Complexity

Volume 2018, Article ID 3580628, 8 pages

https://doi.org/10.1155/2018/3580628

## Fixed-Time Sliding-Mode Fault-Tolerant Control of Waste Heat Power Generator Systems

^{1}Jiangsu Province Laboratory of Electrical and Automation Engineering for Coal Mining, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China^{2}China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Correspondence should be addressed to Caoyuan Ma; nc.ude.tmuc@nauyoacm

Received 3 May 2018; Accepted 31 May 2018; Published 15 July 2018

Academic Editor: Diyi Chen

Copyright © 2018 Zhi Wang et al. 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.

#### Abstract

In order to overcome disturbances such as the instability of internal parameters or the actuator fault, the time-varying proportional-integral sliding-mode surface is defined for coordinated control of the excitation generator and the steam valve of waste heat power generation units, and a controller based on sliding-mode function is designed which makes the system stable for a limited time and gives it good performance. Based on this, a corresponding fault estimation law is designed for specific faults of systems, and a sliding-mode fault-tolerant controller is constructed based on the fixed-time control theory so that the systems can still operate stably when an actuator fault occurs and have acceptable performance. The simulation results show that the tracking error asymptotically tends to be zero, and the fixed-time sliding-mode fault-tolerant controller can obviously improve the dynamic performance of the system.

#### 1. Introduction

Observations on the long-term operation of waste heat power generation systems have shown that coordinated control of the excitation generator and the steam valve can significantly improve the limit of the transient stability of the system [1, 2], effectively improving the dynamic response following system failure, and reducing low-frequency oscillations. Therefore, generator control technology must be developed according to the coordinated control of the steam valve and excitation, and this project requires a combination of valve control and generator excitation control for its implementation.

However, it is very difficult to achieve precise control for synchronous generators with strong nonlinearity. Many kinds of control schemes have been put forward on the basis of a large number of domestic and foreign research. The control system of a turbogenerator set has been developed to a certain extent, but there is no perfect control means, and there are always shortcomings to some extent.

In research on the theory of intelligent control, a series of new techniques of coordinated control have been proposed. After years of theoretical and technical advancements, they have achieved good results in the context of the coordinated control of the excitation generator and steam valve. Neural networks are very appropriate for coordinated control because of its self-learning and self-organization. In [3], neural networks are used to design the controller, which can quickly solve the coordinated control problem. In [4], the neural network inverse system has been applied to the system by analyzing its reversibility. Fuzzy control has also been widely applied to coordinated control [5–7]. But on the whole, using intelligent methods in coordinated control is not so mature, which needs more study.

Sliding-mode variable-structure control has the advantages of fast response, simple design process, good robustness, and some advantages in dealing with nonlinear problems [8]. It has been applied to a nonholonomic mobile manipulator system [9], where the system had uncertainties in the kinematic mobile manipulator model and drive motor parameters, external disturbances, and nonmodeled dynamics, which is typical for many nonlinear uncertain systems. It is robust against the internal and external parameters of disturbance systems [10].

However, at present, the sliding-mode variable-structure control is mostly finite-time sliding-mode variable-structure control, which is limited to the limitation of finite-time stability. The stability of the system is affected by the initial conditions of the system, and it is generally difficult to obtain accurate parameters in the power system at the initial conditions, with the fixed-time stability being a generalization of the finite-time stability. It has all the advantages of finite-time control, and fixed-time control can not only guarantee the upper bound of the stable time range, but also have stronger robustness and anti-interference ability. The most important thing is the global uniform asymptotic stability of the system under any initial condition.

Technical experience has indicated, however, that a conventional feedback control design for complex systems such as waste heat power generator systems results in unsatisfactory performance, or even instability, in the event of malfunctions in actuators [11]. To circumvent such weaknesses, new approaches to control system design have emerged with the purpose of tolerating component malfunctions while maintaining desirable stability and performance properties. These types of control systems are often known as fault-tolerant control systems (FTCSs). FTCSs are control systems that possess the ability to accommodate component faults automatically. The main task to be tackled in achieving fault tolerance is the design of a controller with a suitable structure to guarantee stability and satisfactory performance, not only when all components are operational but also in the event that actuators malfunction [11–14]. Ideally, in the presence of faults, the closed-loop system should be capable of maintaining its safe and reliable operation while preserving system stability [15].

On the basis of summarizing the current research status of residual heat generator control, this paper makes a comprehensive and in-depth study on the excitation and valve coordination control system of a waste heat generating set, establishes a mathematical model of the residual heat generator set, and uses a fixed-time fault-tolerant control based on a sliding-mode variable structure for the existing actuator failures. The controller controls the excitation and valve-coordinated control system, thus restraining the disturbance and enabling the system to run stably on the premise of acceptable performance.

#### 2. Excitation and Steam Valve Coordination Control System Model for Thermal Generator Units

Consider the mathematical model of an excitation and steam valve coordination control system model as follows:
where *δ* is the power angle, *ω* is the rotating speed, is the transient emf, *P _{H}* is the high-pressure mechanical power of the cylinder,

*ω*

_{0}is the generator synchronous angular speed,

*C*is the medium-pressure and low-pressure mechanical power of the cylinder,

_{ML}*P*

_{m0}is the initial input value of the mechanical power of the generator,

*U*is the bus voltage of the infinite capacity system,

_{s}*D*is the damping power coefficient,

*H*is the inertia constant of the generator rotor, is the transient integrated reactance, is the inertia time constant of the excitation winding, and

*T*

_{HΣ}is the time constant of the high-pressure cylinder and the high-pressure oil motive.

Let . The system becomes

Parameter is controlled by the valve control parameter, *u*′, and is unrelated to other state quantities and control quantities. It is related only to the system parameters, which are set as control quantity 1, and the output voltage, *u,* of the excitation winding, which is set as control quantity 2. At the same time, the actuator failures are considered. Then, the system becomes
where *d*_{1} is the fault of the steam valve and *d*_{2} is the fault of the excitation generator.

Let , and , . is a constant, and , thus, the system becomes

*Remark 1. * is bounded and is also bounded; adding does not affect the existence of the solution of the original differential equation. When the actuator fault function is a constant value, and . When the actuator fault function is time-varying, considering , are normal scalars.

If , satisfied the function, then (1), are continuous on ;(2), are on to satisfy the Lipschitz condition.

That is, the continuity of functions is to guarantee the existence of equation solutions, while the Lipschitz condition is to ensure the uniqueness of solutions.

#### 3. Controller Designed for the Excitation and Steam Valve Coordination Control System Model

For the design of the controller, we provide the definition of fixed-time stability and related lemmas.

*Definition 1 (see [16]). *Consider the following dynamic system:
where is the system’s state and is a smooth nonlinear function. If there is a fixed settling time, , under any initial condition, the magnitude of is independent of the initial conditions.

For example, and . If , the system in (5) is said to be fixed-time stable.

Lemma 1 (see [17]). *If there is a continuous function , wherein
*(1)* is positive definite,*(2)*there are real numbers > 0 and 0 < < 1, satisfying**then, when , , the system is stable in
*

Lemma 2 (see [18]). *If there is a continuous unbounded function , where
*(1)*,*(2)*For , when x(t) satisfies the following inequality*

*where represents the upper-right derivative of function , the system is said to be stable, and the time required for it to stabilize is given by*

Lemma 3 (see [7]). *If nonzero positive real numbers , then,
*

This is the process for designing the controller. The switching function is first designed. We set the system output tracking target, and define the controlled magnitude of error as follows:

To extend robustness to the entire system and eliminate the steady state error, an integral term is added to the linear sliding surface to construct the following integral sliding surface:

Once the design of the sliding surface is complete, we need to construct a suitable control law, and design the following: where , , and, are the rated value, , , the feedback gain of the tuning parameter of the terminal attractor > 0, and where is any normal number.

Theorem 1. *Considering the waste heat power generator system (3), using the designed switching function (12) and control law (13), and applying the fixed-time stability theory, the system will reach the stable state in a fixed time and become steady.*

*Proof 1. *The Lyapunov function is selected as follows:
Thus, according to Lemma 3,
Therefore, it follows from Lemma 1, where , , and *k* is any constant greater than zero. The proof is complete.

From Lemma 2, we can get a stable fixed time:

For the designed control law, the following theorem is given.

Theorem 2. *Considering the waste heat power generator system (3), using the designed switching function (12) and control law (13), and applying the change of the sliding surface to estimate the fault, the system will reach the stable state in a limited time and become steady.*

*Proof 2. *Let one select the Lyapunov function as follows:
where,
Definition:
then,
then,

When , choosing and that are large enough to make small enough in this case will be , then,

In the same way,

#### 4. Numerical Simulation

To verify the effect of the controller, numerical simulations were performed on MATLAB. In the simulations, the system parameters are shown in Table 1.