Abstract
A sliding mode control- (SMC-) based limit cycle oscillation (LCO) regulation method is presented, which achieves asymptotic LCO suppression for UAVs using synthetic jet actuators (SJAs). With a focus on applications involving small UAVs with limited onboard computational resources, the controller is designed with a simplistic structure, requiring no adaptive laws, function approximators, or complex calculations in the control loop. The control law is rigorously proven to achieve asymptotic regulation of both pitching and plunging displacements for a class of systems in a dual-parallel underactuated form, where a single scalar control signal simultaneously affects two states. Since dual-parallel underactuated systems cannot be expressed in a strict feedback or cascade form, standard backstepping-based control techniques cannot be applied. This difficulty is mitigated through careful algebraic manipulation in the regulation error system development, along with innovative design of the sliding surface. A detailed model of the UAV LCO dynamics is utilized, and a rigorous analysis is provided to prove asymptotic regulation of the pitching and plunging displacements. Numerical simulation results are provided to demonstrate the performance of the control law.
1. Introduction
Limit cycle oscillation (LCO) (flutter) is a self-excited aeroelastic phenomenon that can create difficulties in aircraft tracking control and could potentially result in structural damage and even catastrophic failures. Even in the low Reynolds number (low-Re) regimes, characteristic of small unmanned aerial vehicle (UAV) systems, LCO can be adversarial in flight control systems and can lead to unsafe UAV operating conditions [1]. Motivated by these challenges, automatic control methods for LCO regulation systems have been widely investigated in recent controls literature [2–5]. LCO suppression control systems are usually designed using moving deflection surfaces (e.g., elevators, rudders, and ailerons) to deliver the required control force or moment. However, for applications involving smaller, lighter weight UAVs, the use of heavy mechanical actuators might not be practical. As these practical considerations motivate the need for smaller, low power-consumptive control actuators, synthetic jet actuators (SJAs) have emerged as a popular alternative to mechanical deflection surfaces for UAV control applications [1, 6–14].
SJAs are practical tools for UAV LCO suppression control systems due to their low cost, small size, and low power consumption properties. SJAs utilize a vibrating diaphragm to create trains of vortices (or jets) of air through the periodic ejection and suction of air through a small orifice (see Figure 1). The resulting trains of air vortices impart linear momentum to a flow system, and this momentum transfer enables SJAs to deliver an equivalent control force or moment when implemented in UAV wings. Since SJAs only use the surrounding air of the flow system to generate the vortices, they can deliver a control force or moment with zero net mass injection across the flow boundary. This is a key benefit in small UAV applications: SJAs do not require space for a fuel supply. The boundary layer flow field near the surface of a UAV wing can be altered by using the trains of air vortices generated by the SJAs. By utilizing this SJA-commanded modification of the boundary layer flow field, SJAs can be employed to achieve automatic regulation control of LCO in UAV wings. SJA-based control design is complicated, however, due to the inherent nonlinearity in the equations governing the SJA dynamics. Specifically, the virtual surface deflection delivered by the SJAs is a nonlinear (nonaffine) function of the voltage signal commanded to the SJAs. An additional challenge in control design using SJAs is that the nonlinear actuator dynamic model includes parametric uncertainty. This SJA nonlinear SJA dynamic model was determined empirically through repeated experiments, and it is well-accepted in SJA-based control literature (e.g., see [10, 12, 14]).
In addition to the control design challenges involved in compensating for actuator uncertainties, significant challenges arise in situations where multiple actuators lose control effectiveness. When the number of control actuators is less than the number of degrees of freedom to be controlled, the system becomes underactuated, and control design for underactuated systems presents a nontrivial challenge. The integrator backstepping technique is often utilized to address the challenge of control design for underactuated systems [15]. However, the backstepping-based control design approach can only be applied to systems in strict feedback or cascade form. Backstepping techniques cannot be utilized for systems in a dual-parallel underactuated form, where a single scalar control input simultaneously affects two states. Several open problems remain in control design for dual-parallel underactuated systems.
To compensate for the nonlinearity and parametric uncertainty inherent in the SJA dynamic model, recent approaches typically employ adaptive parameter estimation, neural networks (NN), fuzzy logic rule sets, or complex fluid dynamics computations in the feedback loop (e.g., see [10, 14, 16]). These types of SJA-based control design approaches have been shown to achieve good closed-loop performance; however, control designs involving complex calculations or function approximation methods can incur an increased computational requirement, which might not be practical for applications involving small UAVs. An adaptive inverse control method is presented in [10, 14], which achieves asymptotic trajectory tracking for an aircraft dynamic model equipped with SJAs. To achieve the results in [10, 14], a series of adaptive parameter estimation laws are utilized along with rigorous Lyapunov-based stability analyses. Inspired by these results and motivated by the desire to investigate a more computationally minimal control strategy, our previous result in [17] was the first SJA-based tracking control approach to achieve asymptotic trajectory tracking for an aircraft using a simple (single-loop) feedback control law without the use of parameter adaptation or function approximators in the control law. A question that remains to be answered is as follows: Can a computationally minimal nonlinear SJA-based control law achieve asymptotic regulation of LCO for UAV systems in a dual-parallel underactuated form?
The contribution of this paper is the development and rigorous analysis of a sliding mode control law, which achieves asymptotic regulation of both pitching and plunging LCO in UAV wings using a single, scalar SJA as the control input (i.e., a dual-parallel underactuated system). To the best of the authors’ knowledge, this is the first SJA-based nonlinear control result to rigorously prove asymptotic regulation of both pitching and plunging LCO states in a dual-parallel underactuated UAV system. Moreover, to address a practical UAV scenario where onboard computational resources are limited, the result presented here compensates for the inherent SJA actuator nonlinearity and parametric uncertainty without the use of adaptive laws or function approximators. To achieve the result, a sliding mode control strategy is utilized, which employs a periodic switching law. A detailed mathematical model of the UAV dynamics is utilized to develop the regulation error dynamics, and a rigorous Lyapunov-based stability analysis is presented to prove asymptotic regulation of the pitching and plunging displacements. Numerical simulation results are also provided to complement the theoretical development.
2. Dynamic Model and Properties
The equation describing dynamics of LCO in an airfoil can be expressed as [18]where the coefficients denote the structural mass and damping matrices and is a nonlinear stiffness matrix. In (1), , where denote the plunging and pitching displacements, respectively. Figure 2 illustrates the pitching and plunging displacements in a standard airfoil.
Also in (1), the structural linear mass matrix is defined as [18]where the parameter denotes the nondimensional distance measured from the elastic axis to the center of mass, is the semichord of the wing [m], is the mass of the wing section, and is the mass moment of inertia of the wing about the elastic axis (see Figures 2 and 3).
The structural linear damping matrix is described aswhere the parameters are the structural damping coefficient in plunge due to viscous damping and structural damping coefficient in pitch due to viscous damping ·, respectively. The nonlinear stiffness matrix utilized in this study iswhere is the structural spring constant in plunge and in · is the torsion stiffness coefficient described in terms of a polynomial asThe right hand side of (1) is explicitly given by [18]where is the velocity , is the density of air , is the wing span , is the lift coefficient per angle of attack, is the moment coefficient per control surface deflection, is the lift coefficient per control surface deflection, is the moment coefficient per control surface deflection, and is the nondimensional distance from the midchord to elastic axis. The term denotes the surface deflection angle of the wing (see Figure 3). By rearranging (1), the dynamics equations can be expressed aswhere is the state vector with , , , and . In (7), is a nonlinear state matrix (containing nonlinearities due to the torsion stiffness coefficient introduced in (5)), and is the control input gain matrix. By separating the constant elements from the state-dependent elements of the matrix , the dynamic model (7) can be rewritten aswhere and denote known constant parameters, which are dependent on the physical parameters of the wing section. The explicit definitions of the constant parameters are unwieldy and are omitted here for brevity. In (8), represents the virtual surface deflection delivered by the SJA (or SJA array) (i.e., the term introduced in (7)). Also in (8), the nonlinear vector function is defined in terms of the stiffness coefficient asThe constant control input gain terms introduced in (8) are explicitly defined as
3. SJA Dynamics
The dynamics of the SJA are nonlinear and they contain parametric uncertainty. Figure 1 represents the basic structure of a SJA [10, 14]. This SJA has a piezoelectrically driven diaphragm in its cavity that generates time varying pressure gradients across a small orifice. The periodic ingestion and expulsion of the air through the small orifice results in the formation of vortex rings which formulates a steady turbulent jet of air.
Based on empirical data, the dynamics of a SJA can be expressed as [10, 14]where denotes the virtual airfoil surface deflection (i.e., the control input), denotes the peak-to-peak SJA voltage, and are uncertain physical parameters.
To compensate for the SJA nonlinearity and parametric uncertainty in (11), a robust inverse control design structure will be utilized for the voltage input signal [17]. The robust inverse controller can be expressed aswhere denote constant feedforward estimates of the uncertain parameters and ; and is a subsequently defined auxiliary control signal.
4. Control Development
The objective is to design the scalar control signal to asymptotically regulate the plunging and pitching dynamics (i.e., and ) to zero. By leveraging the result in [19], will be designed using a sliding mode control law with a periodic switching function asBased on the dynamic equations in (8) and the subsequent stability analysis, the sliding surface in (13) is designed aswhere are positive constant control gains and are introduced in (8).
4.1. Stability Analysis
Theorem 1. The robust control law in (13) ensures asymptotic convergence to the sliding manifold .
Proof. Proof of Theorem 1 can be found in [19] and is omitted here to avoid distraction from the main contribution of the current result.
Theorem 2 (main result). Convergence to the sliding manifold results in asymptotic regulation of both pitching and plunging displacements in the sense thatas
Proof (plunging regulation). It follows directly from the definition of the sliding surface in (14) and the LCO dynamic equations in (8) thatBy using the state definition , the expression in (16) can be used to show that, on the sliding manifold , the dynamics are governed bywhere , are introduced in (14). Since , the ODE in (17) is Hurwitz, and (17) can be used to prove that and . Thus, convergence to the sliding manifold directly results in asymptotic (exponential) regulation of the plunging displacement to zero.
Proof (pitching regulation). By substituting (14) into (8) and using the fact that , it can be shown that convergence to the sliding manifold results in the pitching dynamicswhere and are known constant parameters introduced in (8); , ; and is defined in (5). By using the state definition , the expression in (18) can be rewritten as the second-order nonlinear ODEAfter utilizing the definition of given in (5), the ODE in (19) can be expressed in the form [20]Noting that , the auxiliary functions in (20) are explicitly defined asNote. It can be shown that . The expressions in (21) and (22) satisfy the following properties:To prove asymptotic regulation of the pitching displacement to zero, consider the positive definition function (i.e., Lyapunov function candidate) [20]After taking the time derivative of (25) and utilizing (20), can be expressed asThe Lyapunov derivative in (26) is negative semidefinite based on inequality (23). We now use LaSalle’s invariance principle to state that as as , and thus, as . It further follows that as from (21). Since as , (20) can be used along with property (24) to prove that as .
4.2. Results
In this section, a numerical simulation was created to demonstrate the performance of the control law developed in (13) and (14). The simulation is based on the dynamic model given in (1). The dynamic parameters utilized in the simulation are summarized in Table 2 and were obtained from [18].
The control gains , , , and were manually selected as described in Table 1. The actual SJA parameters and and estimates and were selected as and , respectively.
The control law was simulated for three different cases of initial conditions: Case 1: (Figures 4 and 5), Case 2: (Figures 6 and 7), and Case 3: (Figures 8 and 9). Figures 4, 6, and 8 show the time evolution of the pitching and plunging displacements and during closed-loop controller operation for Cases 1, 2, and 3, demonstrating the rapid convergence of the pitching and plunging displacements to zero. In addition, Figures 5, 7, and 9 show that the pitching and plunging velocities and converge to zero rapidly during closed-loop operation. Further, the top plots in Figures 4, 6, and 8 show the control effort used during closed-loop controller for Cases 1, 2, and 3.
5. Conclusion
In this paper, a sliding mode control law is presented, which achieves asymptotic LCO regulation in UAV wings using SJAs. Moreover, the control law achieves asymptotic regulation of both the pitching and plunging displacements for UAV LCO dynamics in a dual-parallel underactuated form, where a single scalar control signal simultaneously affects both displacements. To achieve the result, a sliding mode control strategy is utilized, which employs a periodic switching function and a novel sliding surface, which is derived based on the nonlinear dynamics of the UAV LCO system. The inherent nonlinearity and parametric uncertainty in the SJA dynamic model is addressed by means of a robust inverse control structure, which utilizes constant feedforward estimates of the uncertain SJA parameters. The use of constant estimates as opposed to time varying adaptive parameter estimates is motivated by the desire to develop a practical LCO regulation method that can be implemented on small UAVs with limited onboard computational resources. A rigorous analysis is presented to prove the theoretical result, and numerical simulation results are provided to complement the theoretical development.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research is supported by NSF Award number 1335405.