Mathematical Problems in Engineering

Volume 2017, Article ID 6901894, 10 pages

https://doi.org/10.1155/2017/6901894

## Lyapunov Based Estimation of Flight Stability Boundary under Icing Conditions

Air Force Engineering University, Xi’an, Shaanxi 710038, China

Correspondence should be addressed to Binbin Pei; moc.361@0491nib

Received 4 November 2016; Accepted 4 January 2017; Published 30 January 2017

Academic Editor: Francisco Gordillo

Copyright © 2017 Binbin Pei 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

Current fight boundary of the envelope protection in icing conditions is usually defined by the critical values of state parameters; however, such method does not take the interrelationship of each parameter and the effect of the external disturbance into consideration. This paper proposes constructing the stability boundary of the aircraft in icing conditions through analyzing the region of attraction (ROA) around the equilibrium point. Nonlinear icing effect model is proposed according to existing wind tunnel test results. On this basis, the iced polynomial short period model can be deduced further to obtain the stability boundary under icing conditions using ROA analysis. Simulation results for a series of icing severity demonstrate that, regardless of the icing severity, the boundary of the calculated ROA can be treated as an estimation of the stability boundary around an equilibrium point. The proposed methodology is believed to be a promising way for ROA analysis and stability boundary construction of the aircraft in icing conditions, and it will provide theoretical support for multiple boundary protection of icing tolerant flight.

#### 1. Introduction

Aircraft icing has always been a familiar but unsolved factor threatening flight safety. The formation of ice on aircraft in flight alters the original smooth aerodynamic shape of the aircraft, which has a negative effect on aircraft performance, stability, and control. The shrinking flight envelope caused by aircraft icing can easily lead to flight accident. Obviously, ice avoidance is the most reliable way to increase flight safety; however this conservative way is unnecessary. To improve the operation ability of the aircraft and to maintain revenues and schedules, ice tolerance is the preferred choice for icing conditions.

The goal of the ice tolerance flight is to make sure that the aircraft operate in the safe flight envelope. For this purpose, great efforts have been made by many researchers around the world. Bragg et al. proposed a Smart Icing System conception [1, 2]; the system could sense and characterize the presence of ice, notify the pilot, and ensure the safety of the aircraft. Sharma et al. researched the envelope protection theory when operating in autopilot mode as one part of SIS research [3]. Gingras et al. [4, 5] have developed the Icing Contamination Envelope Protection (ICEPro) system which is mainly designed to provide flight and control limit information to the pilot through cues and messaging on his display to improve flight safety. All above achievements provide systematic approaches for ice tolerance flight. In addition, researches related to one aspect of ice tolerance have also been carried out to perfect the theory and methodology.

Melody et al. [6] compared three different parameter identification algorithms in the context of icing detection and pointed out that method can provide a timely and accurate icing indication. Caliskan et al. [7] adopted Kalman filter and neural network technique to study problems on aircraft icing identification, detection, and reconfigurable control. Dong and Ai proposed using time-varying case of the algorithm and probabilistic neural network to provide inflight parameter identification and icing location detection [8].

The definition of the flight boundaries under icing conditions is one of the key aspects of ice tolerance flight. However, current flight boundaries are mainly defined by critical values of several state parameters. For example, the decreased stall angle of attack due to icing is commonly used to restrict the incidence angle within its safe envelope. Such method has two main limitations: on the one hand, no matter single parameter limit or several parameters limit, the interrelation of each key parameter is not taken into consideration, while loss of control (LOC) may happen caused by the coupling of state parameters although each of them is within its own limitation. One the other hand, the method is feasible when neglecting the outside disturbance, whereas the aircraft can always be perturbed due to an upset condition or wind gust which may lead to system instability even if the state parameter does not exceed its extreme value. To keep the passengers comfort and safety flight under icing conditions, it is of great importance to make sure that the aircraft operates in the stable flight domain.

The paper presented here aims at developing a methodology to conduct a preliminary analysis of the stability boundary of the aircraft under icing conditions. The remainder of the paper is organized as follows. Section 2 proposes a nonlinear icing effect model for a wide range of angle of attack to incorporate the icing effect into flight dynamics. The process of derivation of the longitudinal axis polynomial model of the aircraft is presented in Section 3. Section 4 describes the underlying theory of region of attraction (ROA) analysis for nonlinear polynomial system based on sum-of-square (SOS) theory. Comparison between ROA analysis results and the phase diagram of different icing severity presented in Section 5 demonstrate that the boundary of the calculated ROA can be regarded as the approximate flight stability boundary of the aircraft under icing conditions. Concluding remarks and recommendations for the future work are presented at the end of this paper.

#### 2. Nonlinear Icing Effect Model

Modeling of abnormal aerodynamics caused by ice accretion has been studied for many years [9–12]. Most of these researches only focus on linear interval of incidence angle, that is, within stall angle. To give an insight into flight dynamic characteristics near the flight boundary especially for poststall region, nonlinear icing effect model for a wide range of angle of attack should be established. In this section, a nonlinear icing effect model is proposed to include the adverse effect of ice accretion on aerodynamic parameters.

##### 2.1. Linear Icing Effect Mode

Current estimation of iced aerodynamics model normally based on linear stability and control derivatives, namely, aerodynamic force or moment, is linear function of flight state parameters and control value. For longitude aerodynamic parameters, the aerodynamic force coefficients acting along the body axes, , and the pitch moment coefficients are calculated by

Lift coefficient and drag coefficient can be available via a rotation

For linear icing effect model, the effect of ice accretion is reflected by the change of the aerodynamic derivatives. The icing effect model proposed by Bragg et al., which combines aircraft specific property and icing conditions together, is physically representative and has been widely used in icing research [12]. According to the theory, the icing effect theory is based on the following equation:where is any arbitrary aerodynamic coefficient that is affected by ice accretion. represents icing severity which is only the function of the atmospheric conditions. is a constant value which denotes the modification of the aerodynamic coefficients; it is dependent on the coefficient being modified and on the properties of the specific aircraft.

##### 2.2. Nonlinear Icing Effect Model

###### 2.2.1. Establishment of the Model

The adopted equation form of the longitude nonlinear aerodynamics model is [2, 13]where , , and are unitless aerodynamic coefficients computed from look-up tables.

Compared with the linear aerodynamic model, both of the two models have similar formation. For example, as for , the term in linear aerodynamic model (see (1)) is corresponding to the term in nonlinear aerodynamic model (see (1)). And , terms are corresponding to terms, respectively. According to linear icing effect model shown as (3), each term of in the nonlinear aerodynamics model can be calculated bywhere subscription iced and clean denotes the flight condition with and without ice, respectively.

###### 2.2.2. Modification of Nonlinear Icing Effect Model

According to the subscale, complete airplane wind tunnel test for a wide range of angle of attack, the differences between clean and iced configuration are gradually narrowing and finally disappear with increasing angle of attack in poststall area for longitudinal aerodynamic coefficients [14–17]. Typical alteration characteristics especially in high angle of attack are summarized as follows:(i)A delayed nose-down break for pitch moment coefficient in the stall region.(ii)A slight reduction in lift curve slope and a sizable reduction in maximum lift.(iii)A “flattening” of the lift curve slope in the stall region.(iv)Lift coefficient decrease or increase again at almost linear lift curve slope in the poststall region.

However, the direct calculated results through (11) shown by blue line in Figure 1 cannot reflect those phenomena, and the disagreement is due to the invariant value of . In fact, is no longer a constant value when it is in high angle of attack and it is also the function of state parameters like [2]. Based on above analysis, modification on iced coefficients can be divided into following intervals:(1)For linear change interval, is the same as linear icing effect model.(2)In the stall region, value of is depend on the corresponding aerodynamic coefficient, such as curvature of lift coefficient curve in stall region decrease as the severity of ice increase.(3)In poststall region, the variation of for iced aerodynamic coefficient is determined by the approximate linear relationship of aerodynamic coefficient and angle of attack. When the angle of attack reaches the certain value, turn out to be zero, which means the differences between clean and iced configuration disappear through point.