Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 938246, 7 pages

http://dx.doi.org/10.1155/2015/938246

## Integral Sliding Mode Control for Helicopter via Disturbance Observer and Quantum Information Technique

^{1}College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA

Received 6 August 2014; Accepted 19 September 2014

Academic Editor: Ke Zhang

Copyright © 2015 Qiang Qu 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

A novel self-repairing control scheme is proposed for a helicopter with unknown disturbance. Firstly, a disturbance observer is introduced to observe the disturbance of the system, which can produce corresponding control signals according to the disturbance signals. Secondly, an integral sliding mode controller is designed to compensate the unobserved disturbance and uncertainties. All of the closed-loop poles can be arbitrarily placed and the output errors converge to zero effectively through the controller. Besides, a robust closed-loop system against disturbance and parameter uncertainties is achieved. In addition, quantum information technique is used to increase the self-repairing control accuracy of helicopter. Finally, simulation results demonstrate the effectiveness and feasibility of the proposed self-repairing control scheme.

#### 1. Introduction

A helicopter is a complicated aircraft and its performance is seriously influenced by environmental changes. Because of its nonlinearity, heavy coupling, varying parameters, and model uncertainty, a helicopter is very difficult to control. We can classify the characteristics of the helicopter into three categories: nonlinearity, uncertainty, and instability. The control of the helicopter, therefore, represents a challenge for control system design [1–3]. In recent years, lots of helicopter flight control schemes have been proposed, such as adaptive neural network control [4], fuzzy control [5], robust control [6], and adaptive back-stepping control.

Fuzzy control is a nonlinear control method, essentially. By using linguistic information, it possesses several advantages such as model-free, robustness, rule-based algorithm, and universal approximation theorem. It is also easy to use, simple to design, and strong in the anti-interference function. Recently, fuzzy control has been successfully and widely applied to many nonlinear systems [7–10].

Sliding mode control (SMC) is a robust method that is used to control nonlinear and uncertain systems. The SMC does not rely on an accurate aircraft mathematical model, and it can overcome model uncertainties and disturbance in the system, allowing the system track reference model with high precision. It also can stabilize some complex nonlinear systems which are difficult to be stabilized by the state feedback laws [11–15]. Integral sliding mode control (ISMC) is presented in [16], and it not only makes the uncertainties and disturbance rejected, but also achieves zero steady state error. Moreover, ISMC is more robust than conventional SMC in the application of electrohydraulic servo control systems.

Considering the existence of disturbance, quantum information technique is introduced in this paper. The study of quantum information technique [17, 18] has been a hot research topic; therefore, the scope of the applied research on quantum information technique is very wide [19, 20]. In this paper, quantum information technique is used to increase the self-repairing control accuracy of the helicopter and improve the ability of anti-interference.

It is such a challenge to control a helicopter with its nonlinearity, heavy coupling, varying parameters, and model uncertainty. The main content of this paper is to design a direct self-repairing flight control system for the helicopter. The purpose of the system is to eliminate the influence of the parameter uncertainties and external disturbance. In this paper, an integrative method is proposed by combining fuzzy control, sliding mode control, and quantum information technique together, which therefore possesses the advantages of all three techniques. This method is both effective and feasible, which will be verified by simulation.

The proposed method can achieve the following superiorities.(i)All of the closed-loop poles can be arbitrarily assigned.(ii)The control system can be asymptotically stable and the output errors can converge to zero effectively.(iii)Robust system [21, 22] against system parameter uncertainties and external disturbance can be achieved.

This paper not only theoretically proves the stability of the proposed method, but also verifies the effectiveness of this method through the simulation of the helicopter control system. Simulation results show that the proposed method provides a feasible method for the actual design of the helicopter control system.

This paper is organized as follows. Firstly, a disturbance observer is introduced to observe the disturbance of the system, which can produce control signals according to the disturbance signals. Secondly, ISMC with fuzzy tuning is designed. All of the closed-loop poles can be arbitrarily placed, and the design procedure includes the sliding function definition, the control law formulation, and the stability proof for the system. The sliding function involves the integral of the state as well as the output errors, so the output errors can converge to zero effectively. An additional fuzzy tuning control here is introduced to accelerate the reaching time and reduce chattering by utilizing fuzzy logic judge. Furthermore, quantum information technique is used to increase the self-repairing control accuracy of helicopter.

#### 2. Description of Mathematical Model

The helicopter is a nonlinear system with strong coupling. In this paper, the linear model which is linearized about equilibrium point is only considered. The state equation of the model is as follows: where and are the state and the output of the system, respectively. is the control vector. , , and are the parameter matrices.

Some assumptions have been made to achieve the method which is applied in the flight control system.(1)The pair is completely controllable and is completely observable.(2)The disturbance is unknown and bounded.(3)Denote as the th row of the matrix in (17). Let , where , . For the disturbance , the following conditions are satisfied: , .

#### 3. Quantum Information Technique

Quantum information is a new discipline, which is combined by quantum mechanics and information science. In recent years, based on the incomparable advantages on data transmission security, sensor measurement sensitivity and accuracy, and quantum computing parallelism, it has attracted widespread attention and development. Currently, the central issue of quantum information technique applied research is quantum cryptography, quantum communication, quantum computation, quantum simulation, quantum metrology, the physical basis of quantum information, and so on. By using quantum information technique, the researchers can simplify the modeling pattern so that the problems would be much easier.

In quantum computation, and denote the two basic states of microparticles, which are named as quantum bit (qubit). Arbitrary single-qubit state can be expressed as the linear combination of two basic states. The state of qubit is not only and , but also a linear combination of the state, usually called as superposition state; namely, where and are a pair of complex, called the probability amplitude of quantum state; namely, the measurement results in quantum state collapsing with a probability of or collapsing with a probability of and satisfying Therefore, quantum state can be also denoted by the probability amplitude of quantum state in the form of . Obviously, if , , is the basic state , which can be described by , then , , is the basic state , which can be described by . Generally speaking, quantum state is the unit vector of two-dimensional complex vector space.

Due to the collapse of quantum states caused by observation, there is a continuous state between the quantum bits and , until it has been observed. The existence of continuous state qubit and behavior has been confirmed by a large number of experiments. And there are many different physical systems that can be used to realize quantum bits.

Similar to the single-qubit state, double-quantum-bit state can be expressed as with the probability amplitude satisfying

Similarly, three-qubit state can be expressed as

And the probability amplitude satisfying

To increase the self-repairing control accuracy of helicopter, quantum information technique is added in the approach. The probability amplitudes of two quantum bits for the module can be seen in Table 1.