ℒ 1 adaptive control of a nonlinear aeroelastic system despite gust load

Abstract

The development of a control system for the suppression of aeroelastic vibration of a two-dimensional nonlinear wing-flap system based on the ℒ₁ adaptive control theory is the subject of this paper. The prototypical aeroelastic wing section model considered here includes structural nonlinearity, parameter uncertainties and gust loads. For the purpose of control, a single trailing-edge control surface is used. The uncontrolled aeroelastic model exhibits limit cycle oscillations beyond a critical free-stream velocity. An ℒ₁ adaptive law is developed for the suppression of aeroelastic oscillations using the pitch angle and pitch rate feedback. The control system includes a state predictor. The adaptation gain and the parameter of a filter are properly chosen to satisfy desirable performance bounds on the system trajectories. Simulation results are presented which show that the control system suppresses the oscillatory responses of the system in the presence of large parameter uncertainties and triangular, sinusoidal, and exponential gust loads.

Keywords

aeroelastic model flutter suppression gust load pitch angle and plunge displacement

1. Introduction

The new generation of highly flexible and combat aircraft with long slender wings are prone to undesirable aeroelastic instabilities. Aeroelastic phenomena such as flutter and limit cycle oscillations (LCOs) impose constraints on the flight vehicle performance. As such the analysis of aeroelastic dynamics and the design of active control systems for avoiding instability in aeroelastic systems continues to be an active research area (Mukhopadhyay, 2003; Dowell, 2004). Mukhopadhyay (2003) has provided a historical perspective on analysis and control of aeroelastic systems and a large number of references. The stability analysis using the μ method for uncertain aeroelastic models has been attempted (Lind and Brenner, 1999). Aeroelastic behavior of aircraft with freeplay nonlinearity has been considered (Wang et al., 2011a). For the benchmark active control technology (BACT) wind-tunnel model constructed at the NASA Langley Research Center, controllers for flutter suppression have been designed (Waszak, 2001). Researchers have designed flutter control systems using the classical, minmax, pacification methods (Mukhopadhyay, 2000) and the gain scheduling technique (Barker and Balas, 2000). A passivity-based robust controller for the BACT model has been designed (Kelkar and Joshi, 2000). A controller including a neural network has been proposed (Scott and Pado, 2000).

At the Texas A&M University, a two-degree-of-freedom plunging and pitching experimental apparatus has been constructed for examining the aeroelastic behavior and the performance of control systems (Ko et al., 1997). The apparatus includes a nonlinear spring along the pitch axis and it exhibits limit cycle oscillations beyond a critical speed (Sheta et al., 2002). For this model, linear and feedback linearizing control systems have been designed (Block and Strganac, 1998). A variety of adaptive control laws for suppressing oscillations have been developed (Ko et al., 1999; Xing and Singh, 2000; Bhoir and Singh, 2004; Platanitis and Strganac, 2004; Behal et al., 2006; Reddy et al., 2007). Based on the immersion and invariance principle, adaptive laws for the aeroelastic model have been also proposed (Lee and Singh, 2009, 2010). Marzocca et al. (2001) have examined the aeroelastic responses of a plunging–pitching model to gust and blast pressure signatures. Recently, Wang et al. (2011b) have designed an adaptive control law using leading- and trailing-edge flaps, and demonstrated effectiveness of the controller in attenuating the effect of gust loads given by Marzocca et al. (2001).

Recently, a new ℒ₁ adaptive control theory has been developed for the control of uncertain nonautonomous nonlinear systems (Cao and Hovakimyan, 2008; Hovakimyan and Cao, 2010). This design uses a state predictor similar to indirect adaptive systems; however, the control input is obtained by filtering the estimated control signal. This low-pass filter allows the low-frequency control signal to pass. The use of the filter for generating the actuating control signal provides several advantages over the traditional adaptive systems. The ℒ₁ adaptive controller can achieve fast parameter adaptation and yields quantifiable performance bounds. This design approach has been applied for the wing-rock control (Cao and Hovakimyan, 2006), missile control (Cao and Hovakimyan, 2009), and the other aerospace vehicles. Gregory et al. (2007) have considered the control of a flexible wing using four control surfaces. This control system includes an ℒ₁ adaptive controller wrapped around a nominal controller with a proportional and integral feedback structure. (Readers may refer to Hovakimyan and Cao (2010) for additional references.) Several control systems for the prototypical wing section of Block and Strganac (1998) have been designed in the past, but the application of ℒ₁ adaptive theory for this model has not been attempted. As such, it is of interest to develop an ℒ₁ adaptive flutter controller for the prototypical plunge–pitch 2-D aeroelastic system in the presence of gust loads.

In this paper, the design of a control system for the control of an aeroelastic system, in the presence of parameter uncertainties and gust loads, is considered. The design is based on the ℒ₁ adaptive control theory (Hovakimyan and Cao, 2010). This aeroelastic model has two degrees of freedom and governs the nonlinear plunge and pitch motion of the wing section (Ko et al., 1997). This type of model has been traditionally used for the theoretical as well as experimental analysis of two-dimensional (plunge and pitch) aeroelastic behavior. The model has pitch polynomial type structural nonlinearities and uses a single control surface for the purpose of control. It is assumed that all of the system parameters, except the sign of a single control input coefficient, are not known. The aeroelastic model has quasi-steady linear aerodynamic aerodynamics; however, it is noted that one can extend this approach to the case of nonlinear aerodynamics as well. This aeroelastic model exhibits limit cycle oscillations when the free-stream velocity exceeds a critical value. An ℒ₁ adaptive control system for the pitch angle trajectory control is derived. The control law is synthesized by feeding the pitch angle and its derivative. The control system includes a state predictor for the derivation of the update law. In the closed-loop system, the pitch angle and plunge displacement trajectories are stabilized. Unlike the traditional adaptive systems, the designed system has good robustness properties, and can achieve quantifiable performance bounds on the system trajectories and the control input by the choice of adaptation and feedback gains. Simulation results for the control of the oscillatory responses of the aeroelastic system are obtained. It is seen that the controller suppresses the oscillatory motion of the system, despite large parameter uncertainties and triangular, exponential and sinusoidal gust loads.

The organization of the paper is as follows. Section 2 presents the aeroelastic model. A control law including a state predictor is designed in Section 3. Finally, Section 4 presents the simulation results.

2. Aeroelastic model and control problem

Figure 1 shows the aeroelastic model. A laboratory model of this has been developed at the Texas A&M University for performing experiments. The model has two degrees of freedom. The differential equations describing the pitch angle (α) and the plunge displacement h of the wing including the gust load are governed by second-order differential equations (Ko et al., 1997; Platanitis and Strganac, 2004; Wang et al., 2011b). These are

[I_{α} m_{w} x_{α} b m_{w} x_{α} b m_{t}] [\overset{··}{α} \overset{··}{h}] + [c_{α} 00 c_{h}] [\overset{\cdot}{α} \overset{\cdot}{h}]

+ [k_{α} (α) 00 k_{h 0}] [α h] = [M + M_{g} - L - L_{g}]

(1)

where m_w is the mass of the wing, m_t is the total mass, b is the semichord of the wing, I_α is the moment of inertia, x_α is the nondimensionalized distance of the center of mass from the elastic axis, c_α and c_h are the pitch and plunge damping coefficients, respectively, and M and L are the aerodynamic moment and lift. It is assumed that the quasi-steady aerodynamic force and moment are of the form

L = ρ_{a} U^{2} b c_{l_{α}} s_{p} [α + \overset{\cdot}{h} U + (1/2 - a) b \overset{\cdot}{α} U] + ρ_{a} U^{2} b c_{l_{β}} s_{p} β M = ρ_{a} U^{2} b^{2} c_{m_{α} - eff} s_{p} [α + \overset{\cdot}{h} U + (1/2 - a) b \overset{\cdot}{α} U] + ρ_{a} U^{2} b^{2} c_{m_{β} - eff} s_{p} β

(2)

c_{m_{α} - eff} = (1/2 + a) C_{l_{α}} + 2 C_{m_{α}}, c_{m_{β} - eff} = (1/2 + a) C_{l_{β}} + 2 C_{m_{β}}

where a is the nondimensionalized distance from the midchord to the elastic axis, s_p is the span, c_{l
_α} and c_{m
_α} are the lift and moment coefficients per angle of attack, and c_l_{_β} and c_{m
_β} are lift and moment coefficients per control surface deflection β.

The aerodynamic force and moment due to the wind gust is modelled as (Marzocca et al., 2001)

L_{g} = ρ_{a} U^{2} b s_{p} c_{l_{α}} w_{G} (τ) / U = ρ_{a} Ub s_{p} c_{l_{α}} w_{G} (τ) M_{g} = (0.5 - a) {bL}_{g}

(3)

where w_G(τ) denotes the disturbance velocity and τ is a dimensionless variable defined as τ = Ut/b. Later different waveforms of disturbance velocity given by Marzocca et al. (2001) are used for simulation.

The nonlinear function k_α(α) has a polynomial form of second degree and is given by

α k_{α} (α) = α (k_{α_{0}} + k_{α_{1}} α + k_{α_{2}} α^{2}) \equiv k_{α_{0}} α + k_{n_{α}}

Define the matrix

M_{s} = [I_{α} m_{w} x_{α} b m_{w} x_{α} b m_{t}]^{- 1}

(4)

Premultiplying equation (1) by M_s and using equation (2) gives

\overset{··}{a}

and

\overset{··}{h}

of the form

\overset{··}{α} = a_{11} α + a_{12} \overset{\cdot}{α} + a_{13} (α) + a_{14} h + a_{15} \overset{\cdot}{h} + b_{1} u + d_{g 1}

\overset{··}{h} = a_{21} α + a_{22} \overset{\cdot}{α} + a_{23} (α) + a_{24} h + a_{25} \overset{\cdot}{h} + b_{2} u + d_{g 2}

(5)

where u = β, the nonlinear vector function is [a₁₃(α), a₂₃(α)]^T = M_s[k_nα, 0]^T, [b₁, b₂]^T = ρU²bs_pM_s[bc_{m_β−eff}, −c_{l
_β}]^T, disturbance input [d_g1, d_g2]^T = M_s[M_g, −L_g]^T. The remaining parameters a_ik are defined in equation (5). The model equation (1) with d_gi(t) = 0 exhibits LCOs beyond a critical free-stream speed.

Let α be the controlled output variable. We are interested in the design of an ℒ₁ adaptive control system for the stabilization of the pitch angle and plunge displacement trajectories using α and $\overset{\cdot}{α}$ feedback.

3. ℒ₁ adaptive law

In this section, an adaptive control system is designed for the suppression of the plunge and pitch oscillatory motion of the aeroelastic model using the trailing-edge control surface. The design of the control system is based on the ℒ₁ adaptive control theory (Hovakimyan and Cao, 2010). Here the derivation of the controller for the aeroelastic model is briefly presented. (The proof of stability can be completed by following the steps given of Hovakimyan and Cao (2010).)

First of all, for the derivation of the control system, a state transformation is introduced. Define new variables

z_{1} = h - α b_{2} b_{1} z_{2} = {\overset{\cdot}{z}}_{1} = \overset{\cdot}{h} - \overset{\cdot}{α} b_{2} b_{1}

(6)

Also define

x = (x_{1}, x_{2})^{T} = (α, \overset{\cdot}{α})^{T} \in R^{2}

and z = (z₁, z₂)^T ∈ R². For obtaining a state variable representation of equation (5) using new coordinates, one eliminates the variables h and

\overset{\cdot}{h}

to obtain

\overset{··}{α} = a_{31} α + a_{32} \overset{\cdot}{α} + a_{33} (α) + a_{34} z_{1} + a_{35} z_{2} + b_{1} u + d_{g 1} {\overset{··}{z}}_{1} = a_{41} α + a_{42} \overset{\cdot}{α} + a_{43} (α) + a_{44} z_{1} + a_{45} z_{2} + d_{g 3}

(7)

where the constant parameters a_ik and the nonlinear functions a_jk(α) are easily obtained using equations (5) and (7). The disturbance input for the plunge motion is

d_{g 3} = d_{g 2} - b_{2} b_{1}^{- 1} d_{g 1}

and a₄₃(α) = a₂₃ − (b₂/b₁)a₁₃.

Then the equations of motion can be expressed in a compact form as

\overset{\cdot}{x} = (01 a_{31} a_{32}) x + (0 b_{1}) (u + f (x, z, t)) ≐ A_{x} x + b_{x} (u + f (x, z, t)), x (0) = x_{0}

(8)

\overset{\cdot}{z} = (01 a_{44} a_{45}) z + (01) g (x, t) ≐ A_{z} z + b_{z} g (x, t), z (0) = z_{0}

(9)

where t denotes the dependence of the functions on the exogenous signals d_gi(t) and

g = a_{41} α + a_{42} \overset{\cdot}{α} + a_{43} (α) + d_{g 3}

Assumption 1:

The parameters a_ik and the nonlinear functions f(x, z, t) and g(x, t) are not known.

Let b_m be a nominal value of the uncertain parameter b₁. Define b₁ = b_1mw, where b_1m is a known nominal value and w is an unknown parameter satisfying

0 < w_{l} \leq w \leq w_{u}

(10)

where w_l, w_u are lower and upper bounds for the uncertain parameter w.

Consider a Hurwitz matrix given by

A_{m} = (01 a_{m 1} a_{m 2})

where a_mi are design parameters. Then adding and subtracting A_mx in the x-dynamics of equation (8) and collecting terms gives

\overset{\cdot}{x} = A_{m} x + b_{m} (wu + f_{n} (x, z, t))

(11)

where b_m = [0, b_1m]^T and f_n(x, z, t) is a new unknown nonlinear function. We shall be interested in the solution of equation (11) for the initial conditions ‖x₀‖_∞ ≤ ρ₀, where ρ₀ is a sufficiently large number. Let X = [x^T, z^T]^T and f_n(X, t) = f_n(x, z, t). For the aeroelastic model with polynomial nonlinearity, the following assumptions are made.

Assumption 2:

There exists B > 0 such that f_n(0, t) ≤ B for all t > 0.

Assumption 3:

For ‖X‖_∞ ≤ δ, where δ > 0 is an arbitrary number, the partial derivatives of f_n(X, t) are bounded as

‖ \partial f_{n} (X, t) \partial X ‖_{1} \leq d_{f_{nx}} (δ), | \partial f_{n} (X, t) \partial t | \leq d_{f_{nt}} (δ)

Assumption 4:

The flow velocity and the parameter a are such that A_z is Hurwitz.

In view of Assumption 4, for bounded x(t) and d_gi(t), the trajectory z(t) of equation (9) is bounded; that is, the z-dynamics are bounded input bounded state stable with respect to z₀, x, and d_g3. Let $Φ_{z} (t) = e^{A_{m} t}$ be the transition matrix associated with A_m. Then solving equation (9) gives

z (t) = Φ_{z} (t) z (0) + \int_{0}^{t} Φ_{z} (t, τ) b_{z} g (x (τ), τ) d τ

(12)

which can be bounded as

‖ z_{t} ‖_{L_{\infty}} \leq L_{z 1} ‖ x_{t} ‖_{L_{\infty}} + L_{z 2}

(13)

where L_z1 and L_z2 are positive numbers; and for any vector function y(τ), the truncated function y_t is defined as

y_{t} (τ) = {y (τ) if τ \leq t 0 if τ > t

In view of equation (12), the function norm ‖z_t‖_{ℒ_∞} can be bounded as

‖ z_{t} ‖_{L_{\infty}} \leq ‖ Φ_{z} (t) ‖_{L_{\infty}} ‖ z (0) ‖_{\infty} + ‖ Φ_{z} (t) b_{z} ‖_{L_{1}} ‖ g (x_{t}, t) ‖_{L_{\infty}}

(14)

The function g in equation (12) can be written as

g = [g_{α} (α), a_{42}] x + d_{g 3} ≐ v_{g}^{T} (α) x + d_{g 3}

(15)

where g_α is a polynomial function of α. In view of equation (13), using equation (14) gives

L_{z 1} = ‖ Φ_{z} (t) b_{z} ‖_{L_{1}} v_{gm}

L_{z 2} = ‖ Φ_{z} (t) b_{z} ‖_{L_{1}} ‖ d_{g 3} ‖_{L_{\infty}} + ‖ Φ_{z} (t) ‖_{L_{\infty}} ‖ z (0) ‖_{\infty}

(16)

where

v_{gm} = \max {‖ v_{g} (α (t)) ‖_{L_{1}}}

The matrix A_m has stable eigenvalues. As such there exists a ρ_in > 0 such that ‖x_in‖_{ℒ_∞} ≤ ρ_in, where x_in(s) = (sI − A_m)⁻¹x₀. For the design of the controller, consider a strictly proper transfer function D(s) such that

C (s) = kwD (s) 1 + kwD (s)

(17)

is a strictly proper stable with C(0) = 1 for all w_l ≤ w ≤ w_u, where k > 0. For a given δ, define L_δ as

L_{δ} = \bar{δ} (δ) δ^{- 1} d_{f_{nx}} (\bar{δ} (δ)), \bar{δ} (δ) = max {δ + \bar{γ_{1}}, L_{1} + (δ + \bar{γ_{1}}) + L_{2}}

(18)

where

\bar{γ_{1}}

is an arbitrary positive number. For the design of control law, one must choose k and D(s) to ensure that for a given ρ₀, there exists ρ_r > ρ_in, such that the ℒ₁-condition

‖ G (s) ‖_{L_{1}} < ρ_{r} - ρ_{in} L_{ρ_{r}} ρ_{r} + B

(19)

is satisfied, where G(s) = H(s)(1 − C(s)), H(s) = (sI − A_m)⁻¹b_m, and L_{ρ_r} is defined according to equation (15) (Hovakimyan and Cao, 2010).

Let $ρ = ρ_{r} + {\bar{γ}}_{1}$ . It has been shown that for ‖x_τ‖_{ℒ_∞} ≤ ρ, and bounded u and $\overset{\cdot}{u}$ , the x-dynamics can be expressed as

\overset{\cdot}{x} = A_{m} x + b_{m} [wu + θ (t) ‖ x_{t} ‖_{L_{\infty}} + σ (t)]

(20)

for some unknown functions θ(t) and σ(t), where t ∈ [0, τ]. The derivation of the adaptation law is based on a state predictor given by

\overset{\cdot}{\overset{\land}{x}} (t) = A_{m} \overset{\land}{x} + b_{m} [\overset{\land}{w} u + \overset{\land}{θ} (t) ‖ x_{t} ‖_{L_{\infty}} + \overset{\land}{σ} (t)], \overset{\land}{x} (0) = x (0)

(21)

where

\overset{\land}{θ}

\overset{\land}{w}

and

\overset{\land}{σ}

denote the estimates of θ, w and σ, respectively. The prediction error

\tilde{x} = \overset{\land}{x} - x

satisfies

\overset{\cdot}{\tilde{x}} = A_{m} \tilde{x} + b_{m} [\tilde{w} u + \tilde{θ} (t) ‖ x_{t} ‖_{L_{\infty}} + \tilde{σ} (t)]

(22)

where

\tilde{w} = \overset{\land}{w} - w

\tilde{θ} = \overset{\land}{θ} - θ

and

\tilde{σ} = \overset{\land}{σ} - σ

are the parameter errors. Now consider a Lyapunov function

V (\tilde{x}, \tilde{w}, \tilde{θ}, \tilde{σ}) = {\tilde{x}}^{T} P \tilde{x} + Γ^{- 1} ({\tilde{w}}^{2} + {\tilde{θ}}^{2} + \tilde{σ})

(23)

where the positive-definite symmetric matrix P (denoted as P > 0) satisfies the Lyapunov equation

A_{m}^{T} P + P A_{m} = - Q

(24)

for a chosen matrix Q > 0.

Then for cancelling unknown terms in the derivative of V along the solution of equation (22), the adaptation law is chosen as

\overset{\cdot}{\overset{\land}{w}} = Proj (\overset{\land}{w}, - Γ {\tilde{x}}^{T} P b_{m} u), \overset{\land}{w} (0) = {\overset{\land}{w}}_{0} \overset{\cdot}{\overset{\land}{θ}} = Proj (\overset{\land}{θ}, - Γ {\tilde{x}}^{T} P b_{m} ‖ x_{t} ‖_{L_{\infty}}), \overset{\land}{θ} (0) = {\overset{\land}{θ}}_{0} \overset{\cdot}{\overset{\land}{σ}} = Proj (\overset{\land}{σ}, - Γ {\tilde{x}}^{T} P b_{m}), \overset{\land}{σ} (0) = {\overset{\land}{σ}}_{0}

(25)

where the projection operator is used so that

\overset{\land}{w}

∈ [w_l, w_u],

\overset{\land}{θ} \in [- L_{ρ}, L_{ρ}]

, and

| \overset{\land}{σ} | \leq Δ

, where

ρ = ρ_{r} + {\bar{γ}}_{1}

and Δ is a computable bound (Hovakimyan and Cao, 2010).

In view of equation (25), one finds that the adaptation law is a function of $\tilde{x} = \overset{\land}{x} - x$ , where $\overset{\land}{x}$ is obtained from the state predictor equation (21). Apparently, the state predictor plays a key role in the synthesis of the adaptation law. It is also pointed out that the state predictor (equation (21)) and control law (equation (26)) use feedback of the signal ‖x_t‖_{ℒ_∞} and, therefore, $x = (α, \overset{\cdot}{α})^{T}$ must be available for synthesis.

The estimated parameters are used to compute the control law of the form

u = - kD (s) [\overset{\land}{w} u + \overset{\land}{θ} ‖ x_{t} ‖_{L_{\infty}} + \overset{\land}{σ} (t)]

(26)

For the evaluation of performance bounds, consider a reference system given by

{\overset{\cdot}{x}}_{ref} = A_{m} x_{ref} + b_{m} [{wu}_{ref} + f_{n} (x_{ref}, z, t)], x_{ref} (0) = x_{0} u_{ref} = - w^{- 1} C (s) f_{n} (x_{ref}, z, t)

(27)

It is seen that ideal control signal u_ref is obtained by passing the inverse control law signal through the low pass filter C(s). It has been shown by Hovakimyan and Cao (2010) that in the closed-loop system, the following bounds on the signals hold:

‖ x ‖_{L_{\infty}} \leq ρ

(28)

‖ \tilde{x} ‖_{L_{\infty}} \leq γ_{0}

(29)

‖ x - x_{ref} ‖_{L_{\infty}} \leq γ_{1}

(30)

‖ u - u_{ref} ‖_{L_{\infty}} \leq γ_{2}

(31)

where

γ_{1} = ‖ C (s) ‖_{L_{1}} 1 - ‖ G (s) ‖_{L_{1}} L_{ρ_{r}} γ_{0} + d_{0}

and γ₂ = ‖w⁻¹C(s)‖_ℒ₁ L_{ρ_r} γ₁ + ‖H₁(s)‖_ℒ₁ γ₀ with d₀ and γ₀ arbitrary small positive constants. The transfer function

H_{1} (s) = C (s) ({wc}_{0}^{T} H (s))^{- 1} c_{0}^{T}

is BIBO (bounded input bounded output) proper stable by the choice of vector c₀. It is interesting to note that the performance bounds γ₀, γ₁ and γ₂ can be made arbitrarily small by increasing the adaptation gain Γ. This feature is not present in the aeroelastic adaptive systems published in literature. Although, following Hovakimyan and Cao (2010), Γ can be computed to satisfy the prescribed performance bounds, its value is selected by observing the simulated responses in the next section. Although the ℒ₁ adaptive control theory allows use of large Γ, from the viewpoint of implementation, small values are preferred, especially in the presence of sensor noise. It is noted that the state prediction error dynamics (equation (22)) used for the derivation of the adaptation law are independent of the control input. It has been found by simulation in the next section that even with saturating control, larger Γ improves the tracking performance.

Here for synthesis, a smooth projection algorithm is used for each parameter $\overset{\land}{w}, \overset{\land}{θ}$ and $\overset{\land}{σ}$ . Suppose that a parameter p belongs to a set Ω_p defined by

Ω_{p} = {p : | p - μ | < p_{m}}

where μ and p_m are given constants. Consider a function ψ of p given by

ψ (p) = 2 ε [(p - μ p_{m})^{2} - 1 + ε]

where 0 < ε < 1. Then a smooth projection (termed Proj) is defined as (Pomet and Praly, 1992)

Proj (p, y) = y if ψ (p) \leq 0 y if ψ (p) \geq 0 and \partial ψ \partial p (p) y \leq 0 y - ψ (p) \partial ψ \partial p (p) y ‖ \partial ψ \partial p (p) ‖^{2} \partial ψ \partial p (p)^{T} if not

where

\partial ψ \partial p (p) = 4 ε p - μ p_{m}

4. Simulation results

In this section, numerical results for the model given by (Platanitis and Strganac, 2004) are obtained. The system parameters are given in the appendix. The chosen initial conditions h(0) = 0 (m), α(0) = 5.729°, $\overset{\cdot}{α} (0) = \overset{\cdot}{h} (0) = 0$ are identical to those used by Wang et al. (2011b). The initial state vector of the predictor is $\overset{\land}{x} (0) = (5 . 729^{\circ}, 0)^{T}$ . The norm of the truncated state vector defined as ‖x_t‖_{ℒ_∞} = max{|x₁(τ)|, |x₂(τ)|}, τ ∈ [0, t]} is used for simulation. For the free-stream velocity U ∈ [8, 13.28] (m/s), the parameter b₁ lies in the interval [−48.78, −17.7] . Here the nominal value of b₁ is arbitrarily selected as b_1m = −43.9086. Later responses will be obtained for U = 8 and 13.28 (m/s) with a = −0.6719. The conservative lower and upper bounds for the uncertain parameters w, θ, and σ are selected as (w_l, w_u) = (0.1, 20), (θ_l, θ_u) = (−20, 20), and (σ_l, σ_u) = (−20, 20). The design parameters a_m1 and a_m2 are a_m1 = −15, a_m2 = −8. The initial estimates of the unknown parameter vector w, θ, and σ are arbitrarily set as $\overset{\land}{w} (0) = 0, \overset{\land}{θ} (0) = 0, \overset{\land}{σ} (0) = 0$ . This is rather a poor choice of parameter estimates but is made to examine the robustness of the controller. Selecting Q = diag{100, 100}, equation (24) gives

P = (126.6667 3.3333 3.3333 6.6667)

The transfer function of the filter is

D (s) = 1 s

and the value of the gain is k = 100.

The poles of the linearized system for U = 8 m/s and a = −0.6719 are (−0.5065 ± 10.2038i, −0.8934 ± 14.5665i), and the zeros of the transfer function relating α and u are −1.6403 ± 16.8671i. The poles of the linearized system for U = 13.28 m/s and a = −0.6719 are (1.3071 ± 12.9398i, −2.8512 ± 12.4120i), and the zeros of the transfer function relating α and u are −1.8089 ± 16.8499i. Thus the open-loop system is stable for U = 8 m/s, and unstable for U = 13.28 m/s, but the linearized system has minimum phase transfer function relating α and u for these free-stream velocities.

The velocity distributions of gust given by Marzocca et al. (2001) and Wang et al. (2011b) are used for simulation. The selected velocity distributions (w_G(τ) ) for (a) triangular gust, (b) exponential (graded) gust, (c) sinusoidal gust, and (d) sum of sinusoidal and persistent random gust are shown in Figure 2. The triangular and sinusoidal gusts have finite duration. The responses of the open-loop system (u = 0) for the velocity distributions of Figure 2 are plotted in Figure 3. For triangular w_G(τ), the pitch angle and plunge displacement converge to zero for U = 8 m/s (Figure 3(a) and (b)). However, for U = 13.28 m/s, the system exhibits LCOs after initial transience in all cases (Figure 3(c)–(f)), except for the random velocity distribution. For the random w_G, the oscillatory motion is nonperiodic (Figure 3(g) and (h)).

Now the closed-loop responses for the model equation (1) including the control law equation (26), adaptation law equation (25) and gust load are obtained. Closed-loop responses are obtained using triangular gusts with intensity w₀ = 0.5 and 2.0. For a realistic simulation, similar to Wang et al. (2011b), control surface deflections are allowed to saturate at 15°.

Case A. Adaptive control: triangular gust, U = 8 m/s

For examining the performance of the controller, the closed-loop system including the triangular gust for U = 8 m/s and a = −0.6719 is simulated. The velocity distribution shown in Figure 2(a) is

w_{G} (τ) = 2 w_{0} τ τ_{G} (H (τ) - H (τ - τ_{G} 2)) + 2 w_{0} (τ τ_{G} - 1) \times (H (τ - τ_{G}) - H (τ - τ_{G} 2))

where H(·) denotes the unit step function, w₀ = 0.5, τ_G = Ut_G/b and t_G = 0.5 s. The selected responses are shown in Figure 4. It is observed that the plunge and pitch angle trajectories converge to zero. Compared with the open-loop response in Figure 3(a) and (b), it is seen that the closed-loop system has a faster response time. The convergence time for the open-loop system is about 7 seconds, but for the closed-loop system, it is about 3 seconds. The control input saturates in the transient period and estimated parameters converge to certain constant values. The values of the bounding parameters L_zi for the z-dynamics in equation (13) are computed numerically using the transition matrix according to equations (14)–(16). The steady-state value of the signal in Figure 4(e) provides the bounding parameter. For the chosen triangular velocity distribution, it is seen that L_z1 = 34.9787 and L_z2 = 0.5329.

Case B. Adaptive control: triangular gust, U = 13.28 m/s

Now the closed-loop system with triangular gust of larger intensity w₀ = 2 and U = 13.28 m/s is simulated. Selected responses are shown in Figure 5. It is observed that in spite of the stronger gust, the pitch angle converges to zero in 2 seconds, but the plunge displacement is regulated to zero in about 2.8 seconds. It is pointed out that the control input gain b₁ increases with the free-stream velocity; and, therefore, control effectiveness of the flap improves.

Case C. Adaptive control: sinusoidal gust, U = 13.28 m/s

The simulation is done using the sinusoidal w_G(τ) of the form

w_{G} = w_{0} \sin (6 π τ / τ_{G}) (H (τ) - H (τ - τ_{G}))

where τ_G = Ut_G/b, t_G = 1, w₀ = 0.07 (also given in Figure 2(c)). The responses are shown in Figure 6. We observe the suppression of the oscillations in α (in 2 seconds) and in h (in less than 3 seconds). The estimated parameters remain bounded and converge to constant values. Again the control input saturates in the transient period.

Case D. Adaptive control: exponential gust, U = 13.28 m/s

For simulation, exponential (graded) gust given by

w_{G} (τ) = H (τ) w_{0} (1 - e^{- 0.75 τ 3})

(shown in Figure 2(b)) is used, where w₀ = 0.07. The free-stream velocity is U = 13.28 m/s and a = −0.6719. Selected responses are shown in Figure 7. Again the oscillations in the system are suppressed. The convergence time for the plunge and pitch angle trajectories is about 2 seconds.

Simulation is also done for the exponential gust of Figure 7, but a larger value of w₀ = 1 is assumed. Remaining parameters of Figure 7 are retained. The selected responses are shown in Figure 8. It is seen that the oscillations in α and h are completely suppressed in about 2 seconds. For this case the clamped value of the control input is 18° instead of 15°. The pitch angle converges to zero, but the plunge displacement converges to the equilibrium value of −0.0141 (m) in the steady state.

Case E. Adaptive control: combined sinusoidal and persistent random gust, U = 13.28 m/s

Now the effect of the persistent random gust combined with the sinusoidal gust shown in Figure 2(d) is examined. The net velocity

w_{G} (τ) = w_{0} \sin (6 π τ / τ_{G}) (H (τ) - H (τ - τ_{G})) + d_{n} H (τ)

where τ_G = Ut_G/b, t_G = 1, w₀ = 0.07. The random velocity distribution d_nH(τ) is generated by passing a white noise with unit variance through a transfer function F_d(s) = 10⁻⁵/(s + 5). Simulated responses are shown in the Figure 9. It is seen that the pitch angle converges to zero in about 2 seconds. The plunge displacement trajectory amplitude is significantly attenuated by the controller, but tiny fluctuations (of the peak value of about 0.002 m) around h = 0 persist in the steady state. This should not be surprising because the controller has been designed for the regulation of only the pitch angle.

Extensive simulation has been performed for various kinds of wind gust. In each case, it has been observed that the ℒ₁ adaptive control law accomplishes control of the oscillatory motion of the aeroelastic system, in spite of the uncertainties in the parameters and wind gusts of different shapes.

Now it is appropriate to compare the ℒ₁ adaptive law using a single trailing-edge control surface derived here and the direct adaptive control law designed using trailing- and leading-edge (two) control surfaces by Wang et al. (2011b). In their paper, unknown functions are approximated by multi-layer neural networks. As such a large number of parameters (weights) (82 parameters) are needed for obtaining good approximation of the unknown functions. In this ℒ₁ control system, only three parameters are estimated; and, therefore, the adaptation law is simple. Of course, a state predictor of dimension two is needed here. It is pointed out that similar to Wang et al. (2011b), control saturation has been introduced here to limit the flap deflection. It is seen that the convergence characteristics of the closed-loop system of this paper are somewhat similar to those of Wang et al. (2011b). Figure 8 obtained for U = 13.28 (m/s) and exponential w_G(τ) with a larger value w₀ = 1 shows that the ℒ₁ control system suppresses the oscillatory motion in two seconds; but it is about one second for the controller of Wang et al. (2011b). The control saturation level in Figure 8 is 18°, which is slightly larger compared with 15° used by Wang et al. (2011b). Also similar to Wang et al. (2011b), h converges to a nonzero value, but this terminal magnitude (0.0141 (m)) of h is larger than the value 0.008 m of Wang et al. (2011b). Of course, the ℒ₁ adaptive law utilizing a single flap cannot suppress the oscillations in h in the presence of persistent (long lasting) sinusoidal gusts. For the model with two control surfaces used by Wang et al. (2011b), zero dynamics do not exist and it is possible to control both α and h responses for all U and a.

5. Conclusions

In this paper, a controller for the control of a nonlinear uncertain two-dimensional aeroelastic system based on the ℒ₁ adaptive control theory was developed. The plunge–pitch dynamics of the model included uncertain parameters as well as gust load. A single trailing-edge flap was used for control. The ℒ₁ adaptive law was developed for the pitch angle control. The control system included a state predictor and filtered estimated control signal was synthesized using pitch angle and pitch rate feedback. In the closed-loop system, plunge displacement and pitch angle oscillations were suppressed. Unlike the traditional adaptive systems designed for aeroelastic models, the derived controller provides explicit performance bounds on state trajectory and control input. There exists flexibility in the choice of controller parameters. Enhanced performance is possible by choosing large adaptation gain. Simulation results for a variety of gust loads was obtained. It was found that the controller accomplished suppression of the plunge and pitch angle trajectory oscillations, despite uncertainties in the model parameters and triangular, sinusoidal and exponential gust loads. However, in the presence of random gust, tiny fluctuations around zero exist in the plunge response.

Nomenclature

A_b, A_x, A_z, b_x, b_z

System matrices

Nondimensionalized distance from the midchord to the elastic axis

Semichord of the wing

c_h, c_α

Plunge and pitch structural damping coefficient

C(s), D(s), G(s), H(s)

Transfer functions

Filter parameter

Plunge displacement

I _α

Moment of inertia of the wing about the elastic axis

M, M_g, L, L_g

Aerodynamic and disturbance moments and lift

k_h, k_{α_i}

Plunge and pitch structural spring constant

m _t

Mass of the plunge–pitch system

m _w

Mass of the wing

s _p

Span

U, u = β

Free-stream velocity, flap deflection

w, θ , σ

Unknown parameters

\overset{\land}{w}, \overset{\land}{θ}

\overset{\land}{σ}

Parameter estimates

w_G(τ)

Disturbance velocity

x, z, \overset{\land}{x}

State vectors

x _α

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

ρ_a

Density of air

α, Γ

Pitch angle, adaptation gain

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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ℒ 1 adaptive control of a nonlinear aeroelastic system despite gust load

Abstract

Keywords

1. Introduction

2. Aeroelastic model and control problem

3. ℒ1 adaptive law

Assumption 1:

Assumption 2:

Assumption 3:

Assumption 4:

4. Simulation results

Case A. Adaptive control: triangular gust, U = 8 m/s

Case B. Adaptive control: triangular gust, U = 13.28 m/s

Case C. Adaptive control: sinusoidal gust, U = 13.28 m/s

Case D. Adaptive control: exponential gust, U = 13.28 m/s

Case E. Adaptive control: combined sinusoidal and persistent random gust, U = 13.28 m/s

5. Conclusions

Nomenclature

Footnotes

Funding

References

3. ℒ₁ adaptive law