A Time-Varying Gain Design Method for State Feedback Control of Upper Triangular Nonlinear Systems

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.


Introduction
Many physical models can be described by nonlinear systems [1][2][3][4]. erefore, 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 [6][7][8][9][10]. e 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]. e 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 timevarying 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.
is paper uses the time-varying gain design method to study the control problem of upper triangular nonlinear systems. e 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. us, a time-varying gain design method is introduced to achieve the control goals of the concerned system.
Assumption 1. For all (t, ξ(t), u(t)) ∈ R + × R n × R, the following inequalities hold: where θ is an unknown positive constant and c is a known constant which satisfies 0 ≤ c < 1.

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. erefore, compared with the assumption in [19,20], Assumption 1 is more general, and the nonlinear characteristics of system (1) are more diverse. e controller designed in this paper is effective for a class of nonlinear systems as long as the nonlinear terms satisfy (2). e control goal of this paper is to construct a state feedback controller such that all the signals of the closedloop system converge to zero. As long as the context does not cause confusion, the parameters of the function can be simplified.
Proof. Let c � 2c. One presents the coordinate transformations as follows: Based on (1) and (4), it is obtained that where ϵ n+1 � u.

Remark 2.
From the proof of eorem 1, we can see that the constant c is a key design parameter. As long as the parameter c satisfies the condition 0 < c < 0.5, one guarantees that equation (13) holds, and then, one handles the effects of the unknown parameter θ and the function on time in the nonlinear terms. e parameters in controller (3) consist of two parts. One is the parameter c, which only needs to be satisfied by c � 2c. e other part is the Hurwitz polynomial coefficients β i , i � 1, 2, . . . , n, which are also relatively easy to choose. erefore, the parameters in controller (3) are better selected, which avoid excessive calculation process.
e proof procedure is similar to the proof procedure of eorem 1. One chooses c � 0.5 in (4), and then, controller (15) is designed. In order to avert repetition, the detailed proof is omitted.
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 eorem 2, when c � 0, one can choose c � 0.5. en, controller (15) can ensure the convergence performance of the states. In fact, when c � 0, as long as the constant c < 1 is selected, the effectiveness of controllers (15) and (16) can be ensured.

Conclusion
is 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
e author declares no conflicts of interest.