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.
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.
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 aswhere is the transient internal voltage of the armature, is the bus voltage at infinity, is the direct axis transient reactance, 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 aswhere is the transfer coefficient of turbine flow on the head, is the transfer coefficient of turbine torque on the main servomotor stroke, , and 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
Here, the parameters of system (7) are, respectively, , , , , , , , , , , , , and . For convenience, we use to replace , , , , and the random disturbances are considered. The fractional order HGS (8) can be rewritten as
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.
(a) State trajectory of
(b) State trajectory of
(c) State trajectory of
(d) State trajectory of
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 th fractional order Riemann-Liouville integration of function is defined bywhere and is the Gamma function.
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. is the well-known Euler’s gamma function which is defined asand the following identity holds:
Definition 3 (see [37]). The Riemann-Liouville fractional derivative of order of a continuous function is defined as the derivative of fractional integral (9) of order :where is the smallest integer larger than or equal to and denotes the Gamma function.
Definition 4 (see [37]). The Caputo fractional derivative of order of a continuous function at time instant is defined as the fractional integral (9) of order of the derivative of :where is the smallest integer number larger than or equal to and 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 and ; 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
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 as instead of .
3.2. Frequency Distributed Model Transformation
For the convenience of mathematical analysis, the -dimensional fractional order system is equally written aswhere is the order of the system, is the system state vector, and is the nonlinear term.
Then, an auxiliary time and frequency domain function is defined as
Theorem 6. It follows from (19) that the fractional order system (18) can be equivalently written aswhere ,
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 getStep 2. According to definition (11) of Euler function and definition (9) of fractional calculus, there isDefine the variablewith .
According to (23) and (24), one getsIntroducing the auxiliary function (19), one hasNoteBased on (12), one can getThen (26) can be written asBased on Properties 2 and 3 and (29), one getsThis completes the proof.
3.3. Controller Design
Lemma 7 (see [39]). Consider the -dimensional fractional order system (18); assume that there exists a positive constant , such thatand ; if , then the fractional order nonlinear system (18) will be stable in the finite time .
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:where are the system states and , , are the given sliding surface parameters, with , , . 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 asSelect Lyapunov function asTaking its time derivative, one getsAccording to the definition of saturation function , there is the following.
Case 1 (). In this case, one hasBecause of and is a given positive constant, one hasOne can easily getCase 2 (). In this case, one hasIt is clear thatConsidering both Cases and , there isAccording 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 aswhere are the state variables, are the external random disturbances, are the control inputs, and 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 in a finite timewhere present bounded values of the external disturbances, , , are given positive constants with , .
Proof. Select Lyapunov function , and one getsSubstituting into (49), there isConsidering , one hasIntroducing control law (48) to (51), one can getAccording to , one getsTaking integral of both sides of (53) from 0 to There isAnd let ; after calculation we can get the finite timeAccording to Lyapunov stability theory, the state trajectories of the fractional order HGS (47) will converge to asymptotically. And we can easily get the reaching time . 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:
According to the sliding surface (32), one can get
Based on the control law (48), there is
Figure 3 shows the control results of fractional order HGS (8) with initial condition . 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.
(a) State trajectory of
(b) State trajectory of
(c) State trajectory of
(d) State trajectory of
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).