Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components

Definition 1.1. Let Φ₁,   Φ₂, …,   Φ_N : ℛ^m → ℛⁿ be a given finite number of functions such that they have positive values in an open subset D of ℛ^m. Then, a reciprocally convex combination of these functions over D is a function of the form

where the real numbers α_i satisfy α_i > 0 and ∑_iα_i = 1.

Lemma 1.2 (See [10]). Let f₁, f₂, …, f_N : ℛ^m → ℛ have positive values in an open subset D of ℛ^m. Then, the reciprocally convex combination of f_i over D satisfies

subject to In the following, we present a new stability criterion by a convex polyhedron method and Lemma 1.2.

Theorem 2.1. System (1.5) with delays d₁(t) and d₂(t) satisfying (1.6) is asymptotically stable if there exist symmetric positive definite matrices  P,  Q₁,  Q₂,  Q₃,  Q₄,  Q₅,  Q₆, Z,  Z₁,  Z₂ and any matrices  S₁₂,  N,  M,  L,  S,  P₁,  P₂ with appropriate dimensions, such that the following LMIs hold:

where

Proof. Construct a new Lyapunov functional candidate as

Remark 2.2. Our paper fully uses the information about d(t),  d₁(t), and d₂(t), but [15, 16] only use the information about d₁(t) and d₂(t), when constructing the Lyapunov functional V(x(t)). So the Lyapunov functional in our paper is more general than that in [15, 16], and the stability criteria in our paper may be more applicable.

The time derivative of V(x(t)) along the trajectory of system (1.5) is given by

The $$ is upper bounded by where the inequality in (2.12) comes from the Jensen inequality lemma, and that of (2.13) from Lemma 1.2 as where α = (d − d(t))/d,  β = d(t)/d. Note that when d(t) = d or d(t) = 0, one can obtain ζ^T(t)(e₂  −  e₃) = 0 or ζ^T(t)(e₁ −  e₂) = 0, respectively. So the relation (2.13) also holds.

By the Jensen inequality lemma, it is easy to obtain

where

From the Leibniz-Newton formula, the following equations are true for any matrices N,  M, L,  S,  P₁,  P₂ with appropriate dimensions

Hence, according to (2.8)–(2.19), we can obtain where If $$, then there exists a scalar ε > 0, such that The $$ leads for d₁(t) → d₁ to E₁ < 0 and leads for d₁(t) → 0 to E₂ < 0, where It is easy to see that E₁ results from $$, where we deleted the zero row and the zero column. Define The LMI (2.23) and (2.24) imply (2.22) because and $$ is convex in d₁(t) ∈ [0, d₁].

LMI (2.23) leads for d₂(t) → d₂ to LMI (2.2) and for d₂(t) → 0 to LMI (2.3). It is easy to see that $$ results from $$, where we deleted the zero row and the zero column. The LMI (2.2) and (2.3) imply (2.23) because

and E₁  is convex in d₂(t) ∈ [0, d₂], where $$and $$ are defined in Theorem 2.1.

Similarly, the LMI (2.4) and (2.5) imply (2.24) because

and E₂  is convex in d₂(t) ∈ [0, d₂], where $$and $$are defined in Theorem 2.1. According to the above analysis, one can conclude that the system (1.5) with delays d₁(t) and d₂(t) satisfying (1.6) is asymptotically stable if the LMIs (2.1)–(2.5) hold.

From the proof of Theorem 2.1, one can obtain that $$ is negative definite in the rectangle 0 ⩽ d₁(t) ⩽ d₁, 0 ⩽ d₂(t) ⩽ d₂, only if it is negative definite at all vertices. We call this method as the convex polyhedron method.

Remark 2.3. To avoid the emergence of the reciprocally convex combination in (2.12), similar to [9], the integral terms in (2.10) can be upper bounded by

which results in a convex combination on γ. However, Theorem 2.1 directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma 1.2, which achieves performance behavior identical to the approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.

Remark 2.4. Compared to some existing ones, the estimation of $$ in the proof of Theorem 2.1 is less conservative due to the convex polyhedron method is employed. More specifically, $$ is retained, while $$ is divided into $$ and $$. When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term $$ as $$. The convex polyhedron method is employed to verify the negative definiteness of $$. Therefore, Theorem 2.1 is expected to be less conservative than some results in the literature.

Remark 2.5. The case in which only two additive time-varying delay components appear in the state has been considered, and the idea in this paper can be easily extended to the system (1.3) with multiple additive delay components satisfying (1.4). Choose the Lyapunov functional as

Then, the corresponding stability result can be easily derived similar to the proof of Theorem 2.1. The result is omitted due to complicated notation.

Remark 2.6. The stability condition presented in Theorem 2.1 is for the nominal system. However, it is easy to further extend Theorem 2.1 to uncertain systems, where the system matrices A and A_d contain parameter uncertainties either in norm-bounded or polytopic uncertain forms. The reason why we consider the simplest case is to make our idea more lucid and to avoid complicated notations.

Example 3.1. Consider system (1.5) with the following parameters:

Our purpose is to calculate the upper bound d₁ of delay d₁(t), or d₂ of delay d₂(t), when the other is known, below which the system is asymptotically stable. By combining the two delay components together, some existing stability results can be applied to this system. The calculation results obtained by Theorem 2.1, in this paper, Theorem  1 in [6, 12, 15, 16], [14, Theorem  2] for different cases are listed in Table 1. It can be seen from the Table 1 that Theorem 2.1, in this paper, yields the least conservative stability test than other results.

Example 3.2. Consider system (1.5) with the following parameters:

We assume condition 1: $$, $$; condition 2: $$, $$, and under the two cases above, respectively. Table 2 lists the corresponding upper bounds of d₂ for given d₁. This numerical illustrates the effectiveness of the derived results.