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Generalized composite noncertainty-equivalence adaptive control of a prototypical wing section with torsional nonlinearity

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Abstract

The paper presents a generalized composite noncertainty-equivalence adaptive control system for the control of a prototypical aeroelastic wing section using a single trailing-edge control surface. The plunge–pitch (two-degree-of-freedom) dynamics of this aeroelastic system include torsional pitch-axis nonlinearity. The open-loop system exhibits limit cycle oscillations beyond a critical free-stream velocity. It is assumed that parameters of the model are not known. The objective is to suppress the oscillatory responses of the system. Based on the immersion and invariance approach, a generalized composite noncertainty-equivalence adaptive (NCEA) control system for regulation of the pitch angle is designed. The control system consists of a control module and a composite parameter identifier—designed independently. The composite integral parameter estimation law is based on (1) the immersion and invariance (I&I) theory, (2) gradient-based adaptation algorithm, and (3) classical certainty-equivalence adaptive (CEA) update rule. Besides the composite integral component, the full parameter estimate also includes a judiciously chosen nonlinear algebraic function. This composite identifier inherits stronger stability properties. Using the Lyapunov analysis, asymptotic suppression of the limit cycle oscillations and the boundedness of system trajectories are established. Interestingly, in the closed-loop system including the composite update rule, there exist two attractive manifolds to which the system’s trajectories converge. Simulation results are presented which show the suppression of the oscillatory plunge displacement and pitch angle responses despite uncertainties in the model parameters. Furthermore, the performance and stability properties of this composite NCEA control system—including the gradient-based adaptation and the update rule of the CEA system—are better than the simple NCEA system.

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Abbreviations

a :

Nondimensionalized distance from the midchord to the elastic axis

b :

Semichord of the wing

\(c_h\) :

Structural damping coefficient in plunge due to viscous damping

\(c_\alpha \) :

Structural damping coefficient in pitch due to viscous damping

\(k_{1}, (\gamma _g,\Gamma )\) :

Feedback gain, adaptation gains

h :

plunge displacement

\(I_\alpha \) :

Mass moment of inertia of the wing about the elastic axis

\(k_h\) :

Structural spring constant in plunge

\(k_{\alpha _i}\) :

Structural spring constants in pitch

\(m_t\) :

Mass of the plunge–pitch system

\(m_w\) :

Mass of the wing

\(M_s, M_d, B_0, g_0,b_0 \) :

System matrices

M, L :

Moment and lift

\(s_p\), s :

Span, \(\dot{{\tilde{\alpha }}}+\lambda _1 {\tilde{\alpha }}\)

U, u :

Free-stream velocity and flap deflection

\(V_g, V_c\) :

Lyapunov functions

x :

State vector (\(h, \alpha , \dot{h}, {\dot{\alpha }})^T\)

\(x_\alpha \) :

Nondimensionalized distance measured from the elastic axis to the center of mass

z :

Parameter error \({\hat{\theta }}-\theta \)

\(\alpha \), \(\alpha _r\), \(\tilde{\alpha }\) :

Pitch angle, reference angle, \(\alpha -\alpha _r\)

\(u=\beta ,\) \(\beta _f\) :

Flap deflection angle, filtered \(\beta \)

\((\phi , W)\), \(\phi _f\) :

Regressor vectors, filtered \(\phi \)

\(\lambda _1\), (\(\lambda _f\), \(k_f\)):

Gain in s, filter parameters

\(\mu \) :

Algebraic component of \({\hat{\theta }}\)

\(\rho \) :

Density of air

\(\theta \) :

Unknown parameter vector

\({\hat{\theta }}\), \(\hat{\theta }_{Igc}\) :

Estimate of \(\theta \), integral part of \({\hat{\theta }}\)

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Authors and Affiliations

  1. Department of Electronic Engineering, Catholic Kwandong University, Gangneung-si, Gangwon-do, 25601, Republic of Korea

    Keum W. Lee

  2. Department of Electrical and Computer Engineering, University of Nevada Las Vegas, Las Vegas, NV, 89154, USA

    Sahjendra N. Singh

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  1. Keum W. Lee
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Correspondence to Sahjendra N. Singh.

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Appendix

Appendix

System parameters

$$\begin{aligned}&\!\!\!b = 0.135[\hbox {m}], m_w = 2.049[\hbox {kg}], \\&\!\!\!c_h = 27.43[\hbox {Ns/m}], c_{\alpha }=0.036[\hbox {Ns}]\\&\!\!\!\rho = 1.225[\hbox {kg}/\hbox {m}^3], c_{l \alpha } = 6.28, c_{l \beta }=3.358, \\&\!\!\!c_{m \alpha }=(0.5+a) c_{l \alpha }\\&\!\!\!c_{m \beta }=-0.635, m_t = 12.387[\hbox {kg}],\\&\!\!\!I_{\alpha }=0.0517+m_wx_{\alpha }^2b^2 [\hbox {kg} \cdot \hbox {m}^2]\\&\!\!\!x_{\alpha } = [0.0873 - (b + ab)]/b\\&\!\!\!k_{\alpha }=2.82*(1-22.1\alpha \\&\qquad +1315.5 \alpha ^2 -8580 \alpha ^3 +17289.7 \alpha ^4)[\hbox {N} \cdot \hbox {m/rad}]\\&\!\!\!k_{h0}=2844.4[\hbox {N/m}].\\ \end{aligned}$$

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Lee, K.W., Singh, S.N. Generalized composite noncertainty-equivalence adaptive control of a prototypical wing section with torsional nonlinearity. Nonlinear Dyn 103, 2547–2561 (2021). https://doi.org/10.1007/s11071-021-06227-3

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