Robust Stability of Switched Delay Systems with Average Dwell Time under Asynchronous Switching

Notations. Throughout this paper, ℜⁿ denotes the n-dimensional Euclidean space and ℜ^n×n refers to the set of all n × m real matrices. For real symmetric matrices X and Y, the notation X ≥ Y (resp., X > Y) mean that the matrix X, Y are positive semidefinite, (resp., and positive definite). I is the identity matrix with appropriate dimensions. * represents the elements below the main diagonal of a symmetric matrix. The superscripts ⊺ and −1 stand for matrix transposition and matrix inverse, respectively; ∥·∥_c denotes the Euclidean norm. λ_max(P) and λ_min(P) denote the maximum and minimum eigenvalues of matrix P, respectively. The shorthand diag {M₁, …, M_n} denotes a block diagonal matrix with diagonal blocks being the matrices M₁, …, M_n. In this paper, if not explicit, matrices are assumed to have compatible dimensions.

Definition 2.1 (see [29].)The unforced system is said to be exponential stable if there exist constants ν > 0 and ϑ > 0 such that

Definition 2.2 (see [30].)For γ > 0, the switched system (2.1) is said to have weighted L₂-gain, if under zero initial condition ϕ(t),  t ∈ [−τ_M, 0], it holds that

Definition 2.3 (see [30].)For any T₂ > T₁ ≥ 0, let N_σ(T₁, T₂) denote the switching number of discontinuities of σ(t) during on an intercal (T₁, T₂). If N_σ(T₁, T₂) ≤ N₀ + (T₂ − T₁)/T_a holds for N₀ ≥ 0 and T_a > 0, then N₀ and T_a are called chattering bound and average dwell time, respectively. Here we assume N₀ = 0 for simplicity as commonly used in the literature.

Lemma 2.4 (see [29].)For any given symmetric positive definite matrix X ∈ ℜ^n×n, and scalars α > 0,   0 ≤ d₁ < d₂, if there exists a vector function $$ such that the following integration is well defined, then

Lemma 2.5 (see [28].)Let ϱ ≥ 0 and θ > δ > 0. If there exists a real-value continuous function x(t) ≥ 0, t ≥ t₀, such that the differential inequality

Lemma 2.6 (see [9].)If a real scalar function x(t) satisfies the following differential inequality:

Theorem 3.1. For given scalars 0 ≤ τ₀ ≤ τ_M, α > 0,   β > 0, then the system (2.1) is exponentially stable, if there exist positive-definite matrices P_i,   P_ij,   Q_ki,   Q_kij (k = 1,2, 3), and R_li,   R_lij (l = 1,2) such that the following LMIs hold:

In this case, for any switching signal with the average dwell-time satisfying

Proof. When t ∈ Λ₁, we consider the following Lyapunov-Krasovskii functional:

Taking the time derivative of V_i(t, x(t)) for t ∈ [0, ∞) along the trajectory of the system (2.1) turns out to be

On the other hand, according to Lemma 2.4, we get that

Then we can get

In view of Schur complement, (3.3) implies that $$. Then we have $$ for all ψ(t) ≠ 0.

Then during the matched period, by Lemma 2.5, V_i(t, x(t)) satisfy

When t ∈ Λ₂, we consider the following Lyapunov-Krasovskii functional:

Similarly we have that

Through Lemma 2.5, we also have

When t ∈ [t_ℑ, t_ℑ + Δ_ℑ),  ℑ = 1,2, …, we have the relationship $$, and it follows that

From (3.7), we can obtain

Then through (3.23) and (3.24), we can easily get

By (3.9) and (3.25), then we have

Similarly, when t ∈ Λ₁, we can also have

For convenience, let a = max {a₁, a₂}, b = min _{i≠j,i,j∈ℙ}λ_min(P_i, P_ij), through (3.27) and (3.29), we have

Theorem 3.2. For given scalars 0 ≤ τ₀ ≤ τ_M,   α > 0,   β > 0, then the system (2.1) is exponentially stable with L₂-gain, if there exist positive-definite matrices P_i,   P_ij,   Q_ki,   Q_kij (k = 1,2, 3), and R_li, R_lij (l = 1,2) such that the following LMIs hold:

Proof. For all nonzero ω(t) ∈ L₂[0, ∞) and a scalar γ > 0, then we establish system (2.1) with L₂-gain performance $$. For convenience, denoting $$.

When t ∈ [t₀, t₁)∪[t_ℑ−1 + Δ_ℑ−1, t_ℑ),  k = 2,3, …, by the system (2.1), we can obtain

From (3.32), we can easily get

Integrate this inequality during [t₀, t], it is known that

When t ∈ [t_ℑ, t_ℑ + Δ_ℑ),  ℑ = 2,3, …, by the system (2.1), by the same way, we can obtain

Then we have

When t ∈ [t_ℑ, t_ℑ + Δ_ℑ),  ℑ = 1,2, …, it follows that

Under the zero initial condition, Let t₀ = 0, (3.40) implies

Integrate (3.41) during [0, ∞), then we can obtain

When t ∈ [t₀, t₁)∪[t_ℑ−1 + Δ_ℑ−1, t_ℑ),  ℑ = 2,3, …, by the same mathematical operations, we have $$.

From which we can get $$. This proof is completed.

Remark 3.3. If μ₁ = μ₂ = 0, which implies that P_i = P_ij = P,   Q_ki = Q_kij = Q_k,   R_li = R_lij = R_l,   i, j ∈ ℙ, by (3.3)-(3.4) and (3.32)-(3.33), we have T_a = 0, then it requires a common Lyapunov functional for all subsystems, and the switching signals can be arbitrary. If μ_k → ∞ (k = 1,2), we get from (3.3)-(3.4) and (3.32)-(3.33) that there is no switching, that is, switching signal will have a great dwell-time on the average.

Example 4.1. Consider the system (2.1) with parameters as follows:

τ₀ is fixed and assumed to be 0.2. The initial condition is assumed to be x(0) = [9,    − 9] ^⊺, ω(t) = 0.5sin t. Then by solving the LMIs in Theorem 3.2, different T_a and τ_M for different μ₁μ₂, α, and β can be obtained in Table 1. It can be seen that, for the given τ₀, the upper bounds of the time delay τ_M and the minimal average dwell-time T_a are dependent on α, β, and μ₁μ₂. Then the simulation result of the system is shown in Figures 1, 2, and 3. The switching signal σ(t) with average dwell-time T_a is shown in Figure 1. Figures 2–3 indicate that the state response of the switched system without asynchronous switching and with asynchronous switching, respectively.