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This paper studies the problem of fixed-time stability of hydraulic turbine governing system with the elastic water hammer nonlinear model. To control and improve the quality of hydraulic turbine governing system, a new fixed-time control strategy is proposed, which can stabilize the water turbine governing system within a fixed time. Compared with the finite-time control strategy where the convergence rate depends on the initial state, the settling time of the fixed-time control scheme can be adjusted to the required value regardless of the initial conditions. Finally, we numerically show that the fixed-time control is more effective than and superior to the finite-time control.

Hydropower, as a low-cost, zero-polluting, and renewable energy source, has been deeply developed since the twentieth century [

At present, the control method commonly used in water turbine control system mainly includes the following: nonlinear control [

The definition of the fixed-time stability was firstly proposed by Polyakov in [

In [

For the convenience of analysis, we give some necessary definitions and lemmas in advance.

Consider the following nonlinear dynamic system:

Suppose there exists a continuous function such that

there exist real numbers

If there exists a continuous radically unbounded function

any solution

Then the origin is globally fixed-time stable and the following estimate holds:

If

Here, we use the nonlinear model of the water turbine strike system proposed in [

From system (

The hydraulic turbine governing system (

Here, we use the two-step method of two steps to prove that the system is stable for the fixed time; for the sixth subsystem of system (

To sum up, when

In this section, numerical results are provided to verify the theoretical results. The system parameters and controller parameters in this paper are

Figures

The response of the system without controller.

Figures

The response of the fixed-time controllers

The response of the system with fixed-time controllers.

In order to make a fair comparison between fixed-time control and finite-time control, the parameters, initial conditions, and the tuning parameters of the terminal attractor are the same in this paper. Figures

Comparison of converging speed of fixed-time and finite-time controllers.

To explore the relationship between the settling time and the values of the parameters

Response of

Response of

In order to explore the effect of the initial state of hydropower system with fixed-time controllers, we compared the response of three different initial conditions of the hydropower system. Figures

The response of the system with the different

In this paper, to ensure the safe and stable operation of hydraulic turbine governing system, a new control method based on the fixed-time theory is proposed. Compared with the finite-time control method, the hydraulic turbine governing system under the fixed-time controllers has more advantages: better robustness, fast response ability, and the setting time to reach the stable state being regardless of the initial state. Finally, the effectiveness and superiority of the proposed control method are verified by the simulation results. Note that time delay may influence the dynamic behavior of the system; the fixed-time control of hydraulic turbine governing system with time delay is our future direction.

The authors declare no conflicts of interest regarding the publication of this paper.

Caoyuan Ma, Yongzheng Sun, and Chuangzhen Liu conceived and designed the experiments; Chuangzhen Liu, Xuezi Zhang, Wenbei Wu, and Jin Xie performed the experiments and analyzed the data; Chuangzhen Liu and Xuezi Zhang wrote the paper.

This work is supported by the Fundamental Research Funds for the Central Universities (Grant no. 2017XKZD11).