Further results on delay-dependent robust H ∞ control for uncertain systems with interval time-varying delays

This work focuses on the delay-dependent robust H ∞ control for uncertain linear systems with interval time-varying delays. The key features of the method include employment of a tighter integral inequality and construction of an appropriate Lyapunov–Krasovskii functionals (LKF). Meanwhile, the delay-dependent conditions with less conservatism are derived owing to the consideration of the information of the lower bound of time delay. Based on the obtained criteria, robust H ∞ controller design and performance analysis for the uncertain system are presented. Some numerical examples are also provided to show the effectiveness and advantage of the main results.


Introduction
The phenomenon of time delays and parameter uncertainty frequently appear in many practical systems such as communication systems, engineering systems, chemical engineering systems, nuclear reactors, population ecology and other fields (Park et al., 2015;Zhang et al., 2019;Zhou, 2016). It is well recognized that the existence of delays and uncertainty often causes poor performance, oscillation or even system instabilities (Qian et al., 2017;Wang et al., 2017). Therefore, the stability analysis of uncertain systems with time-varying delays has become an important issue in theory and application, and many related problems also have been researched, such as robust stability (Cheng et al., 2016;Li & Liao, 2018;Lu et al., 2019), H ∞ performance (Ding et al., 2017;Kwon et al., 2016;Meng et al., 2020), passivity (Peng & Jian, 2018), control (Niamsup & Phat, 2020). During the past decades, lots of fruitful results have been achieved. See the references Lee et al. (2014), Cheng et al. (2016), Bai et al. (2016), Qian et al. (2019) and Zheng et al. (2020).
As is well-known, H ∞ control is a worst-case design which is suitable and effective for solving system robustness against disturbances with no prior knowledge other than being energy bounded and parametric uncertainties. The maximum delay and H ∞ performance level are two key indexes to judge the conservatism of the obtained criterion. In order to continuously reduce the conservatism, the considerable research efforts have been made on two aspects, one is the selection of appropriate Lyapunov-Krasovskii functional, and the other is CONTACT Haibo Liu liuhaibo09@hpu.eu.cn to estimate the derivative of LKF more accurately, such as delay partitioning approach (Ding et al., 2017), freematrix-based integral inequality , the augmented LKF approach (Meng et al., 2020;Peng & Jian, 2018), Wirtinger-based inequality (Li & Liao, 2018), reciprocally convex combination (Kwon et al., 2016;Li & Xue, 2016;Zhang et al., 2017), auxiliary function-based integral inequality (Zhang et al., 2018) and Bessel-Legendre inequality (Seuret & Gouaisbaut, 2018).
Recently, considering the disturbances and parameter uncertainty in modelling, the robust H ∞ control has been studied in Bai et al. (2016), Sun et al. (2018), Meng et al. (2020) and Niamsup and Phat (2020). In Meng et al. (2020), based on the sliding mode observer framework, a sufficient condition and novel integral sliding surface function with the compensator were proposed to investigate robust H ∞ asymptotic stabilization for uncertain neutral-type systems. A new delay-dependent sufficient condition for admissibility of the system with nondifferentiable delay was presented, and state feedback controllers was designed which ensure the descriptor closed-loop system admissible with a maximum H ∞ disturbance attenuation level in Niamsup and Phat (2020). In Bai et al. (2016), the free weighting matrix approach was used to obtain some delay-dependent robust H ∞ performance analysis for uncertain linear system. In Seuret and Gouaisbaut (2018), in order to integrate the LKF with the estimating technique effectively, a new augmented vector and LKF with triple integral terms were constructed. However, it should be noted that, the criteria derived in Bai et al. (2016) and Qian et al. (2019) require that the lower bound of time delay is restricted to be 0, and the information of the lower bound of time delay is not considered. In addition, some more slack variables introduced would increase computation complexity and some useful terms were also neglected. Therefore, there is still space for improvement in integrating LKF construction and inequality scaling techniques effectively, and some new methods should be explored.
Motivated by the above discussion, we study further the stability and robust H ∞ conrol for uncertain linear time-varying delay systems with disturbances. The main contributions of the paper can be summarized as follows: (1) For estimating the derivative of LKF more accurately, the derivatives of triple integrals are separated elaborately to use the delay information ignored in existing methods and are estimated by employing improved Wirtinger inequality and reciprocally convex method, which can lead to less conservative results.
(2) Based on the improved stability criteria, a delaydependent condition for the existence of a state feedback controller is obtained, which ensures asymptotic stability and a prescribed H ∞ performance level of the closed-loop system for all admissible uncertainties.

Problem formulations
Consider the following uncertain systems with timevarying delays and disturbance: (1) where x(t) ∈ R n is the state vector, u(t) ∈ R m is the control vector, ω(t) ∈ R p denotes the disturbance input such thatω(t) ∈ L 2 [0, ∞), z(t) ∈ R l denotes the control output, the initial condition φ(t) is a continuously differentiable vector-valued function; A, A d , B, B ω , C, C d , D and D ω are known constant matrices with appropriate dimensions; A(t), A d (t) and B(t) are the uncertainties of system matrices of form (2) in which E a ,E d ,E b and H are known constant matrices and the time-varying nonlinear function F(t) satisfies F T (t)F(t) ≤ I. The delay h(t), is time-varying continuous function that satisfies where 0 ≤ h 1 < h 2 , and μ are constant values. The purpose of this paper is to study the robust stability analysis and H ∞ performance for systems (1). In order to obtain our main results, we need the following definition and lemmas.
Definition 1: [Qian et al., 2019]. Given a scalar γ > 0, system (1) is said to be asymptotically stable with the H ∞ performance level γ , if it is asymptotically stable and satisfies the H ∞ -norm constraint Lemma 1: For any constant symmetric matrix S ∈ R n×n , real scalars a, b satisfying a < b, and vector-valued function ω ∈ [a, b] → R n , the following integral inequality holds Lemma 2: [Qian et al., 2019]. For any positive definite matrix S ∈ R n×n , real scalars a, b satisfying a < b, and vector-valued function x ∈ [a, b] → R n , the following integral inequality holds b a b βẋ Lemma 3: [Zhang et al., 2018]. Let R 1 , R 2 ∈ S m + , ζ 1 , ζ 2 ∈ R m , and a scalar α ∈ (0, 1). If there exist matrices X 1 , X 2 ∈ S m and Y 1 , Y 2 ∈ R m×m such that

Remark 1:
As is known, an appropriate LKF is crucial to reduce the conservatism of the system. A newly augmented LKF containing single, double and triple forms is constructed, and combined the improved Wirtinger inequality and extended reciprocally convex. Since the information of lower bound of time delay is considered in LKF, Theorem 1 can deal with the case that the lower bound of time delay is not restricted to be 0, but some recent results cannot deal with this case Qian et al., 2018). It will be shown Theorem 1 can provide less conservatism results through some numerical examples.
Remark 2: It should be noted that, auxiliary-functionbased double integral inequalities are used to estimate t+θẋ (s)dsdθin theV 4 (t), and a new extended relaxed integral inequality is employed to tackling with the single integral terms. Meanwhile, by utilizing the improved Wirtinger inequality in Lemma 1 and extended reciprocally convex inequalities, a much tighter lower bound is obtained and the conservatism of the proposed method is further reduced efficiently.
Further, considering the external disturbance, the robust H ∞ performance analysis and controller for the system (1) are developed.

Remark 3:
It is worth pointing out that, although Wirtinger inequality and LKF with triple integral terms have been used to obtain less conservative stability results of time-delay systems (Qian et al., 2017;Wang et al., 2017;Zhang et al., 2018), the most of existing results only discussed stability analysis, not consider in controller design for time-delay systems. Therefore, our method is expected more effective than the existing methods in (Kwon et al., 2016;Sun et al., 2018).
For a given h 2 , the minimum γ that satisfies (19) for Theorem 2 can be obtained by solving a constrained optimization problem. For example, the minimum γ with parametric uncertainties can be obtained by solving the following constrained optimization problem (24), (25) and (26).

Numerical simulations
In this section, we use some numerical examples to demonstrate the effectiveness and advantages of the main results in this paper.
Example 1: Consider the following systeṁ The above system has been used in the literature for concerning delay-dependent stability analysis to  Bai et al. (2016), An et al. (2014) and Hien and Trinh (2015). Furthermore, it can be seen that our result has a smaller number of variables than that of An et al. (2014).
Example 2: Consider the uncertain system (4) with u (t) = 0 The purpose of this example is to find the admissible upper bounds h 2 , which ensure robust asymptotic stability of the above system when the different lower bound h 1 and μ are given. The maximum upper bounds h 2 for different h 1 and μ obtained from Theorem 1 are shown in Table 2. For comparison, Table 2 also lists the upper bounds obtained from the criteria in Wu et al. (2014), Yan, Zhang and Meng (2016) and Sun et al. (2018). From  Table 3. For comparison, Table 3 also lists the upper bounds obtained from the criteria in (). Obviously, the maximum upper bounds obtained from Theorem 2 are larger than those of Sun et al. (2018) in the same conditions, which further implies that the main results in this paper are less conservative than the existing results. Example 4: To verify the effectiveness of robust H ∞ controller design results derived by Theorem 3, consider the following uncertain system Similar to Sun et al. (2018), this paper assumes h(t) satisfies 1.2 ≤ h(t) ≤ 1.8 and μ = 0.3. The minimum allowable γ = 0.2465 for a prescribed delay bound could be derived by solving the convex optimization problem (27) It can be seen that compared with the reference Sun et al. (2018), the optimal disturbance attenuation level γ obtained by Theorem 3 is smaller. Therefore, it can be concluded that Theorem 3 is less conservative than the results derived in Sun et al. (2018).

Conclusions
In this paper, we have investigated the delay-dependent robust H ∞ control for uncertain linear system with interval time-varying delays and disturbance. By utilizing the improved Wirtinger inequality and reciprocally convex approach, the improved robust stability criteria are obtained based on the new augmented Lyapunov-Krasovskii functional. Since the proposed method contains more time-delay terms and the information of the lower bound of time delay, the main results have less conservative. Based on the improved criteria, robust H ∞ controller design and performance analysis for uncertain system have been presented. The effectiveness and advantage of the proposed method have been demonstrated through numerical examples.

Disclosure statement
No potential conflict of interest was reported by the author(s).