Comparative Control Strategies of an Underactuated Aircraft Wing Model

Flexible aircraft structures offer improved efficiency, but their stability margins can be compromised. This instability results from the interaction of aerodynamic, inertial, and elastic forces. Nonlinearities in the wing structure, such as coupled bending, torsion, and elasticity, can lead to undesirable outcomes such as flutter and limit cycle oscillations [1-3]. Various models, including plates or shells with complex cross sections or an equivalent system with beams and flexible elements, are feasible [4]. This paper discusses the development of advanced nonlinear control strategies for aeroelastic systems. It describes the construction of a basic model that can be adapted using online feedback estimation techniques. The model aims to accurately capture the dynamic characteristics of the system while remaining simple. The aeroservoelastic (ASE) model is based on aerodynamic and structural principles for lifting surfaces, with an emphasis on simplifying assumptions of causality, linearity, and small perturbations. Complex aeroelastic models are not scalable for repeated online computations, so a linearized baseline model is sufficient. An ASE plant model utilizes a combination of steady, quasi-steady, or unsteady aerodynamics and small fluctuations in structural dynamics and can be represented in either integral or operational linear form. The basic plant structural model predicts aerodynamic lift and pitch moments via simple Newtonian or Lagrange techniques. It is important to either find a closed-form solution of the unstable aerodynamics or perform an inversion of certain fundamental integral equations. However, accurate aerodynamic prediction may require the solution of nonlinear partial differential equations [4]. Therefore, this study focuses on modeling an aeroelastic wing based on an equivalent system with three degrees of freedom: pitch, aileron, and control surface coordinates. Using quasi-steady aerodynamics, the dynamics related to the control surface coordinate can be eliminated, and thus two DOF reduced wing models are developed.

A control system is important to attenuate wing vibrations. Vibration damping control can be classified into passive and active control strategies. Passive control schemes focus on modifications in wing geometry and aerodynamic characteristics, while active control requires actuators and sensors to regulate the wing oscillations. This research emphasizes the active control category. Miscellaneous control strategies have been used for attenuating the wing oscillations, such as adaptive decoupled fuzzy sliding control [5], control based on the tensor product model [6], output feedback control [7-9], the state-dependent Riccati equation method [10], linear quadratic Gaussian LQG [11], terminal sliding mode control [12, 13], synchronization theory [14], and adaptive neural control [15, 16]. In general, in most aeroservoelastic analyses and applications of optimal control theory, the unsteady generalized forces are linearized, combined with controller dynamics and gust spectral models, and converted into linear state-space equations [17]. However, the linear control theory (e.g., PID control [18], pole placement [19], LGR/G [11, 20], and robust control [21]) might be insufficient to control the wing dynamics under nonlinear conditions such as free play, time delay, and bifurcation due to nonlinear structural stiffness [22]. Therefore, Kurdila et al. [23] provided a review of nonlinear control strategies for damping limit cycle oscillations of a nonlinear aeroelastic model. They focused on the use of partial feedback linearization control and MRAC, showing their characteristics and shortcomings. They concluded that the nonlinear MRAC seems to be more robust than the adaptive feedback linearization control under high flow velocities. However, other nonlinear control methods have been reported in the literature, such as gain scheduling scheme [24], inversion dynamic control [25], incremental inversion control [26], and recurrent neural control for system identification [27].

However, most of the above control techniques may not focus on the underactuation behavior of aircraft wings excepting the work of Ko and colleagues [23, 28, 29]. The number of control surfaces can affect the situation, leading to three potential actuation scenarios: an underactuated system, a fully actuated system, and an overactuated system [30]. This investigation delves into a challenging, underactuated case. Underactuated systems, which are mechanical systems with fewer control inputs than degrees of freedom, are found in a variety of applications, including flexible mobile and locomotive systems, robotics, aircrafts, and marines. The phenomenon of underactuation in systems can be because of actuator failure, engineering and robotic control design intentionally, and the internal dynamics of the system [31]. Therefore, this study develops and designs four control strategies for flexible wing structures to suppress aeroelastic instabilities: PFLC, EC, FC, and SCFC. Each method presents certain benefits concerning certain parts of the system’s dynamic behavior; thus, they are examined in relation to their performance in oscillation control and system management of underactuation. The details on each control scheme are presented below. To our knowledge, literature reported using PFLC for regulation of oscillations of underactuated wings [28, 29], while the EC, FC, and the SCFC have been used in robotic systems extensively.

(1) The PFLC. An efficient approach eliminates the nonlinearities through the design of a suitable control law feedback loop. This method divides the degrees of freedom (DOFs) into active and passive. Active DOFs are controlled while passive joints may remain internally stable as regulated by the internal dynamics. This is very important where the number of inputs is less than that of the degrees of freedom. Having this controller for the design of the active DOFs control assures that the active DOFs are controlled with stability and accuracy. This technique has been used in regulating oscillations of most underactuated wing models [23, 28, 29] and also in several other robotic systems such as inverted pendulums, floating base and flexible base robots, and mobile robots [31-34]. The control of many systems at present is associated with the theory of feedback linearization, which is exact in most cases. Computed torque control in robotics is developed based on the concept of input-output linearization. Nonetheless, there are limitations when using such approaches in practice related to singularities or non-linear mistake cancellation, and when the system is not in the minimal phase and has zero dynamics, which can lead to unstable control of the closed loops. The maximal linearization issue concerns the notion of flatness and the ability to construct an output with a relatively high degree.

(2) The EC. A universal stabilizing strategy adjusts the energy of underactuated manipulators using state feedback in order to reach closed-loop systems with desired stable states. It includes two major approaches: the controlled Lagrange method, which generates certain kinetic or potential energy from the Lie group, and the IDA-PBC method, which extends the controlled Lagrange methodology. The controlled Lagrange method is considered a complicated procedure where the simultaneous reconstruction of kinetic and potential energy is not undertaken. On the contrary, IDA-PBC is capable of reconstructing both energies with ease and has a relatively less sophisticated construction mechanism with its control law integrating energy reconstruction and input modulation. Over the recent years, the developments in the IDA-PBC methodology have offered better theoretical approaches as well as extended usability [35-38]. Despite this improvement, the two methods still consider a feedback control law that involves solving a series of partial differential equations to achieve it, which is problematic for handling systems with such simple nonlinear features and limits their use as well. In general, it encompasses the management of mechanical energy, the control of active variables, and the limitation of passive ones. For this, no more complicated mathematical formulation is needed than the one envisaging potential and kinetic energy for the concrete dynamical system at hand. Furthermore, this is not a case of partial feedback linearization since the latter only attempts to control the output in the presence of the active control surfaces. The simplicity of the controller being determined from the mechanical energy of the system is one of the benefits of energy-based methods. The controller is constructed from the energy storage function and guarantees the asymptotic stability of the system’s equilibrium. Nevertheless, the non-actuated coordinate, which is not controlled directly by the system, will eventually tend to zero [39, 40].

(3) The FC. It is an improved method in system and control theory focusing on the control systems that are non-linear and where the control system of interest is not defined. It is based on the idea of “flat outputs” in the system; outputs depend on inputs, and all states and inputs are observable without integration. This is useful because the behavior of the system in question is determined by its outputs, which allows for precision in trajectory generation. When trajectory generation is required to be very accurate differential systems are used in order to connect the trajectories to the appropriate inputs [41]. In the study [42], a series of flat outputs are used to model the crane system dynamics with respect to certain constraints in fast positioning of the trolley and reduced swing angle. The ideal set of flat output parameters are obtained, and a tracking controller is designed to correct for any disturbances from wind. The backstepping controller, which is a control strategy based on flatness output, only adjusts some output of the system and thus can be applied to underactuated cranes since the output space and the flat output space can be connected through differentiable homeomorphisms. For more information on control to flattening, see the studies [43, 44].

(4) The SCFC. The blended use of feedforward and feedback control loops in the process of trajectory tracking in multibody systems offers a good practical solution to the problem of performance versus stability. These permit the engineers to perform correct and fast trajectory tracking, even for complex systems with elastic elements. The feedforward control design aims at improving the tracking accuracy and response time of the system; in contrast, the feedback control design aims at stabilizing the system and making it robust to disturbances. That synergy is important in ensuring that trajectory tracking performance of multibody systems is enhanced at all times. In addition, simulated control is not always possible because an underactuated system can have more degrees of freedom than actuators, which will give rise to right half-plane zeros. This poses a research challenge as lightweight structures gain popularity for energy efficiency. The technique of inversion is widely used in the literature with various adaptations; see the studies [45-48] for more information. For the reason that underactuated systems have a smaller number of actuators, the overall dynamic model is presented in terms of partitioning coordinates into actuated and unactuated subsystems. The output path is therefore described as a superposition of actuated and unactuated coordinates with the aid of differential and algebraic phase for calculations. The algebraic part employs the dynamic sub-model with actuation coordinates, and suitable tuning parameters are used to maintain stability in internal dynamics. In the control design of multibody systems, it is common to use feedforward approaches to control partially controllable systems, which yields differential algebraic equations (DAEs). The servo-constraint method has been successfully applied to the dynamics of rigid bodies and is promising in differentially flat systems such as cranes and airplanes, as well as non-differentially flat systems such as passive joints and soft arms. Inverse modeling focuses on the system's internal dynamics, which can be examined with differential-geometric nonlinear control theory. Nowadays PFLC and EC approaches are not adequate for dealing with underactuated aeroelastic systems, due specifically to the presence of strong nonlinearities.

The present study is significant in that it makes contributions to the problem of underactuated aeroelastic wings by designing a simpler but still effective model and testing four advanced control strategies via simulations. The results are intended to guide the future evolution of aircraft structures that are lightweight and energy efficient but have to counteract aerodynamic instability. Lightweight, flexible airplanes exhibit aeroelastic behaviors that affect their performance when subjected to aerodynamic instabilities. Thus, this research proposes the implementation and assessment of advanced nonlinear control techniques in underactuated aeroelastic systems. The study focuses on four control schemes for a quasi-steady 2D aeroelastic wing model: (i) Partial Feedback Linearization Control (PFLC), (ii) Energy Control (EC), (iii) Flatness-based Control (FC), and (iv) Servo Constraints Feedforward Control (SCFC). PFLC utilizes the active degrees of freedom in control while employing passive dynamics to retain stability, although the method needs the inverse inertia matrix. EC Control incorporates the desired energy in the control in practice but has a problem of several computations. While FC removes the nonlinearity in the system under control, the control method has problems associated with external noise due to high-order derivatives of system states. SCFC introduced an output-dependent kinematic constraint, which subsequently changes the form of the motion equations and looks at the concept of stability. MATLAB/SIMULINK investigations showed that all techniques generate comparable overdamped responses; however, SCFC could give a fast response due to the presence of a feedforward term.

The remainder of the paper is organized as follows: In Section 2, the dynamics of the quasi-steady 2D aeroelastic wing model is detailed. Control schemes are discussed in Section 3. Section 4 describes the simulation setups, results, and pertinent discussion. Finally, in Section 5, the conclusions and recommendations are specified.

3.1 Partial feedback linearization control

This method classifies the Degrees of Freedom (DOFs) as either active or passive and addresses control of active DOFs only, while the passive joints remain intact due to their internal dynamics. This is useful for underactuated systems that have more degrees of freedom than control inputs. Careful controller synthesis guarantees the capability of active DOF control with high stability and accuracy. Hence, once Eq. (9) is separated into active and passive coordinates, one can obtain the following relations:

$\ddot{\varphi}_1+B_a(\varphi, \dot{\varphi})+g_a(\varphi)=f_3 V^2 u$                (10)

$\ddot{\varphi}_2+B_p(\varphi, \dot{\varphi})+g_p(\varphi)=0$                       (11)

where,

$B_a(\varphi, \dot{\varphi})=\left[c_1+c_2\left(\frac{f_4}{f_3}\right)\right] \dot{\varphi}_1+\left(\frac{c_2}{f_3}\right) \dot{\varphi}_2$,

$g_a(\varphi)=\frac{1}{f_3} q_u\left(\varphi_1, \varphi_2\right)\left(\varphi_2+f_4 \varphi_1\right)$,

$B_p(\varphi, \dot{\varphi})=-\left[c_1 f_4+c_2\left(\frac{f_4^2}{f_3}\right)-c_3 f_3-c_4 f_4\right] \dot{\varphi}_1-\left[c_2\left(\frac{f_4}{f_3}\right)-c_4\right] \dot{\varphi}_2$,

$g_p(\varphi)=-\left(f_4 k_1-f_3 k_3\right) \varphi_1-\frac{1}{f_3}\left[f_4 q_u\left(\varphi_1, \varphi_2\right)-f_3 p_u\left(\varphi_1, \varphi_2\right)\right]\left(\varphi_2+f_4 \varphi_1\right)$.

Limitation on accelerations in Eq. (11) implies no applied torques at the passive joints. The principle of inverse dynamics control seeks to accomplish the opposite, which is to linearize the nonlinear system in Eq. (10) and Eq. (11) so that linear control techniques can be applied. For applying the known control input, it is easy to construct the control law from Eq. (10) when internal zero dynamics of the unactuated coordinate are considered in Eq. (11). The subsequent control law is thus chosen:

$u=\frac{1}{f_3 V^2}\left(u_0+B_a(\varphi, \dot{\varphi})+g_a(\varphi)\right)$                (12)

$u_0=\ddot{\varphi}_{d 1}+K_d\left(\dot{\varphi}_{d 1}-\dot{\varphi}_1\right)+K_p\left(\varphi_{d 1}-\varphi_1\right)$                    (13)

where, the subscript d refers to desired references, K_d and K_p are positive feedback gains ensuring stability of the system. Substituting Eq. (12) and Eq. (13) into Eq. (10) yields the following closed loop dynamics.

$\left(\ddot{\varphi}_{d 1}-\ddot{\varphi}_1\right)+K_d\left(\dot{\varphi}_{d 1}-\dot{\varphi}_1\right)+K_p\left(\varphi_{d 1}-\varphi_1\right)=0$                    (14)

The system of Eq. (14) is stable as long as the gains K_p and K_d are positive. For details on stability of internal dynamics related to passive coordinates, see the studies [29, 53, 54].

3.2 Energy-based control

Partial feedback linearization enables us to meet our control objective to follow a predetermined reference trajectory for the controlled variables. However, is it feasible to improve this by controlling additional variables? The total body energy is another significant factor that is intimately associated with the movement of a system. In the studies [55, 56], the control of an underactuated mechanical system was considered with the main goal of controlling the total mechanical energy and some actuated variables of interest. The central question is whether it is possible to generate a control input that achieves the desired outcome.

$\lim _{t \rightarrow 0} E_d-E=0$,            (15)

$\lim _{t \rightarrow 0} \varphi_{1 d}-\varphi_1=0, \lim _{t \rightarrow 0} \dot{\varphi}_1=0$                (16)

The task involves stabilizing actuated variables φ₁ and $\dot{\varphi}_1$ in Eq. (16) while controlling total mechanical energy E in Eq. (15), limiting unactuated variables φ₂ and $\dot{\varphi}_2$. To design a suitable controller based on the total mechanical energy of the system, we propose the following Lyapunov function candidate:

$V=\frac{1}{2}\left(E-E_d\right)^2+\frac{1}{2} K_d \dot{\varphi}_1{}^2+\frac{1}{2} K_p \tilde{\varphi}_1{ }^2$                       (17)

Deriving the above along the trajectories of Eq. (9), we get:

$\dot{V}=\dot{\varphi}_1\left[\left(E-E_d\right) u+K_d \ddot{\varphi}_1+K_p \tilde{\varphi}_1\right]$                  (18)

Let us select the control input, u, as:

$\left(E-E_d\right) u+K_d \ddot{\varphi}_1+K_p \tilde{\varphi}_1=-\gamma \dot{\varphi}_1$                  (19)

Then Eq. (18) becomes:

$\dot{V}=-\gamma \dot{\varphi}_1{}^2 \leq 0$.                (20)

From Eq. (10), we have:

$\ddot{\varphi}_1=f_3 V^2 u-B_a(\varphi, \dot{\varphi})+g_a(\varphi)$                 (21)

Substituting Eq. (21) into Eq. (19) yields:

$\left(E-E_d\right) u+K_d\left(f_3 V^2 u-B_a(\varphi, \dot{\varphi})-g_a(\varphi)\right)+K_p \tilde{\varphi}_1=-\gamma \dot{\varphi}_1$                   (22)

According to Eq. (22), we can get the following control law:

$u=\frac{1}{\left(E-E_d\right)+K_d f_3 V^2}\left(K_d B_a(\varphi, \dot{\varphi})+K_d g_a(\varphi)-\gamma \dot{\varphi}_1-K_p \tilde{\varphi}_1\right)$                  (23)

Remark 1. The total energy of the 2D wing model is calculated as:

$E= kinetic \,\,energy + potential \,\,energy =\frac{1}{2} \dot{q}^T M \dot{q}+\frac{1}{2} q^T K q$                      (24)

where, $q=\left[\begin{array}{l}\delta \\ \theta\end{array}\right], M=\left[\begin{array}{cc}m & m x_\theta a \\ m x_\theta a & I_\theta\end{array}\right], K=\left[\begin{array}{cc}k_\delta & 0 \\ 0 & k_\theta\end{array}\right]$.

Therefore, the desired reference system energy can be computed in terms of the desired coordinates in Eq. (24).

3.3 Differential flatness-based control

It is a sophisticated strategy in control theory, especially advantageous for the nonlinear control systems of an unknown type. It is based on a notion of “flat outputs” in a system, where outputs correspond to the inputs, and all the states and inputs can be obtained without any integration. This is useful in generating trajectories accurately because the trajectory has output for the system. When trajectory generation accuracy is required, differential systems arise, allowing trajectory input correspondence instead [41]. Eq. (10) and Eq. (11) can be reformulated considering linear structural stiffness as follows:

$\begin{gathered}\ddot{\varphi}_1+a_{11} \dot{\varphi}_1+a_{12} \dot{\varphi}_2+a_{13} \varphi_1+a_{14} \varphi_2=b_{11} u \\ \ddot{\varphi}_2+a_{21} \dot{\varphi}_1+a_{22} \dot{\varphi}_2+a_{23} \varphi_1+a_{24} \varphi_2=0\end{gathered}$                 (25)

Let $\varphi_1=x_1, \dot{\varphi}_1=x_2, \varphi_2=x_3, \dot{\varphi}_2=x_4$, then Eq. (25) can be expressed in a state space form as:

$\dot{x}=A x+B u$                   (26)

where, $A=\left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ -a_{13} & -a_{11} & -a_{14} & -a_{12} \\ 0 & 0 & 0 & 1 \\ -a_{23} & -a_{21} & -a_{24} & -a_{22}\end{array}\right], B=\left[\begin{array}{c}0 \\ b_{11} \\ 0 \\ 0\end{array}\right]$.

The flat output y of the linear system in (26) can be obtained as follows [41]:

$y=\left[\begin{array}{llll}0 & 0 & \cdots & 1\end{array}\right]\left[\begin{array}{llll}B, & A B & \cdots & A^{n-1} B\end{array}\right]^{-1} x$               (27)

$\begin{gathered}y=\left(a_{21} x_4+a_{21}^2 x_1-\left(a_{23}-a_{21} a_{22}\right)\right) / d, \\ d=b_{11}\left(a_{24} a_{21}^2-a_{22} a_{21} a_{23}+a_{13}^2\right)\end{gathered}$                 (28)

The output y clearly parametrizes all system variables. To demonstrate this, we compute the time derivatives of y in succession until the control input u appears as follows:

$\dot{y}=-\frac{a_{21} a_{23}}{d} x_1-\frac{a_{21} a_{24}}{d} x_3-\frac{a_{23}}{d} x_4$                  (29)

$\ddot{y}=\frac{a_{23}^2}{d} x_1+\frac{a_{23} a_{24}}{d} x_3+\left(\frac{a_{23} a_{22}-a_{21} a_{24}}{d}\right) x_4$                 (30)

$\dddot{y}=d_1 x_1+d_2 x_2+d_3 x_3+d_4 x_4$                (31)

$y^{(4)}=\left(-d_2 a_{13}-d_4 a_{23}\right) x_1+\left(d_1-d_2 a_{21}-d_4 a_{21}\right) x_2+\left(-d_2 a_{14}-d_4 a_{24}\right) x_3+\left(d_2 a_{12}+d_4 a_{22}\right) x_4+d_2 b_{11} u$              (32)

The control input can be determined from the last equation.

Remark 2. Due to the imposition of linear structural stiffness, Eq. (25) is linear in terms of state space form. Besides, the equations are regulated about set points of pitch and plunge coordinates that encourages using linearized model for the target system (in case some nonlinearity is included).

3.4 Servo constraints-based feedforward control

Servo constraints provide an effective way to model inversion through equations of constraints in output trajectory tracking. This is particularly beneficial in multibody system dynamics, where they extend traditional geometric constraints. A servo constraint can be incorporated into the equation of motion in Eq. (9) for tracking output variables. Consequently, the vibration of the aeroelastic wing model will be described as:

$D(\varphi) \ddot{\varphi}+B(\varphi, \dot{\varphi})+g(\varphi)=A u$                  (33a)

$h(\varphi)-y_d=0$                     (33b)

There are two main methods to solve Eq. (33): projection and coordinate transformation methods (see the studies [46-48] for details). This section will focus on coordinate transformation for feedforward control design. Consider the following coordinate transformation:

$\bar{\varphi}=\left[\begin{array}{c}y=h(\varphi) \\ \varphi_2\end{array}\right]=\alpha(\varphi)$                 (34)

Differentiation Eq. (34) twice leads to:

$\dot{\bar{\varphi}}=H \dot{\varphi}$               (35a)

$\ddot{\bar{\varphi}}=H \ddot{\varphi}+\dot{H} \dot{\varphi}$                   (35b)

Substituting Eq. (33a) into Eq. (35b) leads to:

$\ddot{\bar{\varphi}}=\left[\begin{array}{c}\ddot{y} \\ \ddot{\varphi}_2\end{array}\right]=\left[\begin{array}{c}\left(H D^{-1} B\right) u-H D^{-1}((\varphi, \dot{\varphi})+g(\varphi))+\dot{H} \dot{\varphi} \\ {\left[\begin{array}{ll}0 & 1\end{array}\right] D^{-1}(B u-((\varphi, \dot{\varphi})+g(\varphi)))}\end{array}\right]$               (36)

The above equation represents the equation of motion for an underactuated aeroelastic airfoil denoted by the output variable y and the underactuated coordinate φ₂. The control law can be designed from the first row of Eq. (36) for a given desired output as follows:

$u=\left(H D^{-1} B\right)^{-1}\left(H D^{-1}((\varphi, \dot{\varphi})+g(\varphi))-\dot{H} \dot{\varphi}+\ddot{y}_d\right)$                (37)

whereas the internal dynamics can be determined from the second row of Eq. (36) with inserting Eq. (37) into it to get:

$\ddot{\varphi}_2=\left[\begin{array}{ll}0 & 1\end{array}\right] D^{-1}\left(B\left(H D^{-1} B\right)^{-1}\left(H D^{-1}((\varphi, \dot{\varphi})+g(\varphi))-\dot{H} \dot{\varphi}+\ddot{y}_d\right)-((\varphi, \dot{\varphi})+g(\varphi))\right)$                (38)

Remark 3. (i) Other possible control techniques include coupling control [57] and backstepping control [58] to stabilize underactuated dynamic systems. (ii) It should be noted that the aforementioned control techniques presume the knowledge of system dynamics. Therefore, work on these techniques in adaptive versions is important for practice. As far as adaptive control and the schemes for its extensions are concerned, refer to [58-62].