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

Abstract

This paper is concerned with delay-dependent stability for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

1. Introduction

Robust stability of dynamic interval systems covering interval matrices and interval polynomials has attracted considerable attention over last decades. Reference [1] presents some necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. It is well known that time delay frequently occurs in many industrial and engineering systems, such as manufacturing systems, telecommunication, and economic systems, and is a major cause of instability and poor performance. Over the past decades, much efforts have been invested in the stability analysis of time-delay systems [2–16]. Reference [2] deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete time systems with state delay. Reference [3] deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed linear dynamic systems. However, almost all the reported results mentioned above on time-delay systems are based on the following basic mathematical model:𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝑑(𝑡)),(1.1) where 𝑑(𝑡) is a time delay in the state 𝑥(𝑡), which is often assumed to be either constant or time-varying satisfying certain conditions, for example,0⩽𝑑(𝑡)⩽𝑑<∞,𝑑̇(𝑡)⩽𝜏<∞.(1.2)

Sometimes in practical situations, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. Thus, in recent papers [15, 16], a new model for time-delay systems with multiple additive time-varying delay components has been proposed: 0⩽𝑥̇𝑑(𝑡_𝑖)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥𝑡−∑^𝑛_𝑖=1𝑑_𝑖(𝑡),(1.3)(𝑡)⩽𝑑_𝑖<∞,𝑑̇_𝑖(𝑡)⩽𝜏_𝑖<∞.(1.4) To make the stability analysis simpler, we proceed by considering the system (1.3) with two additive delay components as follows: 𝑥̇(𝑡)=𝑥𝐴(𝑥𝑡)(=𝑡)+𝜙(𝐴𝑡)_𝑑,𝑥𝑡𝑡−∈𝑑[₁−(𝑑𝑡),0−]𝑑.₂(𝑡),(1.5) Here, 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝑑₁(𝑡) and 𝑑₂(𝑡) represent the two delay components in the state, and we denote 𝑑(𝑡)=𝑑₁(𝑡)+𝑑₂(𝑡); 𝐴, 𝐴_𝑑 are system matrices with appropriate dimensions. It is assumed that00⩽⩽𝑑𝑑₁₂(𝑡)⩽𝑑₁<∞,𝑑̇₁(𝑡)⩽𝜏₁<∞,(𝑡)⩽𝑑₂<∞,𝑑̇₂(𝑡)⩽𝜏₂<∞,(1.6) and 𝑑=𝑑₁+𝑑₂, 𝜏=𝜏₁+𝜏₂. 𝜙(𝑡) is the initial condition on the segment [−𝑑,0].

The purpose of our paper is to derive new stability conditions under which system (1.5) is asymptotically stable for all delays 𝑑₁(𝑡) and 𝑑₂(𝑡) satisfying (1.6). One possible approach to check the stability of this system is to simply combine 𝑑₁(𝑡) and 𝑑₂(𝑡) into one delay 𝑑(𝑡) with0⩽𝑑(𝑡)⩽𝑑₁+𝑑₂<∞,𝑑̇(𝑡)⩽𝜏₁+𝜏₂<∞.(1.7) Then, the system (1.5) becomes 𝑥̇(𝑥𝑡)(𝑡=)=𝐴𝑥𝜙((𝑡𝑡))+,𝐴𝑡_𝑑∈𝑥([𝑡−−𝑑𝑑,0(]𝑡).),(1.8) By using some existing stability conditions, the stability of system (1.8) can be readily checked. As discussed in [15, 16], however, since this approach does not make full use of the information on 𝑑₁(𝑡) and 𝑑₂(𝑡), it would be inevitably conservative for some situations. Recently, some new delay-dependent stability criteria have been obtained for system (1.5) in [15, 16], by making full use of the information on 𝑑₁(𝑡) and 𝑑₂(𝑡). However, the stability result is conservative because of overly bounding some integrals appearing in the derivative of the Lyapunov functional. On the one hand, the integral −∫^{𝑡𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 in [15] was enlarged as −∫^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠, with −∫^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 discarded. On the other hand, some integrals were estimated conservatively. Take −∫^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 as an example, by introducing0=2𝜁^𝑇𝑆𝑥(𝑡)−𝑥𝑡−𝑑₁(𝑡)−^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇(𝑠)𝑑𝑠(1.9) with an appropriate vector 𝜁(𝑡) and a matrix 𝑆, respectively, it was estimated as2𝜁^𝑇(𝑡)𝑆𝑥(𝑡)−𝑥𝑡−𝑑₁(𝑡)+𝜁^𝑇(𝑡)𝑑₁𝑆𝑍¹⁻¹𝑆^𝑇𝜁(𝑡)(1.10) with 𝑑₁(𝑡)𝑆𝑍¹⁻¹𝑆^𝑇 enlarged as 𝑑₁𝑆𝑍¹⁻¹𝑆^𝑇.

The problem of delay-dependent stability criterion for continuous systems with two additive time-varying delay components has been considered in this paper. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to indicate less conservatism of the stability results.

Definition 1.1. Let Φ₁,Φ₂,…,Φ_𝑁∶ℛ^𝑚→ℛ^𝑛 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 1𝛼₁Φ₁+1𝛼₂Φ₂+⋯+1𝛼_𝑁Φ_𝑁∶𝐷⟶𝑅^𝑛,(1.11) where the real numbers 𝛼_𝑖 satisfy 𝛼_𝑖>0 and ∑_𝑖𝛼_𝑖=1.

The following Lemma 1.2 suggests a lower bound for a reciprocally convex combination of scalar positive functions Φ_𝑖=𝑓_𝑖.

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 min_{𝛼𝑖∣𝛼𝑖∑>0,𝑖𝛼𝑖=1}_𝑖1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)}_𝑖≠𝑗𝑔_𝑖,𝑗(𝑡)(1.12) subject to ⎧⎪⎨⎪⎩𝑔_𝑖,𝑗∶𝑅^𝑚⟶𝑅,𝑔_𝑗,𝑖(𝑡)Δ𝑔_𝑖,𝑗(𝑡),⎡⎢⎢⎣𝑓^𝑖𝑔(𝑡)𝑔_𝑖,𝑗(𝑡)_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)⎤⎥⎥⎦⩾0⎫⎪⎬⎪⎭.(1.13) In the following, we present a new stability criterion by a convex polyhedron method and Lemma 1.2.

2. Main Results

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 matrices𝑃,𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅,𝑄₆, 𝑍,𝑍₁,𝑍₂ and any matrices𝑆₁₂,𝑁,𝑀,𝐿,𝑆,𝑃₁,𝑃₂ with appropriate dimensions, such that the following LMIs hold: ⎡⎢⎢⎣𝑍𝑆₁₂∗𝑍⎤⎥⎥⎦⩾0,(2.1)𝐸₁₃=⎡⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑁−𝑑₂𝐿∗∗𝑍₁−𝑑0₂𝑍₂⎤⎥⎥⎥⎥⎦<0,(2.2)𝐸₁₄=⎡⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑁−𝑑₂𝑆∗∗𝑍₁−𝑑0₂𝑍₂⎤⎥⎥⎥⎥⎦<0,(2.3)𝐸₂₃=⎡⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑀−𝑑₂𝐿∗∗𝑍₁−𝑑0₂𝑍₂⎤⎥⎥⎥⎥⎦<0,(2.4)𝐸₂₄=⎡⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑀−𝑑₂𝑆∗∗𝑍₁−𝑑0₂𝑍₂⎤⎥⎥⎥⎥⎦<0,(2.5) where E=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣E¹¹E₁₂𝑆^𝑇120000E₁₈∗E₂₂E₂₃0000𝐴^𝑇𝑑𝑃^𝑇2∗∗∗∗E∗₃₃E0₄₄0000∗∗∗∗−0𝑄₄000∗∗∗∗∗E0₆₆00∗∗∗∗∗∗−0𝑄₆0∗∗∗∗∗∗∗E0₈₈+𝑁+𝐿00𝑀−𝑁−𝑀𝑆−𝐿−𝑆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦0+_𝑇𝑁+𝐿00𝑀−𝑁−𝑀𝑆−𝐿−𝑆0𝐸₁₁,𝐸₁₈=𝑄₁+𝑄₂+𝑄₃+𝑄₄+𝑄₅+𝑄₆−𝑍+𝑃₁𝐴+𝐴^𝑇𝑃^𝑇1,𝐸₁₂=−𝑆^𝑇12+𝑍+𝑃₁𝐴_𝑑,𝐸=𝑃−𝑃₁+𝐴^𝑇𝑃^𝑇2,𝐸₂₂=−(1−𝜏)𝑄₁−2𝑍+𝑆₁₂+𝑆^𝑇12,𝐸₂₃=−𝑆^𝑇12+𝑍,₃₃𝐸₈₈=−𝑄₂−𝑍,𝐸₄₄=−1−𝜏₁𝑄₃,𝐸₆₆=−1−𝜏₂𝑄₅,=𝑑²𝑍+𝑑₁𝑍₁+𝑑₂𝑍₂−𝑃₂−𝑃^𝑇2.(2.6)

Proof. Construct a new Lyapunov functional candidate as 𝑉𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡)),₁𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡),₂(𝑥(𝑡))=+^{𝑡𝑡−𝑑(𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑}_1(𝑡)𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠_{𝑡𝑡−𝑑1}𝑉𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑}_2(𝑡)𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑}₂𝑥^𝑇(𝑠)𝑄₆𝑥(𝑠)𝑑𝑠,₃(𝑥(𝑡))=𝑑^0−𝑑^{𝑡𝑡+𝜃}𝑉𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝜃,₄(𝑥(𝑡))=^0−𝑑₁^{𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^0−𝑑₂^{𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃.(2.7)

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 𝑉𝑉₁̇̇(𝑥(𝑡))=2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡),(2.8)₂(𝑥(𝑡))=−𝑥^𝑇𝑥_𝑇(𝑡)𝑄₁+𝑄₂+𝑄₃+𝑄₄+𝑄₅+𝑄₆𝑥(𝑡)−(1−𝜏)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₁𝑥(𝑡−𝑑(𝑡))−𝑥_𝑇(𝑡−𝑑)𝑄₂𝑥(𝑡−𝑑)−1−𝜏₁𝑥^𝑇𝑡−𝑑₁(𝑡)𝑄₃𝑥𝑡−𝑑₁(𝑡)−𝑥_𝑇𝑡−𝑑₁𝑄₄𝑥𝑡−𝑑₁−1−𝜏₂𝑥^𝑇𝑡−𝑑₂(𝑡)𝑄₅𝑥𝑡−𝑑₂(𝑡)𝑉̇𝑡−𝑑₂𝑄₆𝑥𝑡−𝑑₂,(2.9)₃𝑉̇(𝑥(𝑡))=𝑑²𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−𝑑^{𝑡−𝑑(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠−𝑑^{𝑡𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠,(2.10)₄(𝑥(𝑡))=−𝑥̇^𝑇(𝑡)𝑑₁𝑍₁+𝑑₂𝑍₂𝑥̇(𝑡)−^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠−^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠_{𝑡−𝑑2(𝑡)𝑡−𝑑2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡𝑡−𝑑}_2(𝑡)𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠.(2.11) The 𝑉₃̇(𝑥(𝑡)) is upper bounded by 𝑉₃̇(𝑥(𝑡))⩽𝑑²𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−𝑑𝑑−𝑑(𝑡)𝜁^𝑇−𝑑(𝑡)𝑒₂−𝑒₃𝑍𝑒₂−𝑒₃^𝑇𝜁(𝑡)𝑑(𝑡)𝜁^𝑇⩽𝑑₂(𝑡)𝑒₁−𝑒₂𝑍𝑒₁−𝑒₂^𝑇𝜁(𝑡)(2.12)𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−𝜁^𝑇(𝑡)⎡⎢⎢⎢⎣𝑒^𝑇2−𝑒^𝑇3𝑒^𝑇1−𝑒^𝑇2⎤⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝑍𝑆₁₂𝑆^𝑇12𝑍⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑒^𝑇2−𝑒^𝑇3𝑒^𝑇1−𝑒^𝑇2⎤⎥⎥⎥⎦𝜁(𝑡),(2.13) where the inequality in (2.12) comes from the Jensen inequality lemma, and that of (2.13) from Lemma 1.2 as −𝜁^𝑇(𝑡)⎡⎢⎢⎢⎢⎣𝛽𝛼𝑒²−𝑒₃^𝑇−𝛼𝛽𝑒¹−𝑒₂^𝑇⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝑍𝑆₁₂𝑆^𝑇12𝑍⎤⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝛽𝛼𝑒²−𝑒₃^𝑇−𝛼𝛽𝑒¹−𝑒₂^𝑇⎤⎥⎥⎥⎥⎦𝜁(𝑡)⩽0,(2.14) where 𝜁^𝑇(𝑡)=𝑥𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑥^𝑇(𝑡−𝑑)𝑥^𝑇𝑡−𝑑₁(𝑡)𝑥^𝑇𝑡−𝑑₁𝑥^𝑇𝑡−𝑑₂(𝑡)_𝑇𝑒𝑡−𝑑₂𝑥̇^𝑇(𝑡)₁,=𝐼0000000^𝑇,𝑒₂=0𝐼000000^𝑇,𝑒₃=00𝐼00000^𝑇,(2.15)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 −−^{𝑡𝑡−𝑑}_{1(𝑡)𝑡−𝑑1(𝑡)𝑡−𝑑1}𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠⩽−𝑑₁(𝑡)𝑈^𝑇1𝑍₁𝑈₁,−_{𝑡𝑡−𝑑2(𝑡)}𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠⩽−𝑑₁−𝑑₁(𝑡)𝑈^𝑇2𝑍₁𝑈₂,−_{𝑡−𝑑2(𝑡)𝑡−𝑑2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠⩽−𝑑₂(𝑡)𝑈^𝑇3𝑍₂𝑈₃,𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠⩽−𝑑₂−𝑑₂(𝑡)𝑈^𝑇4𝑍₂𝑈₄,(2.16) where 𝑈₁=1𝑑₁(𝑡)^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇(𝑠)𝑑𝑠,𝑈₂=1𝑑₁−𝑑₁(𝑡)^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁𝑈𝑥̇(𝑠)𝑑𝑠,₃=1𝑑₂(𝑡)^{𝑡𝑡−𝑑}_2(𝑡)𝑥̇(𝑠)𝑑𝑠,𝑈₄=1𝑑₂−𝑑₂(𝑡)^{𝑡−𝑑2(𝑡)𝑡−𝑑}₂lim_{𝑑1(𝑡)→0}𝑥̇(𝑠)𝑑𝑠,(2.17)1𝑑₁(𝑡)^{𝑡𝑡−𝑑}_1(𝑡)lim_{𝑑1(𝑡)→𝑑1}𝑥̇(𝑠)𝑑𝑠=𝑥̇(𝑡),1𝑑₁−𝑑₁(𝑡)^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁lim_{𝑑2(𝑡)→0}𝑥̇(𝑠)𝑑𝑠=𝑥̇𝑡−𝑑₁,1𝑑₂(𝑡)^{𝑡𝑡−𝑑}_2(𝑡)lim_{𝑑2(𝑡)→𝑑2}𝑥̇(𝑠)𝑑𝑠=𝑥̇(𝑡),1𝑑₂−𝑑₂(𝑡)^{𝑡−𝑑2(𝑡)𝑡−𝑑}₂𝑥̇(𝑠)𝑑𝑠=𝑥̇𝑡−𝑑₂.(2.18)
From the Leibniz-Newton formula, the following equations are true for any matrices 𝑁,𝑀, 𝐿,𝑆,𝑃₁,𝑃₂ with appropriate dimensions 22𝜁𝜁^𝑇_𝑇(𝑡)𝑁𝑥(𝑡)−𝑥𝑡−𝑑₁(𝑡)−𝑑₁(𝑡)𝑈₁=0,2𝜁_𝑇(𝑡)𝑀𝑥𝑡−𝑑₁(𝑡)−𝑥𝑡−𝑑₁−𝑑₁−𝑑₁(𝑡)𝑈₂=0,2𝜁_𝑇(𝑡)𝐿𝑥(𝑡)−𝑥𝑡−𝑑₂(𝑡)−𝑑₂(𝑡)𝑈₃=0,2𝑥(𝑡)𝑆𝑥𝑡−𝑑₂(𝑡)−𝑥𝑡−𝑑₂−𝑑₂−𝑑₂(𝑡)𝑈₄=0,_𝑇(𝑡)𝑃₁+𝑥̇^𝑇(𝑡)𝑃₂−𝑥̇(𝑡)+𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝑑(𝑡))=0.(2.19) Hence, according to (2.8)–(2.19), we can obtain 𝑉̇(𝑥(𝑡))⩽𝜉^𝑇(𝑡)𝐸𝜉(𝑡),(2.20) where 𝜉^𝑇(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇1𝑈^𝑇2𝑈^𝑇3𝑈^𝑇4,𝐸=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁(𝑡)𝑁−𝑑₁−𝑑₁(𝑡)𝑀−𝑑₂(𝑡)𝐿−𝑑₂−𝑑₂(𝑡)𝑆∗(∗𝑡)𝑍₁−𝑑₁000∗∗−∗𝑑₁(𝑡)𝑍₁−𝑑₂00∗∗∗(∗𝑡)𝑍₂−𝑑₂0−𝑑₂(𝑡)𝑍₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.21) If 𝐸<0, then there exists a scalar 𝜀>0, such that 𝑉̇(𝑥(𝑡))⩽𝜉^𝑇(𝑡)𝐸𝜉(𝑡)⩽−𝜀𝜉^𝑇(𝑡)𝜉(𝑡)⩽−𝜀𝑥^𝑇(𝑡)𝑥(𝑡)<0,∀𝑥(𝑡)≠0.(2.22) The 𝐸<0 leads for 𝑑₁(𝑡)→𝑑₁ to 𝐸₁<0 and leads for 𝑑₁(𝑡)→0 to 𝐸₂<0, where 𝐸₁=⎡⎢⎢⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑁−𝑑₂(𝑡)𝐿−𝑑₂−𝑑₂(𝑡)𝑆∗∗𝑍₁−𝑑₂00∗∗(∗𝑡)𝑍₂−𝑑₂0−𝑑₂(𝑡)𝑍₂𝐸⎤⎥⎥⎥⎥⎥⎥⎦<0,(2.23)₂=⎡⎢⎢⎢⎢⎢⎢⎣𝐸−𝑑¹∗−𝑑₁𝑀−𝑑₂(𝑡)𝐿−𝑑₂−𝑑₂(𝑡)𝑆∗∗𝑍₁−𝑑₂00∗∗(∗𝑡)𝑍₂−𝑑₂0−𝑑₂(𝑡)𝑍₂⎤⎥⎥⎥⎥⎥⎥⎦<0.(2.24) It is easy to see that 𝐸₁ results from 𝐸_{|𝑑1(𝑡)=𝑑1}, where we deleted the zero row and the zero column. Define 𝜉𝜉^𝑇1_𝑇2(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇1𝑈^𝑇3𝑈^𝑇4,(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇2𝑈^𝑇3𝑈^𝑇4,(2.25) The LMI (2.23) and (2.24) imply (2.22) because 𝑑₁(𝑡)𝑑₁𝜁^𝑇1(𝑡)𝐸₁𝜁₁(𝑡)+𝑑¹−𝑑₁(𝑡)𝑑₁𝜁^𝑇2(𝑡)𝐸₂𝜁₂(𝑡)=𝜉^𝑇(𝑡)𝐸𝜉(𝑡)⩽−𝜀𝑥^𝑇(𝑡)𝑥(𝑡)(2.26) and 𝐸 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 𝐸₁₃ results from 𝐸_{1|𝑑2(𝑡)=𝑑2}, where we deleted the zero row and the zero column. The LMI (2.2) and (2.3) imply (2.23) because 𝑑₂(𝑡)𝑑₂𝜁^𝑇13(𝑡)𝐸₁₃𝜁₁₃(𝑡)+𝑑²−𝑑₂(𝑡)𝑑₂𝜁^𝑇14(𝑡)𝐸₁₄𝜁₁₄(𝑡)=𝜉^𝑇1(𝑡)𝐸₁𝜉₁(𝑡)<0(2.27) and 𝐸₁is convex in 𝑑₂(𝑡)∈[0,𝑑₂], where 𝜉𝜉^𝑇13_𝑇14(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇1𝑈^𝑇3,(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇1𝑈^𝑇4(2.28)𝐸₁₃and 𝐸₁₄ are defined in Theorem 2.1.
Similarly, the LMI (2.4) and (2.5) imply (2.24) because 𝑑₂(𝑡)𝑑₂𝜁^𝑇23(𝑡)𝐸₂₃𝜁₂₃(𝑡)+𝑑²−𝑑₂(𝑡)𝑑₂𝜁^𝑇24(𝑡)𝐸₂₄𝜁₂₄(𝑡)=𝜉^𝑇2(𝑡)𝐸₂𝜉₂(𝑡)<0(2.29) and 𝐸₂is convex in 𝑑₂(𝑡)∈[0,𝑑₂], where 𝜉𝜉^𝑇23_𝑇24(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇2𝑈^𝑇3,(𝑡)=𝜁^𝑇(𝑡)𝑈^𝑇2𝑈^𝑇4.(2.30)𝐸₂₃and 𝐸₂₄are 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 𝐸 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 −𝑑^{𝑡𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠⩽−−[𝑥[𝑥(𝑡()𝑡−−𝑥𝜏((𝑡𝑡−))−𝜏(𝑥𝑡)()𝑡]_𝑇−𝑑)]^𝑇𝑍[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−𝑑)]−(1−𝛾)[𝑥(𝑡−𝜏(𝑡))−𝑍𝑥[(𝑥𝑡(−𝑡)𝑑−)]𝑧_𝑇(𝑡−𝜏(𝑡))]−𝛾[𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))]_𝑇𝑍[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−𝑑)]𝑍[𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))](2.31) 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 𝑉̇(𝑥(𝑡)) in the proof of Theorem 2.1 is less conservative due to the convex polyhedron method is employed. More specifically, −∫^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 is retained, while −∫^{𝑡𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 is divided into −∫^{𝑡𝑡−𝑑}_1(𝑡)𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠 and −∫^{𝑡−𝑑1(𝑡)𝑡−𝑑}₁𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)𝑑𝑠. When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term 𝑑₁(𝑡)𝑆𝑍¹⁻¹𝑆^𝑇 as 𝑑₁𝑆𝑍¹⁻¹𝑆^𝑇. The convex polyhedron method is employed to verify the negative definiteness of 𝑉̇(𝑥(𝑡)). 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 𝑉𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡)),₁𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡),₂(𝑥(𝑡))=^{𝑡𝑡−𝑑(𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^{𝑡𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠+^𝑛_𝑖=1^{𝑡𝑡−𝑑}_𝑖(𝑡)+𝑥^𝑇(𝑠)𝑄_3𝑖𝑥(𝑠)𝑑𝑠_𝑛_𝑖=1^{𝑡𝑡−𝑑}_𝑖𝑉𝑥^𝑇(𝑠)𝑄_4𝑖𝑥(𝑠)𝑑𝑠,₃(𝑥(𝑡))=𝑑^0−𝑑^{𝑡𝑡+𝜃}𝑉𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝜃,₄(𝑥(𝑡))=^𝑛_𝑖=1^0−𝑑_𝑖^{𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑍_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃.(2.32) 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.

3. Illustrative Example

Example 3.1. Consider system (1.5) with the following parameters: 𝐴=⎡⎢⎢⎣−02−00.9⎤⎥⎥⎦,𝐴_𝑑=⎡⎢⎢⎣−−11−01⎤⎥⎥⎦,assuming𝑑̇₁(𝑡)⩽0.1,𝑑̇₂(𝑡)⩽0.8.(3.1)
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.

Table 1 

Calculated delay bounds for different cases.

Example 3.2. Consider system (1.5) with the following parameters: 𝐴=⎡⎢⎢⎣−01.0.0−12.0.0⎤⎥⎥⎦,𝐴_𝑑=⎡⎢⎢⎣−01.0.010..00⎤⎥⎥⎦.(3.2) We assume condition 1: 𝑑̇₁(𝑡)⩽0.2, 𝑑̇₂(𝑡)⩽0.5; condition 2: 𝑑̇₁(𝑡)⩽0.2, 𝑑̇₂(𝑡)⩽0.3, 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.

Table 2 

Allowable upper bound of 𝑑₂ for various 𝑑₁.

4. Conclusions

This paper has investigated the stability problem for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors would like to thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by research on the model and method of parameter identification in reservoir simulation under Grant PLN1121.