Memory State-Feedback Stabilization for a Class of Time-Delay Systems with a Type of Adaptive Strategy

Abstract

Stabilization of a class of systems with time delay is studied using adaptive control. With the help of the “error to error” technique and the separated “descriptor form” technique, the memory state-feedback controller is designed. The adaptive controller designed can guarantee asymptotical stability of the closed-loop system via a suitable Lyapunov-Krasovskii functional. Some sufficient conditions are derived for the stabilization together with the linear matrix inequality (LMI) design approach. Finally, the effectiveness of the proposed control design methodology is demonstrated in numerical simulations.

1. Introduction

Time delay is one of the instability sources for various systems in practice. With the aid of memoryless state-feedback controllers, Choi and Chung [1] extended the Riccati equation approach to uncertain dynamic systems with time-varying delay in both the system state and control. Since that, considerable attention has been devoted to the problem of delay-dependent stability analysis and controller design for time-delay systems; see, for example, Fridman [2, 3], where the functional was based on the “descriptor form,” Liu et al. [4] and He et al. [5], where free weighting matrix technique and Leibniz-Newton formula were utilized to reduce the conservativeness, Park and Jeong [6], where delay-upper-bounded state was exploited, and Tian and Zhou [7], where a less conservative conclusion was investigated on neural networks by integral inequality and taking delay upper bound into account. By constructing the whole state-space trajectory solution, De la Sen [8] investigated the stabilization problems for time-delay time-invariant systems and switched dynamic systems with incommensurate point delays. Some robust controllers for uncertain time-delay systems can also be seen in Tsai et al. [9], Han [10], Chen et al. [11], and Zheng et al. [12], where some different kinds of functional were employed to design controllers. Apparently, the memory state-feedback controller is less conservative than the memoryless one [4], as the former can take delayed states into account. However, the memory control results in the mentioned results require precise information of the time delay; that is, the time-delay parameter must be known exactly. If this crucial piece of information is not available, the memory control schemes developed so far cannot be implemented. For these papers mentioned above, only stability analysis can be pursued for systems with unknown time-delay parameter. To the best of the authors’ knowledge, few results have been reported to design memory controllers with unknown time-delay parameter. Jiang et al. [13] provided a type of memory controller with adaptation to the unknown delay parameter; however, the estimate value of the unknown delay was limited to be larger than its real value, which results in the adaptive regulation being unavailable in practice. The bound of the unknown delay was not used to construct the adaptive controller, which is an important factor in leading conservatism. In addition, the LMIs seem unsolvable. Specifically, unknown matrixes like 𝑃 and 𝑃⁻¹ exist in the same LMI. On the other hand, the aforementioned results have not considered the time-delay effect which is actually very common in input. The problem of memory state-feedback controllers for systems with control input delay and available adaptation to unknown time-delay parameter remains open, which motivates the research in this paper.

The main contributions of this paper can be illustrated in the following two aspects: (1) for the unknown time-delay parameter, a new adaptive strategy design method is proposed, considering both the current estimate value and the bound of the time-delay; (2) this work simultaneously constructs a memory controller for a class of systems, which are delayed in the system state, control input, and system matrix.

This paper is organized as follows: in Section 2, we first formulate the problem of the memory controller and adaptive regulation for a class of time-delay systems of a particularly complex nature, since the unknown delay parameter exists in the system state, control input, and system matrix. A lemma is provided to introduce the novel kind of adaptive idea, which can be described as the error to error adaptive technique. In Section 3, for the stabilization problem, by using the function based on “descriptor form” and the technique of separation, the memory controller and the adaptive regulation for the unknown delay parameter can be obtained despite the delayed control input. The bound of the unknown delay parameter is considered in the adaptive strategy, which guarantees the less conservatism. The estimate value is not limited to be larger than the real value; what is more is that it is contained in the memory controller, and thus the conservatism can be reduced further. All the necessary matrixes can be obtained by calculating a solvable LMI. In Section 4, a numerical example is presented to demonstrate the effectiveness of the design method. Finally, several formulations are collected in the appendices.

2. Problem Statements

Consider the following class of time-delay systems as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴𝜏₁𝑥𝑡−𝜏₁+𝐵𝑢𝑡−𝜏₂,𝑥(𝑡)=𝜙(𝑡),∀𝑡∈−max𝜏₁,𝜏₂,0, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector and 𝑢(𝑡)∈𝑅^𝑛1 is the control input vector. 𝐴, 𝐵 are known constant matrices with appropriate dimensions, and 𝐴(𝜏₁) is the matrix depending on 𝜏₁. 𝜏₂≥0 is the time delay in control input with known value. 𝜏₁≥0 is the time delay in system state which is not known exactly, but the upper bound 𝜏^∗₁ and the lower bound 𝜏_1∗ are available. 𝜙(𝑡)∈𝐶[−𝜏,0] is a given continuous vector-valued initial function of system (1). Moreover, there exists a positive constant ℎ₁ such that 0<𝜏^∗₁≤ℎ₁≤𝜏_1∗ holds. Generally, the value ℎ₁ can be chosen as the mean value between 𝜏^∗₁ and 𝜏_1∗; that is, ℎ₁=(𝜏^∗₁+𝜏_1∗)/2.

Assumption 1. The system matrix 𝐴(𝜏₁) is composed of a constant matrix 𝐴₁ and a matrix 𝐴₂(𝜏₁) which is linear with 𝜏₁; that is, 𝐴𝜏₁=𝐴₁+𝜏₁𝐴₂𝜏₁. (2)

Remark 2. The system matrix 𝐴(𝜏₁) is a function of the state delay, which can be seen in the model of measuring intensity for nuclear physics system obtained by [14]. So the problem of stabilization for this type of systems is of some practical significance, and the difficulty in constructing controllers is obvious.

We consider the following feedback controller with memory, with all time delays known: 𝑢(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑥𝑡−𝜏₁+𝜏₁𝐾₃𝑥𝑡−𝜏₁. (3) If the time-delay constant 𝜏₁ of system (1) is not known exactly, which has been introduced above, our main result on memory feedback controller with adaptation to delay parameter for system (1) is presented as follows: 𝑢(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑥𝑡−𝑎₁𝜏₁(𝑡)−𝜏₁(𝑡)−ℎ₁²+𝑎₁𝜏₁(𝑡)+𝜏₁(𝑡)−ℎ₁²×𝐾₃𝑥𝑡−𝑎₁𝜏₁(𝑡)−𝜏₁(𝑡)−ℎ₁², (4) where 𝜏₁(𝑡) is the estimate value of the unknown delay parameter 𝜏₁, satisfying 𝜏̇₁(𝑡)[2(𝜏₁(𝑡)−ℎ₁)+𝑎₁]≤0, for all 𝑡≥0. By using past state information, (𝑎₁𝜏₁(𝑡)+(𝜏₁(𝑡)−ℎ₁)²)𝐾₃𝑥(𝑡−𝑎₁𝜏₁(𝑡)−(𝜏₁(𝑡)−ℎ₁)²) in the controller (4) is designed for 𝜏₁𝐴₂(𝜏₁) in system (1); thus the controller allows for the property of the time-delay system. The constants 𝑎₁, ℎ₁ and the matrices 𝐾_𝑖(𝑖=1,2,3) wait to be determined.

The objective of this paper is to stabilize the system (1) by using the controller (4), obtaining the adaptation law for 𝜏₁(𝑡), which is everywhere time differentiable, at the same time. In order to prove our results, we introduce the following lemmas.

Lemma 3 (see [15]). Given matrices 𝑋 and 𝑌 with the appropriate dimensions, 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝑋^𝑇𝑇𝑋+𝑌^𝑇𝑇⁻¹𝑌,∀𝑇>0. (5)

Lemma 4. Considering the following gain adaptive law for the estimator 𝜏̇₁(𝑡)=−2𝜏₁(𝑡)−ℎ₁+𝑎₁𝑚(𝑡), (6) where 𝑎₁ and ℎ₁ are positive constants and 𝑚(𝑡)≥0 is a positive derivative function, which will be determined later. If we choose 𝑎₁ and ℎ₁ with the following equations: ℎ₁=ℎ₁+ℎ²₁,𝑎₁=2ℎ₁+ℎ²₁−ℎ₁, (7) then 𝜏₁(𝑡) is bounded, and 𝑎₁𝜏₁(𝑡)+(𝜏₁(𝑡)−ℎ₁)² can be bounded with ℎ₁.

Proof. From (6), it is obvious that 𝜏₁(𝑡) satisfies 𝜏̇₁(𝑡)[2(𝜏₁(𝑡)−ℎ₁)+𝑎₁]≤0, for all 𝑡≥0.
Let us prove the boundedness of 𝜏₁(𝑡), for which a Lyapunov function can be constructed as 𝑉(𝜏₁(𝑡))=𝜏²₁(𝑡)/2. The time derivative of 𝑉(𝜏₁(𝑡)) along the adaptive strategy of (6) is 𝑉̇‌𝜏₁(𝑡)=−𝜏₁(𝑡)2𝜏₁(𝑡)−ℎ₁+𝑎₁𝑚(𝑡). (8) If 𝜏₁(𝑡)>(2ℎ₁−𝑎₁)/2, it results in 𝑉̇‌𝜏₁(𝑡)<𝜏₁(𝑡)𝑚(𝑡)≤0, (9) which implies the boundedness of 𝜏₁(𝑡) at (2ℎ₁−𝑎₁)/2. If 𝜏₁(𝑡)≤(2ℎ₁−𝑎₁)/2, together with (8), we have 𝑉̇‌𝜏₁(𝑡)≥𝜏₁(𝑡)𝑚(𝑡)≥0. (10) Therefore, 𝜏₁(𝑡) is increased only when it is less than (2ℎ₁−𝑎₁)/2. Once 𝜏₁(𝑡)=(2ℎ₁−𝑎₁)/2, 𝜏₁(𝑡) will be fixed at this value. Then from (7) yields 𝜏₁(𝑡)=2ℎ₁−𝑎₁2=2ℎ₁+ℎ²₁−2ℎ₁+ℎ²₁−ℎ₁2=ℎ₁. (11) It can also be obtained that if 𝜏₁(𝑡) is fixed at (2ℎ₁−𝑎₁)/2, the value of 𝑎₁𝜏₁(𝑡)+(𝜏₁(𝑡)−ℎ₁)² can also be bounded as follows: 𝑎₁𝜏₁(𝑡)+𝜏₁(𝑡)−ℎ₁²=𝑎₁2ℎ₁−𝑎₁2+2ℎ₁−𝑎₁2−ℎ₁²=4𝑎₁ℎ₁−𝑎²₁4=𝑎₁4ℎ₁−𝑎₁4=ℎ²₁−ℎ²₁=ℎ₁+ℎ²₁−ℎ²₁=ℎ₁. (12) This completes the proof.

Remark 5. 𝜏₁(𝑡) is the estimate value of the unknown delay constant 𝜏₁, satisfying 𝜏̇₁(𝑡)[2(𝜏₁(𝑡)−ℎ₁)+𝑎₁]≤0. Then 𝜏̇₁(𝑡)[𝜏₁(𝑡)−ℎ₁]≤0 can be obtained by fixing ℎ₁ and 𝑎₁ in Lemma 4. Apparently, the difference between 𝜏₁(𝑡) and ℎ₁ determines the variation of 𝜏₁(𝑡). So the adaptive strategy for 𝜏₁(𝑡) is based on a novel type of adaptive idea, which can be described as error to error adaptive technique. This adaptive strategy imposes no limitation on the estimate value. It also guarantees 𝜏₁(𝑡) always in its bound; that is, 0<𝜏^∗₁≤𝜏₁(𝑡)≤𝜏_1∗, for 𝜏₁(𝑡) can be fixed at ℎ₁.

3. Main Results

Using the controller (4), the closed-loop system (1) can be written as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴₁+𝜏₁𝐴₂𝜏₁𝑥𝑡−𝜏₁+𝐵𝐾₁𝑥𝑡−𝜏₂+𝐵𝐾₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁𝑡−𝜏₂−ℎ₁²𝐵𝐾₃𝑥×𝑡−𝜏₂−𝑎₁𝜏₁𝑡−𝜏₂−𝜏₁𝑡−𝜏₂−ℎ₁²,𝑥(𝑡)=𝜙(𝑡),∀𝑡∈[−𝜏,0], (13) where 𝜏=max(𝜏^∗₁,𝜏²₁+2𝜏^∗₁(ℎ₁+ℎ²₁−ℎ₁)+𝜏₂). Considering the “descriptor form” in [2, 3], we can rewrite (13) as 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)+𝐴=𝑦+𝑧=𝐴𝑥₁+𝜏₁𝐴₂𝜏₁𝑥𝑡−𝜏₁, (14) where 𝑧(𝑡)=𝐵𝐾₁𝑥𝑡−𝜏₂+𝐵𝐾₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁𝑡−𝜏₂−ℎ₁²𝐵𝐾₃×𝑥𝑡−𝜏₂−𝑎₁𝜏₁𝑡−𝜏₂−𝜏₁𝑡−𝜏₂−ℎ₁² (15a)which yields consequently𝑦(𝑡)+𝑧(𝑡)=𝐴+𝐴₁+𝜏₁𝐴₂𝜏₁𝑥(𝑡)−𝐴+𝐵𝐾₁+𝜏₁𝐴₂𝜏₁^𝑡_𝑡−𝜏1𝑦(𝑠)+𝑧(𝑠)𝑑𝑠−𝐵𝐾₁^𝑡_𝑡−𝜏2𝑦(𝜉)+𝑧(𝜉)𝑑𝜉−𝐵𝐾₂^𝑡_𝑡−𝜏3𝑦(𝑠)+𝑧(𝑠)𝑑𝑠−𝜏₄𝐵𝐾₃^𝑡_𝑡−𝜏4𝑦(𝑠)+𝑧(𝑠)𝑑𝑠 (16a)with 𝐾=𝐾₁+𝐾₂+𝜏₃𝐾₃, and 𝜏₃=𝜏₄=𝜏₂+𝑎₁𝜏₁(𝑡−𝜏₂)+(𝜏₁(𝑡−𝜏₂)−ℎ₁)².

Remark 6. By separating the “descriptor form” 𝑥̇(𝑡) into two parts 𝑦(𝑡) and 𝑧(𝑡), the specific adaptive regulation constructed later can be obtained in spite of the control input delay.

For systems (15a) and (16a), we consider the following Lyapunov-Krasovskii functional as 𝑉𝑥_𝑡=𝑉₁𝑥_𝑡+𝑉₂𝑥_𝑡+𝑙22𝜏₁𝑡−𝜏₂−ℎ₁+𝑎₁², (17) where 𝑉₁(𝑥_𝑡)=𝑥^𝑇𝑃𝑥=𝑥^𝑇(𝑦+𝑧)^𝑇𝐸𝑃^𝑇𝑥_𝑦+𝑧, 𝐸=^𝐼0₀₀, 𝑃=^𝑃𝑃1_0𝑃2, 𝑉₂𝑥_𝑡=³_𝑖=1⁰_−𝜏𝑖^𝑡_𝑡+𝜃𝑦(𝑠)+𝑧(𝑠)^𝑇×𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑦(𝑠)+𝑧(𝑠)𝑑𝑠𝑑𝜃+𝜏₄⁰_−𝜏4^𝑡_𝑡+𝜃𝑦(𝑠)+𝑧(𝑠)^𝑇×𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑦(𝑠)+𝑧(𝑠)𝑑𝑠𝑑𝜃, (18) with 𝐴₁=𝐴₁+𝜏₁𝐴₂(𝜏₁), 𝐴₂=𝐵𝐾₁, 𝐴₃=𝐵𝐾₂, 𝐴₄=𝐵𝐾₃, 𝐴₅=𝜏₄𝐴₄, 𝑙>0, ℎ₁>0, and 𝑎₁>0 being constants, and matrices 𝑃>0, 𝑄_𝑖>0(𝑖=1,2,3,4) are waiting to be determined.

Remark 7. It is easy to see that 𝑉₁(𝑥_𝑡)=𝑥^𝑇𝑃𝑥>0 and 𝑉₂(𝑥_𝑡)≥0. From (13), (15a), and (16a), it is not difficult to observe that if the norm of 𝑥(𝑡) diverges to infinity, then 𝑉₁(𝑥_𝑡) will also diverge to infinity. If 𝐴_𝑖 for 𝑖=1,2,3,4 are singular, thus 𝑉₂(𝑥_𝑡)=0. Otherwise, the norms of 𝑦(𝑡) and 𝑧(𝑡) are also infinite when the norm of 𝑥(𝑡) is unbounded, resulting in 𝑉₂(𝑥_𝑡) diverging to infinity. Thus the Lyapunov-Krasovskii functional 𝑉(𝑥_𝑡) defined in (17) can be guaranteed to be radially unbounded, that is, a well-posed candidate Lyapunov functional.

Hence, the derivative of 𝑉₁(𝑥_𝑡) along the systems (15a) and (16a) is given by 𝑉̇₁𝑥_𝑡=2𝑥^𝑇𝑃𝑥𝑦+𝑧=2^𝑇𝑦+𝑧^𝑇𝑃𝑃₁0𝑃₂0𝑥𝑦+𝑧=2^𝑇𝑦+𝑧^𝑇𝑃𝑃₁0𝑃₂×𝑦+𝑧+0𝐴𝑥−𝑦−𝑧𝜏₁𝐴₂𝜏₁𝑥−³_𝑖=10𝐴_𝑖^{𝑡𝑡−𝜏𝑖}𝑦(𝑠)(𝑠)−0+𝑧𝑑𝑠𝐴₅^{𝑡𝑡−𝜏4}𝑦(𝑠)(𝑠)𝑥+𝑧𝑑𝑠=2^𝑇𝑦+𝑧^𝑇𝑃𝑃₁0𝑃₂𝑦+𝑧−𝐴𝑥−𝑦−𝑧^5𝑖=1𝜂_𝑖, (19) where 𝐴=𝐴+𝐴₁+𝐵𝐾, 𝜂₅(𝑡)=−2𝑥^𝑇(𝑦+𝑧)^𝑇𝑃⁰_𝐼𝜏₁𝐴₂(𝜏₁)𝑥, 𝜂_𝑖(𝑡)=−2^𝑡_{𝑡−𝜏𝑖}𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝐴_𝑖×𝑦(𝑠)(𝑠)(),𝜂+𝑧𝑑𝑠𝑖=1,2,3₄(𝑡)=−2^𝑡_𝑡−𝜏4𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝐴₅×𝑦(𝑠)+𝑧(𝑠)𝑑𝑠. (20) By means of Lemma 3, we have 𝜂_𝑖≤𝜏_𝑖𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝑅_𝑖0𝐼𝑃^𝑇𝑥+𝑦+𝑧_{𝑡𝑡−𝜏𝑖}𝑦(𝑠)+𝑧(𝑠)^𝑇𝐴^𝑇_𝑖𝑅_𝑖⁻¹𝐴_𝑖𝑦(𝑠)(𝑠)𝜂+𝑧𝑑𝑠,₄≤𝜏²₄𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝑅₄0𝐼𝑃^𝑇𝑥𝑦+𝑧+𝜏₄^𝑡_𝑡−𝜏4𝑦(𝑠)+𝑧(𝑠)^𝑇𝐴^𝑇₄𝑅₄⁻¹𝐴₄𝑦(𝑠)(𝑠𝜂+𝑧)𝑑𝑠,₅≤𝜏₁𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝑄₅0𝐼𝑃^𝑇𝑥𝑦+𝑧+𝜏₁𝑥^𝑇𝐴₂𝜏₁^𝑇𝑄₅⁻¹𝐴₂𝜏₁𝑥. (21)(22)(23) Note that 𝑥^𝑇𝑦+𝑧^𝑇𝑃0𝐼𝑅_𝑖0𝐼𝑃^𝑇𝑥𝑦+𝑧=𝑥^𝑇𝑃₁𝑅_𝑖𝑃^𝑇₁𝑥+2𝑥^𝑇𝑃₁𝑅_𝑖𝑃^𝑇₂+𝑦+𝑧𝑦+𝑧^𝑇𝑃₂𝑅_𝑖𝑃^𝑇₂𝑦+𝑧=𝑥^𝑇𝑃₁𝑅_𝑖𝑃^𝑇₁𝑥+2𝑥^𝑇𝑃₁𝑅_𝑖𝑃^𝑇₂𝑦+𝑧+2𝑧^𝑇𝑃₂𝑅_𝑖𝑃^𝑇₂𝑦+𝑦^𝑇𝑃₂𝑅_𝑖𝑃^𝑇₂𝑦+𝑧^𝑇𝑃₂𝑅_𝑖𝑃^𝑇₂𝜏𝑧,𝑖=3,4,₃=𝜏₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁𝑡−𝜏₂−ℎ₁²=142𝜏₁𝑡−𝜏₂−ℎ₁+𝑎₁²−𝑎²₁+4ℎ₁𝑎₁+𝜏₂,𝜏²₄=𝜏₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁(𝑡−𝜏₂)−ℎ₁²²=𝜏²₂+𝑎²₁𝜏₁𝑡−𝜏₂²+𝜏₁𝑡−𝜏₂−ℎ₁⁴+2𝜏₂𝑎₁𝜏₁𝑡−𝜏₂+2𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁²+2𝑎₁𝜏₁𝑡−𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁²=𝑎²₁𝜏₁(𝑡−𝜏₂)²+𝜏₁(𝑡−𝜏₂)−ℎ₁⁴+2𝑎₁𝜏₁𝑡−𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁²+𝜏₂122𝜏₁𝑡−𝜏₂−ℎ₁+𝑎₁²−𝑎²₁+4ℎ₁𝑎₁. (24)(25) Besides, 𝑑∫⁰_−𝜏𝑖∫^𝑡_𝑡+𝜃𝑥̇(𝑠)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃=𝜏̇𝑑𝑡_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖}𝑥̇(𝑠)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑥̇(𝑠)+𝑑𝑠_0−𝜏𝑖𝑥̇(𝑡)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑥̇(𝑡)−𝑥̇(𝑡+𝜃)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑥̇(𝑑∫𝑡+𝜃)𝑑𝜃,𝑖=1,2,3,_0−𝜏4∫^𝑡_𝑡+𝜃𝜏₄(𝑡)𝑥̇(𝑠)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(𝑠)𝑑𝑠𝑑𝜃=𝜏̇𝑑𝑡₄(𝑡)^𝑡_𝑡−𝜏4𝜏₄(𝑡)𝑥̇(𝑠)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(𝑠)+𝑑𝑠_0−𝜏4𝜏₄(𝑡)𝑥̇(𝑡)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(𝑡)−𝑥̇(𝑡+𝜃)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(+𝑡+𝜃)𝑑𝜃_{0−𝜏4𝑡𝑡+𝜃}𝜏̇₄(𝑡)𝑥̇(𝑠)^𝑇𝐴^𝑇₄𝑄_𝑖⁻¹𝐴₄𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (26) Since 𝜏̇₁(𝑡)[2(𝜏₁(𝑡)−ℎ₁)+𝑎₁]≤0, that is, 𝜏̇₁(𝑡−𝜏₂)[2(𝜏₁(𝑡−𝜏₂)−ℎ₁)+𝑎₁]≤0, and 𝜏̇₃(𝑡)=𝜏̇₄(𝑡)=𝑑𝜏₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁𝑡−𝜏₂−ℎ₁²𝑑𝑡=𝑎₁𝜏̇₁𝑡−𝜏₂+2𝜏₁𝑡−𝜏₂−ℎ₁𝜏̇₁𝑡−𝜏₂=𝜏̇₁𝑡−𝜏₂𝑎₁+2𝜏₁𝑡−𝜏₂−ℎ₁, (27) we can have 𝑉̇₂𝑥_𝑡≤³_𝑖=1𝜏_𝑖𝑦(𝑡)+𝑧(𝑡)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖𝑦(𝑡)(𝑡−+𝑧)_{𝑡𝑡−𝜏𝑖}𝑦(𝑠)+𝑧(𝑠)^𝑇𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖×𝑦(𝑠)+𝑧(𝑠)𝑑𝑠+𝜏²₄(𝑡)𝑥̇(𝑡)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(𝑡)−𝜏₄(𝑡)⁰_−𝜏4𝑥̇(𝑡+𝜃)^𝑇𝐴^𝑇₄𝑄₄⁻¹𝐴₄𝑥̇(𝑡+𝜃)𝑑𝜃. (28) Let 𝑅_𝑖=𝑄_𝑖(𝑖=1,…,4), according to (17)–(28), the following inequalities are obvious by means of Schur complement: 𝑉̇𝑥_𝑡≤𝑥(𝑡)^𝑇Ξ₀𝑥(𝑡)+𝑎₁+2𝜏₁𝑡−𝜏₂−ℎ₁×𝑙𝜏̇₁𝑡−𝜏₂+14𝑎₁+2𝜏₁𝑡−𝜏₂−ℎ₁×𝑧^𝑇𝑃₂𝑄₃+2𝜏₂𝑄₄𝑃^𝑇₂𝑧, (29) where 𝑥(𝑡)^𝑇=𝑥^𝑇𝑦^𝑇𝑧^𝑇,Ξ₀=Ξ₁₁Ξ₁₂Ξ₁₃∗Ξ₂₂Ξ₂₃∗∗Ξ₃₃,Ξ₁₁=𝑃₁𝐴+𝐴^𝑇𝑃^𝑇₁+⁵_𝑖=1𝜏_𝑖𝑃₁𝑄_𝑖𝑃^𝑇₁+𝜏₁𝐴₂𝜏₁^𝑇𝑄₅⁻¹𝐴₂𝜏₁,Ξ₁₂=𝑃−𝑃₁+𝐴^𝑇𝑃^𝑇₂+⁵_𝑖=1𝜏_𝑖𝑃₁𝑄_𝑖𝑃^𝑇₂=Ξ₁₃,𝜏_𝑖=𝜏_𝑖𝑖=1,2,3,𝜏₄=𝜏²₄,𝜏₅=𝜏₁,Ξ₂₃=⁵_𝑖=1𝜏_𝑖𝑃₂𝑄_𝑖𝑃^𝑇₂−𝑃₂−𝑃^𝑇₂+⁴_𝑖=1𝜏_𝑖𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖=Ξ₂₂,Ξ₃₃=²_𝑖=1𝜏_𝑖𝑃₂𝑄_𝑖𝑃^𝑇₂+𝜏₁𝑃₂𝑄₅𝑃^𝑇₂−𝑃₂−𝑃^𝑇₂+³_𝑖=1𝜏_𝑖𝐴^𝑇_𝑖𝑄_𝑖⁻¹𝐴_𝑖+144ℎ₁𝑎₁−𝑎²₁+𝜏₂𝑃₂,𝑄₃𝑃^𝑇₂+𝜏₆𝑃₂𝑄₄𝑃^𝑇₂+𝜏₄𝐴^𝑇₄𝑄₄⁻¹𝐴₄,𝜏₆=𝜏₄−𝜏₂122(𝜏₁(𝑡−𝜏₂)−ℎ₁)+𝑎₁². (30) Results can be obtained as follows.

Theorem 8. Consider the time-delay system (1) with unknown time-delay parameter 𝜏₁; the system (1) can be stabilized by the state-feedback controller (4) if there exist matrices 𝑈_𝑖(𝑖=1,2,3) and positive-definite matrices 𝑋, 𝑄_𝑖(𝑖=1,…,4) such that the linear matrix inequalities (32) hold, with the parameters 𝑎₁ and ℎ₁ selected as (7) in Lemma 4. Moreover, the adaptive strategy about the unknown delay constant 𝜏₁ can be obtained from (31), and the feedback gains of the controller (4) are given by 𝐾_𝑖=𝑈_𝑖𝑋⁻¹(𝑖=1,2,3).

Proof. Consider the following adaptive control: 𝜏̇₁𝑡−𝜏₂=−1𝑎4𝑙¹+2𝜏₁𝑡−𝜏₂−ℎ₁𝑧^𝑇𝑃₂×𝑄₃+2𝜏₂𝑄₄𝑃^𝑇₂𝜏̇𝑧.Thatis,₁(𝑡)1=−𝑎4𝑙¹+2𝜏₁(𝑡)−ℎ₁𝑚(𝑡), (31) where 𝑚(𝑡)=𝑧(𝑡+𝜏₂)^𝑇𝑃₂(𝑄₃+2𝜏₂𝑄₄)𝑃^𝑇₂𝑧(𝑡+𝜏₂) satisfying the adaptive strategy as (6) in Lemma 4. Thus by using (29), we have 𝑉̇(𝑥_𝑡)≤𝑥^𝑇(𝑡)Ξ₀𝑥(𝑡). So if 𝑆=^defΞ₀<0, under the action of the controller (4), the system (1) will be asymptotically stable. The most important work of the memory feedback control problem is how to solve the matrix inequality 𝑆<0. Obviously, there exists 𝑆(𝜏₁)≤𝑆(𝜏^∗₁), for 𝜏₁≤𝜏^∗₁. So 𝑆(𝜏^∗₁)<0 can guarantee 𝑉̇(𝑥_𝑡)<0 satisfied, which means that the time-delay system (8) is asymptotically stabilizable by using feedback controller (4). Let Ξ=^def𝑆(𝜏^∗₁), and consider the Lyapunov matrix 𝑃 with 𝐸𝑃^𝑇=𝑃𝐸. In this case, we can suggest that the 𝑃₁ and 𝑃₂ in 𝑃 can be substituted as 𝑃₁=𝑛₁/𝑛₂𝑃 and 𝑃₂=1/𝑛₂𝑃, where 𝑛₁ and 𝑛₂ are real scalars. In this way, we can solve the above problem; furthermore, by making 𝑛₁ and 𝑛₂ line search parameters (i.e., plain search), we anticipate that less conservative conditions are given. Now before and after multiplying both sides of Ξ<0 by diag𝑋₁𝑋₂𝑋₃, where 𝑋₁=𝑃₁⁻¹=𝑛₂/𝑛₁𝑋, 𝑋₂=𝑋₃=𝑃₂⁻¹=𝑛₂𝑋, and 𝑋=𝑃⁻¹. After substituting ℎ₁=ℎ₁+ℎ²₁ and 𝑎₁=2(ℎ₁+ℎ²₁−ℎ₁) into Ξ<0, the following linear matrix inequalities can be obtained by applying Schur complement.
Consider Ξ=ΞΞ₁⋯Ξ₅Ξ^𝑇₁M₁Ξ⋮⋱_𝑇5M₅<0, (32) where Ξ^𝑇₁=0𝑛₂𝐴₁𝑋𝑛₂𝐴₁𝑋, Ξ^𝑇_𝑖+1=0𝑛₂𝐵₂𝑈_𝑖𝑛₂𝐵₂𝑈_𝑖0₁⋯0_𝑖,  𝑖=1,2,3, 𝑀_𝑖=−(𝜏^∗_𝑖)⁻¹𝑄_𝑖, 𝑖=1,2,3, 𝜏^∗₂=𝜏₂,  𝑀₄=−(𝜏^∗₄)⁻¹𝑄₄, 𝑀₅=−(𝜏^∗₁)⁻¹𝑄₅, Ξ^𝑇₅=𝑛₂/𝑛₁𝐴₂(𝜏₁)𝑋0₁⋯0₆, 𝑈_𝑗=𝐾_𝑗𝑋, 𝑗=1,2,Ξ=^{Ξ11Ξ12Ξ13∗Ξ}₂₂^Ξ_23∗∗Ξ33, Ξ₁₁ = 𝑛₂/𝑛₁(𝐴+𝐴₁)𝑋 + 𝑛₂/𝑛₁∑³_𝑖=1𝐵₂𝑈_𝑖 + 𝑛₂/𝑛₁𝑋(𝐴+𝐴₁)^𝑇 + 𝑛₂/𝑛₁∑³_𝑖=1(𝐵₂𝑈_𝑖)^𝑇 + ∑⁵_𝑖=1𝜏^∗_𝑖𝑄_𝑖, Ξ₁₂ = 𝑛₂/𝑛₁(𝑛₂−𝑛₁)𝑋 + 𝑛₂/𝑛₁𝑋(𝐴+𝐴₁)^𝑇 + 𝑛₂/𝑛₁∑³_𝑖=1(𝐵₂𝑈_𝑖)^𝑇 + ∑⁵_𝑖=1𝜏^∗_𝑖𝑄_𝑖 = Ξ₁₃, Ξ₂₃ = ∑⁵_𝑖=1𝜏^∗_𝑖𝑄_𝑖 − 2𝑛₂𝑋 = Ξ₂₂, and Ξ₃₃ = ∑²_𝑖=1𝜏^∗_𝑖𝑄_𝑖 + 𝜏^∗₅𝑄₅ − 2𝑛₂𝑋 + (ℎ₁+𝜏₂)𝑄₃ + 𝜏^∗₆𝑄₄. 𝜏_𝑖 = 𝜏_𝑖,  𝑖 = 1,2,3,  𝜏₅ = 𝜏₁, 𝜏₄ = 𝜏²₄, and 𝜏₆ = 𝜏₄ − 𝜏₂(1/2)[2(𝜏₁(𝑡−𝜏₂)−ℎ₁)+𝑎₁]².
The linear matrix inequality (32) can be directly solved by LMI toolbox in MATLAB software, and the matrices 𝑈_𝑖(𝑖=1,2,3) and positive-definite matrices 𝑋, 𝑄₃, and 𝑄₄ can also be acquired. Consequently, we have 𝐾_𝑖=𝑈_𝑖𝑋⁻¹(𝑖=1,2,3).

Remark 9. For the value of 𝜏^∗₃, 𝜏^∗₄, and 𝜏^∗₆ see Appendices. The LMI provided here is solvable, while, as for the results in Jiang et al. [13], unknown matrixes like 𝑃 and 𝑃⁻¹ exist in the same LMI, resulting in the LMI unsolvable.

4. Numerical Example

We consider a system with the same structure as (1), and the model matrices are 𝐴=−414−15−3,𝐴₁=,𝐴−100−1₂=81.−100−1,𝐵= (33) The known time delay in control input is 𝜏₂=0.008. We consider the following uncertainty in the time delay 𝜏₁: 𝜏₁∈[0.2,0.3]; that is, the upper bound 𝜏^∗₁=0.3, the lower bound 𝜏_1∗=0.2, and ℎ₁ is selected as (𝜏^∗₁+𝜏_1∗)/2=0.25. According to Lemma 4, we can select ℎ₁=ℎ₁+ℎ²₁=0.559, and 𝑎₁=2(ℎ₁+ℎ²₁−ℎ₁)=0.618. By applying Theorem 8, the feasible solution can be obtained with 𝐾₁=0.5501−1.6962, 𝐾₂=1.0𝑒−006∗0.0806−0.1576, 𝐾₃=1.0𝑒−005∗0.0575−0.1365, and 𝑃₂(𝑄₃+𝑄₄)𝑃^𝑇₂=^0.01260.0026_0.00260.0006. Moreover 𝜏^∗₄=𝜏^∗=max{𝜏(𝜏₁(𝑡−𝜏₂)={𝜏^∗₁,𝜏_1∗})}=0.0623 can be obtained from Appendix B. And the global minimum for LMI (32) is 𝑡_min=−2.4706𝑒−007, while by, the controller proposed in Jiang et al. [13], the global minimum 𝑡_min=−2.4292𝑒−008, which means that our result is less conservation. If the initial conditions are chosen as 𝜏₁(−𝜏)𝜙=0.2,₁(𝑡)𝜙₂(𝑡)=2sin4𝜋(𝑡−𝜏)𝜏−3sin4𝜋(𝑡−𝜏)𝜏,−𝜏≤𝑡≤0, (34) where 𝜏=max(𝜏^∗₁,𝜏²₁+𝜏^∗₁𝑎₁+𝜏₂)=𝜏^∗₁=0.3, and the parameter 𝑙 is chosen as 𝑙=0.4, then the system state under adaptive memory controller is shown in Figure 1. At this time, the estimate value of the unknown time-delay parameter, that is, 𝜏₁(𝑡), is shown in Figure 2.

Figure 1 

The system state under adaptive memory controller.

Figure 2 

The estimate value of the unknown time delay.

Remark 10. In Jiang et al. [13], 𝜏₁(𝑡) was limited to be larger than the real unknown value 𝜏₁. However, since 𝜏₁ is unknown, it is difficult to satisfy the limitation. So the memory controller with such 𝜏₁(𝑡) cannot be implemented as it was described. Besides, 𝜏₁(𝑡) remains decreasing until the system is stabilized. If the memory controller does not perform well, 𝜏₁(𝑡) will remain decreasing, which deteriorates the function of the controller. In this paper, 𝜏₁(𝑡) is maintained between the lower bound 𝜏_∗ and the upper bound 𝜏^∗, which is much easier to be implemented. With the error to error adaptive technique, 𝜏₁(𝑡) will always stay between 𝜏_∗ and 𝜏^∗, so the memory controller with such 𝜏₁(𝑡) can allow for more information of the system, which reduces the conservativeness.

5. Conclusions

In this paper, the problem of memory feedback controller with adaptation to unknown time delay parameter is addressed. The system investigated is with time delay in system state, control input, and system matrix, and additionally the state time-delay is unknown. By using a novel type of adaptive strategy with the idea of error to error and separated “descriptor form” functional technique, the estimate value of the time-delay constant can always be reflected by the feedback controller. Since more information in the system is presented, the controller proposed in this paper is much less conservative. Moreover, the adaptive strategy about time-delay parameter can achieve that no limitation is imposed on the estimate value, so it is more simple and convenient than the existing adaptive controllers. The sufficient condition for stabilization is presented in the form of LMI. To illustrate efficiency of the proposed technique, a numerical example has been provided.

Appendices

A. The Value of 𝜏^∗₃

𝜏₃=𝜏₄=𝜏₂+𝑎₁𝜏₁(𝑡−𝜏₂)+(𝜏₁(𝑡−𝜏₂)−ℎ₁)² as derivative of 𝜏₃(𝑡) can be obtained for 𝜏₁(𝑡−𝜏₂)∈[𝜏_1∗,𝜏^∗₁], and we have 𝑑𝜏₃(𝜏₁(𝑡−𝜏₂))/𝑑(𝜏₁(𝑡−𝜏₂))=2𝜏₁(𝑡−𝜏₂)−2ℎ₁+𝑎₁=0, so 𝜏₃(𝑡) can achieve extremum when 𝜏₁(𝑡−𝜏₂)=(2ℎ₁−𝑎₁)/2. Furthermore, as 𝜏₃^′′((2ℎ₁−𝑎₁)/2)=2>0, so 𝜏₃(𝑡) can achieve minimum when 𝜏₁(𝑡−𝜏₂)=(2ℎ₁−𝑎₁)/2=ℎ₁. As a result, the maximum for 𝜏₃(𝑡), that is, 𝜏^∗₃=max{𝜏₃(𝜏₁(𝑡−𝜏₂)={𝜏^∗₁,𝜏_1∗})}.

B. The Value of 𝜏₄

Consider 𝜏₄=𝜏²₄=𝜏₂+𝑎₁𝜏₁𝑡−𝜏₂+𝜏₁𝑡−𝜏₂−ℎ₁²²=𝜏²₂+𝑎²₁𝜏₁𝑡−𝜏₂²+𝜏₁𝑡−𝜏₂−ℎ₁⁴+2𝜏₂𝑎₁𝜏₁𝑡−𝜏₂+2𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁²+2𝑎₁𝜏₁𝑡−𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁². (B.1)

As it is well known that the value of third power of a variable is difficult to obtain, since 𝜏₁−ℎ₁≤𝜏^∗₁−ℎ₁=𝜏^∗₁−ℎ₁+ℎ²₁<𝜏^∗₁−ℎ₁≤𝜏₁, (B.2) then substituting (B.2) into (B.1), we have 𝜏²₄≤𝜏²₂+𝑎²₁𝜏₁𝑡−𝜏₂²+𝜏⁴₁+2𝜏₂𝑎₁𝜏₁𝑡−𝜏₂+2𝜏₂𝜏²₁+2𝑎₁𝜏₁𝑡−𝜏₂𝜏₁𝑡−𝜏₂−ℎ₁²≜𝜏. (B.3)

By the similar deduction in Appendix A, we have the following conclusion under two kinds of situations.

(1) If 2ℎ₁ℎ₁+ℎ²₁>2ℎ₁+ℎ²₁+3𝜏₂, we can obtain the maximum of 𝜏 which is 𝜏^∗=max𝜏𝜏₁𝑡−𝜏₂=𝜏^∗₁,𝜏_1∗,𝜏₂, (B.4) where 𝜏₂=(4ℎ₁−𝑎₁−𝑎²₁−8𝑎₁ℎ₁+4ℎ²₁−12𝜏₂)/6.

(2) If 2ℎ₁ℎ₁+ℎ²₁≤2ℎ₁+ℎ²₁+3𝜏₂, we can obtain that 𝜏^∗=max𝜏𝜏₁𝑡−𝜏₂=𝜏^∗₁,𝜏_1∗. (B.5)

C. The Value of 𝜏₆

From Appendices A and B, we can obtain that 𝜏^∗₆=𝜏^∗₄−2𝜏₂(𝜏_3∗−𝜏₂−(1/4)ℎ₁).

Acknowledgments

This work was supported in part by US NSF under Grant no. HRD-0932339, National Natural Science Foundation of China (61273119, 61174076, and 61004046) and (60835001: Key Program), Natural Science Foundation of Jiangsu Province of China (BK2011253), and Research Fund for the Doctoral Program of Higher Education of China (20110092110021).