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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.

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 [

Lots of studies have been conducted to recover control performance under aircraft failures or damages. NASA has launched several aircraft safety projects like IFCS [

Generally, the adaptive control algorithms are quite popular in this field. Reference [

Except for ANN adaptive laws, multivariable MRAC with various structures were also studied in this field. Reference [

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

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 [

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 (

Since the value of angular speed remains unchanged, the symbol

With the sensors unmoved, the definitions of airspeed

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

Based on the above analysis, the model of damaged aircraft is rewritten in a polytope form:

Parameter

Compared to (

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 [

First, a parameter grid is generated on the concerned damage cases. For the left outboard wing tip damage case, a grid of

For every specific case in the concerned damage cases, the model

To accurately describe the original model, a relatively intense grid has to be generated. The algorithm based on HOSVD method from [

New models

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,

The curves of the interpolation coefficients to wing tip loss ratios.

The accuracy of the processing procedure can be measured by the 2-norm of the matrix error

The accuracy of processed model.

It can be shown that the error norm is relatively small, which concludes that the resulting polytope model is close to the original ones. The subsequent designs are based on the obtained polytope model.

The control algorithm proposed in this paper is not of LMI designs in [

Assume the polytope model of GTM is written as

The angular rates are to be controlled. Rewrite the state-space equations to treat angular rates as states, while the other signals related to angular rates are treated as measurable disturbances. Dividing the original states into two vectors yields

Reformulate the state-space equation as follows:

Finally, the polytope model with angular rates as states is shown below:

The model is explained as follows. The states are defined as the angular rates. The state transition matrix is

For the specific case of GTM, the linear model of the undamaged case is shown below:

The linear model of 33% semispan left outboard wing tip loss is written as follows:

It can be seen from the above results that the attitude angles, namely,

Although the input matrix

For the convenience of design, the

Model reference scheme with state-tracking method is adopted to recover the control performance after the aircraft structural damage. The reference model is designed as follows:

Hurwitz state transition matrix

Considering the angular motion of an aircraft can be directly altered by control surfaces, input matrix

Define

Thus,

Rewrite the polytope constraint

It is assumed that every state of the aircraft can be measured. Matrix

However,

The tracking error is defined as

The adaptive laws are designed using Lyapunov direct method. The Lyapunov function

The adaptive laws are selected as

Thus, the derivate

The design process of the adaptive law in this paper is quite similar to the classical ones using the Lyapunov method. Properties like stability and asymptotic error convergence can also be proved using same procedures. All states in the system are bounded due to the positivity of

The MRAC theme with control law (

Thanks to the popularity of the standard Lyapunov design procedure, lots of modifications can be applied to the control law, such as robust modifications [

The proposed method updates the interpolation coefficients of the feedback gains. The gains are designed beforehand for the vertices of the polytope model, which integrates prior information of different cases of damage into the control law. With prior knowledge of various damage cases, the performance of the design is expected to be improved. The HOSVD and model simplification process reduce the number of vertices and the amount of parameters. Less stimulation is required for the input signal [

The linearization of the original 6DOF nonlinear model and the vertex simplification using HOSVD method result in an approximated damaged aircraft model. There always exists modeling error between the real model of various damage cases and the developed polytope model. Similar to common adaptive algorithms, the update law has to be modified to ensure closed-loop stability [

It should be noted that the maximum deflection of control surfaces is finite in real aircrafts, which poses a constraint on

However, it is feasible to maintain a straight cruise under some attack angle

In this section, the developed adaptive law under the obtained linear polytope model is validated by simulations on a NASA GTM nonlinear model with damage. GTM is a 5.5% dynamically scaled twin-turbine powered aircraft model, designed and manufactured in the NASA AvSP program, dedicated to flight testing of research control laws in adverse flight conditions [

The damage case with loss of 20% semispan of the left wing tip is selected. Simulations are conducted on the nonlinear 6DOF GTM model, with maximum deflection of the control surfaces limited within ±30 degrees. The aircraft is flying normally at the beginning, injected with wing tip damage at 5 s during simulation. The simulation results with minor step commands are shown in Figure

Commands and responses in simulation with step input signal.

The step input starts at 1 second with value

Control surface deflections in simulation with step input signal.

Figure

Response of the other states in simulation with step input signal.

It can be seen that the responses of the other states are relatively small, especially for the angle of attack

Further demonstration of tracking performance can be shown in sinusoidal command signals, as in Figure

Commands and responses in simulation with sinusoidal input signal.

The sinusoidal inputs are arbitrarily chosen as

The deflections of control surfaces are shown in Figure

Control surface deflections in simulation with sinusoidal input signal.

It can be seen that the ailerons saturate shortly in the oscillations. However, the ailerons come back to about 18 degrees after the transient, and the saturation does not affect the overall performance of the design. The responses of

Response of the other states in simulation with sinusoidal input signal.

It can be seen that the airspeed and air flow angles vary in the same sinusoidal pattern as that in the angular rates. The overall magnitude of state

In summary, the simulations show that the proposed MRAC method guarantees error convergence and stability in both step input and sinusoidal input. It can be concluded that the proposed method is validated by simulation.

In this paper, an MRAC method with prior model knowledge for asymmetric damaged aircraft has been developed. The method is designed for a series of linearized GTM models with the same operation points but with different left outboard wing tip damage degrees. A polytope model for damaged aircraft with small amount of vertices is obtained first. By representing the vast models of various damage cases in a compact tensor-product form, the number of vertices is reduced by omission of smaller singular values using HOSVD algorithm. An extra constant offset term to the derivatives of states is introduced since the actual trimming values of damaged aircraft are unknown. By interpolating the vertex controllers, the model reference control can be achieved. An adaptive law is deduced to update the interpolation coefficients and the trimming values, finalizing the controller design. The method is theoretically proved with closed-loop stability and asymptotic tracking error convergence. Simulations on a nonlinear GTM with damage validate the design.

Some problems remain unsolved in this paper. It was assumed that the

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was funded by National Natural Science Foundation of China (Grant no. 61273099).