A Hierarchical Stability Criterion for Time-Delay Systems based on the Affine Bessel-Legendre Inequality

Ⅰ. Introduction

Time-delays exist in various practical systems such as mechanical systems, remote control systems, cyber-physical systems, and networked control systems, and they often lead to the degradation of performance, oscillation or even instability of the system[1]. Therefore, the stability analysis of systems with time delays has attracted a lot of attention and numerous researches have been investigated in the past few decades[2]-[14].

When deriving the negativity condition of the time-derivative of Lyapunov-Krasovskii functionals for stability anlaysis of delayed systems, it is important to estimate bounds of the integral terms therein using integral inequalities. To solve the problem, various integral inequalities have been provided in the literature such as Jensen’s inequality[2], Wirtinger-based inequality[3], auxiliary function based integral inequalities[4], Bessel-Legendre(BL) inequality[5], free-matrix-based integral inequalities[6]-[8], and affine BL inequality[9]. Especially, in [9], effective stability criteria have been obtained via the affine BL inequality with the certain degrees(N = 1, 2). It is mentioned that if the degree N of the affine BL inequality increases, much more precise bounds of the quadratic integral function can be obtained, but to achieve this, the appropriate Lyapunov-Krasovskii functionals should be designed along with the degree N[9]. On the other hand, in [10], the design method of Lyapunov-Krasovskii functionals with an arbitrary degree N of the new integral inequality has been proposed. By utilizing the proposed Lyapunov-Krasovskii functional, hierarchical stability conditions of delayed systems have been obtained.

In this paper, with the help of the idea of [10], a new generalized Lyapunov-Krasovskii functional with an arbitrary degree N of the affine BL inequality is proposed. Furthermore, hierarchical stability conditions along with the degree N is derived via the proposed Lyapunov-Krasovkii functional. By utilizing the new hierarchical stability conditions, the stability criteria for various degrees of the affine BL inequality can easily be obtained. It is also worth noting that the proposed approach can be applied to derive hierarchical stability criteria based on not only the affine BL inequality but recently proposed integral inequalities[7][8]. A numerical example verifies the generality and the efficiency of the proposed stability conditions.

Notations. Throughout the paper, $$ {\mathbb{R}}^n$$ denotes the n-dimensional Euclidean space. I_n, 0_n, and 0_{m × n} mean n × n identity matrix, n × n zero matrix and m × n zero matrix, respectively. [X]_{m × n} denotes that X is a m × n matrix with its (i, j)-th component X_ij. The notation He (A) and col{x₁,x₂, ⋯, x_n} stand for A+A^T and $$ {\left[x_1^T,x_2^T,\cdots ,x_n^T\right]}^T$$, respectively. X>0(X≥0) represents that X is a real symmetric positive definite(positive semidefinite) matrix, and ⊗ represents a Kronecker product.

Ⅱ. Problem Formulation and Preliminaries

The following linear system with a time-varying delay is considered:

where and ϕ(t) is the initial condition.

To end this section, we quote two lemmas needed to acquire our main results.

Lemma 1 [9]. Let $$ x\left(s\right)\in {\mathbb{R}}^n$$ be a continuously differentiable function for s ∈ [a,b]. For a positive definite matrix R = R^T > 0, any matrix X, and any integer N ≥ 0, the following inequality holds:

where

Lemma 2 [10]. For any integer i ≥ 0, let x be an integrable function in $$ \left[a,b\right]\to {\mathbb{R}}^n$$. Then, we have

where r₀ = a.

Remark 1. The affine BL inequality in Lemma 1 has successfully removed the reciprocal convexity, which makes it more difficult to derive the stability criterion, arose from the BL inequality. Note that increasing the degree N can give less conservative stability results, but the Lyapunov-Krasovskii functional also should be newly designed. Fortunately in [9], the method for designing Lyapunov-Krasovskii functional with a given degree N has been described but it is still not easy to derive the new stability condition based on the new Lyapunov-Krasovskii functional whenever the degree N changes, which is the motivation of our work.

Ⅲ. Proposed Hierarchical Stability Criterion

This section proposes a hierarchical stability criterion of time-delay systems along with the degree of the affine BL inequality (2) based on a new generalized Lyapunov-Krasovskii functional. The following theorem is obtained by utilizing Lemmas 1 and 2.

Theorem 1. For given scalars h > 0 and μ₁ < μ₂ < 1, the system (1) is asymptotically stable if there exist positive definite matrices $$ {\left[P\right]}_{\left(3+2N\right)n\times \left(3+2N\right)n}$$, and $$ {\left[Q\right]}_{4n\times 4n}, {\left[S\right]}_{4n\times 4n}, {\left[R\right]}_{n\times n}$$, and any matrices $$ {\left[X_1\right]}_{\left(2+N\right)n\times \left(1+N\right)n}$$, and $$ {\left[X_2\right]}_{\left(2+N\right)n\times \left(1+N\right)n}$$ satisfying the following condition for i, j = 1,2:

where

and $$ e_l\in {\mathbb{R}}^{n\times \left(5+2N\right)n}$$ for an positive integer l ∈ [1,5+2N] are elementary matrices, for example $$ e_3=\left[0_{n\times 2n}\hspace{0.25em}{I}_n\hspace{0.25em}{0}_{n\times \left(2+2N\right)n}\right]$$

Proof. Consider the generalized Lyapunov-Krasovskii functional with the arbitrary degree N of the inequality in Lemma 1 such that

where

The time derivative of (4) can be computed as follows:

where

After dividing the range [t-h, t] of the integral term in $$ \dot{V}\left(t\right)$$ into two ranges [t-h, t-h(t)] and [t-h(t), t], applying Lemma 1 to the resulting two integral terms gives

where

Combining (5) into $$ \dot{V}\left(t\right)$$ yields

Note that the condition is affine with respect to h(t) and $$ \dot{h}\left(t\right)$$, thus if the condition is satisfied at the vertices of the polyhedral set, i.e., $$ \left(0,{\mu }_1\right),\left(h,{\mu }_1\right),\left(0,{\mu }_2\right),\left(h,{\mu }_2\right)$$, the negativity of $$ \dot{V}\left(t\right)$$ is ensured for all $$ \left(h\left(t\right),\dot{h}\left(t\right)\right)\in \left[0,h\right]\times \left[{\mu }_1,{\mu }_2\right]$$. Also, by utilizing Schur complement, the negativity condition can be represented as a tractable form of LMIs in (3), which ends the proof.

Remark 2. With the help of the approaches in [10], in Theorem 1, the hierarchical stability criterion is obtained via the generalized Lyapunov-Krasovskii functional in (4) with an arbitrary degree N of the affine BL inequality in Lemma 1. When increasing the degree of the inequality in Lemma 1, $$ {\eta }_{1N}\left(t\right)$$ in the Lyapunov-Krasovskii functional also should be designed appropriately for obtaining a less conservative stability condition. In (4), $$ {\eta }_{1N}\left(t\right)$$ is appropriately designed along with an arbitrary degree N, and from the definition of I_i(a,b) in Lemma 2, its time derivative is calculated by utilizing the following derivation:

where I_-1 (a,b) = x(a)

Remark 3. In [9], it is difficult to generalize the stability conditions since it is difficult to find the relation between ξ_N(t) that comes from the inequality in Lemma 1 and the augmented state vector ξ(t) when the degree N of the inequality is changed. It is worth noting that, in Theorem 1, with the help of introducing the matrices $$ {\Pi }_N^i$$ such that

where $$ {\xi }_N^i\left(t\right)$$ are $$ {\xi }_N\left(t\right)$$ in Lemma 1 for the intervals [t-h(t), t] and [t-h, t-h(t)], the upper bounds in (5) and (6) can be easily derived as the quadratic form of ξ(t) in (6), which successfully yields the hierarchical stability conditions.

Ⅳ. Numerical Example

This section verifies the generality and the efficiency of the proposed result by carrying out the well-known numerical example[2].

Example 1. Consider the delayed system (1) with

where μ₂ = -μ₁. For various μ₁ and μ₂, Table 1 shows the allowable upper bounds of h(t) that guarantee the asymptotic stability of the system and the number of variables(N_v). When the degree N of the inequality increases, the number of variables of Theorem 1 also increases since the dimension of matrices P, X₁ and X₂ increases, and it can be computed by (4N²+12N+25)n²+(N+6)n.

Table 1.

Allowable upper bounds h for different (μ1,μ2)

From Table 1, it can be seen that Theorem 1 yields larger allowable upper bounds h for most cases compared to stability results in the literature. Only where N = 1, Theorem 1 is more conservative than the criterion in [14], but it becomes less conservative where N ≥ 2 both in terms of the performance and the computational complexity. It is worth noting that Theorem 1 successfully yields larger allowable upper bounds h as the degree N of the inequality in Lemma 1 increases, which proves the generality and the effectiveness of Theorem 1.

Remark 4. Note that some recent research also have provided stability criteria with certain degree of the newly proposed inequality such as a generalized free-matrix-based integral inequality(GFMBII)[7] and a generalized integral inequality based on free matrices(GIIBFM)[8]. The proposed approaches in Theorem 1 can be applied to derive hierarchical stability criteria based on those newly proposed integral inequality, which will be our future research direction.

Ⅴ. Conclusion

This paper proposed the hierarchical stability criterion for linear systems with a time-varying delay based on the affine BL inequality. The generalized Lyapunov-Krasovskii functional with an arbitrary degree N of the inequality was designed, and its time-derivative was successfully dealt with to obtain stability conditions as a tractable form of LMIs. Since it is hierarchical condition, increasing the degree N of the inequality gives much less conservative results, which is clearly verified by the numerical example. It would be an interesting subject for future research to apply the proposed approach to recent results based on the new integral inequalities with few additional number of variables.

Acknowledgments

This research was supported by Kumoh National Institute of Technology(202003680001)

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