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

Abstract

This paper studies the application of frequency distributed model 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 sliding mode 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.

Academic Editor:Remi Leandre

Department of Electrical Engineering, Northwest A&F University, Yangling, Shaanxi 712100, China

Received 11 September 2015; Revised 2 December 2015; Accepted 16 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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 [11-13]. In recent years, many scholars try to establish the nonlinear model of HGS [14-16]. 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 [19-22]. 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 [23-25]. Until now, some finite time control techniques such as terminal sliding mode (TSM) have been proposed [26-29].

Besides, as we all know, Lyapunov stability theorem is often used in the analysis of integer order system stability. However, it has not yet received satisfactory results in fractional systems. Reference [30] proposes applying the frequency distributed model (FDM) to Lyapunov's method for some simple linear and nonlinear fractional differential equations. In [31, 32], by using the FDM, FDE initial conditions problem where converted into an equivalent ODE initialization problem for the first time. By applying FDM, stability analysis of sliding mode dynamics systems was studied in [33, 34]. Reference [35] introduces the FDM into the fractional order complex dynamic networks, and a robust nonfragile observer-based controller is designed. The main advantage using FDM is that the approach provides a reference for generalization of integer order system theory to fractional order ones, which is obvious a bridge between fractional order system and integer order system.

That is, both FDM in analyzing the stability of fractional order system and finite time control in improving control quality have potential advantages. Can finite time control of fractional order HGS be implemented via FDM? It is still an open problem. Research in this area should be meaningful and challenging.

In light of the above analysis, there are several advantages which make our study attractive. Firstly, a frequency distributed model is proposed by an auxiliary function and the properties of fractional calculus, which is easier to implement. Secondly, a novel fractional order TSM is firstly proposed and its stability to origin is guaranteed based on the proposed FDM and Lyapunov stability theorem. Then, a robust finite time control law to ensure the occurrence of the sliding motion in a finite time is proposed for stabilization of the fractional order HGS regardless the external disturbances. Lastly, simulation results have demonstrated the robustness and effectiveness of this new approach.

The rest of this paper is organized as follows. In Section 2, the fractional order HGS model is presented. Some definitions of fractional order calculus and relevant properties, the FDM, and controller design are given in Section 3. In Section 4, simulation results are provided. Some conclusions end this paper in Section 5.

2. Modeling of HGS

The physical model of penstock system is shown in Figure 1.

Figure 1: The physical model of penstock system.

[figure omitted; refer to PDF]

The dynamic characteristic of synchronous generator can be represented as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the rotor angle, [figure omitted; refer to PDF] is the damping factor of the generator, [figure omitted; refer to PDF] is the variation of the speed of the generator, [figure omitted; refer to PDF] is the output torque of hydroturbine and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denote the inertia time constant of generator and load, respectively, [figure omitted; refer to PDF] .

Here, [figure omitted; refer to PDF]

The electromagnetic power of the generator can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the transient internal voltage of the armature, [figure omitted; refer to PDF] is the bus voltage at infinity, [figure omitted; refer to PDF] is the direct axis transient reactance, [figure omitted; refer to PDF] is the quadrature axis reactance.

The dynamic characteristics of a hydraulic servo system can be got as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the major relay connector response time.

The output torque of turbine governing system is obtained as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the transfer coefficient of turbine flow on the head, [figure omitted; refer to PDF] is the transfer coefficient of turbine torque on the main servomotor stroke, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the transfer coefficient of turbine torque on the water head.

According to formulae (1) to (6), the mathematical model of HGS can be described as [figure omitted; refer to PDF]

Here, the parameters of system (7) are, respectively, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . For convenience, we use [figure omitted; refer to PDF] to replace [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the random disturbances are considered. The fractional order HGS (8) can be rewritten as [figure omitted; refer to PDF]

The state trajectories of system (8) are illustrated in Figure 2. It is clear that the system exhibits nonlinear irregular oscillations. Therefore, it is necessary to design controller for suppressing the complex even chaotic vibration of HGS.

Figure 2: State trajectories of fractional order HGS (8).

(a) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(b) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(c) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(d) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

3. 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.

Definition 1 (see [37]).

The [figure omitted; refer to PDF] th fractional order Riemann-Liouville integration of function [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the Gamma function.

It can be known that when [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

Remark 2.

[figure omitted; refer to PDF] is the well-known Euler's gamma function which is defined as [figure omitted; refer to PDF] and the following identity holds: [figure omitted; refer to PDF]

Definition 3 (see [37]).

The Riemann-Liouville fractional derivative of order [figure omitted; refer to PDF] of a continuous function [figure omitted; refer to PDF] is defined as the [figure omitted; refer to PDF] derivative of fractional integral (9) of order [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the smallest integer larger than or equal to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denotes the Gamma function.

Definition 4 (see [37]).

The Caputo fractional derivative of order [figure omitted; refer to PDF] of a continuous function [figure omitted; refer to PDF] at time instant [figure omitted; refer to PDF] is defined as the fractional integral (9) of order [figure omitted; refer to PDF] of the [figure omitted; refer to PDF] derivative of [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the smallest integer number larger than or equal to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Property 2.

For the Caputo fractional derivative, the following equality holds: [figure omitted; refer to PDF]

Property 3.

The following equality for the Caputo derivative and the Riemann-Liouville derivative are established: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 5.

Compared with Riemann-Liouville fractional derivative, the Laplace transform of the Caputo definition allows utilization of initial conditions of classical integer order derivatives with clear physical interpretations. And the Caputo fractional derivative has the widespread application in the actual modeling process. Therefore in this paper, the Caputo definition of fractional derivative and integral is selected. To simplify the notation, we denote the Caputo fractional derivative of order [figure omitted; refer to PDF] as [figure omitted; refer to PDF] instead of [figure omitted; refer to PDF] .

3.2. Frequency Distributed Model Transformation

For the convenience of mathematical analysis, the [figure omitted; refer to PDF] -dimensional fractional order system is equally written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the order of the system, [figure omitted; refer to PDF] is the system state vector, and [figure omitted; refer to PDF] is the nonlinear term.

Then, an auxiliary time and frequency domain function is defined as [figure omitted; refer to PDF]

Theorem 6.

It follows from (19) that the fractional order system (18) can be equivalently written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof .

The process of proving is divided into two steps.

Step 1 . Equation (19) can be transformed into the form as follows: [figure omitted; refer to PDF]

Take the derivative of (21) with respect to time, and one can get [figure omitted; refer to PDF]

Step 2 . According to definition (11) of Euler function and definition (9) of fractional calculus, there is [figure omitted; refer to PDF]

Define the variable [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

According to (23) and (24), one gets [figure omitted; refer to PDF]

Introducing the auxiliary function (19), one has [figure omitted; refer to PDF]

Note [figure omitted; refer to PDF]

Based on (12), one can get [figure omitted; refer to PDF]

Then (26) can be written as [figure omitted; refer to PDF]

Based on Properties 2 and 3 and (29), one gets [figure omitted; refer to PDF]

This completes the proof.

3.3. Controller Design

Lemma 7 (see [39]).

Consider the [figure omitted; refer to PDF] -dimensional fractional order system (18); assume that there exists a positive constant [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , then the fractional order nonlinear system (18) will be stable in the finite time [figure omitted; refer to PDF] .

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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the system states and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are the given sliding surface parameters, with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The saturation function sat [figure omitted; refer to PDF] is presented as [figure omitted; refer to PDF]

When the system reaches the sliding mode surface [figure omitted; refer to PDF]

According to (32) and (34), one can obtain [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

Based on Properties 2 and 3, there is [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

Select Lyapunov function as [figure omitted; refer to PDF]

Taking its time derivative, one gets [figure omitted; refer to PDF]

According to the definition of saturation function [figure omitted; refer to PDF] , there is the following.

Case 1 ( [figure omitted; refer to PDF] ) . In this case, one has [figure omitted; refer to PDF]

Because of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a given positive constant, one has [figure omitted; refer to PDF]

One can easily get [figure omitted; refer to PDF]

Case 2 ( [figure omitted; refer to PDF] ) . In this case, one has [figure omitted; refer to PDF]

It is clear that [figure omitted; refer to PDF]

Considering both Cases [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , there is [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the state variables, [figure omitted; refer to PDF] are the external random disturbances, [figure omitted; refer to PDF] are the control inputs, and [figure omitted; refer to PDF] are the fractional orders.

Theorem 9.

Consider fractional order HGS (47) and the sliding surface in (32). If the system is controlled by the control law (48), then the states trajectories of the system will converge to the sliding surface [figure omitted; refer to PDF] in a finite time [figure omitted; refer to PDF] where [figure omitted; refer to PDF] present bounded values of the external disturbances, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are given positive constants with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Proof.

Select Lyapunov function [figure omitted; refer to PDF] , and one gets [figure omitted; refer to PDF]

Substituting [figure omitted; refer to PDF] into (49), there is [figure omitted; refer to PDF]

Considering [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF]

Introducing control law (48) to (51), one can get [figure omitted; refer to PDF]

According to [figure omitted; refer to PDF] , one gets [figure omitted; refer to PDF]

Taking integral of both sides of (53) from 0 to [figure omitted; refer to PDF] [figure omitted; refer to PDF] There is [figure omitted; refer to PDF]

And let [figure omitted; refer to PDF] ; after calculation we can get the finite time [figure omitted; refer to PDF]

According to Lyapunov stability theory, the state trajectories of the fractional order HGS (47) will converge to [figure omitted; refer to PDF] asymptotically. And we can easily get the reaching time [figure omitted; refer to PDF] . This completes the proof.

4. Numerical Simulations

For the fractional order HGS (8), according to the proposed method in Section 3, select the corresponding parameters as follows: [figure omitted; refer to PDF]

According to the sliding surface (32), one can get [figure omitted; refer to PDF]

Based on the control law (48), there is [figure omitted; refer to PDF]

Figure 3 shows the control results of fractional order HGS (8) with initial condition [figure omitted; refer to PDF] . 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.

Figure 3: State trajectories of controlled fractional order HGS (8).

(a) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(b) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(c) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(d) State trajectory of [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

5. 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.

Acknowledgments

This work was supported by the Scientific Research Foundation of the National Natural Science Foundation (Grant nos. 51509210 and 51479173), the "948" Project from the Ministry of Water Resources of China (Grant no. 201436), the 111 Project from the Ministry of Education of China (no. B12007), and Yangling Demonstration Zone Technology Project (2014NY-32).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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