The Scientific World Journal

Volume 2015, Article ID 247301, 10 pages

http://dx.doi.org/10.1155/2015/247301

## MRAC Control with Prior Model Knowledge for Asymmetric Damaged Aircraft

^{1}CNIGC Institution of Navigation and Control, Chedaogou No. 10, Haidian District, Beijing 100191, China^{2}Department of Automation Science and Electrical Engineering, Beihang University, Haidian District, Beijing 100191, China

Received 16 November 2014; Accepted 8 December 2014

Academic Editor: Kemao Peng

Copyright © 2015 Xieyu Xu 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

This paper develops a novel state-tracking multivariable model reference adaptive control (MRAC) technique utilizing prior knowledge of plant models to recover control performance of an asymmetric structural damaged aircraft. A modification of linear model representation is given. With prior knowledge on structural damage, a polytope linear parameter varying (LPV) model is derived to cover all concerned damage conditions. An MRAC method is developed for the polytope model, of which the stability and asymptotic error convergence are theoretically proved. The proposed technique reduces the number of parameters to be adapted and thus decreases computational cost and requires less input information. The method is validated by simulations on NASA generic transport model (GTM) with damage.

#### 1. Introduction

It has been shown that the vehicle damage to airframe and engines had led to quite a few aircraft accidents and fatalities in recent years [1]. Asymmetric aircraft structural damage results in abrupt deterioration on flight performance and handling quality, which impairs safety. Researches to recover aircraft stability and handling qualities, like fault-tolerant flight control, reconfigurable control method, and so forth, have already become a hot spot in related academic fields.

Lots of studies have been conducted to recover control performance under aircraft failures or damages. NASA has launched several aircraft safety projects like IFCS [2] and AvSP [3], which innovate flight control algorithms and experiment on aircraft testbeds. Various flight control laws like the adaptive control designs [4–9], linear quadratic regulator (LQR) designs [6, 10, 11], and robust linear designs [12] were proposed and evaluated by flights. The pilot Cooper-Harper ratings showed that the proposed methods could retain the handling qualities under certain failure cases [4, 13].

Generally, the adaptive control algorithms are quite popular in this field. Reference [14] used an adaptive artificial neural network (ANN) control method to recover the handling qualities of an F-18 model. The method uses a pretrained ANN to provide a model inversion block with aerodynamics and handling characteristics and another online-adjusted ANN to compensate modeling error and failure conditions. Reference [15] extended the result of [14]. It changes the pretrained ANN to an online learning network, which adjusts the inversion block adaptively, resulting in a hybrid direct and indirect adaptive ANN control scheme. Reference [16] designed an angular rate controller using adaptive single hidden layer ANN method for an unmanned airborne vehicle. With a classic linear guidance law, several successful automated landings were made in flight experiments with 25% left wing loss.

Except for ANN adaptive laws, multivariable MRAC with various structures were also studied in this field. Reference [9] combined a direct MRAC method with a parameter estimator using gradient algorithm, obtaining a combined MRAC structure. References [6, 17] proposed a derivative-free MRAC using delayed weight update law. Reference [7] utilized the adaptive law [18] for a flight controller design. Reference [19] analyzed the plant characteristics by LDS decomposition of the high frequency gain matrix (HFGM) of the damaged aircraft. Realizing the signs of the leading principle minors of the HFGM does not change before and after damage. Along with several other results and assumptions, an MRAC law with output feedback structure is proposed. The method is validated through simulations on a high-fidelity GTM model with left wing tip damage. A similar design can be found in [20], which uses a state feedback structure with adaptive gains instead of the original output feedback structure.

The methods mentioned above do not need to take the characteristics of damage conditions into account. As long as the assumptions hold, these methods can be applied to various damage cases and allow for relatively large parameter changes. However, the range of admissible parameter changes can be so large, in a way that is far enough to cover all damage cases. If proper descriptions can be used to model the concerned damage cases, it is possible to design a controller with parameters adjusted in a smaller region, reducing computation workload and improving performance.

Based on the analysis above, this paper models the structural damaged aircraft with polytope LPV models and model reference controllers (MRCs) are given, which explicitly integrates plant characteristics. The parameters updated by adaptive laws are the polytope interpolation coefficients, instead of control gains in traditional MRAC designs. The number of the adaptive parameters is therefore reduced. The paper is organized as follows. Section 1 introduces the background and design philosophy of the proposed method. Section 2 models and analyzes a damaged aircraft, namely, GTM. A polytope linear model is introduced to describe the concerned damage cases. Section 3 designs the control laws and adaptive laws, as well as Lyapunov stability and error convergence proof. Section 4 validates the design by simulations on a 6DOF nonlinear GTM model with left outboard wing tip damage. Section 5 concludes the whole paper.

#### 2. Modeling of an Asymmetric Damaged Aircraft with Prior Knowledge

##### 2.1. Nonlinear 6DOF Modeling

This paper studies the GTM from IRAC project in AvSP program carried out by NASA. Due to the limitation of modeling data, this paper only discusses the damage case of left outboard wing tip loss. With data and methods from [21], the characteristics of aerodynamics, mass, inertia, center of gravity (CG) movement, and so forth, can be modeled. With methods from [22], the rigid body dynamic equations are deduced. It should be noted that the sensors remain unmoved before and after damage, so that the point which the dynamic equations describe remains unchanged. The dynamic equation can be written as follows:

and refer to the total force and moment relevant to original CG, respectively. , , and are the total mass, inertia in original CG, and the offset of CG reference to its origin location. , are the line speed and angular speed described in the origin body frame of the aircraft. Superscript is the skew-symmetric matrix form of a vector, .

The line speed of a specific point generally changes with its location, only except that it moves parallel to the rotation axis of the rigid body. The calculation of the line acceleration has to take the centripetal and the Coriolis force into account, as long as the point is not located on the rotation axis. In short, due to the CG movement by the structural damage, the calculation of line speed becomes quite complicated, coupling with angular speed and angular acceleration. However, if the body frame is selected parallel to the original one, the direction of each axis does not change, and the values of the angular speed are the same wherever the frame is. Thus, the angular speed can be modeled in the body frame at the new CG without changing its value, while the line speed has to be modeled at the same point. Rewrite (1) as follows:

Since the value of angular speed remains unchanged, the symbol is used as before. and are the inertia and moment represented in the new body frame located at CG after damage. The kinetic equations are the same as those of normal aircrafts. is the Euler angle. is the position vector relative to the ground:

With the sensors unmoved, the definitions of airspeed , angle of attack , and sideslip are the same as before. The equations are as follows:

##### 2.2. Polytope Linear Model

Generally, the nonlinear equations of an aircraft can be described as

represent the state and input vectors, respectively. An operation point, or trim point, should be selected before linearization. It should be noted that the actual damage cases are unknown, which renders the trim point unavailable. A general solution is to select a common operation point for all damage cases, for example, the normal point [20]. Because the normal point generally is not the trim point for damaged cases, an extra term is added to the state-space equations, as follows [23]:where , are unknown constant matrices, representing the state transition matrices and control matrices of different patterns and degrees of structural damage, respectively. is the unknown constant disturbance. Various control methods can be designed with (6) [20, 23]. However, are unknown constants with weak constraints like the signs of leading principle minors of high frequency gain matrix that do not change, which could lead to relatively large parameter variations. The actual damage cases may not cover all these variations. This paper believes that if proper definitions of different damage cases are introduced, the parameter variations can be limited, which facilitates the controller designs.

Based on the above analysis, the model of damaged aircraft is rewritten in a polytope form:where , are known constant matrices, the vertices of polytope . There exist nonnegative coefficients ; thus, can be calculated by

Parameter represents different damage patterns and degrees. For any specific damage case, it is assumed that a certain can describe the model of the damaged aircraft. is the extra unknown constant term, for the aircraft is untrimmed under damage. also captures other effects from structural damages and is not modelled in the polytope form. Finally the model is written as

Compared to (6), the only unknown variables in (9) are and . Different damage cases simply differs in weighting coefficient and constant , thus vastly reduces the number of unknown variables.

##### 2.3. Vertex Calculation by High Order SVD Method

As stated before, the less the number of vertices is, the less complicated the controller is. It is crucial to balance between the accuracy and the complexity of the polytope aircraft model with damage of various patterns and degrees. The high order SVD (HOSVD) method is a decomposition method for high order arrays or data tensors. By representing the polytope LTI models into a tensor-product (TP) form, the HOSVD method can be used to reduce the number of vertices [24].

First, a parameter grid is generated on the concerned damage cases. For the left outboard wing tip damage case, a grid of -element vector is used. Each node of the grid represents a different wing tip loss ratio to the semispan. The GTM model with damage is linearized at each node of the grid, written in the system matrix form

For every specific case in the concerned damage cases, the model can be written as the interpolation of the linearized models. That is,wherein function is the interpolation coefficient function. The interpolation calculation can be rewritten in a compact TP form. Stack all the linearized models to make the model tensor and all the coefficients to make the parameter tensor . Thus,

To accurately describe the original model, a relatively intense grid has to be generated. The algorithm based on HOSVD method from [24] is adopted in this paper to simplify the grid. By omission of small singular values, the number of models and corresponding coefficients can be reduced, without losing accuracy to the original ones. With this algorithm, the preliminary TP model can be approximated to

New models and interpolation coefficients , , can be extracted from the system tensor and the coefficient tensor . The algorithm automatically processes the coefficients to build up a polytope model, constrained by , .

The numerical results are shown as follows. The left outboard wing tip loss of GTM ranges from 0% to 33% semispan. A grid size of 10 is selected. State-space models are obtained by linearizing nonlinear GTM models of 1 normal case and 9 damage cases equally spaced along the loss ratios to wing span. After rewriting the models into TP form, the singular values calculated by HOSVD method are given as follows in sorted sequence:

It can be seen that the first 2 singular values are relatively large. By omitting the rest of singular values, the algorithm simplifies the 10 models to 3 vertex models. Thus, any model from the original 10 models, , for example, can be approximated by interpolation of the obtained 3 models as follows:

is the interpolation coefficient. The output models are in accordance with the properties of polytope models. The curves of the interpolation coefficients to wing tip loss ratios are shown in Figure 1.