Abstract
In this paper, a time-varying gain design method is used to investigate the state feedback control problem of upper triangular nonlinear systems. Firstly, the nonlinear term recognizes an incremental rate relying on the unknown constant and the function with respect to time. Then, a time-varying gain design method is utilized to construct a state feedback controller. With the help of a suitable coordinate transformation and a Lyapunov function, one obtains that all the signals of the closed-loop system converge to zero. Finally, two numerical examples are presented to display the effectiveness of the time-varying gain design method.
1. Introduction
Many physical models can be described by nonlinear systems [1ā4]. Therefore, the control problem of these physical models can be transformed into the control problem of nonlinear systems [5]. Compared with linear systems, the behavior of nonlinear systems is more diverse [6ā10]. The research of control algorithms is generally developed for the specific type of nonlinear systems [11, 12].
In general, many results about nonlinear systems have focused on nonlinear systems with triangular structures, that is, lower triangular nonlinear systems [13, 14] and upper triangular nonlinear systems [15]. The common method for studying lower triangular nonlinear systems is the backstepping design method [16], and the common method for considering upper triangular nonlinear systems is the forwarding design method [17]. Although, based on the iterative design algorithm, these methods can effectively deal with strong nonlinearities, the design procedure is more complicated. In the past few decades, the gain design method is a very effective tool to deal with the control problem of upper triangular nonlinear systems [18].
Based on the coordinate transformation, the time-varying gain design method is an effective strategy for dealing with the uncertain parameter of upper triangular nonlinear systems [19]. By introducing a time-varying function in the controller, it can effectively deal with the nonlinear terms of upper triangular nonlinear systems [20]. Compared with the commonly adaptive control strategy, the time-varying gain design method is more concise, the calculation process is less, and a lot of calculation work is reduced. Furthermore, the time-varying gain design method does not require too many design parameters and avoids complicated calculation process.
This paper uses the time-varying gain design method to study the control problem of upper triangular nonlinear systems. The nonlinear characteristics of the system considering here are more obvious, that is, the unknown constant and the function with respect to time are allowed in the nonlinear terms. Compared with the assumption about the nonlinear terms in [19, 20], the assumption in this paper is more general. Thus, a time-varying gain design method is introduced to achieve the control goals of the concerned system.
2. Preliminaries
In this paper, we consider a class of nonlinear systems in the following form:where is the state and is the input; the uncertain continuous functions , , satisfy the following growth condition.
Assumption 1. For all , the following inequalities hold:where is an unknown positive constant and is a known constant which satisfies .
Remark 1. Assumption 1 is a reasonable condition. According to Assumption 1, we can know that the nonlinear terms of system (1) can include the unknown constant and the function with respect to time. Therefore, compared with the assumption in [19, 20], Assumption 1 is more general, and the nonlinear characteristics of system (1) are more diverse. The controller designed in this paper is effective for a class of nonlinear systems as long as the nonlinear terms satisfy (2).
The control goal of this paper is to construct a state feedback controller such that all the signals of the closed-loop system converge to zero. As long as the context does not cause confusion, the parameters of the function can be simplified.
3. Main Results
Theorem 1. When the constant satisfies , all the signals of system (1) can converge to zero by the following controller:where , , are coefficients of the Hurwitz polynomial and .
Proof. Let . One presents the coordinate transformations as follows:Based on (1) and (4), it is obtained thatwhere .
Lettingwhere , , are given in (3), the following equation is satisfied:where , , andBecause , , are coefficients of the Hurwitz polynomial , there is a positive definite matrix satisfying [21]. Letting , one getsBy Assumption 1 and (4), one hasIt follows from (10) thatSubstituting (11) into (9), one getsSince , there is a finite time such thatTherefore, it is verified from (12) thatBased on (14) and the definition of , the states , , converge to zero. By (6), the controller converges to zero. From (4), one has that the states , , converge to zero.
Remark 2. From the proof of Theorem 1, we can see that the constant is a key design parameter. As long as the parameter satisfies the condition , one guarantees that equation (13) holds, and then, one handles the effects of the unknown parameter and the function on time in the nonlinear terms. The parameters in controller (3) consist of two parts. One is the parameter , which only needs to be satisfied by . The other part is the Hurwitz polynomial coefficients , , which are also relatively easy to choose. Therefore, the parameters in controller (3) are better selected, which avoid excessive calculation process.
Theorem 2. In the case of , the states of system (1) can converge to zero by the following controller:where , , are coefficients of the Hurwitz polynomial .
Proof. The proof procedure is similar to the proof procedure of Theorem 1. One chooses in (4), and then, controller (15) is designed. In order to avert repetition, the detailed proof is omitted.
Theorem 3. When the constant satisfies , all the signals of system (1) can converge to zero by the following controller:where , , are coefficients of the Hurwitz polynomial .
Proof. One presents the coordinate transformations as follows:Based on (16) and (17), it is obtained thatLettingwhere , , are given in (3), the following equation is satisfied:where , , andBecause , , are coefficients of the Hurwitz polynomial , there is a positive definite matrix satisfying [21]. Letting , one getsBy Assumption 1 and (4), one hasIt follows from (10) thatSubstituting (11) into (9), one getsSince , there is a finite time such thatTherefore, it is verified from (12) thatBased on (27) and the definition of , the states , , converge to zero. By (20), the controller converges to zero. From (17), one has that the states , , converge to zero.
Remark 3. In this paper, with the help of the Lyapunov function, a new control strategy is proposed for upper triangular nonlinear systems, and state feedback controllers (3), (15), and (16) are designed such that all the signals of the closed-loop system converge to zero. In Theorem 2, when , one can choose . Then, controller (15) can ensure the convergence performance of the states. In fact, when , as long as the constant is selected, the effectiveness of controllers (15) and (16) can be ensured.
4. Simulation Examples
Example 1. One considers the following nonlinear system:One gets that system (28) satisfies Assumption 1 with and . Let , , , and . Based on Theorem 1, a state feedback controller for system (28) is designed asThe initial condition is chosen as , , and . From Figures 1ā3, one has that the states , , and of system (28) converge to zero. The controller is shown in Figure 4.
Example 2. A practical example about the resonant circuit system is investigated as follows [22]In system (30), the meaning of the parameters is found in [22]. Following the coordinate transformation in [22], one getsOne gets that system (30) satisfies Assumption 1 with and . Let , , , and . Based on Theorem 2, a state feedback controller for system (30) is designed asThe initial condition is chosen as , , and . From Figures 5ā7, one has that the states , , and of system (30) converge to zero. The controller is shown in Figure 8.
Example 3. One considers the following nonlinear system:One gets that system (33) satisfies Assumption 1 with and . Let , , and . Based on Theorem 1, a state feedback controller for system (33) is designed asThe initial condition is chosen as , , and . From Figures 9ā11, one has that the states , , and of system (33) converge to zero. The controller is shown in Figure 12.
Remark 4. In the two simulation examples, one can see that the parameters in controllers (29), (32), and (34) are relatively easy to select, which do not require a large amount of calculation to deal with the unknown parameter in systems (28), (30), and (33), respectively. In addition, controllers (29), (32), and (34) are effective for a class of nonlinear systems as long as the nonlinear terms satisfy (2). Compared with the example of the results [18, 22], the unknown constant and the function with respect to time are allowed in the nonlinear terms. Compared with the commonly adaptive control strategy, the time-varying gain design method is more concise, the calculation process is less, and a lot of calculation work is reduced.
5. Conclusion
This paper has investigated the state feedback control problem of upper triangular nonlinear systems. One has assumed that the nonlinear term recognizes an incremental rate relying on the unknown constant and the function with respect to time. A time-varying gain design method has been used to construct a state feedback controller. With the help of a Lyapunov function, one has obtained that all the signals of the closed-loop system have converged to zero. Finally, two numerical examples have been presented to illustrate the effectiveness of the time-varying gain design method.
Data Availability
All figures are made by Matlab.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 61503214 and 61873334).