Abstract

This paper investigates the output feedback stabilization problem of linear time-varying uncertain delay systems with limited measurable state variables. Each uncertain parameter and each delay under consideration may take arbitrarily large values. In such a situation, the locations of uncertain entries in the system matrices play an important role. It has been shown that if a system has a particular configuration called a triangular configuration, then the system is stabilizable irrespective of the given bounds of uncertain variations. In the results so far obtained, the stabilization problem has been reduced to finding the proper variable transformation such that an 𝑀-matrix stability criterion is satisfied. However, it still has not been shown whether the constructed variable transformation enables the system to satisfy the 𝑀-matrix stability condition. The objective of this paper is to show a method that enables verification of whether the transformed system satisfies the 𝑀-matrix stability condition.

1. Introduction

This paper examines the stabilization problem of linear time-varying uncertain delay systems by means of linear memoryless state feedback control. The systems under consideration contain uncertain entries in the system matrices and uncertain delays in the state variables. Each value of uncertain entries and delays may vary with time independently in an arbitrarily large bound. Under this situation, the locations of uncertain entries in the system matrices play an important role. This paper presents investigation of the permissible locations of uncertain entries, which are allowed to take unlimited large values, for the stabilization using linear state feedback control.

It is useful to classify the existing results on the stabilization of uncertain systems into two categories. The first category includes several results which provide the stabilizability conditions depending on the bounds of uncertain parameters. The results in the second category provide the stabilizability conditions that are independent of the bounds of uncertain parameters but which depend on their locations. This paper specifically addresses the second category.

For uncertain systems with delays, the Lyapunov stability approach with the Krasovskii-based or Razumikhin-based method is a commonly used tool. The stabilization problem has been reduced to solving linear matrix inequalities (LMIs) [13]. However, LMI conditions fall into the first category; for this reason, they are often used to determine the permissible bounds of uncertain parameters for the stabilization. When the bounds of uncertain parameter values exceed a certain value, LMI solver becomes infeasible. In such cases, guidelines for redesigning the controller are usually lacking.

On the other hand, the stabilizability conditions in the second category can be verified easily merely by examining the uncertainty locations in given system matrices. Once a system satisfies the stabilizability conditions, a stabilizing controller can be constructed, irrespective of the given bounds of uncertain variations. We can redesign the controller for improving robustness merely by modifying the design parameter when the uncertain parameters exceed the upper bounds given beforehand.

In the second category, the stabilization problem of linear time-varying uncertain systems without delays was studied by Wei [4]. The stabilizability conditions have a particular geometric configuration with respect to the permissible locations of uncertain entries. Using the concept of antisymmetric stepwise configuration (ASC) [4], Wei proved that a linear time-varying uncertain system is stabilizable independently of the given bounds of uncertain variations using linear state feedback control if and only if the system has an ASC. Wei derived the successful result on the stabilization problem of systems without delays, however, his method [4] is inapplicable to systems that contain delays in the state variables.

On the one hand, based on the properties of an 𝑀-matrix, Amemiya and Leitmann [5] developed the conditions for the stabilization of linear time-varying uncertain systems with time-varying delays using linear memoryless state feedback control. The conditions obtained in [5] show a similar configuration to an ASC, but the allowable uncertainty locations are fewer than in an ASC by one step.

The aforementioned results presume that all state variables are accessible for designing a controller. However, it is usual that the state variables of the systems are measured through the outputs and hence only limited parts of them can be used directly. The output feedback stabilization of linear uncertain delay systems with limited measurable state variables has been investigated in [6, 7]. The conditions so far obtained show that if a system has a particular configuration called a triangular configuration, then the system is stabilizable independently of the given bounds of uncertain variations. The conditions in [7] consist of not only the system matrix Δ𝐴 but also input and output coefficient matrices, Δ𝐵 and Δ𝐶, while the conditions in [6] consist of only Δ𝐴.

The results so far obtained were derived using an 𝑀-matrix stability criterion. In [57], the stabilization problem has been reduced to finding the variable transformation such that the 𝑀-matrix stability condition is satisfied. Although the developed conditions called a triangular configuration has been shown in [7], it still has not been shown whether the constructed variable transformation enables the system to satisfy the 𝑀-matrix stability condition. The objective of this paper is to show a method that enables verification of whether the transformed system satisfies the 𝑀-matrix condition. This paper specially examines the functional-order structure of the transformed system in order to verify whether the system has an 𝑀-matrix structure.

This paper is organized as follows. Some notations and terminology are given in Section 2. The systems considered here are defined in Section 3. In Section 4, some preliminary results are introduced to state the present problem. The main results are provided in Section 5. Finally, some concluding remarks are presented in Section 6.

2. Notations and Terminology

First, some notations and terminology used in the subsequent description are given. For 𝑎,𝑏𝑚 or 𝐴,𝐵𝑛×𝑚, every inequality between 𝑎 and 𝑏 or 𝐴 and 𝐵 such as 𝑎>𝑏 or 𝐴>𝐵 indicates that it is satisfied component-wise by 𝑎 and 𝑏 or 𝐴 and 𝐵. If 𝐴𝑛×𝑚 satisfies 𝐴0, 𝐴 is called a nonnegative matrix. The transpose of 𝐴𝑛×𝑚 is denoted by 𝐴. For 𝑎=(𝑎1,,𝑎𝑚)𝑚, |𝑎|𝑚 is defined as |𝑎|=(|𝑎1|,,|𝑎𝑚|). Also for 𝐴=(𝑎𝑖𝑗)𝑛×𝑚, |𝐴| denotes a matrix with |𝑎𝑖𝑗| as its (𝑖,𝑗) entries. Let diag{} denote a diagonal matrix. Let [𝑎,𝑏], 𝑎,𝑏 be an interval in . The set of all continuous or piecewise continuous functions with domain [𝑎,𝑏] and range 𝑛 is denoted, respectively, by 𝒞𝑛[𝑎,𝑏] or 𝒟𝑛[𝑎,𝑏]. We denote it simply by 𝒞𝑛 or 𝒟𝑛 if the domain is .

The notation for a class of functions is introduced below. Let 𝜉(𝜇)𝒞1 and let 𝑚 be a constant. If 𝜉(𝜇) satisfies the conditions limsup|𝜇|||||𝜉(𝜇)𝜇𝑚||||<,limsup|𝜇|||||𝜉(𝜇)𝜇𝑚𝑎||||=(2.1) for any positive scalar 𝑎, then 𝜉(𝜇) is called a function of order 𝑚, and we denote this as follows: Ord(𝜉(𝜇))=𝑚.(2.2) The set of all 𝒞1 functions of order 𝑚 is denoted by 𝑂(𝑚), 𝑂(𝑚)=𝜉(𝜇)𝜉(𝜇)𝒞1,Ord(𝜉(𝜇))=𝑚.(2.3) Also, it is worth to note that 𝑚 can be a negative number and that the following relations between 𝜉1(𝜇)𝑂(𝑚1) and 𝜉2(𝜇)𝑂(𝑚2) hold: 𝜉Ord1(𝜇)±𝜉2𝑚(𝜇)=max1,𝑚2,𝜉Ord1(𝜇)×𝜉2(𝜇)=𝑚1+𝑚2,𝜉Ord1(𝜇)𝜉2(𝜇)=𝑚1𝑚2.(2.4) A real square matrix all of whose off-diagonal entries are nonpositive is called an 𝑀-matrix if it is nonsingular and its inverse matrix is nonnegative. The set of all 𝑀-matrices is denoted by .

3. System Description

Let 𝑛 be a fixed positive integer. The system considered here is given by a delay differential equation defined on 𝑥𝑛 for 𝑡[𝑡0,) as follows: ̇𝑥(𝑡)=𝐴0𝑥(𝑡)+Δ𝐴1(𝑡)𝑥(𝑡)+𝑟𝑖=1Δ𝐴2𝑖(𝑡)𝑥𝑡𝜏𝑖𝐶(𝑡)+(𝑏+Δ𝑏(𝑡))𝑢(𝑡),𝑦(𝑡)=+Δ𝐶(𝑡)𝑥(𝑡),(3.1) with an initial curve 𝜙𝒟𝑛[𝑡0𝜏0,𝑡0]. Here, 𝐴0, Δ𝐴1(𝑡), Δ𝐴2𝑖(𝑡)(𝑖=1,,𝑟) are all real 𝑛×𝑛 matrices, where 𝑟 is a fixed positive integer; also, 𝐴0 is a known constant matrix. Furthermore, Δ𝐴1(𝑡) and Δ𝐴2𝑖(𝑡)(𝑖=1,,𝑟) are uncertain coefficient matrices and may vary with 𝑡[𝑡0,). Other variables are as follows: 𝑢(𝑡) is a control variable, 𝑏𝑛 is a known constant vector, and Δ𝑏(𝑡)𝑛 is an uncertain coefficient vector which may vary with 𝑡[𝑡0,). 𝑦(𝑡)2 is an output variable, 𝐶𝑛×2 is a known constant matrix, and Δ𝐶(𝑡)𝑛×2 is an uncertain coefficient matrix which may vary with 𝑡[𝑡0,).

In addition, all 𝜏𝑖(𝑡)(𝑖=1,,𝑟) are piecewise continuous functions and are uniformly bounded, that is, for nonnegative constant 𝜏0 they satisfy 0𝜏𝑖(𝑡)𝜏0(𝑖=1,,𝑟)(3.2) for all 𝑡𝑡0. The upper bound 𝜏0 can be arbitrarily large and is not necessarily assumed to be known.

It is assumed that all entries of Δ𝐴1(𝑡), Δ𝐴2𝑖(𝑡), Δ𝑏(𝑡), and Δ𝐶(𝑡) are piecewise continuous functions and are uniformly bounded, that is, for nonnegative constant matrices Δ𝐴10, Δ𝐴2𝑖0𝑛×𝑛, Δ𝐶0𝑛×2, and for a nonnegative constant vector Δ𝑏0𝑛, they satisfy ||Δ𝐴1||(𝑡)Δ𝐴10,||Δ𝐴2𝑖||(𝑡)Δ𝐴2𝑖0,||||Δ𝑏(𝑡)Δ𝑏0,||||Δ𝐶(𝑡)Δ𝐶0(3.3) for all 𝑡𝑡0. The upper bound of each entry can independently take an arbitrarily large value, but each is assumed to be known.

Assumption 3.1. Because the system must be controllable, we assume that the pair (𝐴0,𝑏) of the nominal system is a controllable pair and is in the controllable canonical form. Then 𝐴0 and 𝑏 are given as follows: 𝐴0=00101000000010000,𝑏=.(3.4)

Assumption 3.2. Because the system must be observable, we assume that 𝐶=(𝑐1,𝑐2)𝑛×2 is given as follows: 𝑐1=(1,0,,0),𝑐2=(0,,0,1,0,,0),(3.5) where all entries of 𝐶 are equal to zero except that the first entry and the 𝑘th entry of 𝑐1 and 𝑐2 are equal to 1, respectively. 𝑘 has a strong relation to the configuration of uncertain entries and is defined in the subsequent discussion.

Considering a necessary and sufficient condition for linear uncertain systems to be observability invariant [8], we see that the observability of a given system might be lost without Assumption 3.2.

Next, we consider the following system:̇𝑧(𝑡)=(𝐴0𝐿𝐶)𝑧(𝑡)+𝐿𝑦(𝑡)+𝑏𝑢(𝑡),(3.6) where 𝑧(𝑡)𝑛 is an auxiliary state variable, and 𝐿𝑛×2 is a constant matrix. This is an observer in the most basic sense. Our objective is to find a controller for stabilizing the overall 2𝑛-dimensional system consisting of (3.1) and (3.6). Let 𝑒(𝑡) be defined by 𝑒(𝑡)=𝑧(𝑡)𝑥(𝑡).(3.7) Let 𝑢(𝑡) be given by 𝑢(𝑡)=𝑔𝑧(𝑡)=𝑔𝑒(𝑡)+𝑔𝑥(𝑡),(3.8) where 𝑔𝑛 is a constant vector.

Definition 3.3. System (3.1) is said to be delay independently stabilizable if there exists a linear memoryless state feedback control 𝑢(𝑡)=𝑔𝑧(𝑡) such that the equilibrium point 𝑥=0 of the resulting closed-loop system is uniformly and asymptotically stable for all admissible uncertain delays and uncertain parameters.

4. Preliminaries

The 2𝑛-dimensional system consisting of 𝑥(𝑡)𝑛 and 𝑒(𝑡)𝑛 is written as follows: ̇𝐴𝑤(𝑡)=0𝐿𝐶00𝐴0+𝑏𝑔𝑤(𝑡)+Δ𝑏(𝑡)𝑔Δ𝐴1(𝑡)+𝐿Δ𝐶(𝑡)Δ𝑏(𝑡)𝑔𝑏𝑔+Δ𝑏(𝑡)𝑔Δ𝐴1(𝑡)+Δ𝑏(𝑡)𝑔+𝑤(𝑡)𝑟𝑖=10Δ𝐴2𝑖(𝑡)0Δ𝐴2𝑖𝑤(𝑡)𝑡𝜏𝑖(,𝑡)(4.1) where 𝑤(𝑡)=(𝑒(𝑡),𝑥(𝑡))2𝑛.

Because of Assumption 3.1, it is possible to choose 𝑔𝑛 such that all eigenvalues of (𝐴0+𝑏𝑔) are real, negative, and distinct. Likewise, because of Assumption 3.2, it is possible to choose 𝐿𝑛×2 such that all eigenvalues of (𝐴0𝐿𝐶) are real, negative, and distinct. Let 𝑔 and 𝐿 be chosen in such a way. In addition, let 𝜆1,𝜆2,,𝜆𝑛 and 𝜎1,𝜎2,,𝜎𝑛 be the eigenvalues of (𝐴0+𝑏𝑔) and (𝐴0𝐿𝐶), respectively. Let 𝑇 and 𝑆 be Vandermonde matrices constructed from 𝜆𝑖 and 𝜎𝑖, respectively, as follows: 𝜆𝑇=1111𝜆2𝜆𝑛𝜆12𝜆22𝜆𝑛2𝜆1𝑛1𝜆2𝑛1𝜆𝑛𝑛1,𝑆𝑆=100𝑆2,(4.2) where 𝑆1 and 𝑆2 are given by 𝑆1=𝜎1𝑘2𝜎1𝑘3𝜎11𝜎2𝑘2𝜎2𝑘3𝜎21𝜎𝑘2𝑘1𝜎𝑘3𝑘1𝜎𝑘11,𝑆2=𝜎𝑘𝑛𝑘𝜎𝑘𝑛𝑘1𝜎𝑘1𝜎𝑛𝑘𝑘+1𝜎𝑛𝑘1𝑘+1𝜎𝑘+11𝜎𝑛𝑛𝑘𝜎𝑛𝑛𝑘1𝜎𝑛1.(4.3)𝑇 and 𝑆 are well known to be nonsingular in view of the previous assumptions. Define Λ and Σ as follows: Λ=𝑇1𝐴0+𝑏𝑔𝜆𝑇=diag1,𝜆2,,𝜆𝑛,𝐴Σ=𝑆0𝐿𝐶𝑆1𝜎=diag1,𝜎2,,𝜎𝑛.(4.4) Let 𝑃1, 𝑃2, and 𝑃3 be defined as follows: 𝑃1=,𝑃Σ00Λ2=0||𝑆||Δ𝐴30||𝑇||||𝑇1||||𝑏𝑔||||𝑆1||||𝑇1||Δ𝐴30||𝑇||,𝑃3=||𝑆||Δ𝑏0||𝑔||||𝑆1||||𝑆||||𝐿||Δ𝐶0+Δ𝑏0||𝑔||||𝑇||||𝑇1||Δ𝑏0||𝑔||||𝑆1||||𝑇1||Δ𝑏0||𝑔||||𝑇||,(4.5) where Δ𝐴30 is given by Δ𝐴30=Δ𝐴10+𝑟𝑖=1Δ𝐴2𝑖0.(4.6) In addition, let 𝑃 be defined by 𝑃=𝑃1𝑃2𝑃3.(4.7)

Here, we introduce the fundamental lemma which plays a crucial role to lead the main results.

Lemma 4.1 (see [6]). If there exist 𝑇 and 𝑆 which assure 𝑃,(4.8) then system (3.1) is delay independently stabilizable.

Note that our problem has been reduced to finding 𝑇 and 𝑆 that enable 𝑃 to satisfy condition (4.8). In the subsequent discussion, we consider the possibility of choosing 𝑇 and 𝑆 that assure 𝑃.

5. Main Results

First, we introduce a set of matrices Ω(𝑘)(𝑛+1)×(𝑛+1) as follows.

Definition 5.1. Let 𝑘 be an integer satisfying 0𝑘𝑛. For this 𝑘, let Ω(𝑘)={𝐷=(𝑑𝑖𝑗)(𝑛+1)×(𝑛+1)} be a set of matrices with the following properties: (1)if1𝑗𝑘,then𝑑𝑖𝑗=0,for𝑗+1𝑖2𝑘𝑗,(2)if𝑘+1𝑗𝑛+1,then𝑑𝑖𝑗=0,for2𝑘𝑗𝑖𝑗1.

Now, we state the main result.

Theorem 5.2. Construct a matrix Γ(𝑛+1)×(𝑛+1) as Γ=Δ𝑐010Δ𝐴30Δ𝑏0(5.1) by means of system parameters. If for fixed 𝑘, ΓΩ(𝑘),Δ𝑐02=0,(5.2) then system (3.1) is delay independently stabilizable.

System (3.1) is said to have a triangular configuration if the system satisfies condition (5.2). A schematic view of the system having a triangular configuration is shown below. Here, indicates an uncertain entry not necessarily equal to zero457468.fig.001(5.3)

Proof of Theorem 5.2. According to Lemma 4.1, the existence of 𝑇 and 𝑆 which assure 𝑃 is shown in the rest of this section. Here, let 𝜇 be a positive number and let 𝛼𝑖(𝑖=1,,𝑛) be all negative numbers that are different from one another. Likewise, let 𝛽𝑖(𝑖=1,,𝑛) be all negative numbers that are different from one another. Let 𝜇 be chosen larger than all upper bounds of uncertain elements Δ𝐴30, Δ𝑏0, and Δ𝑐01. 𝛼𝑖 and 𝛽𝑖 are used for distinguishing eigenvalues 𝜆𝑖 and 𝜎𝑖 from one another. Let 𝜆𝑖 and 𝜎𝑖 be chosen as follows: 𝜆𝑖=𝛼𝑖𝜇1𝜎𝑂(1)(𝑖=1,,𝑘1),𝑖=𝛽𝑖𝜇1𝜆𝑂(1)(𝑖=1,,𝑘1),𝑖=𝛼𝑖𝜎𝜇𝑂(1)(𝑖=𝑘,,𝑛),𝑖=𝛽𝑖𝜇𝑂(1)(𝑖=𝑘,,𝑛).(5.4) Then, we can write 𝑇 and 𝑆 as follows: 𝑇𝑇=1𝑇2=00𝑖+1𝑖1𝑛+1𝑛1,𝑆(5.5)1=𝑘+2𝑘+𝑗+10𝑆,(5.6)2=𝑛𝑘𝑛𝑘𝑗+10,(5.7) where 𝑇1 and 𝑇2 denote 𝑛×(𝑘1) and 𝑛×(𝑛𝑘+1) matrices, respectively. In addition, 𝑆1 and 𝑆2 denote (𝑘1)×(𝑘1) and (𝑛𝑘+1)×(𝑛𝑘+1) matrices, respectively. In the above notation, 𝑚 and 𝑚 denote a row vector and a column vector, whose all entries are functions of 𝜇 of order 𝑚, respectively. For convenience, we adopt such notation for matrices in the subsequent discussion and neglect further explanation when it is clear. The notation of (5.5) means that all entries of the 𝑖th row of 𝑇1 and 𝑇2 are functions of 𝜇 of order (𝑖+1) and (𝑖1), respectively. The notations of (5.6) and (5.7) mean that all entries of the 𝑗th column of 𝑆1 and 𝑆2 are functions of 𝜇 of order (𝑘+𝑗+1) and (𝑛𝑘𝑗+1), respectively.
Next, from the relations between the roots and the coefficients of the characteristic equations det(𝐴0+𝑏𝑔) and det(𝐴0𝐿𝐶), we find that 𝑔 and 𝐿 have the following structure: 𝑔=𝑛2𝑘+2𝑛𝑘𝑛𝑘+11,(5.8)𝐿=10𝑘+20𝑘+1101020𝑛𝑘+1.(5.9) The notation of (5.8) means that the entry of the 𝑗th column of 𝑔 is a function of 𝜇 of order (𝑛2𝑘+𝑗+1) if 𝑗𝑘1, and of order (𝑛𝑗+1) if 𝑗𝑘. The notation of (5.9) means that the entry of the 𝑖th row and the first column of 𝐿 is a function of 𝜇 of order 𝑖 if 𝑖𝑘1, and the entry of the 𝑖th row and the second column of 𝐿 is of order 𝑖𝑘+1 if 𝑖𝑘.
Considering such structures of 𝑇, 𝑆, 𝑔 and 𝐿, it turns out from the careful calculation that each block matrix in (4.5) is further decomposed into four block matrices as follows: ||𝑇1||||𝑏𝑔||||𝑆1||=𝑘32𝑘𝑛2𝑘+1𝑛+2||𝑇,(5.10)1||Δ𝑏0||𝑔||||𝑆1||=𝑘42𝑘𝑛3𝑘2𝑛1||𝑇,(5.11)1||Δ𝐴30||𝑇||=22𝑘42𝑘+20||𝑇,(5.12)1||Δ𝑏0||𝑔||||𝑇||=22𝑘42𝑘2||𝑆||,(5.13)Δ𝐴30||𝑇||=𝑘𝑘2𝑛2𝑘+1𝑛1||𝑆||||𝐿||,(5.14)Δ𝐶0||𝑇||=𝑘𝑘20000||𝑆||0000,(5.15)Δ𝑏0||𝑔||||𝑇||=𝑘𝑘20000||𝑆||0000,(5.16)Δ𝑏0||𝑔||||𝑆1||=2𝑘𝑛100000000.(5.17) In the above notation, all the entries of each block matrix are functions of 𝜇 of the same order.
Now, let 𝑃2𝑛×2𝑛 in (4.7) be decomposed into four block matrices as follows: 𝑃𝑃=11𝑃12𝑃21𝑃22,(5.18) where 𝑃11||𝑆||=ΣΔ𝑏0||𝑔||||𝑆1||,𝑃12||𝑆||=Δ𝐴30||𝑇||||𝑆||||𝐿||Δ𝐶0||𝑇||||𝑆||Δ𝑏0||𝑔||||𝑇||,𝑃21||𝑇=1||||𝑏𝑔||||𝑆1||||𝑇1||Δ𝑏0||𝑔||||𝑆1||,𝑃22||𝑇=Λ1||Δ𝐴30||𝑇||||𝑇1||Δ𝑏0||𝑔||||𝑇||.(5.19) It is apparent that 𝑃 if and only if 𝑃11𝑃,22𝑃,22𝑃21𝑃111𝑃12.(5.20) The following lemma shown in [9] is useful for verification of whether a given matrix is an 𝑀-matrix.

Lemma 5.3 (see [9]). Let 𝑘 be an integer satisfying 1<𝑘<𝑛. Let 𝐵𝑛×𝑛 be a diagonal matrix whose every entry is positive, and let 𝐶𝑛×𝑛. Let 𝐵 and 𝐶 be decomposed into four block matrices as follows: 𝐵𝐵=1100𝐵22𝐶,𝐶=11𝐶12𝐶21𝐶22.(5.21) Therein, 𝐵11 and 𝐵22 are 𝑘×𝑘 and (𝑛𝑘)×(𝑛𝑘) diagonal matrices, respectively. 𝐶11, 𝐶12, 𝐶21, and 𝐶22 are 𝑘×𝑘, 𝑘×(𝑛𝑘), (𝑛𝑘)×𝑘, and (𝑛𝑘)×(𝑛𝑘) block matrices, respectively. Suppose that all the entries of each block matrix are functions of 𝜇 of the same order. Let all the entries of 𝐵11 and 𝐵22 belong to 𝑂(𝑏11) and 𝑂(𝑏22), respectively. Let all the entries of 𝐶11, 𝐶12, 𝐶21, and 𝐶22 belong to 𝑂(𝑐11), 𝑂(𝑐12), 𝑂(𝑐21), and 𝑂(𝑐22), respectively. For sufficiently large 𝜇, if 𝑏11>𝑐11,𝑏22>𝑐22,𝑏11>𝑐12𝑏22+𝑐21,(5.22) then the matrix 𝐴=𝐵|𝐶| is an 𝑀-matrix.

Using Lemma 5.3, we can deduce whether the matrix whose entry represents the functional order is an 𝑀-matrix.

Taking into account the fact that Σ is a diagonal matrix in which all diagonal entries belong to 𝑂(1) from the first to (𝑘1)th entry or 𝑂(1) from the 𝑘th to 𝑛th entry, we see from (5.17) that 𝑃11.

From (5.14)–(5.16), it follows that 𝑃12=𝑘𝑘2𝑛2𝑘+1𝑛1.(5.23) From (5.10) and (5.11), we have 𝑃21=𝑘32𝑘𝑛2𝑘+1𝑛+2.(5.24) From (5.12) and (5.13), it follows that 𝑃22=Λ22𝑘42𝑘+20.(5.25) Taking into account the fact that Λ is a diagonal matrix in which all diagonal entries belong to 𝑂(1) from the first to (𝑘1)th entry or 𝑂(1) from the 𝑘th to 𝑛th entry, we obtain 1>2,1>0,1>2𝑘412𝑘+2=3.(5.26) According to Lemma 5.3, the inequalities (5.26) show that 𝑃22. From (5.23) and (5.24), it follows that 𝑃21𝑃111𝑃12=22𝑘42𝑘+20.(5.27) Then, from (5.25) and (5.27), we have 𝑃22𝑃21𝑃111𝑃12=Λ22𝑘42𝑘+20.(5.28) Hence, it is apparent from the inequalities (5.26) that 𝑃22𝑃21𝑃111𝑃12. Taking into account the fact that all the conditions of (5.20) hold, we see that 𝑃.

Therefore, using Lemma 4.1, we can conclude that system (3.1) is delay independently stabilizable.

6. Conclusions

The stabilization problem of linear time-varying uncertain delay systems with limited measurable state variables was studied here. Each uncertain parameter and each delay under consideration may take arbitrarily large values. It was shown that if the uncertain entries enter the system matrices in a way to form a particular geometric pattern called a triangular configuration, then the system is stabilizable irrespectively of the given bounds of uncertain parameters and delays. The method that enables verification of whether the transformed system satisfies the 𝑀-matrix stability condition was provided here. Moreover, it was shown that the constructed variable transformation enables the system to satisfy the 𝑀-matrix stability condition. The obtained conditions have a strong similarity to the ones called an antisymmetric stepwise configuration by Wei [4]. To develop the conditions obtained here into the ones of antisymmetric stepwise configurations is a problem to be considered next.

Acknowledgment

This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (Start-up), 20860040.