Mathematical Problems in Engineering

Volume 2015, Article ID 905184, 12 pages

http://dx.doi.org/10.1155/2015/905184

## Extreme Learning Machine Assisted Adaptive Control of a Quadrotor Helicopter

^{1}School of Aeronautics and Astronautics, Zhejiang University, Zhejiang, Hangzhou 310027, China^{2}State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Liaoning, Shenyang 110189, China^{3}Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China

Received 21 August 2014; Revised 10 December 2014; Accepted 14 December 2014

Academic Editor: Yi Jin

Copyright © 2015 Yu Zhang 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

Control of quadrotor helicopters is difficult because the problem is naturally nonlinear. The problem becomes more challenging for common model based controllers when unpredictable uncertainties and disturbances in physical control system are taken into account. This paper proposes a novel intelligent controller design based on a fast online learning method called extreme learning machine (ELM). Our neural controller does not require precise system modeling or prior knowledge of disturbances and well approximates the dynamics of the quadrotor at a fast speed. The proposed method also incorporates a sliding mode controller for further elimination of external disturbances. Simulation results demonstrate that the proposed controller can reliably stabilize a quadrotor helicopter in both agitated attitude and position control tasks.

#### 1. Introduction

Unmanned aerial vehicles (UAVs) have received considerable attention in recent years due to their wide military and civilian applications. Their typical applications include collecting data, monitoring, surveillance, investigation, and inspection [1], which can be used in scenarios such as environmental monitoring, resource exploration, agriculture surveying, traffic control, weather forecasting, aerial photography, disasters search, and rescue. Particularly, unmanned rotorcrafts play an important role in these applications because of their flexibility such as hovering and vertical take-off and landing (VTOL). However, conventional rotorcraft with a main rotor and a tail rotor has extremely complex dynamics. Its maneuverability is greatly limited since it is very difficult to design a controller with high performance. Meanwhile, a special kind of rotorcrafts called quadrotor helicopter which has a compact form is becoming more and more popular than conventional rotorcrafts as they are mechanically and dynamically simpler and easier to control. In spite of this, the quadrotor is still a dynamically unstable system and its controller design is also challenging because of the inherent system characteristics such as nonlinearities, cross couplings due to the gyroscopic effects, and underactuation [2]. Besides, all the applications mentioned above require a vehicle with stable and accurate performance of motion control. Therefore, how to design a high quality controller for quadrotor is an important and meaningful problem.

Many different control theories and methods are employed to design the attitude stabilizer or motion controller for quadrotors [3]. In the early stage, Bouabdallah et al. first apply two different control techniques, PID and linear quadratic (LQ) techniques, to a microquadrotor called OS4 [2]. Subsequently, other research works using PID [4, 5] or LQ [6] methods are also reported. These two kinds of controllers are easy to design and implement. However, they cannot handle the unmolded dynamics and external disturbances. Because of the nonlinearity, feedback linearization technology is adopted to design the controller for quadrotors [7, 8]. However, precise model of quadrotors which is required for linearization is difficult to obtain. Backstepping design method [9–11] is another choice to deal with the nonlinear models of quadrotors [12, 13], but the controller is usually vulnerable to parameters uncertainty. To deal with the dynamic uncertainty and external disturbances, sliding mode control [14] and infinite control [15, 16] algorithms are used to improve the robustness of the system. However, sliding mode controller is likely to have chattering phenomena in both sides of sliding mode surface due to the delay of sensors or actuators, and infinite control method, which requires approximate linearization near the equilibrium point of the system, is not suitable for aggressive control.

The performance of model based controllers above degrades significantly in case of model uncertainties and unknown disturbances. One potential way to solve this problem is intelligent control methods such as fuzzy logic control [17, 18], neural networks control [19, 20], and learning based control [21]. One main challenge of utilizing these techniques is the convergence performance of controllers. Slow convergence may cause failure in real time control system.

In this paper, a novel computational intelligence technique called extreme learning machine (ELM) [22] is introduced to control the quadrotors by compensating the dynamic uncertainties and the external disturbances. ELM theories have been successfully improved recently by Cao et al. [23] and widely used in several control systems [24]. Essentially, it is a learning policy for generalized single hidden layer feedforward networks (SLFNs) whose input weight and hidden layer do not need to be tuned. Compared with backpropagation (BP) method and support vector machines (SVMs), ELM provides better generalization performance at a much faster learning speed and with least human intervention [25]. Thus, ELM can be used to estimate and compensate the uncertainties and disturbances of the systems simultaneously in real time.

There are three main contributions of this paper. First, we employ Lyapunov second method to minimize the cost function of ELM and satisfy the quadrotor control system stability simultaneously under the framework of ELM. So it is a development of ELM theory. Second, traditional neural network based controller is facing two problems. One is that too many parameters need to be initialized and tuned. The other is slow convergence. Since ELM can converge very fast with its input weight and hidden nodes parameters fixed, the two problems above are significantly alleviated when ELM is employed to the quadrotor control. Third, the quadrotor control system is a complicated dynamic system. The stability of the ELM based control system is proved.

This paper is organized as follows: In Section 2, the mathematical model of ELM is presented. Kinematics and dynamics models of quadrotors are described in Section 3. In Section 4, the details of designing an ELM-assisted quadrotor controller are presented. The stability of the proposed controller is also proved in this section. Simulation results are given in Section 5 to demonstrate the performance of the proposed controller. Finally, the paper is concluded in Section 6.

#### 2. Preliminary on Extreme Learning Machine

In this section, the basic idea of ELM is briefly reviewed to provide a background for designing controller for the quadrotor. ELM is a special SLFN whose learning speed can be much faster than conventional feedforward network learning algorithm such as BP algorithm while obtaining better generalization performance [26]. The essence of ELM is that the input weights and the parameters of the hidden layer do not need to adjust during the learning procedure. We take a SLFN with hidden nodes as an example. The output of the SLFN can be modeled aswhere is the output weight connecting the th hidden node to the output node, is the activation function of the th hidden node, and and are the parameters of the activation function which are randomly generated and then fixed afterwards. Furthermore, there are two kinds of hidden nodes. Usually, additive hidden nodes use Sigmoid or threshold activation function as follows:where is the input weight vector for the th hidden node and is the bias of the th hidden node. For RBF hidden nodes, Gaussian or triangular activation function is used for activation which can be given bywhere and are the center and impact factor of the th RBF node, respectively.

Then, sample pairs are used to train the SLFN. If this network can approximate samples with zero error, there must exist , , and such thatThe previous equation can be rewritten compactly aswhereELM aims to minimize not only the training error but also the norm of output weights, which would yield a better generalization performance [25]. In other words, ELM try to minimize the training error as well as the norm of the output weights. So the objective function can be expressed as

Finally, the minimal norm least-square method instead of the standard optimization method was adopted in the original implementation of ELM [25], and the closed form solution is obtained:where is the Moore-Penrose generalized inverse of matrix .

#### 3. Quadrotor Helicopters Model

The quadrotor helicopter has four rotors in cross configuration. As we can see from Figure 1, the two pairs of rotors (1, 3) and (2, 4) always turn in opposite directions. By changing the rotor speed, we can move the vehicles in different directions in 3D space. Initially, suppose all the rotors have the same speed as shown in Figure 1(c); increasing the four rotors speeds together generates upward movement. Then, increasing or decreasing 2 and 4 rotors speed inversely will change the attitude of the quadrotor. It generates roll rotation as well as lateral motion. Changing 1 and 3 rotors speed in the same way produces the pitch rotation as well as the longitudinal movements (see Figure 1(d)). Finally, if the counter-torque resulting from rotor (1, 3) is different from that of rotor (2, 4), the yaw rotations of the quadrotor are generated as shown in Figures 1(a) and 1(b).