Finite Time Control for Fractional Order Nonlinear Hydroturbine Governing System via Frequency Distributed Model

This paper studies the application of frequency distributedmodel for finite time control of a fractional order nonlinear hydroturbine governing system (HGS). Firstly, the mathematical model of HGS with external random disturbances is introduced. Secondly, a novel terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite time stability and slidingmode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional order HGS. Finally, simulation results show the effectiveness and robustness of the proposed scheme.


Introduction
Nowadays, fractional calculus has attracted numerous scientific researchers' attention in various fields.It has been widely used in mechanics [1], electrical engineering, [2] and some other fields [3,4].Since many practical models of engineering applications could be better described by fractional order calculus, like fractional order PMSM system [5,6], chemical processing systems [7], and wind turbine generators [8], fractional calculus still has great potential especially for the description of hereditary and memory attributes of numerous processes and materials [9,10].
The hydroturbine governing system plays a very important role in a hydroelectric station, and its running conditions directly affect the stable operation of hydroelectric stations and electrical systems, which has arisen many researchers' interests [11][12][13].In recent years, many scholars try to establish the nonlinear model of HGS [14][15][16].However, most of the models are on the basis of integer order calculus.As we all know, HGS is a highly coupling, nonlinear as well as nonminimum phase system.For this reason, integer calculus is not suitable for describing complex hydroturbine governing system.According to the history-dependent and memory character of hydraulic servo system, the fractional order hydroturbine governing system that is more in line with actual project is considered in this paper.
Many studies have indicated that the hydroturbine governing system exhibits nonlinear even chaotic vibration in nonrated operating conditions [17,18].So it is very important to design robust controller for suppressing nonlinear even chaotic vibration of HGS.Recently, fractional order nonlinear control has attracted increasing attention.Some control methods have been presented for stability control of fractional order nonlinear or chaotic systems, such as fuzzy control method, sliding mode control, pinning control, and predictive control [19][20][21][22].It is clear that all of the above schemes are focused on the asymptotical stability, which needs infinite time theoretically in order to achieve the control objectives.From the perspective of optimizing the control time, finite time stability theory based control methods should be studied, which has good performance on improving the transition time, overshoot, and oscillation frequency [23][24][25].Until now, some finite time control techniques such as terminal sliding mode (TSM) have been proposed [26][27][28][29].
Besides, as we all know, Lyapunov stability theorem is often used in the analysis of integer order system stability.

Modeling of HGS
The physical model of penstock system is shown in Figure 1.
The dynamic characteristic of synchronous generator can be represented as where  is the rotor angle,  is the damping factor of the generator,  is the variation of the speed of the generator,   is the output torque of hydroturbine and   ,   denote the inertia time constant of generator and load, respectively, Here, The electromagnetic power of the generator can be expressed as where   is the transient internal voltage of the armature,   is the bus voltage at infinity,   Σ is the direct axis transient reactance,  Σ is the quadrature axis reactance.
The dynamic characteristics of a hydraulic servo system can be got as where  is the incremental deviation of the guide vane opening.The hydraulic servo system has significant historical reliance.Since it is a powerful advantage for fractional calculus to describe the function which has significant historical reliance, the fractional order hydraulic servo system is adopted.
According to fractional calculus, the fractional order hydraulic servo system is described as [36] where   is the major relay connector response time.
The output torque of turbine governing system is obtained as where  ℎ is the transfer coefficient of turbine flow on the head,   is the transfer coefficient of turbine torque on the main servomotor stroke,  =  ℎ  ℎ /  −  ℎ , and  ℎ is the transfer coefficient of turbine torque on the water head.

Finite Time Controller Design for
Fractional Order HGS Based on FDM 3.1.Preliminaries.In this section, some basic definitions and properties would be used related to fractional calculus.The two most usually used definitions of fractional derivative are Riemann-Liouville and Caputo definitions.
It can be known that when  approaches to zero, fractional integral (9) would change into the identity operator in the weak sense.In this paper, 0th fractional integral is considered to be the identity operator which is defined as Remark 2. Γ(⋅) is the well-known Euler's gamma function which is defined as and the following identity holds: Definition 3 (see [37]).The Riemann-Liouville fractional derivative of order  > 0 of a continuous function () is defined as the th derivative of fractional integral (9) of order  − : where  is the smallest integer larger than or equal to  and Γ(⋅) denotes the Gamma function.
Definition 4 (see [37]).The Caputo fractional derivative of order  > 0 of a continuous function () at time instant  ≥ 0 is defined as the fractional integral (9) of order  −  of the th derivative of (): where  is the smallest integer number larger than or equal to  and Γ(⋅) denotes the Gamma function.
The next are some useful properties of fractional differential and integral operators which will be used for the controller design [38].
Property 1.The fractional integral meets the semigroup property.Let  > 0 and  > 0; then Property 2. For the Caputo fractional derivative, the following equality holds: Property 3. The following equality for the Caputo derivative and the Riemann-Liouville derivative are established: where  ≥  ≥ 0.

Frequency Distributed Model Transformation.
For the convenience of mathematical analysis, the -dimensional fractional order system is equally written as where  is the order of the system, () ∈   is the system state vector, and () is the nonlinear term.
Proof .The process of proving is divided into two steps.
Step 1. Equation ( 19) can be transformed into the form as follows: Take the derivative of ( 21) with respect to time, and one can get Step 2. According to definition (11) of Euler function and definition (9) of fractional calculus, there is Define the variable with  = 2( − ).
According to ( 23) and ( 24), one gets Introducing the auxiliary function (19), one has Note Based on (12), one can get Then ( 26) can be written as Based on Properties 2 and 3 and ( 29), one gets This completes the proof.
In general, the design process of sliding mode control can be divided into two steps.Firstly, one can select an appropriate sliding surface which represents the required system dynamic characteristics.In this paper, a novel fractional order FTSM is defined as follows: (32) where  = [ 1 ,  2 , . . .,   ]  ∈   are the system states and  1 ,  2 ,  are the given sliding surface parameters, with  1 > 0,  2 > 0, 0 <  < 1.The saturation function sat(⋅) is presented as When the system reaches the sliding mode surface According to (32) and (34), one can obtain Then Based on Properties 2 and 3, there is Theorem 8.If the terminal sliding mode is selected in the form of (32), the sliding mode dynamics system (37) is stable and its state trajectories will converge to zero.
Proof.According to Theorem 6, the sliding mode dynamical system (37) can be described as

Advances in Mathematical Physics
Select Lyapunov function as Taking its time derivative, one gets According to the definition of saturation function sat(⋅), there is the following.

Case 1 (|𝑥| > 𝑘).
In this case, one has Because of  sign() = || and  is a given positive constant, one has One can easily get Case 2 (|| ≤ ).In this case, one has It is clear that Considering both Cases 1 and 2, there is According to Lyapunov stability theory, the state trajectories of the sliding mode dynamics system (37) will converge to zero asymptotically.This completes the proof.
As for the fractional order HGS (8), its controlled form can be briefly represented as where  = [ 1 ,    Taking integral of both sides of (53) from 0 to  1 There is And let ( 1 ) = 0; after calculation we can get the finite time According to Lyapunov stability theory, the state trajectories of the fractional order HGS (47) will converge to () = 0 asymptotically.And we can easily get the reaching time  ≤ (1/(1 − ))ln((|(0)| 1− + )/).This completes the proof.

Numerical Simulations
For the fractional order HGS (8), according to the proposed method in Section 3, select the corresponding parameters as follows: (59) Figure 3 shows the control results of fractional order HGS (8) with initial condition [0, 0, /6, 0].From Figure 3, it is obvious that when the proposed controller (59) is put into system (8), the sliding mode is guaranteed and the state trajectories converge to zero immediately, which implies that the nonlinear vibration of the fractional order HGS (8) is efficiently suppressed in a finite time, regarding the system with external random disturbances.Simulation results have demonstrated the robustness and effectiveness of the proposed method.

Conclusions
A new robust finite time terminal sliding mode control scheme was designed to stabilize a nonlinear fractional order HGS in this paper.An auxiliary time and frequency domain function was introduced to transform the fractional order nonlinear systems into FDM.Then, a novel TSM is proposed and its stability to the origin was guaranteed based on the FDM and Lyapunov stability theory.Furthermore, a robust finite time control law to ensure the occurrence of the sliding motion in a finite time was proposed for stabilization of the fractional order HGS regardless the external disturbances.Numerical simulations were employed to demonstrate the effectiveness and robustness with the theoretical results.

Figure 1 :
Figure 1: The physical model of penstock system.