# Admissibility Analysis of a Sampled-Data Singular System Based on the Input Delay Approach

Authors: Guoquan Chen; Minjie Zheng; Shenhua Yang; Lina Li
Journal: Complexity
 (2022)

---

## Abstract

This study investigates the sampled-data admissibility problem for a singular system. The objective of this paper is to design a sampled-data controller to ensure the admissibility of a singular system and to construct an appropriate Lyapunov–Krasovskii functional (LKF) to get less conservative results for the sampled-data singular system. To accomplish these objectives, the system is converted into a time-delay system by the input delay approach firstly, and both lower and upper bounds of the delay are considered. Secondly, by introducing a suitable LKF, the admissibility criteria are obtained. Then, when estimating the derivative of the LKF, a relaxation variable is introduced by the method of reciprocally convex inequality, and it is proved that the conservatism is reduced. Finally, two numerical examples are given to prove that the designed sampled-data controller can ensure the states of the systems under the influence of external interference.


---

## Body

## 1. Introduction

Singular systems have gained considerable attention recently, especially in the research of admissibility and other related theories. Compared with the state-space system, the stability, the regularity, and the impulsivity (or causality) have to be considered in singular systems. The singular system has extensive applications, which cover a variety of engineering fields and practical systems, such as the circuit system, mechanical engineering system, economic system, aerospace system, and robot. Besides, the singular system has more algebraic equations than the normal systems, so the singular system has better extension performance. In recent years, research on the admissibility problem of singular systems has become a hot subject [1–11]. In [7], the admissibility of singular systems is discussed in view of Lyapunov theory, and stability conditions and controller design methods are given. In [8], the robust admissibility issue for the uncertain discrete switched singular system is studied, and the arbitrary switching law and output feedback controller are designed to guarantee that the system is admissible. In [9], the delta operator model for a singular system is obtained by replacing the traditional discrete model with the generalized discrete model. And the relationship between discrete and delta operator models is established. In [10], the admissibility of the stochastic singular system with the T-S fuzzy model is studied, and a new quadratic Lyapunov function is proposed to obtain more relaxed admissibility conditions than the existing methods. The issue about robust admissibility for fuzzy singular systems is studied in [11]. By establishing the LMI, nonquadratic Lyapunov functions are constructed, which achieve less conservative result.Recently, sampled-data systems have become a hot topic. The system’s characteristic is that the discrete and continuous signals are both existing in one system, and it is difficult to be analyzed and designed. Recently, various viewpoints have been reported for sampled-data systems (see [12–20]). The main design methods for analyzing the sampled-data systems can be divided into three kinds. The first one is the lifting technique (see [21]), which converts a continuous system into a discrete system of equal finite dimension. The second one is jump system methods (see [22]), where the performance analysis of the sampling system is transformed into the solution of two Riccati equations with mutual jump. However, the two methods mentioned above cannot solve the uncertainty problems caused by the sampling time or system matrix. The third one is the input delay method (see [23]), and the system is transformed into a continuous time-varying delay system, and then the stability analysis can be accomplished using the method of time-delay system. Besides, the input delay approach can deal with the uncertainty of system parameters, and it has been widely used in the sampled-data system (see [24, 25]) and practical engineering field such as autonomous airship, high-speed train, and dynamic positioning ship (see [26–28]). In [24], based on the polytopic linear parameter-varying method, the issue of the sampled-data control is discussed for a nonlinear system. And Wirtinger’s inequality is used to obtain less conservative exponential stability condition. In [25], based on the input delay approach, the exponential stability issue for the linear parameter-varying system with aperiodic sampled-data rates is studied. And the distance between actual parameters and measured parameters is considered. In [26], the robust sampled-data control problem of the high-speed train is studied. Through the input delay approach, the high-speed cruise sampled-data controller of the train is designed. It ensures that the train is robust and stable under the interference of external wind. In [27], the sampled-data control issue for an autonomous airship with polyhedral parameter uncertainty is discussed. And a sampled-data controller is designed to guarantee that the system is exponentially stable and meets the H∞ performance index. In [28], a state-derivative control law is provided for a sampled-data dynamic positioning ship. By combining the delay-decomposition technique with Wirtinger’s integral inequalities, less conservative results are obtained.However, although the sampled-data control technology has been well developed in control theory, few results have been reported for the sampled-data singular systems, and the research process on this field is nearly blank. The technical challenges are listed as follows:(1)
The mathematical structure of the sampled-data singular system is more complex(2)
How to improve the LKF to get less conservative results for the sampled-data singular system(3)
How to design a sampled-data controller to guarantee the admissibility of the system and achieve the performance indexAs mentioned above, compared with the state-space system, the singular system has richer connotation and wider range. Besides, many multidimensional and multilevel large-scale complex systems are suitable to be dealt with by singular systems. So, it is important and meaningful to analyze the sampled-data control problem for the singular system, which is the motivation of this paper.In this paper, the admissibility of the sampled-data singular system is studied. Firstly, the system’s model is established, and then it is transformed into a time-delay system by using the input delay method. Then, both lower and upper bounds of the sampling period are considered. LMI and Lyapunov functions are introduced for admissibility analysis. By introducing the convex reciprocal inequality, the less conservative results can be obtained. Then, the design method of the sampled-data controller is provided, which makes the system achieve good performance under external interference. Finally, two examples are given to prove the effectiveness of this method.The main contributions of this article are summarized as follows:(1)
The admissibility condition is established for a sampled-data singular system to ensure the regularity, impulse free, and asymptotical stability. The system has wider application scopes than the existing sampled-data system.(2)
The upper and lower bounds of the variable sampling period are considered, which covers the previous research works as special cases.(3)
Reciprocally convex combination approach is used to estimate the integral terms of the LKF to obtain less conservative results.Notations: the superscript “T” stands for the transpose. ℝm×n denotes the set of m×n real matrices. “∗”denotes the matrix entries implied by symmetry. L20,∞ is the space of square integrable functions on 0,∞.


## 2. Problem Formulation

Consider the singular system as follows:(1)Ex˙t=Axt+But+Bwwt,yt=Cxt+Dut,where xt∈ℝn is the state, ut∈ℝm is the control input, yt∈ℝp is the control output, and wt∈ℝs is the disturbance which belongs to L20,∞ and satisfies that ∫0∞wTtwtdt<∞, and it means that the energy of the disturbance is finite. Matrix E∈ℝn×n is assumed to be singular, and rankE=r≤n. A,B,C,D,andBw are known constant matrices.It is assumed that the control signals are obtained at each sampling time0=t0<t1<⋯<tk<⋯<limk⟶∞tk=+∞, and they have(2)ut=udtk,tk≤t<tk+1,where ud represents the discrete-time control signal. The proposed sampled-data scheme is depicted in Figure 1.Figure 1
The schematic of the sampled-data singular system.The sampling period is assumed such that(3)d1≤tk+1−tk≤d2,∀k≥0,d2>d1≥0,where d1 and d2 are the lower and upper bound of the sampling period. Consider the sampled-data controller as follows:(4)ut=Kxtk,tk≤t<tk+1,where K is a controller gain matrix. By substituting (4) into (1), then(5)Ex˙t=Axt+BKxtk+Bwwt,t>0,yt=Cxt+DKxtk.Remark 1.
It is noted that both continuous and discrete signals exist in a singular system (5). Therefore, compared with the existing methods for analyzing the continuous or discrete singular systems, it is more complex and meaningful to study the sampled-data control issue for singular systems.
To obtain the main results, we list the following definitions.Definition 1 (see [29]).
(1)
If there exists a constants∈ℂ (ℂ represents the complex field) satisfying detsE−A≠0, then the matrix pair E,A is regular(2)
IfdegdetsE−A=rankE, then the matrix pair E,A is impulse freeDefinition 2 (see [30]).
(1)
The singular system(6)Ex˙t=Axt+Bxt−τtis said to be regular and impulse free if the pair (E, A) is regular and impulse free.(2)
System (6) is said to be asymptotically admissible if it is regular, impulse free, and asymptotically stable.
The objective of this paper is designing a sampled-data controller to satisfy that(1)
System (5) withwt=0 is asymptotically admissible(2)
Despite the external disturbances, the output signalyt satisfies that ‖yt|2≤γ‖wt‖2 for all nonzero wt∈L20,∞ under the zero condition, where γ>0
Based on the input delay approach, define(7)τt=t−tk,tk≤t<tk+1,where the time-varying delay τt is piecewise linear satisfying τt∈d1,d2. And the derivative of τt is(8)τ˙t=1,t≠tk.
Substituting the time delayτt into the control input signal ut, sampled-data controller (4) can be converted such that(9)ut=udtk=ut−t−tk=ut−τt.
Substituting (8) into (5), sampled-data system (5) is converted into a singular system with time-varying delay.(10)Ex˙t=Axt+BKxt−τt+Bwwt,t>0,yt=Cxt+DKxt−τt.Remark 2.
Compared with [24, 25, 31] which use the similar input delay approach to deal with the sampled-data problem, it can be seen that if E = I, then system (9) is simplified to the normal sampled-data system such as [24, 25, 31], which shows that the work covers these references as special issues. However, [31] did not consider the variable sampling period and only considered the constant sampling period. [24, 25] considered the upper bound of the sampling period, but the lower bound of the sampling period is omitted, which will limit its application scope, and it will also lead to conservatism to some extent. In this paper, the upper and lower bounds of the variable sampling period are considered. Let d1=d2=d; the sampling period will reduce to be a constant such as [24, 25, 31]. Thus, the proposed method has wider application scopes than the references.
The following lemma will be used for obtaining the main results.Lemma 1 (see [32]).
For given matrixJ,R=RT>0, scalars h1 and h2 satisfying h1≤dt≤h2, and differentiable function x satisfying x˙:−h2,−h1⟶ℝn, the following inequality holds:(11)h1−h2∫t−h2t−h1x˙TsRx˙sds≤≤−ϑTtI−I00I−ITWI−I00I−Iϑt,where(12)ϑTt=xTt−h1xTt−dtxTt−h2,W=−RJ∗−R≤0.


## 3. Main Results

In this section, the sufficient admissibility criteria for sampled-data singular systems are exhibited by establishing the LKF and formulating in terms of LMI.Theorem 1.
For constant delayd1,d2, system (9) is asymptotically admissible with H∞ performance γ if there exist matrices P,R,Qi>0,Zi>0,i=1,2,3, such that(13)ETP=PTE≥0,Z3RRTZ3>0,Π11Π12∗Π22<0,where(14)Π11=Ξ11ETZ1EPTBKETZ2EPTBw∗Ξ22Ξ23ETRE0∗∗Ξ33Ξ340∗∗∗Ξ440∗∗∗∗−γ2I,Π12=C0DK00A0BK0BwT,Π22=diag−I,−Z−1,Ξ11=PTA+ATP+Q1+Q2+Q3−ETZ1E−ETZ2E,Ξ22=−Q1−ETZ1E−ETZ3E,Ξ23=ETZ3E−ETRE,Ξ33=−2ETZ3E+ETRE+ETRTE,Ξ34=ETZ3E−ETRE,Ξ44=−Q3−ETZ2E−ETZ3E,Z=d12Z1+d22Z2+d212Z3,d21=d2−d1.Proof.
Firstly, we prove system (9) is regular and impulse free. From (13), it can be obtained that(15)Q1+Q2+Q3+ATP+PTA−ETZ1E−ETZ2E<0.
AsQ1>0,Q2>0,andQ3>0, then(16)ATP+PTA−ETZ1E−ETZ2E<0.
SincerankE=r≤n, there exist nonsingular matrices M∈ℝn×n and H∈ℝn×n such that(17)E¯=MEH=Ir000.
Similar to (17), it can be defined that(18)A¯=MAH=A11A12A21A22,P¯=M−TPH=P11P12P21P22.
From (17) and (18),(19)P¯=P110P21P22,P11T=P11>0.
Premultiplying and postmultiplyingHT and H with Ξ11<0,(20)A22TP22+P22TA22<0.
From (20), it means that A22 is nonsingular, and the pair (E, A) is regular and impulse free. According to Definition 1, system (9) is regular and impulse free.
Then, we will show that system (9) is asymptotically stable. The LKF is constructed as(21)Vt=∑i=14Vit,t∈tk,tk+1V1t=xtTETPxt,V2t=∫t−d1txsTQ1xsds+∫t−τttxsTQ2xsds+∫t−d2txsTQ3xsds,V3t=d1∫−d10∫t+θtx˙TsETZ1Ex˙sdsdθ+d2∫−d20∫t+θtx˙TsETZ2Ex˙sdsdθ,V4t=d21∫−d2−d1∫t+θtx˙TsETZ3Ex˙sdsdθ.
Calculating the derivative ofVt, it can be obtained that(22)V˙1t=2xtTETPx˙t,V˙2t=xtTQ1xt−xt−d1TQ1xt−d1+xtTQ2xt+xtTQ3xt−xt−d2TQ3xt−d2,V˙3t=d12x˙tTETZ1Ext+d22x˙tTETZ2Ex˙t−d1∫t−d1tx˙TsETZ1Ex˙sds−d2∫t−d2tx˙TsETZ2Ex˙sds,V˙4t=d212x˙tTETZ3Ex˙t−d21∫t−d2t−d1x˙TsETZ3Ex˙sds.
According to Jensen’s inequality, it can be obtained that(23)−d1∫t−d1tx˙TsETZ1Ex˙sds≤−xt−xt−d1TETZ1Ext−xt−d1,−d2∫t−d2tx˙TsETZ2Ex˙sds≤−xt−xt−d2TETZ2Ext−xt−d2,and(24)−d21∫t−d2t−d1x˙TsETZ3Ex˙sds=−d21∫t−d2t−τtx˙TsETZ3Ex˙sds−d21∫t−τtt−d1x˙TsETZ3Ex˙sds≤−d21d2−τtxt−τt−xt−d2T×ETZ3Ext−τt−xt−d2+d21τt−d1xt−d1−xt−τtT×ETZ3Ext−d1−xt−τt.
It can be found from (12) that(25)d2−dtτt−d1xt−d1−xt−τtE−dt−d1d2−τtxt−τt−xt−d2ETZ3RRTZ3d2−τtτt−d1xt−d1−xt−τtE−τt−d1d2−τtxt−τt−xt−d2E≥0.
Thus, according to Lemma1, we obtain from (24) and (25) that(26)−d21∫t−d2t−d1x˙TsETZ3Ex˙sds≤−xt−d1−xt−τtxt−τt−xt−d2TETZ3EETRE∗ETZ3Ext−d1−xt−τtxt−τt−xt−d2=−ϑTtETZ3E−ETZ3E+ETRE−ETRE∗2ETZ3E−ETRT+RE−ETZ3E+ETRE∗∗ETZ3Eϑt,where(27)ϑt=xTt−d1xTt−τtxTt−d2T.
Substituting (26) into (22),(28)V˙t≤ςTtΦ+A0BK0Td12Z1+d22Z2+d212Z3A0BK0ςt,where(29)Φ=Ξ11ETZ1EPTBKETZ2E∗Ξ22Ξ23ETRE∗∗Ξ33Ξ34∗∗∗∗Ξ44,ζt=xTtxTt−d1xTt−τtxTt−d2T.
By the Schur complement, (28) guarantees that(30)Φ+A0BK0Td12Z1+d22Z2+d212Z3A0BK0<0.
Therefore, it can be obtained thatV˙t<−σxt2 when xt≠0andσ>0. Then, system (9) is asymptotically stable.
Then, it will be proved that system (9) satisfies the H∞ performance index γ. Consider the H∞ performance for all nonzero wt∈L20,∞ as follows:(31)Jzw≤∫0∞yTsys−γ2wTswsds,γ>0.
Then,(32)yTtyt−γ2wTtwt+V˙t≤ζTtΘ+C0DK00TC0DK00+A0BK0BwTd12Z1+d22Z2+d212Z3A0BK0Bwζt,where(33)ζt=xTtxTt−d1xTt−τtxTt−d2wTtT,Θ=Ξ11ETZ1EPTBKETZ2EPBw∗Ξ22Ξ23ETRE0∗∗Ξ33Ξ340∗∗∗Ξ440∗∗∗∗−γ2I.
By the Schur complement, (16) guarantees that(34)Θ+C0DK0TC0DK0+A0BKBwTd12Z1+d22Z2+d212Z3A0BKBw<0.
From (34), we can obtain that(35)yTtyt−γ2wTtwt+V˙t<0.
Under zero initial conditions, we haveV0=0 and V∞≥0. From (35), it can be obtained that yt2≤γwt2 for all nonzero wt∈L20,∞.
Then, according to the condition, system (9) is asymptotically stable and satisfies the H∞ performance index γ. This completed the proof.Remark 3.
IfE=I,Q1=Q2=0, and R2=R3=0, then the LKF is similar with the one in [31]. Therefore, the LKF in [31] is a special case of (9), which means that the result in Theorem 1 has wider application scopes than [31].Remark 4.
To reduce the conservativeness, severable methods have been reported, such as the free-weighting matrix method and Wirtinger-based inequality. In the constructed LKF, we directly use tighter interactive convex combinatorial inequalities to estimate the integral terms−d21∫t−d2tkx˙TsETZ3Ex˙sds.
During this process, a matrixR is introduced to deal with the two terms(36)d21d2−τtxt−τt−xt−d2TETZ3Ext−τt−xt−d2,d21τt−d1xt−d1−xt−τtTETZ3Ext−d1−xt−τt.
However, the above two terms are not considered in [33, 34], which will lead to conservatism to some extent. Besides, (25) is free as matrix R could be more than nonnegative. Therefore, the conservatism has been greatly reduced.Remark 5.
It is noted that the number of decision variables is4.5n2+4.5n, and it illustrates that the proposed methodology has lower computational complexity than the free-weighting matrix method or Jensen inequality lemma which increases the computational complexity with too much additional slack variables.
Furthermore, according to the following theorems, we will design the sampled-data controller to stabilize system (9).Theorem 2.
For scalesd1,d2,andγ, system (9) is asymptotically admissible with H∞ performance γ if there exist matrices P,R,Qi>0,Zi>0,i=1,2,3, satisfying(37)ETP=PTE≥0,Z¯3R¯R¯TZ¯3>0,Π¯11Π¯12∗Π¯22<0,where(38)Π11=Ξ¯11ETZ¯1EBK¯ETZ¯2EBw∗Ξ¯22Ξ¯23ETR¯E0∗∗Ξ¯33Ξ¯340∗∗∗Ξ¯440∗∗∗∗∗−γ2I,Π12=CP¯T0D¯K¯00AP¯T0B¯K¯0BwP¯TT,Π22=diag−I,−P¯Z−1P¯,Ξ¯11=AP¯+P¯AT+Q¯1+Q¯2+Q¯3−ETZ¯1E−ETZ¯2E,Ξ¯22=−Q¯1−ETZ¯1E−ETZ¯3E,Ξ¯23=ETZ¯3E−ETR¯E,Ξ33=−2ETZ¯3E+ETR¯E+ETR¯TE,Ξ¯34=ETZ¯3E−ETR¯E,Ξ44=−Q¯3−ETZ¯2E−ETZ¯3E,Z¯=d12Z¯1+d22Z¯2+d212Z¯3,d21=d2−d1.Then, the controller gain matrixK can be obtained such that(39)K=K¯P¯−1.Proof.
By noticing that−P¯Z¯−1P¯≤Z¯−2P¯, let η=diagP−T,P−T,P−T,P−T,I,I,Iand, κ=diagP−T,P−T, and define(40)P¯=P−1,K¯=KP−1,Z¯=P−TZP−1,R¯=P−TRP−1,Z¯i=P−TZiP−1,Q¯i=P−TQiP−1,i=1,2,3.
Pre- and postmultiplying (12) by κ and κT, respectively, we can obtain (33). Pre- and postmultiplying (13) by η and ηT, respectively, (34) can be obtained by the Schur complement. The proof is completed.Remark 6.
It can be seen that the control performance is affected by the design parametersd1,d2, and these parameters are chosen to solve the LMI problem. We can adjust the design parameters to achieve a better solution. The design and implementation step of the proposed method is provided as follows.Step 1.
We solve the LMI problemPs to guarantee the system is admissible.(41)Ps:minimizeγsubject toP,R,Qi>0,Zi>0,i=1,2,3LMIs11,12,13.Step 2.
If the above problem is feasible, then we solve the LMI problemsPo to obtain the sampled-data controller. Otherwise, we adjust the design parameters d1,d2, and then go back to Step 1.(42)Po:minimizeγ,subject toP¯,R¯,Q¯i>0,Z¯i>0,i=1,2,3,LMIs30,31,32.Step 3.
If the above problem is solvable, we can obtain the controller, that is,K=K¯P¯−1. Otherwise, we adjust the design parameters d1,d2, and then go back to Step 2.
The flowchart of the proposed method is provided in Figure2.
Then, whenE=I, the following Corollary 1 can be obtained based on Theorem 2. According to Corollary 1, we can use the proposed method to solve the sampled-data control problem of the practical systems, such as dynamic positioning ship and autonomous airship.Figure 2
The flowchart of the proposed method.Corollary 1.
For scalesd1,d2,andγ, system (9) is asymptotically stable with H∞ performance γ if there exist matrices P,R,Qi>0,Zi>0,i=1,2,3, such that(43)Z¯3R¯R¯TZ¯3>0,Π¯11Π¯12∗Π¯22<0,where(44)Π11=Ξ¯11Z¯1BK¯Z¯2Bw∗Ξ¯22Ξ¯23R¯0∗∗Ξ¯33Ξ¯340∗∗∗Ξ¯440∗∗∗∗−γ2I,Π12=CP¯T0D¯K¯00AP¯T0B¯K¯0BwP¯TT,Π22=diag−I,−P¯Z−1P¯,Ξ¯11=AP¯+P¯AT+Q¯1+Q¯2+Q¯3−Z¯1−Z¯2,Ξ¯22=−Q¯1−Z¯1−Z¯3,Ξ¯23=Z¯3−R¯,Ξ33=−2Z¯3+R¯+R¯T,Ξ¯34=Z¯3−R¯,Ξ44=−Q¯3−Z¯2−Z¯3,Z¯=d12Z¯1+d22Z¯2+d212Z¯3,d21=d2−d1.Then, the controller gain matrixK can be obtained such that(45)K=K¯P¯−1.Proof.
The procedure of the proof is similar with Theorem2, so it is omitted.


## 4. Numerical Examples

In this section, to validate the effectiveness of the given method, two numerical examples are introduced.Example 1.
Consider the following parameters:(46)A=−13.1−13.7−15.4−23.8,B=−18.6−10.4−25.2−16.8,Bw=1.91.8,C=0.4−0.8,E=9362,D=11.
Whenwt=0, for different lower bounds of the time delay d1, Table 1 lists the maximum time delay d2 to guarantee the admissibility of the system by different methods. From Table 1, the upper bound of time delay d2 obtained by Theorem 1 is larger than that in [33–36], which indicates that the proposed method in the paper can obtain lower conservative results than the other references.
Next, consider the situation aboutwt≠0. For different d2>0, Table 2 lists the minimal H∞ performance indicators γ that guarantee the admissibility of the system by Theorem 1 and [37] when d1=0. From Table 2, it is easy to see that, for different upper bounds of time delay d2, H∞ performance γ obtained by Theorem 1 is smaller than that in [37]. Therefore, it is shown that Theorem 1 improves the results of [37].
Assuming the sampling intervald2=1.7, the H∞ performance is obtained such that γmin=0.8625. Then, the controller gain can be computed such that(47)K=−0.1663−0.1109−0.0554−0.0370.
Consider the system’s initial statexst=−2−2 and the external environment disturbance wt=0.5cost and random disturbance, respectively. Under different disturbances, the response curves of the system’s state and the control input ut are shown in Figures 3–6, respectively. From Figures 3 and 5, it is shown that the system’s state x1t and x2t are stable in a short time under different disturbances, which further illustrates that the designed sampled-data controller can make the system’s states stable and have acceptable control performance with external variable disturbances.Table 1
Maximum values of the upper boundd2.
d11.41.61.82.02.2[33]2.11212.14502.28412.43282.5852[34]2.23142.27612.40412.53832.6777[35]2.33602.37042.42422.54252.7007[36]2.33722.37302.49232.61812.7494Theorem12.34182.38362.50132.64082.7826Table 2
Minimum values ofH∞ performance γ (d1=0).
d20.91.11.31.51.7[37]0.27130.35480.47440.63780.8736Theorem10.26270.34680.46420.61740.8625Figure 3
The state response curve of system (9) (wt=0.5cost).Figure 4
Control input for system (9) (wt=0.5cost).Figure 5
The state response curve of system (9) (wt is the random disturbance).Figure 6
Control input for system (9) (wt is the random disturbance).Example 2.
Similar with [38], the motion equations of a sampled-data dynamic positioning ship system are considered as follows:(48)Mυ˙+Dυ=τ+w,where τ denotes the control input, w denotes the external disturbance, M is the inertia matrix, D is the damping coefficient matrix, and υ=p,v,rT includes the velocity of surge, sway, and yaw, respectively. Define the input matrix(49)xt=xa,ya,ψ,p,v,rT,where xa,ya,andψ represent the x-position, y-position, and yaw angle, respectively. The values of M and D are considered such that(50)M=0.7540001.1990.21100.0290.524,D=0.0140000.102−0.02400.1920.095.
LetA=0I0−M−1D,B=0M−1,andBw=0M−1.
Then,(51)A=0001−0.034900000.034910000001000−0.0186000000−0.02080.03420000−0.3653−0.1832,B=Bw=0000000001.32630000.8422−0.33910−0.04661.9272.
The initial states of the ship are assumed such thatxst=10100.1000T. Let wt=sin0.1t. And the values of the sampling period are d1=0.8andd2=1.6; then, the H∞ performance index is obtained according to Corollary 1 such that γmin=2.945. Then, the controller gain is obtained as follows:(52)K=−0.2438−0.0109−0.0008−0.5459−0.01050.00020.0072−0.4429−0.10180.0186−0.5794−0.1722−0.00110.0088−0.2082−0.00220.2673−0.5638.
The responses of the DPS state are shown in Figures7 and 8. Figure 7 shows the responses of the position of x and y and yaw angle of DPS. Figure 8 shows the response of velocities of DPS. It is shown that the DPS can be stabilized by the designed sampled-data controller even if external disturbances exist.Figure 7
The responses of the position ofx and y and yaw angle of DPS.Figure 8
The response of velocities of DPS.


## 5. Conclusion

The sampled-data control issue for singular systems is discussed in this paper. Both lower and upper bounds of the variable period have been considered. Then, the criteria of asymptotical admissibility are given by constructing a suitable LKF. Then, the tighter reciprocally convex inequalities are used to estimate the derivative of the LKF, and less conservative results can be obtained. Two numerical examples are given to illustrate the effectiveness of the proposed method. In the future, we will improve the proposed methods and consider the actuator faults and then discuss the sampled-data fault-tolerant issue for the more complex nonlinear singular systems such as fuzzy singular systems and singular Markov jump systems.



---

*Source: 3151620-2022-02-14.xml*