Abstract
An investigation of the input-to-state practical stability (ISpS) and the integral input-to-state practical stability (iISpS) of nonlinear systems with time delays (NSWTDs) is presented in this paper. The ISpS and iISpS of the systems are obtained by using a continuously differentiable Lyapunov–Krasovskii functional (LKF) within-definite derivative, which generalizes the classical LKF with a positive definite derivative. As a result, the uniform practical asymptotic stability (UPAS) criteria of NSWTD were also given by using the proposed LKF.
1. Introduction
Input-to-state stability (ISS) [1] ensures the asymptotic stability for zero input systems. It was proposed in [1] that a negative derivative of a Lyapunov function (LF) was sufficient for ISS, which was characterized later by Wang et al. [2]. Also, Wang et al. and Khalil [2, 3] showed that uniform asymptotic gain and ISS are equivalent. However, Sontag [4] introduced the notion of integral input-to-state stability (iISS), which is a nonlinear extension of stability. As a result, it was strictly weaker than ISS. As we know, a time-invariant system is iISS if it has an LF with a derivative that is negative definite along the system [5]. There is a widespread phenomenon of time delay in practical control systems. Recent years have seen intensive studies of NSWTD’s ISS property. For the ISS property of NSWTD, the paper [6] showed that the nonlinear small-gain theorem is equivalent to the Razumikhin-type theorem. For impulsive systems in [7], hybrid delayed systems in [8], and NSWTD in [9], LKF methods were used. Authors in [8] explain in detail how delay bounds and dwell times are related to hybrid delayed systems. It is difficult to choose a suitable LKF with a negative definite derivative to verify the ISS property based on the existing results in [7–11]. In many cases, a closed-loop system’s ISS behavior cannot be ensured via feedback because it is too costly or impossible. According to [12], input-to-state practical stability (ISpS) has been proposed as a relaxation of the ISS concept for such applications. There is in fact an extension of the earlier technique of practical asymptotic stability of nonlinear differential equations in the ISpS, as shown in [13–16]. The characterizations of ISpS have been developed for a class of ordinary differential equations [17]. Nonlinear systems are usually stabilized using LFs [14, 18, 19]. In addition, they provide tools for designing a more robust controller and/or verifying its robustness.
In this paper, the ISpS and the iISpS properties of NSWTD are established. This paper introduces a new practical LKF to address the issue raised by time delays and stable scalar functions [14]. We establish some sufficient conditions for ISpS and iISpS by constructing a more general function over [20]. We also present some new sufficient conditions for UPAS.
The following is the organization of this paper. We will discuss the main notations and definitions in the next section. By utilizing a new LKF, Section 3 provides the ISpS and iISpS criteria and provides some sufficient conditions for UPAS. The paper concludes with Section 4.
2. Basic Results
This work assumes , refers to the usual Euclidean norm, and is the space of piecewise continuous function on to . Also, we denote by the space of continuous functions equipped with the norm as follows:
Consider the following NSWTD:where is the time, is the state, is the control which is assumed to be measurable and locally essentially bounded, is the function segment defined by , , with is the time delay, and is piecewise continuous with respect to , locally Lipschitz in , and uniformly continuous in . Then, for a given initial state , and initial time , system (2) exists in a unique solution , which satisfies , for all .
Definition 1. System (2) is said to be ISpS if there exist a class function , a class function , and , such that, for any initial state and any measurable, locally essentially bounded input , the solution exists for all and satisfies the following:
Definition 2. System (2) is said to be iISpS if there exist a class function , a class function , and a class function such that, for any initial state , any measurable, locally essentially bounded input , the solution exists for all and satisfies the following:
Definition 3. System (2) is said to be globally uniformly practically asymptotically stable (GUPAS), if there exist and , such that, for any initial state , any measurable, locally essentially bounded input , the solution exists for all and satisfies the following:
Remark 4. The inequality (4) implies that will be bounded by a small bound , that is, will be small for sufficiently large , by taking an initial condition outside the Ball . In this situation, a robustness result can be obtained if we suppose that depends on a small parameter ; in this case, if approaches to zero as tends to zero, then, approaches the origin exponentially as goes to infinity.
Also, we need the following definition.
Definition 5. The derivative of the continuous function along the solutions of (2) is defined as follows:Here is the definition that is given in [14].
Definition 6. Let . is globally uniformly practically exponentially stable (GUPES) if there is , , and , such that for every ,with .
Remark 7. Indeed, the notion of stable scalar functions was first given in [21, 22] after the author in [14] has extended these definitions to the practical case. It is worth noting that, if the function is GUPES, then the systemis GUPES.
In the sequel, we will use the following auxiliary result called the generalized Gronwall–Bellman inequality, which is taken from [22].
Lemma 8. Let , and is a differentiable function, such that, for all ,
Then, for every , we have
3. Main Results
Now, we are ready to give the main lemma.
Lemma 9. Assume that is an absolutely continuous function, , , and . If for almost all ,with an initial value , and is GUPES, then, for all , we have
Proof. For the inequalityLet us consider the following two cases: Case 1: (13) holds for almost all Case 2: (13) does not hold for all For Case 1, the generalized Gronwall–Bellman inequality gives uswhere . Since, is GUPES, then, there exist , , and , such that for all ,Thus,For Case 2, we have the measure of is greater than zero. Let . Then, either or . If , so for almost , we obtainThus, the estimation (14) holds for all . Now, by the continuity of and the fact that is GUPES, one obtainsFrom the definition of , the last inequality givesIf , then there is a constant for every ,If we let , and we use the continuity of , we get, for every ,Now, combining (16), (19), and (21), we conclude that for all
Remark 10. A new comparison principle is presented in Lemma 9 to determine the upper bounds for NSWTD. It has the advantage that its derivative can be positive during some interval of time in . In contrast, when , the derivative of the LKF requires that the derivative of the differential inequality be negative definite such as [8, 9].
A useful property of a function is presented in the following lemma in .
Lemma 11. Let be a class function. We have for all ,
Applying Lemmas 9 and 11, we obtain the following theorem for the ISpS of system (2) for an indefinite LF.
Theorem 12. Suppose that there is a function functions , a function , a positive constant , and such that for every , , and ,
Then, system (2) is ISpS if is GUPES.
Proof. Since is GUPES, there is , , and , such that for every ,Lemma 9 and the inequality (25) give for all Using the inequalities given in (24), we obtain for all Thus, for all , one hasNow, using Lemma (14), we obtain for all ,Hence, for all , we havewith , , and .
Based on the Definition 2, the system is ISpS.
ISS implies iISpS, as we know. Here, we give an iISpS result for the NSWTD (2) by using the indefinite derivative LKF.
Theorem 13. Suppose that there is a function functions , a function , a positive constant , and such that for and ,
Then, system (2) is iISpS if is GUPES.
Proof. By Lemma 9, we obtain from (33) that for every ,Since is GUPES, there is , , and , such that for every ,We further obtain for all ,From condition (32), we can see that, and . The inequality (37) implies that system (2) is iISpS.
Remark 14. Condition (25) in Theorem 12 implies condition (25) in Theorem 13, On the other hand, the converse might not be true. The following two cases illustrate this statement:(i)Case 1: If , it follows from (25) in Theorem 12 that Following is an estimation that can easily be verified as follows:(ii)Case 2: If , let the function be as follows:for . We consider any function with , for every . ThereforeWhen the two cases are combined, the statement is true.
Theorem 15. Suppose that there is a function functions , a function , a positive constant , and such that for every , , and ,
Then, system (2) is iISpS if is GUPES.
Proof. Setting , then . As in the proof of 3.3, we can calculateWe obtainFrom condition (42), one obtainsTherefore, for all ,where , , and . Following Definition 2, system (2) is iISpS. This completes the proof.
UPAS is implied by ISpS and iISpS properties. UPAS for NSWTD (2) with is given as a by-product of Theorem 12.
With , we give the following uniformly practically stable result for (2).
Corollary 16. Suppose that there is a function functions , a function , a positive constant , and such that for every , , and ,
Then, system (2) with is GUPAS if is GUPES.
Proof. Similar to Theorem 12 we obtain for every ,Thus, for all ,Hence, for all , we havewith and .
Now, we conclude that the system is GUPAS following Definition 5.
Example 1. We consider the following NSWTD:where , is a constant, and .
Now, let us consider the following hypothesis:(i)There is , two constants , and , such that for every , we have(ii)There exists a function , such that for every ,(ii)There exists such that for all ,We suppose that is globally uniformly practically stable and then, let us consider the following LKF:where , , , and , whose time-derivative satisfies (see [23]) the following:where . Using almost the same techniques as in [23], one can check that is GUPES. Hence, the NSWTD (53) is GUPAS.
Consider a class of a SDOF system with a time delay described by the following system [24, 25]:where both the damping coefficient, , and the elastic constant, , are time-varying ([22]), the variable represents the position of the mass with respect to its rest position, we use the notation to denote the derivative of with respect to time (i.e., the velocity of the mass) and to represent the second derivative (acceleration), stands for control gain according to displacement and velocity (called displacement gain and velocity gain for the sake of simplicity), is external excitation, and is the time delay.
Let . Then, system (60) can be rewritten as follows:with , , and is a continuously differentiable function such thatwith and satisfies (57).
Let and , where the scalar is a constant parameter that accounts for the variability, we take .
Consider the following LKF:where , which satisfies inequalities (54) and (55) with , , , , , , and . are, respectively, the smallest and the largest eigenvalue of the matrix). Now, if satisfies (56) such that is globally uniformly practically stable, one obtainswhere . Therefore, the NSWTD (61) is GUPAS.
For simulation, we choose ,Figure 1 shows the convergence of the solutions toward a neighborhood of the origin with initial condition .
4. Conclusion
A nonlinear system with delay employing an indefinite LKF has been investigated for the ISpS and iISpS criteria. Indefinite negative derivatives have been used to generalize the classical LKF. Through the use of the indefinite derivative LKF, some sufficient conditions for verifying the ISpS, iISpS, and UPAS are established.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by “Researchers Supporting Project number (RSPD2023R683), King Saud University, Riyadh, Saudi Arabia.”