Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

Abstract

The robust local regularity and controllability problem for the Takagi-Sugeno (TS) fuzzy descriptor systems is studied in this paper. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on parameter uncertainties for preserving the assumed properties. An example is given to illustrate the application of the proposed sufficient condition.

1. Introduction

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno [1], known as the TS fuzzy model, is a successful approach for dealing with the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to the nonlinear control systems (e.g., [2–19] and references therein). Descriptor systems represent a much wider class of systems than the standard systems [20]. In recent years, some researchers (e.g., [4–6, 8, 21–28] and references therein) have studied the design issue of the fuzzy parallel-distributed-compensation (PDC) controllers for each fuzzy rule of the TS fuzzy descriptor systems. Both regularity and controllability are actually two very important properties of descriptor systems with control inputs [29]. So, before the design of the fuzzy PDC controllers in the corresponding rule of the TS fuzzy descriptor systems, it is necessary to consider both properties of local regularity and controllability for each fuzzy rule [23]. However, both regularity and controllability of the TS fuzzy systems are not considered by those mentioned-above researchers before the fuzzy PDC controllers are designed. Therefore, it is meaningful to further study the criterion that the local regularity and controllability for each fuzzy rule of the TS fuzzy descriptor systems hold [30].

On the other hand, in fact, in many cases it is very difficult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters. These parametric uncertainties may destroy the local regularity and controllability properties of the TS fuzzy descriptor systems. But, to the authors’ best knowledge, there is no literature to study the issue of robust local regularity and controllability for the uncertain TS fuzzy descriptor systems.

The purpose of this paper is to present an approach for investigating the robust local regularity and controllability problem of the TS fuzzy descriptor systems with structured parameter uncertainties. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on structured parameter uncertainties for preserving the assumed properties. A numerical example is given in this paper to illustrate the application of the proposed sufficient criterion.

2. Robust Local Regularity and Controllability Analysis

Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of premise part and the linear dynamic model with parametric uncertainties of consequent part in the exact TS fuzzy control model with parametric uncertainties can be derived from the given nonlinear control model with parametric uncertainties [5]. The TS continuous-time fuzzy descriptor system with parametric uncertainties for the nonlinear control system with structured parametric uncertainties can be obtained as the following form: 𝑅^𝑖:IF𝑧₁is𝑀_𝑖1and…and𝑧_𝑔is𝑀_𝑖𝑔,then𝐸_𝑖𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝑢(𝑡), (2.1) or the uncertain discrete-time TS fuzzy descriptor system can be described by 𝑅^𝑖:IF𝑧₁is𝑀_𝑖1and…and𝑧_𝑔is𝑀_𝑖𝑔,then𝐸_𝑖𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘)+𝐵_𝑖+Δ𝐵_𝑖𝑢(𝑘), (2.2) with the initial state vector 𝑥(0), where 𝑅^𝑖(𝑖=1,2,…,𝑁) denotes the 𝑖th implication, 𝑁 is the number of fuzzy rules, 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 and 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇 denote the 𝑛-dimensional state vectors, 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑝(𝑡)]^𝑇 and 𝑢(𝑘)=[𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑝(𝑘)]^𝑇 denote the 𝑝-dimensional input vectors, 𝑧_𝑖(𝑖=1,2,…,𝑔) are the premise variables, 𝐸_𝑖, 𝐴_𝑖, and 𝐵_𝑖(𝑖=1,2,…,𝑁) are, respectively, the 𝑛×𝑛, 𝑛×𝑛 and 𝑛×𝑝 consequent constant matrices, Δ𝐴_𝑖 and Δ𝐵_𝑖(𝑖=1,2,…,𝑁) are, respectively, the parametric uncertain matrices existing in the system matrices 𝐴_𝑖 and the input matrices 𝐵_𝑖 of the consequent part of the 𝑖th rule due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters, and 𝑀_𝑖𝑗(𝑖=1,2,…,𝑁 and 𝑗=1,2,…,𝑔) are the fuzzy sets. Here the matrices 𝐸_𝑖(𝑖=1,2,…,𝑁) may be singular matrices with rank(𝐸_𝑖)≤𝑛(𝑖=1,2,…,𝑁). In many applications, the matrices 𝐸_𝑖(𝑖=1,2,…,𝑁) are the structure information matrices; rather than parameter matrices, that is, the elements of 𝐸_𝑖(𝑖=1,2,…,𝑁) contain only structure information regarding the problem considered.

In many interesting problems (e.g., plant uncertainties, constant output feedback with uncertainty in the gain matrix), we have only a small number of uncertain parameters, but these uncertain parameters may enter into many entries of the system and input matrices [31, 32]. Therefore, in this paper, we suppose that the parametric uncertain matrices Δ𝐴_𝑖 and Δ𝐵_𝑖 take the forms Δ𝐴_𝑖=^𝑚_𝑘=1𝜀_𝑖𝑘𝐴_𝑖𝑘,Δ𝐵_𝑖=^𝑚_𝑘=1𝜀_𝑖𝑘𝐵_𝑖𝑘, (2.3) where 𝜀_𝑖𝑘(𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚) are the elemental parametric uncertainties, and 𝐴_𝑖𝑘 and 𝐵_𝑖𝑘(𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚) are, respectively, the given 𝑛×𝑛 and 𝑛×𝑝 constant matrices which are prescribed a priori to denote the linearly dependent information on the elemental parametric uncertainties 𝜀_𝑖𝑘.

In this paper, for the uncertain TS fuzzy descriptor system in (2.1) (or (2.2)), each fuzzy-rule-nominal model 𝐸_𝑖𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡) or 𝐸_𝑖𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘), which is denoted by {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖}, is assumed to be regular and controllable. Due to inevitable uncertainties, each fuzzy-rule-nominal model {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is perturbed into the fuzzy-rule-uncertain model {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}. Our problem is to determine the conditions such that each fuzzy-uncertain model {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}for the uncertain TS fuzzy descriptor system (2.1) (or (2.2)) is robustly locally regular and controllable. Before we investigate the robust properties of regularity and controllability for the uncertain TS fuzzy descriptor system (2.1) (or (2.2)), the following definitions and lemmas need to be introduced first.

Definition 2.1 (see [33]). The measure of a matrix 𝑊∈𝐶^𝑛×𝑛 is defined as 𝜇𝑊≡lim_𝜃→0𝐼+𝜃𝑊−1𝜃, (2.4) where ‖⋅‖ is the induced matrix norm on 𝐶^𝑛×𝑛.

Definition 2.2 (see [34]). The system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is called controllable, if for any 𝑡₁>0 (or 𝑘₁>0), 𝑥(0)∈𝑅^𝑛, and 𝑤∈𝑅^𝑛, there exists a control input 𝑢(𝑡) (or 𝑢(𝑘)) such that 𝑥(𝑡₁)=𝑤 (or 𝑥(𝑘₁)=𝑤).

Definition 2.3. The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally regular, if each fuzzy-rule-uncertain model {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}(𝑖=1,2,…,𝑁) is regular.

Definition 2.4. The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally controllable, if each fuzzy-rule-uncertain model {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}(𝑖=1,2,…,𝑁) is controllable.

Lemma 2.5 (see [34]). The system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is regular if and only if rank[𝐸_𝑛𝑖𝐵_𝑑𝑖]=𝑛², where 𝐸_𝑛𝑖∈𝑅^𝑛2×𝑛 and 𝐸_𝑑𝑖∈𝑅^𝑛2×𝑛2 are given by 𝐸_𝑛𝑖=𝐸_𝑖0⋅⋅⋅0,𝐸_𝑑𝑖=𝐴_𝑖𝐸_𝑖𝐴_𝑖𝐸⋅⋅⋅⋅⋅⋅_𝑖𝐴_𝑖. (2.5)

Lemma 2.6 (see [29, 35]). Suppose that the system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is regular. The system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is controllable if and only if rank[𝐸_𝑑𝑖𝐸_𝑏𝑖]=𝑛² and rank[𝐸_𝑖𝐵_𝑖]=𝑛, where 𝐸_𝑑𝑖∈𝑅^𝑛2×𝑛2 is given in (2.5) and 𝐸_𝑏𝑖=diag{𝐵_𝑖,𝐵_𝑖,…,𝐵_𝑖}∈𝑅^{𝑛2×𝑛𝑝}.

Lemma 2.7 (see [33]). The matrix measures of the matrices 𝑊 and 𝑉, namely, 𝜇(𝑊) and 𝜇(𝑉), are well defined for any norm and have the following properties:(i)𝜇(±𝐼)=±1, for the identity matrix 𝐼;(ii)−‖𝑊‖≤−𝜇(−𝑊)≤Re(𝜆(𝑊))≤𝜇(𝑊)≤‖𝑊‖, for any norm‖⋅‖ and any matrix 𝑊∈𝐶^𝑛×𝑛;(iii)𝜇(𝑊+𝑉)≤𝜇(𝑊)+𝜇(𝑉), for any two matrices 𝑊,𝑉∈𝐶^𝑛×𝑛;(iv)𝜇(𝛾𝑊)=𝛾𝜇(𝑊), for any matrix 𝑊∈𝐶^𝑛×𝑛 and any non-negative real number 𝛾,where 𝜆(𝑊) denotes any eigenvalue of  𝑊, and Re(𝜆(𝑊)) denotes the real part of 𝜆(𝑊).

Lemma 2.8. For any 𝛾<0 and any matrix 𝑊∈𝐶^𝑛×𝑛,𝜇(𝛾𝑊)=−𝛾𝜇(−𝑊).

Proof. This lemma can be immediately obtained from the property (iv) in Lemma 2.7.

Lemma 2.9. Let 𝑁∈𝐶^𝑛×𝑛. If 𝜇(−𝑁)<1, then det(𝐼+𝑁)≠0.

Proof. From the property (ii) in Lemma 2.7 and since 𝜇(−𝑁)<1, we can get that Re(𝜆(𝑁))≥−𝜇(−𝑁)>−1. This implies that 𝜆(𝑁)≠−1. So, we have the stated result.

Now, let the singular value decompositions of 𝑅_𝑖=[𝐸_𝑛𝑖𝐸_𝑑𝑖],𝑄_𝑖=[𝐸_𝑑𝑖𝐸_𝑏𝑖], and 𝑃_𝑖=[𝐸_𝑖𝐵_𝑖] be, respectively, 𝑅_𝑖=𝑈_𝑖𝑆_𝑖0_𝑛2×𝑛𝑉^𝐻_𝑖𝑄_𝑖,=𝑈_𝑟𝑖𝑆_𝑟𝑖0_{𝑛2×𝑛𝑝}𝑉^𝐻_𝑟𝑖𝑃_𝑖,=𝑈_𝑐𝑖𝑆_𝑐𝑖0_𝑛2×𝑞𝑉^𝐻_𝑐𝑖, (2.6)(2.7)(2.8) where 𝑈_𝑖∈𝑅^𝑛2×𝑛2 and 𝑉_𝑖∈𝑅^{(𝑛2+𝑛)×(𝑛2+𝑛)} are the unitary matrices, 𝑆_𝑖=diag{𝜎_𝑖1,𝜎_𝑖2,…,𝜎_𝑖𝑛2}, and 𝜎_𝑖1≥𝜎_𝑖2≥⋯≥𝜎_𝑖𝑛2>0 are the singular values of 𝑅_𝑖;𝑈_𝑟𝑖∈𝑅^𝑛2×𝑛2 and 𝑉_𝑟𝑖∈𝑅^{(𝑛2+𝑛𝑝)×(𝑛2+𝑛𝑝)} are the unitary matrices, 𝑆_𝑟𝑖=diag{𝜎_𝑟𝑖1,𝜎_𝑟𝑖2,…,𝜎_{𝑟𝑖𝑛2}} and 𝜎_𝑟𝑖1≥𝜎_𝑟𝑖2≥⋯≥𝜎_{𝑟𝑖𝑛2}>0 are the singular values of 𝑄_𝑖;𝑈_𝑐𝑖∈𝑅^𝑛×𝑛 and 𝑉_𝑐𝑖∈𝑅^{(𝑛+𝑝)×(𝑛+𝑝)} are the unitary matrices, 𝑆_𝑐𝑖=diag{𝜎_𝑐𝑖1,𝜎_𝑐𝑖2,…,𝜎_𝑐𝑖𝑛} and 𝜎_𝑐𝑖1≥𝜎_𝑐𝑖2≥⋯≥𝜎_𝑐𝑖𝑛>0 are the singular values of 𝑃_𝑖;𝑉^𝐻_𝑖, 𝑉^𝐻_𝑟𝑖, and 𝑉^𝐻_𝑐𝑖 denote, respectively, the complex-conjugate transposes of the matrices 𝑉_𝑖, 𝑉_𝑟𝑖, and 𝑉_𝑐𝑖.

In what follows, with the preceding definitions and lemmas, we present a sufficient criterion for ensuring that the uncertain TS fuzzy descriptor system in (2.1) or (2.2) remains locally regular and controllable.

Theorem 2.10. Suppose that the each fuzzy-rule-nominal descriptor system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖} is regular and controllable. The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is still locally regular and controllable (i.e., each fuzzy-rule-uncertain descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖} remains regular and controllable), if the following conditions simultaneously hold^𝑚_𝑘=1𝜀_𝑖𝑘𝜑_𝑖𝑘<1,_𝑚𝑘=1𝜀_𝑖𝑘𝜃_𝑖𝑘<1,_𝑚𝑘=1𝜀_𝑖𝑘𝜙_𝑖𝑘<1, (2.9a)(2.9b)(2.9c)where 𝑖=1,2,…,𝑁, and 𝑘=1,2,…,𝑚: 𝜑_𝑖𝑘=𝜇−𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖𝐼_𝑛2,0_𝑛2×𝑛^𝑇,for𝜀_𝑖𝑘𝑆≥0,−𝜇_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖𝐼_𝑛2,0_𝑛2×𝑛^𝑇,for𝜀_𝑖𝑘𝑅<0,_𝑖𝑘=0_𝑛2×𝑛𝑅_𝑖𝑘∈𝑅^{𝑛2×(𝑛2+𝑛)},𝑅_𝑖𝑘=diag𝐴_𝑖𝑘,…,𝐴_𝑖𝑘∈𝑅^𝑛2×𝑛2,𝜃_𝑖𝑘=𝜇−𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖𝐼_𝑛2,0_{𝑛2×𝑛𝑝}^𝑇,for𝜀_𝑖𝑘𝑆≥0,−𝜇⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖𝐼_𝑛2,0_{𝑛2×𝑛𝑝}^𝑇,for𝜀_𝑖𝑘𝑄<0,_𝑖𝑘=𝐴_𝑖𝑘𝐴_𝑖𝑘⋅⋅⋅𝐴_𝑖𝑘𝐵_𝑖𝑘𝐵_𝑖𝑘⋅⋅⋅𝐵_𝑖𝑘∈𝑅^{𝑛2×(𝑛2+𝑛𝑝)},𝜙_𝑖𝑘=𝜇−𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖𝐼_𝑛,0_𝑛×𝑝^𝑇,for𝜀_𝑖𝑘𝑆≥0,−𝜇⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖𝐼_𝑛,0_𝑛×𝑝^𝑇,for𝜀_𝑖𝑘𝑃<0,_𝑖𝑘=0_𝑛×𝑛𝐵_𝑖𝑘∈𝑅^{𝑛×(𝑛+𝑝)}, (2.10) the matrices 𝑆_𝑖, 𝑈_𝑖,𝑉_𝑖, 𝑆_𝑟𝑖, 𝑈_𝑟𝑖,𝑉_𝑟𝑖, 𝑆_𝑐𝑖, 𝑈_𝑐𝑖, and 𝑉_𝑐𝑖(𝑖=1,2,…,𝑁) are, respectively, defined in (2.6)–(2.8), and 𝐼_𝑛2 denotes the 𝑛²×𝑛² identity matrix.

Proof. Firstly, we show the regularity. Since each fuzzy-rule-nominal descriptor system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖}  (𝑖=1,2,…,𝑁) is regular, then, from Lemma 2.5, we can get that the matrix 𝑅_𝑖=[𝐸_𝑛𝑖𝐸_𝑑𝑖]∈𝑅^{𝑛2×(𝑛2+𝑛)} has full row rank (i.e., rank(𝑅_𝑖)=𝑛²). With the uncertain matrices 𝐴_𝑖+Δ𝐴_𝑖 and 𝐵_𝑖+Δ𝐵_𝑖, each fuzzy-rule-uncertain descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖} is regular if and only if 𝑅_𝑖=𝑅_𝑖+^𝑚_𝑘=1𝜀_𝑖𝑘𝑅_𝑖𝑘 (2.11) has full row rank, where 𝑅_𝑖𝑘=[0_𝑛2×𝑛𝑅_𝑖𝑘]∈𝑅^{𝑛2×(𝑛2+𝑛)} and 𝑅_𝑖𝑘=diag{𝐴_𝑖𝑘,…,𝐴_𝑖𝑘}∈𝑅^𝑛2×𝑛2. Itisknownthatrank𝑅_𝑖=rank𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑉_𝑖. (2.12) Thus, instead of rank(𝑅_𝑖), we can discuss the rank of 𝐼_𝑛2,0_𝑛2×𝑛+^𝑚_𝑘=1𝜀_𝑖𝑘𝑅_𝑖𝑘, (2.13) where 𝑅_𝑖𝑘=𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖, for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚. Since a matrix has at least rank 𝑛² if it has at least one nonsingular 𝑛²×𝑛² submatrix, a sufficient condition for the matrix in (2.13) to have rank 𝑛² is the nonsingularity of 𝐿_𝑖=𝐼_𝑛2+^𝑚_𝑘=1𝜀_𝑖𝑘𝑅_𝑖𝑘, (2.14) where 𝑅_𝑖𝑘=𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖[𝐼_𝑛2,0_𝑛2×𝑛]^𝑇 (for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚).
Using the properties in Lemmas 2.7 and 2.8 and from (2.9a), we get 𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑅_𝑖𝑘=𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖𝐼_𝑛2,0_𝑛2×𝑛^𝑇≤^𝑚_𝑘=1𝜇−𝜀_𝑖𝑘𝑆_𝑖⁻¹𝑈^𝐻_𝑖𝑅_𝑖𝑘𝑉_𝑖𝐼_𝑛2,0_𝑛2×𝑛^𝑇=^𝑚_𝑘=1𝜀_𝑖𝑘𝜑_𝑖𝑘<1. (2.15) From Lemma 2.9, we have that det𝐿_𝑖=det𝐼_𝑛2+^𝑚_𝑘=1𝜀_𝑖𝑘𝑅_𝑖𝑘≠0. (2.16) Hence, the matrix 𝐿_𝑖 in (2.14) is nonsingular. That is, the matrix 𝑅_𝑖 in (2.11) has full row rank 𝑛². Thus, from the Lemma 2.5, the regularity of each fuzzy-rule-uncertain descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖} is ensured.
Next, we show the controllability. Since each fuzzy-rule-nominal descriptor system {𝐸_𝑖,𝐴_𝑖,𝐵_𝑖}(𝑖=1,2,…,𝑁) is controllable, then from Lemma 2.6, we have that the matrix 𝑄_𝑖=[𝐸_𝑑𝑖𝐸_𝑏𝑖] has full row rank (i.e., rank(𝑄_𝑖)=𝑛²) and 𝑃_𝑖=[𝐸_𝑖𝐵_𝑖] has full row rank (i.e., rank(𝑃_𝑖)=𝑛). With the uncertain matrices 𝐴_𝑖+Δ𝐴_𝑖 and 𝐵_𝑖+Δ𝐵_𝑖, each fuzzy-rule-uncertain descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖} is controllable if and only if 𝑄_𝑖=𝑄_𝑖+^𝑚_𝑘=1𝜀_𝑖𝑘𝑄_𝑖𝑘,𝑃_𝑖=𝑃_𝑖+^𝑚_𝑘=1𝜀_𝑖𝑘𝑃_𝑖𝑘 (2.17)(2.18) have full row rank, where 𝑄_𝑖𝑘=𝐴_𝑖𝑘𝐴_𝑖𝑘⋅⋅⋅𝐴_𝑖𝑘𝐵_𝑖𝑘𝐵_𝑖𝑘⋅⋅⋅𝐵_𝑖𝑘∈𝑅^{𝑛2×(𝑛2+𝑛𝑝)}, (2.19) and 𝑃_𝑖𝑘=[0_𝑛×𝑛𝐵_𝑖𝑘]∈𝑅^{𝑛×(𝑛+𝑝)}.
It is known that rank𝑄_𝑖=rank𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑉_𝑟𝑖. (2.20) Thus, instead of ran𝑘(𝑄_𝑖), we can discuss the rank of 𝐼_𝑛2,0_{𝑛2×𝑛𝑝}+^𝑚_𝑘=1𝜀_𝑖𝑘𝑄_𝑖𝑘, (2.21) where 𝑄_𝑖𝑘=𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖, for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚. Since a matrix has at least rank 𝑛² if it has at least one nonsingular 𝑛²×𝑛² submatrix, a sufficient condition for the matrix in (2.21) to have rank 𝑛² is the nonsingularity of 𝐺_𝑖=𝐼_𝑛2+^𝑚_𝑘=1𝜀_𝑖𝑘𝑄_𝑖𝑘, (2.22) where 𝑄_𝑖𝑘=𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖[𝐼_𝑛2,0_{𝑛2×𝑛𝑝}]^𝑇 (for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚).
Applying the properties in Lemmas 2.7 and 2.8 and from (2.9b), we get 𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑄_𝑖𝑘=𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖𝐼_𝑛2,0_{𝑛2×𝑛𝑝}^𝑇≤^𝑚_𝑘=1𝜇−𝜀_𝑖𝑘𝑆⁻¹_𝑟𝑖𝑈^𝐻_𝑟𝑖𝑄_𝑖𝑘𝑉_𝑟𝑖𝐼_𝑛2,0_{𝑛2×𝑛𝑝}^𝑇=^𝑚_𝑘=1𝜀_𝑖𝑘𝜃_𝑖𝑘<1. (2.23) From Lemma 2.9, we have that det𝐺_𝑖=det𝐼_𝑛2+^𝑚_𝑘=1𝜀_𝑖𝑘𝑄_𝑖𝑘≠0. (2.24) Hence, the matrix 𝐺_𝑖 in (2.22) is nonsingular. That is, the matrix 𝑄_𝑖 in (2.17) has full row rank 𝑛².
And then, it is also known that rank𝑃_𝑖=rank𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑉_𝑐𝑖. (2.25) Thus, instead of rank(𝑃_𝑖), we can discuss the rank of 𝐼_𝑛,0_𝑛×𝑝+^𝑚_𝑘=1𝜀_𝑖𝑘𝑃_𝑖𝑘, (2.26) where 𝑃_𝑖𝑘=𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖, for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚. Since a matrix has at least rank 𝑛 if it has at least one nonsingular 𝑛×𝑛 submatrix, a sufficient condition for the matrix in (2.26) to have rank 𝑛 is the nonsingularity of 𝐻_𝑖=𝐼_𝑛+^𝑚_𝑘=1𝜀_𝑖𝑘𝑃_𝑖𝑘, (2.27) where 𝑃_𝑖𝑘=𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖[𝐼_𝑛,0_𝑛×𝑝]^𝑇 (for 𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚).
Adopting the properties in Lemmas 2.7 and 2.8 and from (2.9c), we obtain 𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑃_𝑖𝑘=𝜇−^𝑚_𝑘=1𝜀_𝑖𝑘𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖𝐼_𝑛,0_𝑛×𝑝^𝑇≤^𝑚_𝑘=1𝜇−𝜀_𝑖𝑘𝑆⁻¹_𝑐𝑖𝑈^𝐻_𝑐𝑖𝑃_𝑖𝑘𝑉_𝑐𝑖𝐼_𝑛,0_𝑛×𝑝^𝑇=^𝑚_𝑘=1𝜀_𝑖𝑘𝜙_𝑖𝑘<1. (2.28) From Lemma 2.9, we get that det𝐻_𝑖=det𝐼_𝑛+^𝑚_𝑘=1𝜀_𝑖𝑘𝑃_𝑖𝑘≠0. (2.29) Hence, the matrix 𝐻_𝑖 in (2.27) is nonsingular. That is, the matrix 𝑃_𝑖 in (2.18) has full row rank 𝑛. Thus, from the Lemma 2.6 and the results mentioned above, the controllability of each fuzzy-rule-uncertain descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖} is ensured. Therefore, we can conclude that the uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally regular and controllable, if the inequalities (2.9a), (2.9b), and (2.9c) are simultaneously satisfied. Thus, the proof is completed.

Remark 2.11. The proposed sufficient conditions in (2.9a)–(2.9c) can give the explicit relationship of the bounds on 𝜀_𝑖𝑘(𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚) for preserving both regularity and controllability. In addition, the bounds, that are obtained by using the proposed sufficient conditions, on 𝜀_𝑖𝑘 are not necessarily symmetric with respect to the origin of the parameter space regarding 𝜀_𝑖𝑘(𝑖=1,2,…,𝑁 and 𝑘=1,2,…,𝑚).

Remark 2.12. This paper studies the problem of robust local regularity and controllability analysis. If the proposed conditions in (2.9a)–(2.9c) are satisfied, each rule of the uncertain TS fuzzy descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}is guaranteed to be robustly locally regular and controllable. This implies that, in the fuzzy PDC controller design, if the proposed conditions in (2.9a)–(2.9c) are satisfied, the PDC controller of each fuzzy rule can control every state variable in the corresponding rule of the uncertain TS fuzzy descriptor system {𝐸_𝑖,𝐴_𝑖+Δ𝐴_𝑖,𝐵_𝑖+Δ𝐵_𝑖}. However, here, it should be noticed that although the PDC controller of each control rule can control every state variable in the corresponding rule under the presented conditions being held, the PDC controller gains should be determined using global design criteria that are needed to guarantee the global stability and control performance [5], where many useful global design criteria have been proposed by some researchers (e.g., [4-6, 8, and 21-28] and references therein).

3. Illustrative Example

Consider a two-rule fuzzy descriptor system as that considered by Wang et al. [21]. The TS fuzzy descriptor system with the elemental parametric uncertainties is described by𝑅¹:IF𝑧₁is𝑀₁₁,then𝐸₁𝑥(𝑘+1)=𝐴₁+Δ𝐴₁𝑥(𝑘)+𝐵₁+Δ𝐵₁𝑢(𝑘);𝑅²:IF𝑧₁is𝑀₂₁,then𝐸₂𝑥(𝑘+1)=𝐴₂+Δ𝐴₂𝑥(𝑘)+𝐵₂+Δ𝐵₂𝑢(𝑘), (3.1a)(3.1b)where 𝑥(𝑘)=𝑥₁(𝑘)𝑥₂(𝑘)^𝑇,𝐸₁=𝐸₂=1000,𝐴₁=,𝐴0.84800−0.315₂=−0.236000.113,𝐵₁=𝐵₂=11,Δ𝐴_𝑖=²_𝑘=1𝜀_𝑖𝑘𝐴_𝑖𝑘,Δ𝐵_𝑖=²_𝑘=1𝜀_𝑖𝑘𝐵_𝑖𝑘,𝐴₁₁=00−0.10,𝐴₂₁=000.20,𝐴₁₂=𝐴₂₂=,𝐵0000_𝑖1=00,𝐵_𝑖2=0.10.1,𝑀₁₁=11+exp−0.5𝑧₁,𝑀−0.3₂₁=exp−0.5𝑧₁−0.31+exp−0.5𝑧₁−0.3,𝜀_𝑖1∈[−11.1],𝜀_𝑖2∈[−1.210],inwhich𝑖=1,2. (3.2)

Now, applying the sufficient conditions in (2.9a)–(2.9c) with the two-norm-based matrix measure, we can get the following:(I)for the fuzzy rule 1:(i)²_𝑘−1𝜀_𝑖𝑘𝜑_𝑖𝑘≤0.17460<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()−1.210ii²_𝑘−1𝜀_𝑖𝑘𝜑_𝑖𝑘≤0.15873<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],()−1.210iii²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.03297<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()010iv²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.13861<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],(v)010_2𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.25078<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],()−1.20vi²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.14514<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()−1.20vii²_𝑘−1𝜀_𝑖𝑘𝜙_𝑖𝑘=0<1,for𝜀_𝑖1∈[−11.1],𝜀_𝑖2∈[],()010viii²_𝑘−1𝜀_𝑖𝑘𝜙_𝑖𝑘≤0.1200<1,for𝜀_𝑖1∈[−11.1],𝜀_𝑖2∈[−1.20]; (3.3a)(3.3b)(3.3c)(3.3d)(3.3e)(3.3f)(3.3g)(3.3h)

(i)²_𝑘−1𝜀_𝑖𝑘𝜑_𝑖𝑘≤0.97345<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()−1.210ii²_𝑘−1𝜀_𝑖𝑘𝜑_𝑖𝑘≤0.88496<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],()−1.210iii²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.56719<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()010iv²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.87768<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],(v)010_2𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.99740<1,for𝜀_𝑖1∈[−10],𝜀_𝑖2∈[],()−1.20vi²_𝑘−1𝜀_𝑖𝑘𝜃_𝑖𝑘≤0.168691<1,for𝜀_𝑖1∈[01.1],𝜀_𝑖2∈[],()−1.20vii²_𝑘−1𝜀_𝑖𝑘𝜙_𝑖𝑘=0<1,for𝜀_𝑖1∈[−11.1],𝜀_𝑖2∈[],()010viii²_𝑘−1𝜀_𝑖𝑘𝜙_𝑖𝑘≤0.1200<1,for𝜀_𝑖1∈[−11.1],𝜀_𝑖2∈[−1.20]. (3.4a)(3.4b)(3.4c)(3.4d)(3.4e)(3.4f)(3.4g)(3.4h)

From the results in (3.3a)–(3.3h) and (3.4a)–(3.4h), we can conclude that the uncertain TS fuzzy descriptor system (3.1a) and (3.1b) is locally robustly regular and controllable.

4. Conclusions

The robust local regularity and controllability problem for the uncertain TS fuzzy descriptor systems has been investigated. The rank preservation problem for robust local regularity and controllability of the uncertain TS fuzzy descriptor systems is converted to the nonsingularity analysis problem. Under the assumption that each fuzzy rule of the nominal TS fuzzy descriptor system has the full row rank for its related regularity and controllability matrices, a sufficient criterion has been proposed to preserve the assumed properties when the elemental parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient conditions in (2.9a)–(2.9c) can provide the explicit relationship of the bounds on elemental parameter uncertainties for preserving the assumed properties. One example has been given to illustrate the application of the proposed sufficient conditions. On the other hand, the issue of robust global regularity and controllability with evolutionary computation [36] for the uncertain TS fuzzy descriptor systems will be an interesting and important topic for further research.

Acknowledgment

This work was in part supported by the National Science Council, Taiwan, under Grants nos. NSC 100-2221-E-151-009, NSC 101-2221-E-151-076, and NSC 101-2320-B-037-022.