Abstract
This article investigates the robust stabilization problem for nonlinear time-delay systems with dead-zone input and complex dynamics. By flexibly using the inequality technique, the backstepping control method, and skillfully introducing a new Lyapunov–Krasovskii functional, we obtain a stable controller without using unmeasurable signals in the dynamic subsystem. The control system is guaranteed to be stable finally. Two simulation examples are given to verify the control strategy.
1. Introduction
Many practical models in engineering are nonlinear systems such as the flexible-joint robot [1], the wheeled inverted pendulum [2], and the autonomous underwater vehicle [3]. In the past few years, scholars have focused on studying nonlinear systems, such as [4–11]. For numerous nonlinear systems, the time-delay phenomenon which may lead to system instability often exists and is inevitable [12]. Nonlinear systems sometimes involve complex dynamics, in which the information of the states is not available. For more results on nonlinear systems with complex dynamics, we refer the reader to [13]. Besides the time delay, nonlinear conditions, and complex dynamics, specific control inputs such as the dead-zone input [14, 15] can also have a significant impact on the system. Considering the above facts, the nonlinear time-delay systems with dead-zone input and complex dynamics are investigated in this paper.
In recent years, some complicated linear systems have been studied. For example, Xu and Zhang [16] considered the stochastic large population system and presented a novel strategy for linear-quadratic games. However, different from linear systems, control problems of nonlinear systems are often difficult since they have more complicated dynamics. Specially, Guo [17] studied nonlinear chaotic systems and raised a physically implementable controller to solve the projective synchronization problem. In engineering practice, lots of nonlinear systems can be approximated by using linear systems at the origin; thus, the theory of linear systems can be applied. However, some systems may not be linearized at the origin or can be linearized but have uncontrollable Jacobian linearization [18, 19]. So, it is necessary to study nonlinear control design methods for those systems. Besides, time-delay problems cannot be ignored for the system control design, since ignoring it may make the system unstable. Many scholars have studied the associated control design for systems with time delay (for example, see the adaptive control problem [20], the stabilization problem [21], and the tracking problem [22]).
In recent years, the control design for systems with complex dynamics has been one of the interesting topics. Particularly, with the choice of a state observer, the adaptive control problem of systems containing complex dynamics was solved in [23]. By utilizing a new Lyapunov function, the adaptive tracking problem was studied in [24] for systems with input saturation and complex dynamics. On the other hand, time delay may bring negative effects to the stability of systems. Therefore, scholars have studied control problems for nonlinear systems with time delay. For systems involving time delay and complex dynamics, in [25], by the neural network method and the Lyapunov–Krasovskii functional, the tracking control design was studied. In [26], a modified strategy of adding a power integrator was applied for stochastic delayed nonlinear systems with complex dynamics. Subsequently, this technique was further applied to systems with uncertainty in [27].
In practice, dead zone may exist in the actuator or the control input of the system. There have been some related reports mainly discussing the neural method and the fuzzy method. Specially, the neural control method [28] and the adaptive fuzzy control method [29] were applied to solve adaptive control problems of nonlinear systems. Recently, the Lyapunov–Krasovskii functional control approach has been used to study the nonlinear tracking control of systems containing dead-zone input and implies that the approach is important for systems with time delay, see [30]. However, this method is not extended to solve the robust stabilization problem for systems involving time delay, dead-zone input, and complex dynamics. Also, few studies in the literature considered the robust control problem for the system.
The difficulty and the contribution of this paper are as follows:(i)Considering that the system of this paper involves complicated dynamics, time delay, external disturbances, and input dead zone, the robust stabilization control problem of this paper is more challenging. The existing methods are difficult to solve the problem of this paper. Particularly, the homogeneous domination approach [5] has not proposed a strategy to solve the input dead-zone problem, and the tuning functions-based robust control method [7] has not give a solution to the complicated dynamics. The neural control method [28] and the fuzzy method [29] are difficult to give appropriate bounds for the nonlinear time-delay terms. The continuous control methods in [31, 32] did not provide a strategy to deal with the external disturbances. This article will consider more complicated nonlinear systems and present a control strategy to solve the control problem.(ii)A new robust stabilization strategy is raised. By recursively selecting a Lyapunov–Krasovskii functional and by presenting a modified backstepping control technique, a stable controller without utilizing the unmeasurable states is successfully constructed.
2. Problem Formulation and Preliminaries
Consider the nonlinear system as follows:where is the unmeasurable dynamics and is the state vector which is measurable. . and . is the constant time delay. is the Hurwitz matrix that satisfies , where and are positive definite symmetric matrices. The control coefficients satisfy with and . , are continuous functions. The dead-zone input iswhere , and the are time-varying functions.
To facilitate the problem simply, let , where and are the even integer and the odd integer, respectively. Therefore, define the constants and . Next, we need the following assumptions.
Assumption 1. The nonlinear terms satisfy the following:where and are constants and is a bounded disturbance.
Assumption 2. There are constants , and satisfyingNext, Lemma 1 is provided for control design.
Lemma 1. (see [30]). For given and functions , there holds
3. Main Results
Theorem 1. For system (1), suppose that Assumptions 1-2 hold. Then, under the following transformationthere exists a robust controller:where . The controller ensures that the considered system is globally stable.
Proof. We divide the proof into several parts.
3.1. Part I: Robust Control Design
We construct the controller by using the modified backstepping technique. Step 1: defining and choosing , we have Noting that and , where is the minimal eigenvalue of and is the maximal eigenvalues of . is the minimal eigenvalue of . Then, introduce and define . From Assumption 1 and Lemma 1, we get where and are the positive constants depended on and . Substituting (10) into (9), it yields By Lemma 1 and Assumption 1, there are constants and such that Introduce the transformation , and choose , where , , and . is a constant. Then, it follows from (11) and (12) that Selecting the virtual control and substituting it into (13), it yields that Step (): for step , suppose that there exist transformations (6) and a Lyapunov–Krasovskii functional such that there exist the following inequality: Next, we prove it still holds for . Choosing introducing , and taking the derivative of , it can be deduced that On the basis of and Lemma 1, there is a constant such that Similarly to the proof of (12), one obtains where and are constants. From Assumption 1, we have Utilizing (20) and Lemma 1 and noting that , it follows that where are constants. It can be deduced from (21) and Lemma 1 that where and are constants. Choosing such that and using (19) and (22), it yields that Now, we construct the virtual control . With the help of (17), (18), and (23), it is deduced that This completes the control design for step . Step : in this step, we choose and , and using (24), some simple computations lead to
Similar to (18) and (23), by Lemma 1, we obtain thatwhere the constants , and . Now, we choose the virtual control , which, and (25)–(27), give that
Finally, we choose the control input as (7), Then, by using (2), it follows thatwhich further renders that . Then, using (28), we get
3.2. Part II: Stability Analysis
Since and , we have , which leads to
Defining and , it follows from Lemma 1 that which renders thatSubstituting (31) and (32) into (28), it yields thatwhere and . By the definition of , , it follows thatwhich lead towhere the function and . On the other hand, there exists a function such that
In view of (28), (35), and (36), and using Lemma 3.2.4 in [33], we obtainwhich indicates that and are bounded. From , the states of are bounded, which indicates the system composed of (1) and (7) is globally stable.
4. Simulation Example
Example 1. Consider the following nonlinear time-delay system:where is the unmeasurable state, are the measurable states, is the dead-zone input given in (2) with , and is the input of the system. We see that Assumptions 1-2 are satisfied for system (32) with , and . Applying the above control method, we choose . Then, the actual controller is constructed asIn the simulation, choose . Figures 1–4 give the trajectories of and the control input . All signals in systems (38) and (39) are bounded. Hence, the validity of the presented control method is verified.
Example 2. Consider the following system:where is the unmeasurable state, are the system states, is the control input, and is the dead-zone input defined in (2) with . It can be deduced that , , , and . In this example, , and . Thus, all conditions of Assumptions 1-2 are satisfied. Using the control design method in Section 3, we can design the actual controller as follows:where .
In the simulation, the initial conditions are selected as . Figures 5–9 show the trajectories of , and the control input . It can be seen that all signals in systems (34) and (35) are bounded. Hence, the presented control method is effective.
5. Conclusions
The robust stabilization for nonlinear systems with dead-zone input and time delay has been studied. Because the system involves the dead-zone input, time-delay, disturbance, and unmeasurable states, the stabilization control in this work is more challenging. A robust stable controller has been designed via the Lyapunov–Krasovskii functional and the backstepping technique. Another interesting problem is as follows: When the considered system includes uncertainty parameters, and only the system output is measurable, how can we design the adaptive controller via the output feedback control method?
Data Availability
The data used to support the findings are included within this article.
Conflicts of Interest
The authors declare no conflicts of interest in preparing this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 61903239), Youth Science and Technology Research Foundation of the ShanXi Science and Technology Department of China (Grant no. 201801D221167), and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (Grant no. 2019L0492).