Flutter analysis including structural uncertainties using a relaxed LMI-based approach

Abstract

This paper proposes a new approach to investigate the flutter phenomenon considering structural frequencies as uncertain parameters. An affine parameter state space model is used to describe the aeroelastic system and linear matrix inequalities (LMI) are employed to solve the stability analysis with a relaxation on the inequalities. Uncertainties are present during the design phases of an aircraft as well as when considering an operational set of similar aircraft. In the first case, uncertain aeroelastic models have great potential to reduce computational costs during the design phases, and, in the second one, to provide robust aeroelastic predictions under uncertain operational quantities of a set of aircraft. In this work, uncertain aeroelastic predictions for a three-degrees-of-freedom typical section obtained from the relaxed approach are compared to predictions considering the quadratic Lyapunov approach. In the first case, the structural modal frequency of a single degree-of-freedom is considered uncertain. In the second one, all degrees-of-freedom are uncertain. In all cases, the numerical results show that the relaxed approach proposed in this work is less conservative than the quadratic Lyapunov approach, providing a useful tool for flutter analysis including structural uncertainties.

Appendices

Appendix A state space matrices

This appendix presents the submatrices shown in Eq. (6). They are given by

$$\begin{aligned} \bar{\textbf{A}}_1= & {} \begin{bmatrix} q\textbf{Q}_{\phi 3}&\cdots&q\textbf{Q}_{\phi 2+n_{lag}} \end{bmatrix} \end{aligned}$$

(A1)

$$\begin{aligned} \bar{\textbf{A}}_2= & {} \begin{bmatrix} \textbf{I}&\cdots&\textbf{I}&\cdots&\textbf{I} \end{bmatrix}^T \end{aligned}$$

(A2)

which is $(n_{lag}+1)m \times m$ dimensional, where $\textbf{I}$ is $m \times m$ identity matrix. Also,

$$\begin{aligned} \bar{\textbf{A}}_3 = \left[ \textbf{0}^{} \right] ^{(n_{lag}+1) \times 1} \end{aligned}$$

(A3)

where $\textbf{0}$ is a $m \times m$ matrix of zeros, and

$$\begin{aligned} \bar{\textbf{A}}_4 = - \left( \frac{V}{b} \right) \begin{bmatrix} \textbf{0} &{} \cdots &{} \textbf{0} \\ \gamma _1 \textbf{I} &{} \cdots &{} \textbf{0} \\ \textbf{0} &{} \ddots &{} \cdots \\ \textbf{0} &{} \textbf{0} &{} \gamma _{n_{lag}} \textbf{I} \end{bmatrix} \end{aligned}$$

(A4)

Matrix $\textbf{E}$ is given by

$$\begin{aligned} \textbf{E} = \begin{bmatrix} \textbf{M}_{a \phi } &{} \tilde{\textbf{0}}^T \\ \tilde{\textbf{0}} &{} \tilde{\textbf{I}} \end{bmatrix} \end{aligned}$$

(A5)

where $\tilde{\textbf{0}}$ is a $[ (n_{lag}+1)m \times m]$ matrix of zeros, $\tilde{\textbf{I}}$ is an identity matrix with dimension $m(n_{lag}+1)$, and $\textbf{M}_{a \phi }$ and $\textbf{B}_{a \phi }$ are given by

$$\begin{aligned} \textbf{M}_{a \phi }&= \textbf{M}_{\phi } - q \frac{b^2}{V^2} \textbf{Q}_{\phi 2} \end{aligned}$$

(A6)

$$\begin{aligned} \textbf{B}_{a \phi }&= \textbf{B}_{\phi } - q \frac{b}{V} \textbf{Q}_{\phi 1} \end{aligned}$$

(A7)

Appendix B proof of LMI relaxation

This proof is adapted from [22] to consider that the uncertain aeroelastic system is described by an affine parameter model. Let $V_L$ the Lyapunov function, considering the uncertain parameters $\alpha _j$ ($j=1,...,m$) time invariant, $\dot{p}_i=0$ $\forall \; i=1,...,2^m$. Then, $\dot{V}_L$ is given by $\dot{V}_L = 2 \sum _{i = 1}^{2^m} p_i \dot{\textbf{x}}^T \textbf{P}_i \textbf{x}$ or, alternatively, $2 \sum _{i = 1}^{2^m} p_i \textbf{x}^T \textbf{P}_i \dot{\textbf{x}}$ because $\textbf{P}_i$ is symmetric.

Consider $0 = 2\left( \textbf{x}^T\textbf{M}_1+\dot{\textbf{x}}^T\textbf{M}_2\right) \left[ \dot{\textbf{x}} - \textbf{E}^{-1}\left( \textbf{A}_{e0} + \sum _{i=1}^{2^m} p_i \textbf{A}_{ev_i} \right) \textbf{x} \right]$, if $\textbf{A}_{e0}$ is null matrix, this equation corresponds to the proposal of [22], i.e., a polytopic model-based approach to obtain the relaxed LMI. However, considering the new affine parameter model (Eq. 13), the time derivative of $V_L$ is given by $\dot{V}_L = 2 \sum _{i=1}^{2^m}p_i \textbf{ x}^T \textbf{P}_i \dot{\textbf{ x}}$. Then, the following equation can be written

$$\begin{aligned}{} & {} \dot{V}_L = 2 \sum _{i=1}^{2^m}p_i \textbf{ x}^T \textbf{P}_i \dot{\textbf{ x}} + 2\left( \textbf{x}^T\textbf{M}_1+\dot{\textbf{x}}^T\textbf{M}_2\right) \nonumber \\{} & {} \quad \left[ \dot{\textbf{x}} - \textbf{E}^{-1}\left( \textbf{A}_{e0} + \sum _{i=1}^{2^m} p_i \textbf{A}_{ev_i} \right) \textbf{x} \right] \end{aligned}$$

(B8)

Note that it is added zero to $\dot{V}_L$. The following equivalences are employed

$$\begin{aligned}{} & {} 2 \sum _{i = 1}^{2^m} p_i \dot{\textbf{x}}^T \textbf{P}_i \textbf{x}= \sum _{i = 1}^{2^m} p_i \left( \dot{\textbf{x}}^T \textbf{P}_i \textbf{x} + \textbf{x}^T \textbf{P}_i \dot{\textbf{x}} \right) \nonumber \\{} & {} \quad 2 \textbf{x}^T \textbf{M}_1 \dot{\textbf{x}} = \sum _{i = 1}^{2^m} p_i \left( \textbf{x}^T \textbf{M}_1 \dot{\textbf{x}} + \dot{\textbf{x}}^T \textbf{M}_1^T \textbf{x} \right) \nonumber \\{} & {} \quad 2 \dot{\textbf{x}}^T \textbf{M}_2 \dot{\textbf{x}} = \sum _{i = 1}^{2^m} p_i \left( \dot{\textbf{x}}^T \textbf{M}_2 \dot{\textbf{x}} + \dot{\textbf{x}}^T \textbf{M}_2^T \dot{\textbf{x}} \right) \nonumber \\{} & {} \quad 2\dot{\textbf{ x}}^T \textbf{M}_2 \sum _{i=1}^{2^m}p_i \left( \textbf{E}^{-1}\textbf{A}_{ev_i}\right) \textbf{ x} = \nonumber \\{} & {} \quad \;\;\; \sum _{i=1}^{2^m}p_i \left[ \dot{\textbf{ x}}^T \textbf{M}_2 \left( \textbf{E}^{-1}\textbf{A}_{ev_i}\right) \textbf{ x} + \textbf{ x}^T \left( \textbf{E}^{-1}\textbf{A}_{ev_i}\right) ^T \textbf{M}_2^T \dot{\textbf{ x}} \right] \nonumber \\{} & {} \quad 2 \textbf{ x}^T \textbf{M}_1 \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) \textbf{ x} = \nonumber \\{} & {} \quad \;\;\; \sum _{i = 1}^{2^m} p_i \left[ \textbf{x}^T \textbf{M}_1 \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) \textbf{x} + \textbf{x}^T \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) ^T\textbf{M}_1^T \textbf{x} \right] \nonumber \\{} & {} \quad 2 \dot{\textbf{ x}}^T \textbf{M}_2 \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) \textbf{ x} = \nonumber \\{} & {} \quad \;\;\; \sum _{i = 1}^{2^m} p_i \left[ \dot{\textbf{ x}}^T \textbf{M}_2 \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) \textbf{ x} + \textbf{ x}^T \left( \textbf{E}^{-1}\textbf{A}_{e0}\right) ^T\textbf{M}_2^T \dot{\textbf{ x}} \right] \end{aligned}$$

(B9)

Rearranging this equation and defining $\textbf{z} = \{\textbf{x}^T \;\; \dot{\textbf{x}}^T \}^T$ it is possible to write $\dot{V}_L = \sum _{i=1}^{2^m} \textbf{z}^T \left( \textbf{Z}_i + \textbf{Z}_{\textbf{A}_0}\right) \textbf{z}$, and demonstrate that $\textbf{Z}_i$ is given by Eq. (18) and (19). The relaxed stability is verified if $\dot{V}_L<0$, which is achieved for any $\textbf{z}$ if, and only if, $\textbf{Z}_i \prec - \textbf{Z}_{\textbf{A}_0}$ and $\textbf{P}_i \succ 0$, $i=1,...,2^m$.