Abstract
Nonlinear systems constantly suffer from time-varying delays, which cover slow delays and nonslow delays. The existing results were used to impose slow delay conditions and were used to study the control problems, but few pieces of research studies have discussed the case of systems with nonslow delay. In this work, we remove the slow time-delay condition and consider the nonlinear delayed system with complicated polynomial terms. By proposing a dynamic gain method and constructing a new Lyapunov–Razumikhin (L-R) function, we successfully construct a stable controller, which guarantees that the plant is globally asymptotically stable (GAS). An example is utilized to verify the raised control scheme.
1. Introduction
Nonlinear systems can be easily found in engineering such as robotic systems [1], chemical systems [2], and circuit systems [3]. Such systems have many special characteristics and complicated structures where lots of states are coupled together, which lead to many challenging systems control issues. Up to now, plenty of splendid studies have been reported (see [4–7]). The raised strategies gave positive answers to many engineering problems and developed the theory of nonlinear system control. However, there are still some difficult problems, especially for nonlinear time-delay systems.
Time delay arises in numerous practical plants and constantly plays the role of destroyer in system stability and responses. Time delay covers constant delays and time-varying. When systems have constant delays, the control issues are often solved via the Lyapunov–Krasovskii (L-K) functional approach (see [8, 9]). In detail in [8], we considered the feedforward system with constant delays via a dynamic control method [9] and further extended the method to study systems with input delay. When systems contain time-varying delays, the system control mainly covers the L-K functional method and the L-R function method. Most of the reported L-K functional control methods need a slow delay condition [10–13], i.e., the derivatives of delay functions should be smaller than one (see Remark 1). However, the L-R function method can remove the abovementioned condition. Due to the difficulty of constructing an appropriate L-R function and estimating the complicated nonlinear terms, it was not an easy work to propose an L-R function method. In the last decades, many works have discussed the related problems under the slow delay condition via L-K functional method. For example, by employing the sampled-data control approach, the authors in [14] investigated global stabilisation regulation of upper-triangular delayed systems. By constructing the state observer and presenting an adaptive regulation method, the authors in [15] discussed the control issue for switched time-delay systems. By utilizing the L-K functional method and double dynamic gains, the authors in [16] studied a large-scale delayed model and designed a stable controller. When system delays do not satisfy this condition, i.e., they behave as nonslow delays, the existing methods cannot be applied to the system control. Up to now, there are very few results discussing this problem. Specially, the authors in [17] considered a class of delayed systems and presented an L-R function-based adaptive backstepping method, and the authors in [18] considered stochastic time-delay nonlinear systems and presented some outstanding control strategies on the basis of the L-R stability theory. It should be mentioned that the control issue is still challenging for nonlinear systems with nonslow delay conditions. Naturally, an interesting problem is that
Can we remove the slow delay condition and construct a stable controller for nonlinear systems with complicated polynomial nonlinear terms and time-varying delay?
For the abovementioned issue, this work will give a positive answer. The main contributions are the following:(i)More general nonlinear delayed systems are studied. In this work, the nonlinear terms possess polynomial forms. Besides, the slow delay condition which was imposed by many existing studies has been successfully removed.(ii)For time-delay nonlinear systems, a novel dynamic-based L-R control strategy is raised. By introducing a new transformation, we obtain a dynamic system. By skillfully utilizing the homogeneous domain idea and the L-R stability theory and designing an appropriate dynamic gain, we finally construct a dynamic stable controller for the considered system.
2. Problem Formulation
We consider the triangular time-delay system aswhere and represent the states and the input, respectively, and . is the time delay with ( is a constant). For , is bounded and satisfies for , and is a continuous function. is the initial condition where is a specified continuous function, and is a constant, and the states are available.
Remark 1. When the time delay is a constant, the normal method is to construct the L-K functional for the control design. The L-K functional has the form and , where is a continuous or smooth function. When the time delay is a time-varying function , then the slow delay condition is assumed, i.e., should satisfy for a constant . Besides, also needs to satisfy . Subsequently, the control design for systems with time-varying delay can be performed using the L-K functional , where . However, the slow delay condition is somewhat stronger and sometimes does not hold. So, in this paper, we remove the condition and try to present a L-R function-based control method.
Assumption 1. There exists a continuous function that renderswhere , , and satisfies for a positive constant .
Remark 2. Assumption 1 indicates that system (1) is in a triangular form and the nonlinear terms satisfy a polynomial growing condition, which covers linear conditions in [19] and high-order conditions in [2, 5, 13]. Also, are functions, which are more general than the constant gain in [2, 13, 19]. Besides, it needs to mention that the time delay in may not satisfy , which makes the control issue of this paper not be solved via the existing approaches. So far, it is still challenging to construct a stable controller for system (1).
We provide the useful lemmas in this work which are as follows.
Lemma 1 (see [7]). Here, we hold the inequalitywhere are constants and are functions.
Lemma 2 (see [13]). Let and , . For , there holds
We select a constant such that , and we define . It is deduced that
We introduce the transformationswhere , and will be defined later. Then, the derivative of satisfieswhich implies that
Now, in the following, we start to design the controller.
Step 1. We choose the function , where . Then, the derivative of satisfieswhere . By Assumption 1 and (2), it yields thatwhere is a constant. We define , and by using (6), one haswhere are positive constants. By choosing the virtual control , it follows thatwhere .
2.1. Recursive Design (Step )
Suppose that there exist transformationswhere is a constant, and a positive continuous differentiable function in step renderswhere are positive constants. In step , we select the function as , whose derivative along the system satisfies the following equation:
By Lemma 1, we getwhich leads towhere is a constant. From Assumption 1 and (2), it follows thatwhere is a constant. By using (6) and (13), it yields from thatwhere are positive constants. Similarly, we havewhere are constants. By utilizing Lemma 1, there is a constant such that
Using (8), it is deduced that
By Lemmas 1 and 2, there exists such that
From Lemma 1, there is a constant such that
Noting that , it leads towhere . By choosingwe obtainwhere , and is selected such that .
When , by utilizing the abovementioned method, we choose and select , which gives thatwhere are positive constants. When , there holds , where is a constant. Thus, we get
Also, there holds
Considering , we getwhere is a continuous function. Similarly, we getwhere is a continuous function. For simplicity, we assume that and . By using (31), (32), and (28), it yields thatwhere and . Now, we design thatwhich yields that
3. Main Results
Next, we supply the results of this work.
Theorem 1. Suppose Assumption 1 holds, then there is a dynamic controllerfor system (1). In the controller (36), the parameter satisfies , where and . is defined as with being defined in (13). We have the function , where and are defined in (31) and (32). is computed using (26) when . is computed by (27) when . is defined as , where is defined in step such that , and is a constant satisfying . The controller (36) ensures the boundedness and convergence of all system states and guarantees that the system is GAS.
Proof. By the definition of , there hold , where and are class functions. In addition, when , we deduce from (35) that , where is a class function. Thus, system (1) is GAS, which indicates that are bounded on [−σ0, +∞) and . Noting that, is a continuous function, is also bounded. Hence, there is a constant such that . Suppose that there is a finite time on [−σ0, +∞) such that . Then, (34) indicates that . Therefore, is bounded on [T0, +∞). Otherwise, we have . That is, . Thus, is bounded on [−σ0, +∞), and show that is also bounded. By (3), we see that is bounded. Besides, there hold . Similarly, by , we show that . Since is bounded, it yields that . Therefore, .
Remark 3. We need to explain the following two points (i) the system time delay cannot be arbitrarily time-varying functions. In the existing studies, is generally assumed to be bounded, i.e., , where is a constant. For systems with arbitrary delay functions, many problems will arise such as weak robustness, poor system responses, or system instability. Some scholars also discussed the problems for systems with arbitrary delay functions (see [20, 21] for instance). (ii) Under Assumption 1, forward-time completeness of the open-loop system (1) is not guaranteed. Actually, the system can have an arbitrarily small escape time. In this work, with the designed controller (36), the forward-time completeness of the closed-loop system which is composed by (1) and (36) will be guaranteed and the system cannot escape in a finite time. This can be proved by a contradiction. Suppose that there exists a state such that for a finite time , then following the control design and the proof of Theorem 1, we obtain that and are bounded. Using (6) and (13), we can prove that is bounded for . This is a contradiction. Hence, under the designed control, the forward-time completeness of the closed-loop system (1) and (36) can be guaranteed.
Remark 4. The designed controller is in fact a state feedback controller. In this paper, we assume that all the states are available and present a state feedback controller (36), and the controller needs ( depends on ). Besides, the controller (36) is a delay-independent and allows the system (1) to have fast time-varying delays. It adopts a dynamic gain to regulate the nonlinear terms and has a simple form despite the growing conditions provided in Assumption 1 being complicated.
Remark 5. For time-delay nonlinear systems, there are mainly two control design strategies. The first one is to employ the L-K stability theorem, which needs to construct the L-K functional for the control design. However, this method requires the derivative of time-varying delay to satisfy . The second strategy is the L-R stability theorem-based method, which does not need the delay’s derivative and allows for fast time-varying delays. With the second strategy, every solution of system (1) does not depend on the delay’s derivative. Considering this fact, we try to present a control method using the second strategy. It should be mentioned that the detailed control design is not a direct and easy work despite that we know that the L-R stability theorem is helpful in dealing with the time-varying delays. For instance, which kind of L-R function can be constructed? How can we estimate the nonlinear delay terms? How can we design a bounded dynamic gain? All the obstacles should be skillfully overcome to obtain the final controller.
4. Illustrative Example
Example 1. We consider the mass-spring mechanical model [12], which is described bywhere is the mass of the block, is the spring’s restoring force, represents the resistive force, and denotes the force playing the role of control input. Defining , , and , we obtainSuppose that and , then Assumption 1 holds.
Step 1. We introduce the transformationswhere , , and . It follows thatwhere . We choose the function , where . So, we can deduce thatBy choosing , we getwhere .
Step 2. We choose and it yields thatBy utilizing Lemma 1, we getwhere . With the help of Lemmas 1, it yields thatwhere . By defining and , it is easy to obtain the following equation:We define , and we getBy defining , we haveAlso, it follows thatwhere and .
Following the deductions, we choose . By defining , we can construct the controller asBy choosing and , it follows thatIn the example, we select and . The left parameters are , , , , and . Figures 1–3 provide the system responses. As can be seen, with the raised strategy, the designed controller (50) guarantees that all the signals and are bounded, besides, and . Hence, the proposed method is valid.
5. Conclusions
The studied system in this work has polynomial terms and time-varying delays. We remove the slow delay condition and consider the global stabilisation problem. First, we introduce a new transformation and obtain a dynamic system. By choosing an appropriate L-R function and using a recursive control design method, we construct a new dynamic stable controller. All states are bounded and convergent. Naturally, there are also other interesting topics to be considered. For instance, can the results be extended to uncertain systems? Is it possible to construct some state observers so that the controller can be constructed via an output feedback strategy?
Data Availability
The data supporting the current study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the Fundamental Research Program of Shanxi Province (Number 202203021221004).