Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7324508, 14 pages

http://dx.doi.org/10.1155/2016/7324508

## Design and Experimental Verification of Robust Motion Synchronization Control with Integral Action

^{1}School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China^{2}Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1

Received 24 June 2016; Accepted 19 September 2016

Academic Editor: Steffi Knorn

Copyright © 2016 Chen Peng 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 robust attitude motion synchronization problem is investigated for multiple 3-degrees-of-freedom (3-DOF) helicopters with input disturbances. The communication topology among the helicopters is modeled by a directed graph, and each helicopter can only access the angular position measurements of itself and its neighbors. The desired trajectories are generated online and not accessible to all helicopters. The problem is solved by embedding in each helicopter some finite-time convergent (FTC) estimators and a distributed controller with integral action. The FTC estimators generate the estimates of desired angular acceleration and the derivative of the local neighborhood synchronization errors. The distributed controller stabilizes the tracking errors and attenuates the effects of input disturbances. The conditions under which the tracking error of each helicopter converges asymptotically to zero are identified, and, for the cases with nonzero tracking errors, some inequalities are derived to show the relationship between the ultimate bounds of tracking errors and the design parameters. Simulation and experimental results are presented to demonstrate the performance of the controllers.

#### 1. Introduction

In the field of multivehicle cooperative control, robust consensus tracking under model uncertainties and exogenous disturbances has received increasing attention in recent years, where the output (or state) of each vehicle is required to robustly track a common, desired trajectory. For instance, switching controllers or sliding-mode controllers were designed in works [1–3] to reject input disturbances; a sliding-mode disturbance observer is combined with a consensus tracking algorithm in recent work [4] to improve the robustness and control accuracy of a multimotor system. Alternative robust control approaches include the ones based on uncertainty and disturbance estimators as in works [5–7], adaptive control approach [8], and the output-regulation approach [9].

In practice, integral control (IC) is widely used to attenuate disturbances in various (single) vehicle systems. This mainly owes to its structural simplicity and the well-known performance property that IC can asymptotically reject constant input disturbances. Noting these facts, many researchers begin to study IC-based robust control schemes for multivehicle systems (MVSs) as in [10–12]. In particular, the recent work [10] shows that PI controllers successfully attenuate constant disturbances in the network of multiple single-integrator dynamics or the network of multiple double-integrator dynamics.

Another practical issue encountered in many control systems is the lack of sensors. As a result, state observers are often needed to generate the estimates of some necessary states. State observers can be roughly classified into two types: model-dependent ones and model-independent ones. As two representative model-dependent observers, Luenberger observer and Kalman filter suffer from the limitation that the estimation accuracy cannot be guaranteed when the system model suffers from severe uncertainties. To deal with this problem, many model-independent observers are proposed. For instance, some higher-order sliding modes (HOSM) observers (differentiators) were designed in work [13, 14] to ensure finite-time convergence even in the presence of input disturbances.

The main objective of this paper is to use integral control to improve the robustness of a distributed control algorithm for consensus tracking without velocity measurements. The effectiveness of the approach is proved by showing that (a) the resulting tracking errors are ultimately bounded for any input disturbances satisfying a simple Lipschitz-constant condition, and (b) zero-error asymptotic tracking is achieved for a constant input disturbance. The key technical differences between this paper and work [10] are summarized as follows:(1)The paper [10] considers the consensus problem without a common reference. In contrast, this paper considers a consensus tracking problem, where a common desired trajectory exists (and is supposed to be accessible only to partial vehicles in the group). In [10], velocity signals were used in the control design for second-order systems. We here assume that neither the velocity of leader nor the velocity of neighboring vehicles is accessible for control design.(2)Concerning the robustness improvement owing to integral control, the discussion in work [10] is restricted to the rather special case with constant input disturbances. This is not the case in this paper. Actually, we will use the concepts of input-to-state stability to study the general cases where the disturbances are nonconstants and are not completely rejected.(3)In work [10], the controller performance is verified by numerical simulation. In this paper, both numerical simulation results and experimental results on three 3-DOF helicopters are presented, to demonstrate the performance improvement owing to the use of integral control.

The experimental platform of “three 3-DOF helicopters” used in this paper is shown in Figure 1. The single laboratory 3-DOF helicopter with a so-called active disturbance system (ADS) is the same as that in [5] and is shown in Figure 2. This experimental apparatus was developed by Quanser Consulting Inc. for the purpose of control education and research [15–21].