condition equivalency). The asymptotic stability condition of the autonomous part of (5.22) is the same as the asymptotic stability condition of the error reference model as follows: 𝐞̇_𝑎𝑞2(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖𝐞_𝑎𝑞2(𝑡), (5.30) where the estimation error of the unmeasurable part of state variables is 𝐞_𝑎𝑞2(𝑡)=𝐪_𝑎2(𝑡)−𝐪_𝑎2𝑒(𝑡). (5.31)

Proof. Substituting (5.3) in (5.31) gives 𝐞_𝑎2(𝑡)=𝐪_𝑎2(𝑡)−𝐩_2𝑒(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐯(𝑡). (5.32) Defining, in analogy with (5.1)–(5.3), the reference variable 𝐩₂(𝑡) is as follows: 𝐩₂(𝑡)=𝐪_𝑎2(𝑡)−^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐉_𝑖𝐯(𝑡),𝐩̇₂(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘_2𝑖(𝑡),𝐪^∘_2𝑖(𝑡)=𝐀_𝑎𝑒𝑖𝐩₂(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡), (5.33)(5.34)(5.35) and substituting (5.33) in (5.32) gives 𝐞_𝑎𝑞2(𝑡)=𝐩₂(𝑡)−𝐩_2𝑒(𝑡)=𝐞_𝑝2(𝑡),𝐞̇_𝑎𝑞2(𝑡)=𝐞̇_𝑝2(𝑡)=𝐩̇₂(𝑡)−𝐩̇_2𝑒(𝑡), (5.36)(5.37) respectively. Thus, inserting (5.1), (5.2) and (5.34), (5.35) into (5.37) results in 𝐞̇_𝑎𝑞2(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐩₂(𝑡)−𝐩_2𝑒(𝑡), (5.38) and with (5.5), (5.36) then (5.38) implies (5.30).
Since (5.22) and (5.36) are associated with the same system matrix 𝐀_𝑎𝑒𝑖, this concludes the proof.

Note, the form of the time derivative (5.13) is given by the definition.

Corollary 5.5 (error reference model). The equalities (5.17), (5.33) can be compactly written as 𝐩₁(𝑡)𝐩₂(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐓_𝑖𝐀_𝑎𝑖𝐪_𝑎𝑖(𝑡)+𝐁_𝑎𝑖𝐮(𝑡), (5.39) where 𝐓_𝑖=𝐈_𝑚𝟎−𝐉_𝑖𝐈_𝑛−𝑚,𝐓_𝑖𝐁_𝑎𝑖𝐮(𝑡)=𝐈_𝑚𝟎−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎1𝑖𝐁_𝑎2𝑖𝐮(𝑡),𝐓_𝑖𝐀_𝑎𝑖𝐪_𝑎(𝑡)=𝐀_𝑎11𝑖𝐀_𝑎12𝑖𝐀_𝑎21𝑖−𝐉_𝑖𝐀_𝑎11𝑖𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖𝐯(𝑡)𝐪_𝑎2(𝑡). (5.40)
Ones explaining the variable 𝐩₂(𝑡) as a function of 𝐪_𝑎2(𝑡) using (5.39), then the time derivative of 𝐩₂(𝑡) can be obtained by the substitution (5.33), that is, 𝐩̇₂(𝑡)=𝐩₂(𝑡)𝐪_𝑎2(𝑡)_{‖𝐪𝑎2(𝑡)≡𝐩2(𝑡)+∑}^𝑠_{𝑗=1ℎ𝑗(𝜽(𝑡))𝐉𝑗𝐯(𝑡)}. (5.41)

Evidently, (5.39)–(5.41) can be adequately exploited to obtain the time derivative 𝐩̇_2𝑒(𝑡) in the dependency on 𝐩_2𝑒(𝑡), 𝐯(𝑡), and 𝐮(𝑡).

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

6. Estimation of Actuator Faults

To obtain an actuator fault estimation structure based on reduced-order TS fuzzy observer, the matching condition (3.21) has to be satisfied. This implies that a special observer gain matrices have to be chosen.

Theorem 6.1. The estimation error dynamics of the reduced-order TS fuzzy observer (5.1), (5.2) is not affected by actuator faults if, with the matching condition (3.21), there exists a symmetric positive definite matrix 𝐏^∘∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} such that 𝐏^∘𝐉=𝐁^⟂_𝑎𝑓1, (6.1) where 𝐁^⟂_𝑎𝑓1 is the orthogonal complement to 𝐁_𝑎𝑓1, and 𝐉_𝑖=𝐉 for all 𝑖=1,2,…,𝑠.

Proof. The system with an actuator fault is described as 𝐪̇_𝑎𝑓𝑖(𝑡)=𝐀_𝑎𝑖𝐪_𝑎𝑓𝑖(𝑡)+𝐁_𝑎𝑖𝐮(𝑡)+𝐁_𝑎𝑓𝐮_𝑓(𝑡). (6.2) Since (3.13), (5.1), (5.2), and (5.34), (5.35) now implies 𝐁_𝑎𝑓=𝐂^𝑇_𝑎𝐁_𝑎𝑓1=𝐈_𝑚𝟎𝐁_𝑎𝑓1=𝐁_𝑎𝑓1𝟎𝐩̇_2𝑒,(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘_{2𝑒𝑓𝑖}(𝑡),𝐩̇₂(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪^∘_2𝑖(𝑡),𝐪^∘_{2𝑒𝑓𝑖}(𝑡)=𝐀_𝑎𝑒𝑖𝐩_2𝑒(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡)+𝐁_𝑎𝑓𝐮_𝑓(𝑡𝐪),_∘2𝑖(𝑡)=𝐀_𝑎𝑒𝑖𝐩₂(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡), (6.3)(6.4)(6.5)(6.6) the dynamics of the error (5.37) can be rewritten as 𝐞̇_𝑎𝑞2(𝑡)=𝐩̇₂(𝑡)−𝐩̇_2𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐞_𝑎𝑞2(𝑡)+^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑓𝐮_𝑓(𝑡) (6.7) Defining the quadratic positive definite Lyapunov function as follows: 𝑣𝐞_𝑎𝑞2(𝑡)=𝐞^𝑇_𝑎𝑞2(𝑡)𝐏^∘𝐞_𝑎𝑞2(𝑡), (6.8) where 𝐏^∘=𝐏^∘𝑇>0, 𝐏^∘∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)}, after evaluation of derivative of (6.7) with respect to 𝑡 it is obtained 𝑣̇𝐞_𝑎𝑞2(𝑡)=𝐞̇^𝑇_𝑎𝑞2(𝑡)𝐏^∘𝐞_𝑎𝑞2(𝑡)+𝐞^𝑇_𝑎𝑞2(𝑡)𝐏^∘𝐞̇_𝑎𝑞2(𝑡). (6.9) From the expression (6.7) it follows that 𝑣̇𝐞_𝑎𝑞2(𝑡)=𝐞^𝑇_𝑎𝑞2(𝑡)𝐏^∘𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐞_𝑎𝑞2(𝑡)+𝐞^𝑇_𝑎𝑞2(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀^𝑇_𝑎𝑒𝑖𝐏^∘𝐞_𝑎2(𝑡)+𝐞^𝑇_𝑎𝑞2(𝑡)𝐏^∘𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑓𝐮_𝑓(𝑡)+𝐮^𝑇_𝑓(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐁^𝑇_𝑎𝑓−𝐉_𝑖𝐈_𝑛−𝑚^𝑇𝐏^∘𝐞^𝑇_𝑎𝑞2(𝑡), (6.10) and with respect to the matching condition (3.21), it can be set 𝐏^∘−𝐉_𝑖𝐈_𝑛−𝑚𝐁_𝑎𝑓=𝐏^∘−𝐉_𝑖𝐁_𝑎𝑓1𝟎_𝑛−𝑚=𝟎, (6.11) which results in the equality: 𝐏^∘𝐉_𝑖𝐁_𝑎𝑓1=𝟎,∀𝑖. (6.12) Evidently, (6.12) can be satisfied if and only if 𝐉_𝑖=𝐉 for all 𝑖. With such 𝐉, the equality (6.12) will be satisfied if (6.1) is satisfied. This concludes the proof.

Evidently, 𝐁_𝑎𝑓1 may not be a square matrix.

Theorem 6.2. The estimation error dynamic (6.7) is asymptotically stable, if there exists a symmetric positive definite matrix  𝐏^∘∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} such that for 𝑖=1,2,…,𝑠𝐏^∘=𝐏^∘𝑇𝐀>0,_{𝑇𝑎22𝑖}𝐏^∘+𝐏^∘𝐀_𝑎22𝑖−𝐀^𝑇_𝑎12𝑖𝐁^⟂𝑇_𝑎𝑓1−𝐁^⟂_𝑎𝑓1𝐀_𝑎12𝑖<0, (6.13)(6.14) where 𝐁^⟂_𝑎𝑓1 is the orthogonal complement to 𝐁_𝑎𝑓1.
If the above conditions hold, the observer gain matrix is given as 𝐉=𝐏^∘−1𝐁^⟂_𝑎𝑓1. (6.15)

Proof. Satisfying (6.11) then (6.10) implies 𝑣̇𝐞_𝑎𝑞2(𝑡)=𝐞^𝑇_𝑎𝑞2(𝑡)^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐏^∘𝐀_𝑎𝑒𝑖+𝐀^𝑇_𝑎𝑒𝑖𝐏^∘𝐞_𝑎𝑞2(𝑡)<0, (6.16) where 𝐀_𝑎𝑒𝑖 is defined in (5.5). It is evident that (6.16) is negative if for all 𝑖𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖^𝑇𝐏^∘+𝐏^∘𝐀_𝑎22𝑖−𝐉_𝑖𝐀_𝑎12𝑖<0. (6.17) Using (6.1) then (6.17) implies (5.19). This concludes the proof.

Corollary 6.3. Using (5.20), then one has −𝐉𝐈_𝑛−𝑚𝐁_𝑎𝑓=𝐏^∘−1−𝐁^⟂_𝑎𝑓1𝐏^∘𝐂^𝑇𝐁_𝑎𝑓1=𝐏^∘−1−𝐁^⟂_𝑎𝑓1𝐁_𝑎𝑓1𝟎=𝟎. (6.18) Equality given above implies that neither estimation error (6.7), nor reduced-order TS fuzzy observer equation (6.5) is affected by actuator faults.

Corollary 6.4. Since 𝐉_𝑖=𝐉 for all 𝑖 and ∑^𝑠_𝑗=1ℎ_𝑗(𝜽(𝑡))=1, (5.12) and (5.19) take the form 𝐩_2𝑒(𝑡)=𝐪_𝑎2𝑒(𝑡)(𝑡),𝐀−𝐉𝐯_𝑎𝑣𝑖=𝐀_𝑎21𝑖−𝐉𝐀_𝑎11𝑖+𝐀_𝑎22𝑖−𝐉𝐀_𝑎12𝐉. (6.19)(6.20)

Remark 6.5. If 𝐁_𝑎𝑖=𝐁_𝑎=𝐁_𝑎𝑓 for all 𝑖, (6.18) implies that the reduced-order TS fuzzy observer will be independent on the input 𝐮(𝑡) and will exploit only the vector variable 𝐯(𝑡).

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.

Theorem 6.6. Designed with respect to 𝐏^∘ satisfying (5.18)–(5.20), the reduced-order TS fuzzy observer (5.1), (5.2) asymptotically estimates actuator faults.

Proof. Since (3.11), (3.14) implies 𝐪_𝑒(𝑡)=𝐓^𝑇_𝑎𝐪_𝑎𝑒(𝑡)=𝐓^𝑇_𝑎2𝐪_𝑎2𝑒(𝑡)+𝐓^𝑇_𝑎1𝐯(𝑡), (6.21) substituting (6.19) in (6.21) leads to 𝐪_𝑒(𝑡)=𝐓^𝑇_𝑎2𝐩_2𝑒(𝑡)+𝐓^𝑇_𝑎1+𝐓^𝑇_𝑎2𝐉(𝑡),𝐪̇𝐖𝐲_𝑒(𝑡)=^𝑠_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪_𝑒(𝑡)+𝐁_𝑖𝐮(𝑡)+𝐁_𝑓𝐮_𝑓(𝑡). (6.22)(6.23) Thus, using Moore-Penrose pseudoinverse 𝐁_𝑓^⊖1 of 𝐁_𝑓, 𝐮_𝑓𝑒(𝑡)=𝐁_𝑓^𝑠⊖1_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪̇_𝑒(𝑡)−𝐀_𝑖𝐪_𝑒(𝑡)−𝐁_𝑖𝐮(𝑡). (6.24) Explaining 𝐮_𝑓(𝑡) as follows: 𝐮_𝑓(𝑡)=𝐁_𝑓^𝑠⊖1_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪̇(𝑡)−𝐀_𝑖𝐪_𝑒(𝑡)−𝐁_𝑖𝐮(𝑡), (6.25) then for 𝐞_𝑢𝑓(𝑡)=𝐮_𝑓(𝑡)−𝐮_𝑓𝑒(𝑡) it yields 𝐞_𝑢𝑓(𝑡)=𝐁_𝑓^𝑠⊖1_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐞̇_𝑞(𝑡)−𝐀_𝑖𝐞_𝑞(𝑡). (6.26) Owing to that reduced-order observer is stable, 𝐞_𝑢𝑓(𝑡) converges asymptotically to zero. This concludes the proof.

Remark 6.7. Taking the actuator fault reconstructor as given by (6.24), it is necessary to note that 𝐪̇_𝑒(𝑡) has to be computed numerically from (6.22), since (5.7) is affected by actuator faults and so cannot be used to the first state vector component derivative evaluation.

Note, matrix pseudoinverse in (6.24) is the reason that 𝐁_𝑓 has to be the same in all linear submodels of (2.4).

7. Illustrative Example

Referring to [32], a nonlinear hydrostatic transmission system is considered in this section for simulating the real environment. The proposed design method is applied to design an actuator fault estimation scheme based on a reduced-order fuzzy observer using TS model of this model. The hydrostatic transmission system is represented by the nonlinear state-space model of the form 𝑞̇₁(𝑡)=−𝑎₁₁𝑞₁(𝑡)+𝑏₁₁𝑢₁(𝑡)𝑞̇₂(𝑡)=−𝑎₂₂𝑞₂(𝑡)+𝑏₂₂𝑢₂(𝑡),𝑞̇₃(𝑡)=𝑎₃₁𝑞₁(𝑡)𝑝(𝑡)−𝑎₃₃𝑞₃(𝑡)−𝑎₃₄𝑞₂(𝑡)𝑞₄(𝑡),𝑞̇₄(𝑡)=𝑎₄₃𝑞₂(𝑡)𝑞₃(𝑡)−𝑎₄₄𝑞₄(𝑡), (7.1) where 𝑞₁(𝑡) is the normalized hydraulic pump angle, 𝑞₂(𝑡) is the normalized hydraulic motor angle, 𝑞₃(𝑡) is the pressure difference [bar], 𝑞₄(𝑡) is the hydraulic motor speed [rad/s], 𝑢₁(𝑡) is the normalized control signal of the hydraulic pump, 𝑢₂(𝑡) is the normalized control signal of the hydraulic motor, and the external signal 𝑝(𝑡) represents speed of hydraulic pump [rad/s]. It is supposed that the external variable 𝑝(𝑡) and all state variables except 𝑞₃(𝑡) are measurable and the model parameters are 𝑎₁₁=7.6923,𝑎₂₂=4.5455,𝑎₃₃=7.6054∗10⁻⁴,𝑎₃₁=0.7877,𝑎₃₄=0.9235,𝑏₁₁=1.8590∗10³,𝑎₄₃=12.1967,𝑎₄₄=0.4143,𝑏₂₂=1.2879∗10³. (7.2) Since the variables 𝑝(𝑡)∈⟨105,300⟩ and 𝑞₂(𝑡)∈⟨0,1⟩ are bounded on the prescribed sectors, the vector of premise variables was chosen as follows: 𝜽(𝑡)=𝜃₁(𝑡)𝜃₂(𝑡)=𝑞₂(𝑡)𝑝(𝑡), (7.3) where the set of nonlinear sector functions: 𝑤₁₁𝑞₂(𝑡)=𝑏₁−𝑞₂(𝑡)𝑏₁−𝑏₂,𝑤₁₂𝑞₂(𝑡)=1−𝑤₁₁𝑞₂(𝑡),𝑏₁=0,𝑏₂𝑤=1,₂₁𝑝(𝑡)=𝑐₁−𝑝(𝑡)𝑐₁−𝑐₂,𝑤₂₂𝑝(𝑡)=1−𝑤₂₁𝑝(𝑡),𝑐₁=105,𝑐₂=300 (7.4) implies the next set of normalized membership functions: ℎ₁(𝜽(𝑡))=𝑤₁₁𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₂(𝜽(𝑡))=𝑤₁₁𝑞₂(𝑡)𝑤₂₂𝑝(𝑡),ℎ₃(𝜽(𝑡))=𝑤₁₂𝑞₂(𝑡)𝑤₂₁𝑝(𝑡),ℎ₄(𝜽(𝑡))=𝑤₁₂𝑞₂(𝑡)𝑤₂₂𝑝(𝑡). (7.5) The overall TS fuzzy model (2.4), (2.5) with an actuator fault is represented as follows: 𝐪̇(𝑡)=⁴_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑖𝐪(𝑡)(𝐮(𝑡)(𝑡,𝐲(𝑡)(𝑡),+𝐁+𝐟))=𝐂𝐪 (7.6) where 𝐀₁=,𝐀−7.69230000−4.54550082.70860−0.00080000−0.4143₂=,𝐀−7.69230000−4.545500236.31030−0.00080000−0.4143₃=,𝐀−7.69230000−4.54550082.70860−0.0008−0.92350012.1967−0.4143₄=,𝐁−7.69230000−4.545500236.31030−0.0008−0.92350012.1967−0.4143_𝑖=𝐁=𝐁_𝑓=1.8590001.28790000×10³,𝑖=1,2,3,4,𝐂=.100001000001 (7.7) Note that rank(𝐂)>rank(𝐁) and Proposition 3.5 are satisfied.

Using SVD of the output matrix 𝐂, (3.10) implies 𝐔=𝐒=𝐖⁻¹=𝐈₃,𝐕=𝐓_𝑎=diag𝐈₂=,𝐓01101000010000010010_𝑎1=100001000001,𝐓_𝑎2=0010, (7.8) and with such defined 𝐓_𝑎, for 𝑖=1,2,3,4, it yields 𝐂_𝑎=100001000010,𝐁_𝑎1=1.8590001.287900×10³,𝐁_𝑎2=,𝐀00_𝑎211=82.708600,𝐀_𝑎212=,𝐀236.310300_𝑎213=82.70860−0.9235,𝐀_𝑎214=,𝐀236.31030−0.9235_𝑎121=𝐀_𝑎122=000,𝐀_𝑎123=𝐀_𝑎124=00,𝐀12.1967_𝑎11𝑖=−7.6923000−4.5455000−0.4143,𝐀_𝑎22𝑖=−0.0008,𝑖=1,2,3,4. (7.9) Considering the conditions given in Theorem 6.1, the reduced-order observer (5.1)–(5.5) takes now the form 𝐩̇_2𝑒(𝑡)=⁴_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐀_𝑎𝑒𝑖𝐩_2𝑒(𝑡)+𝐀_𝑎𝑣𝑖𝐯(𝑡)+−𝐉𝐈_𝑛−𝑚𝐁_𝑎𝑖𝐮(𝑡), (7.10) where 𝐀_𝑎𝑣𝑖=𝐀_𝑎21𝑖−𝐉𝐀_𝑎11𝑖+𝐀_𝑎22𝑖−𝐉𝐀_𝑎12𝑖𝐉,𝐀_𝑎𝑒𝑖=𝐀_𝑎22𝑖−𝐉𝐀_𝑎12𝑖, (7.11) and 𝐉∈ℝ^1×3 is given by (6.15) as follows: 𝐉=𝐏^∘−1𝐁^⟂_𝑎1, (7.12) where 𝐁^⟂_𝑎1 is an orthogonal complement to 𝐁_𝑎1.

The scalar LMI variable 𝑃^∘ can be found by using the convex optimization techniques if 𝐁^⟂_𝑎1 is defined as a structured LM variable of the form 𝐁^⟂_𝑎1=00𝑍,𝑍∈ℝ, (7.13) where 𝑍 is an LMI variable. Note that a structured matrix variable can be specified only by including LMI matrix variables multiplied by a natural number or zero.

Thus, solving (6.13), (6.14) with respect to the LMI variables 𝑃^∘, 𝑍 using Self-Dual-Minimization (SeDuMi) package for Matlab [33], the reduced observer gain design problem was feasible with the results 𝑃^∘=1.0832,𝐁^⟂_𝑎1=,𝐀000.0410,𝐉=000.0378_𝑎𝑒2=𝐀_𝑎𝑒1=𝐀_𝑎22−𝐉𝐀_𝑎121=−7.6053×10⁻⁴,𝐀_𝑎𝑒4=𝐀_𝑎𝑒3=𝐀_𝑏22−𝐉𝐀_𝑎123=−0.4610. (7.14) It is evident that the design of the stable reduced-order observer with suppressed input fault signals is now completely specified, and the system state can be reconstructed, using (6.22), from the estimated vector 𝐩_2𝑒(𝑡) and the output vector 𝐲(𝑡) as 𝐪_𝑒(𝑡)=0010𝐩_2𝑒(𝑡)++0010𝐲(𝑡)100010000001000.0378100010001. (7.15) Evidently, the final form of the state reconstruction equation is 𝐪_𝑒(𝑡)=0010𝐩_2𝑒(𝑡)+𝐲(𝑡)100010000.0378001, (7.16) and actuator faults, if were occurred, can be computed by (6.24) as 𝐮_𝑓𝑒(𝑡)=𝐁^4⊖1_𝑖=1ℎ_𝑖(𝜽(𝑡))𝐪̇_𝑒(𝑡)−𝐀_𝑖𝐪_𝑒(𝑡)−𝐁𝐮(𝑡), (7.17) where 𝐁^⊖1=0.537900000.776500×10⁻³, (7.18) and 𝜽(𝑡), ℎ_𝑖(𝜽(𝑡))(𝐪̇_𝑒(𝑡)), 𝐀_𝑖, and 𝐁 are above specified. The derivative of the system state estimation 𝐪̇_𝑒(𝑡) was computed by standard numerical method from the obtained 𝐪_𝑒(𝑡). Since the reduced-order observer is used only, for system without uncertainties no extra computation consumption is needed, comparing, for example, with the sliding mode approach.

For simulation purposes only, the equilibrium of the system was stabilized by the fuzzy feedback controller 𝐮(𝑡)=−⁴_𝑗=1ℎ_𝑗(𝜽(𝑡))𝐊_𝑗𝐪(𝑡), (7.19) where, using the method proposed in [34], offering the possibility to design the linear state controller for TS fuzzy system, the gain matrices were computed as 𝐊=𝐊_𝑗=0.23860.00000.03500.00750.00000.02070.00000.0000,𝑗=1,2,3,4. (7.20)

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 1 and 2. Thus, Figure 1 presents the fault signal reflecting single actuator fault in the the second actuator, starting at the time instant 𝑡=20s and continuing during the time 10 s, and Figure 2 illustrates the signals 𝑢¹_𝑓, 𝑢²_𝑓 obtained from (6.26) as a reconstruction of the single fault. Note that equivalent results are obtained for the system working in a forced regime.

Figure 1 

The second actuator fault signal.

Figure 2 

The reconstruction of the actuator fault signal.

From the simulation results of Figures 1 and 2, it can be found that the errors between the signals reflecting a single actuator fault and the observer approximate ones tend to zero. Moreover, the states of the system converge to the equilibrium when the actuator fault disappeared, via the used fuzzy controller.

8. Concluding Remarks

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.

Acknowledgment

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.