Robust H ∞ Control For Uncertain Singular Neutral Time-Delay Systems

The present paper attempts to investigate the problem of robust H∞ control for a class of uncertain singular neutral time-delay systems. First, a linear matrix inequality (LMI) is proposed to give a generalized asymptotically stability condition and an H∞ norm condition for singular neutral time-delay systems. Second, the LMI is utilized to solve the robust H∞ problem for singular neutral time-delay systems, and a state feedback control law verifies the solution. Finally, four theorems are formulated in terms of a matrix equation and linear matrix inequalities.


Introduction
Singular systems are more convenient than regular ones for describing many practical systems because a singular system involves both differential equations and algebraic equations.Applications of singular systems can be found in circuit systems, chemical systems, biological systems, robot systems, and power systems [1].Therefore, many scholars have paid attention to the study of singular systems, and a number of important results have been reported (see, e.g., [2][3][4]).
As is known to all, a time delay frequently arises in practical systems and is often the cause of instability and poor performance.Hence, the stability problem for a singular system with a time delay has attracted many researchers' attention in the past several decades (see, e.g., [5][6][7][8][9][10]).
The robust H ∞ control problem for uncertain singular time-delay systems was investigated by Ji et al. in [24], where the LMI condition was obtained by constructing a degenerate Lyapunov function on the basis of [23].However, the condition does not satisfy A d22 < 1, which renders the design procedure of the LMI law comparatively untenable.Moreover, the problem for singular neutral time-delay systems was not investigated in [24], and some information about the condition itself cannot be revealed even if the method can be applied to a singular neural time-delay system.Also, because of the continuity of the function, it is more difficult to study the neural time-delay system than it is to study singular time-delay systems.Consequently, it is of more theoretical and practical significance to study singular neutral time-delay systems as compared with time-delay systems.
The present paper derives a sufficient condition for the existence of the H ∞ controller on the basis of the LMI approach combined with a class of novel augmented Lyapunov functions, which thus facilitate the attainment of the H ∞ controller using the Matlab LMI toolbox combined with a matrix equation.

Problem Statement and Preliminaries
Consider the following uncertain singular neutral time-delay system: where x(t) ∈ R n is the state vector; u(t) ∈ R m is the control input vector; ω(t) ∈ R p is the disturbance input vector belonging to L 2 [0, +∞); z(t) ∈ R q is the control output vector; τ > 0 is a constant time delay; Φ(t) is a vector-valued initial function belonging to are constant matrices with appropriate dimensions, where E may be singular and is assumed to be rankE = r < n; and ∆A, ∆A τ , ∆B, B ω are unknown matrices representing time-varying parameter uncertainties and can be described as where G and N a , N τ , N b , N c are known constant matrices and F : R + → R m×n is a known matrix with Lebesgue measurable elements and satisfies It is assumed in the present paper that Γ ẋ(t) ≤ Γx(t) for the arbitrary positive-definite matrix Γ.
The parametric uncertainties ∆A, ∆A τ , ∆B, ∆C are said to be admissible if Equations ( 2) and (3) both hold.
Next is a discussion of the system in Equation (1) with no force counterpart item.First, the system is described as Equation ( 4), The following definitions and lemmas are very useful for deriving the main results of this paper.
(1) : The pair (E, A) is known as regular if det(sE − A) is not identically zero.
(2) : The pair (E, A) is known as impulse free if det(sE − A) = rank(E).

Definition 2 ([24]
).The singular neutral time-delay system (Equation ( 4)) is known as regular and impulse free if the pair (E, A) is regular and impulse free.
Remark 1.The regularity and impulses of the pair (E, A) ensure the system (Equation ( 4)) with τ = 0 to be regular and impulse free, and they further ensure the existence of a unique solution to the system in Equation ( 4) on [−τ, +∞).
Since (E, A) is regular and impulse free, there exist two nonsingular matrices Q and P such that the system in Equation ( 4) is equivalent to with the coordinate transformation where n 1 + n 2 = n.Obviously, the system in Equation ( 5) has a unique solution on [−τ, +∞).

Definition 3 ([29]
).If a matrix X satisfies the Penrose condition AXA = A, then there exists a solution to the generalized inverse for AXA = A or {1} inverse of A, and thus, the matrix X is denoted by X = A (1) or X ∈ A{1}, where A{1} denotes the set of all {1} inverse of A.
Therefore, Lemma 3 can be obtained by using a method similar to that in J. Lee (1994).
Lemma 3.For given matrices Q = Q T , H, E, and F of appropriate dimensions, hold simultaneously.Thus, can be obtained.Similarly, there exist positive numbers ε 2 , ε 3 such that the following inequalities also hold
Robust H ∞ control problem.The present paper attempts to address the robust H ∞ control problem by considering the linear state feedback control law as to construct K such that u(t) in Equation ( 6) will (a) stabilize the resultant closed-loop system and (b) guarantee the )dt < 0 under the zero-initial condition of x(t) and ẋ(t) for any nonzero ω(t) ∈ L 2 [0, ∞) and for all admissible parameter uncertainties satisfying Equations (2) and (3).

Results
In the following, the problem of robust H ∞ control is considered for the singular neutral system in Equation (1) with F(t) = 0 and u(t) = 0. Theorem 1.Consider the system in Equation (1) with F(t) = 0 and u(t) = 0.For a given scalar γ > 0, the system in Equation (1) is regular, impulse free, and stable, and the H ∞ norm from ω(t) to z(t) is less than γ, if there exist symmetric positive-definite matrices P, Q, R, L and matrices S, S τ , S ω such that the following linear matrix inequality holds: where r) is any matrix that has full column rank and satisfies E T V = 0.
Proof.The nonlinear singular system (Equation (1)) is proved below to be regular and impulse free.Since rank(E) = r≤ n, there exist two nonsingular matrices F and G ∈ R n×n such that Then, V can be parameterized as can be defined.Since Σ 11 < 0 and Q > 0, the following inequality can be formulated easily: Pre-and post-multiplying Ω < 0 by F T and F, respectively, yields From [17], the following matrix inequalities can be formulated easily: and thus, A 22 is nonsingular.
Then, it can be proved that which implies that det(sE − A) is not identically zero and deg(det(sE − A)) = r = rank(E).Then, the pair (E, A) is regular and impulse free, which implies that the system in Equation ( 1) is regular and impulse free.
In the following, the system in Equation (1) with u(t) = 0 and F(t) = 0 is proved to be asymptotical with the condition of ω(t) = 0 and an H ∞ performance under the zero-initial condition of x(t) and ẋ(t) for any nonzero ω(t) ∈ L 2 [0, ∞).Construct a Lyapunov-Krasovskii function candidate as follows: where P> 0, Q> 0, R> 0, and L> 0. From this follows the derivation of V 0 (t, x t ) with respect to t along the trajectory of the system in Equation (1) with the condition of F(t) = 0 and u(t) = 0 that For the system in Equation (1), the following holds where where S is any matrix with appropriate dimensions.Noting the zero-initial condition of x(t), V(x 0 ) = 0, and V(x ∞ ) > 0, then By substituting Equations (10), (11), and (12) into (13), the following can be obtained: , with there exists a scalar λ > 0 such that J ≤ −λ x(t) 2 ; thus, according to [3], the system in Equation (1) with u(t) = 0 and F(t) = 0 is asymptotically stable.By Lemma 1, Θ < 0 is equivalent to Σ < 0.
It is easy to obtain from the result of Theorem 1 the following conclusion about the H ∞ performance analysis.Theorem 2. Consider the system in Equation (1) with u(t) = 0.For a given scalar γ > 0, the system is regular, impulse free, and stable, and the H ∞ norm from ω(t) to z(t) is less than γ if there exist symmetric positive-definite matrices P, Q, R, L and matrices S, S τ , S ω , and ε > 0 such that the following linear matrix inequality holds: where Σ ij is as defined in Theorem 1.
Proof.It follows from Equation (14) by Lemma 1 that where Σ is as defined in Theorem 1, and It follows from Equation (15) by Lemma 3 that where ω R, and V ∈ R n×(n−r) is any matrix that has full column rank and satisfies E T V = 0.
In the following, the robust H ∞ synthesis problem of the system in Equation (1) is to be considered for the system in Equation (1) with F(t) = 0. Theorem 3. Consider the system in Equation (1) with F(t) = 0.For a given scalar γ > 0, if there exist symmetric positive-definite the matrices P, Q, R, L and matrices S, Y 1 , Y 2 such that the matrix equation and the linear matrix inequality in the following hold simultaneously, then, the control law (where Y is an arbitrary matrix of appropriate dimension, I is a unit matrix, V ∈ R n×(n−r) is any matrix with full column rank and satisfies E T V = 0, and ) stabilizes the singular neutral system and guarantees the H ∞ norm bound within γ in the closed-loop system.
Proof.Substituting the state feedback control law u(t) = Kx(t) into the system in Equation (1) with F(t) = 0, the closed-loop system can be obtained.Since det(sE − (A + BK)) = det(sE T − (A + BK) T ), the pair (E, (A + BK)) is the same as the pair (E T , (A + BK) T ) in that they are both regular and impulse free.Therefore, the solutions of det(sE − (A + BK) − A τ e −sτ − Cse −sτ ) = 0 are equivalent to the solutions of det(sE T − (A + BK) T − A T τ e −sτ − C T se −sτ ) = 0.According to the definition the H ∞ norm, the H ∞ norm of the system in Equation (20) can be given as which is equal to Hence, it can be shown that the regularity, impulse-free state, asymptotic stability, and H ∞ performance of the system in Equation (19) are equivalent to the following system regularity, impulse-free state, asymptotic stability, and H ∞ performance; that is, Then, by replacing A by (A+BK) T ,A τ by A T τ ,D by B T ω ,E by E T ,C by C T in Equation ( 7) and setting S τ = 0, S ω = 0, Y 1 = K(PE T + VS T ), Y 2 = KR, Matrix Equation (17) and Linear Matrix Inequality (18) can be directly obtained.Now, the result for the problem of robust H ∞ control for the system in Equation ( 1) is given.According to Theorem 3, the robust H ∞ performance of the system (Equation (1)) will be stated as follows.
Theorem 4. Consider the uncertain singular neutral time-delay system (Equation ( 1)).For a given scalar γ > 0, if there exist symmetric positive-definite matrices P, Q, R, L and matrices S, Y 1 , Y 2 and ε 1 > 0, ε 2 > 0, ε 3 > 0 such that the matrix equation and the linear matrix inequality in the following hold simultaneously, where where Y is an arbitrary matrix of appropriate dimension, I is a unit matrix, V ∈ R n×(n−r) is any matrix with full column rank and satisfies E T V = 0, and stabilizes the uncertain singular neutral system and guarantees the H ∞ norm bound within γ in the closed-loop system.
Proof.By replacing A by A + GF(t)N a , A τ by A τ + GF(t)N τ , B by B + GF(t)N b , and C by C + GF(t)N c in Theorem 3, the following matrix inequality can be obtained.
where Ξ is as defined in Equation (18), and By Lemma 3, it can be proved that the inequality above is satisfied if there exist scalars ε 1 > 0, ε 2 > 0, and ε 3 > 0 such that which is equal to Equation ( 21) under the condition of Equation (20).

Numerical Illustration
The following numerical example is presented to illustrate the usefulness of the proposed theoretical results.With the zero-initial condition and the parameters given above, Figure 1 gives the simulations for the trajectory z(t) of the system in Equation ( 1) under the control law in Theorem 4. Figure 1 demonstrates the effectiveness of the proposed control method.

Conclusions
The problem of robust H ∞ control for an uncertain singular neutral system is investigated.A new approach is introduced in order to ensure the singular system (Equation (1)) is regular and impulse free.On that basis, the matrix equation and an LMI ensure that the system, which is asymptotic and guarantees the H ∞ norm bound within γ in the closed-loop system for all admissible parameter uncertainties, can be obtained.The needed controller can be constructed by solving the matrix equation and the LMI.It should be emphasized that the controller has a generalized inverse form, which is different from the result of [17].Also, this method can be applied to some practical systems.

Figure 1 .
Figure 1.The trajectory of z(t) of the system in Equation (1).
For a given symmetry matrix A = A 11 A 12 A 21 A 22 , where A 11 , A 12 , A 21 , A 22 have appropriate dimensions, A 21 = A T 12 .