Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

Definition 2.1 (see [33].)The measure of a matrix $$ is defined as

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) ∈ Rⁿ, and w ∈ Rⁿ, 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_ni B_di] = n², where $$ and $$ are given by

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_di E_bi] = n² and rank [E_i B_i] = n, where $$ is given in (2.5) and $$.

Lemma 2.7 (see [33].)The matrix measures of the matrices $$ and $$, namely, $$ and $$, are well defined for any norm and have the following properties:

• (i)

  μ(±I) = ±1, for the identity matrix I;

• (ii)

  $$, for any norm ∥·∥ and any matrix $$;

• (iii)

  $$, for any two matrices $$;

• (iv)

  $$, for any matrix $$ and any non-negative real number γ,

where $$ denotes any eigenvalue of  $$, and $$ denotes the real part of $$.

Lemma 2.8. For any γ < 0 and any matrix $$.

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

Lemma 2.9. Let $$. If $$, then $$.

Proof. From the property (ii) in Lemma 2.7 and since $$, we can get that $$. This implies that $$. So, we have the stated result.

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

where i = 1, 2, …, N, and k = 1, 2, …, m: 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 $$ 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 $$ 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

has full row rank, where $$ and $$. Thus, instead of $$, we can discuss the rank of where $$, 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 where $$ (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

From Lemma 2.9, we have that Hence, the matrix L_i in (2.14) is nonsingular. That is, the matrix $$ 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_di E_bi] has full row rank (i.e., rank (Q_i) = n²) and P_i = [E_i B_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

have full row rank, where and P_ik = [0_n×n B_ik] ∈ R^n×(n+p).

It is known that

Thus, instead of $$, we can discuss the rank of where $$, 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 where $$ (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

From Lemma 2.9, we have that Hence, the matrix G_i in (2.22) is nonsingular. That is, the matrix $$ in (2.17) has full row rank n².

And then, it is also known that

Thus, instead of $$, we can discuss the rank of where $$, 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 where $$ (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

From Lemma 2.9, we get that Hence, the matrix H_i in (2.27) is nonsingular. That is, the matrix $$ 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).