1.5 Survey on the control of RWVTOL vehicles
1.5.1 Linear control using linear approximations
⎢⎢
⎣
˙ x mv˙
R˙ Jω˙
⎤
⎥⎥
⎦=
⎡
⎢⎢
⎣
Rv
−mS(ω)v −T e3+ ΣRΓ +Rmge3+FaeB RS(ω)
−S(ω)J ω+ Γ +TΣT + ΓBae
⎤
⎥⎥
⎦, (1.16)
with T ∈ R and Γ ∈ R3 the thrust force and torque control inputs; FaeB ∈ R3 and ΓBae ∈ R3 the sum of all aerodynamic forces and torques acting on the vehicle, expressed in the body frame; ΣR ∈R3×3 and ΣT ∈R3 constant matrices, representing the coupling between forces and torques created by the force/torque generation mechanism.
The influence of the translational dynamics on the rotational dynamics appears through the term TΣT. In the case of single rotor helicopter, this term is physically justified by the fact that the thrust force vector does not apply at the vehicle’s CoM exactly. How-ever, this term is not bothering because, for most RWVTOL vehicles, it is rather “small”, thus, it can be compensated via full torque actuation. On the other hand, the influence of the rotational dynamics on the translational dynamics is more involved. Generally, ΣR is different from the null matrix, except the case of quadrotor helicopters. For instance, ΣR is given by Eq. (1.12) for ducted fan tailsitters, and by Eq. (1.8) for single rotor helicopters. It is not null either for tandem helicopters (see e.g. (Dzul et al., 2002) for details). However, as long as small body forces ΣRΓremain small, and the control law is adequately designed, these forces are not problematic. In turn, the issue of robustness with respect to aerodynamic perturbations, i.e. the terms FaeB and ΓBae, is more important. In practice, it is extremely difficult to obtain a well-tuned model of these aerodynamic ef-fects which can be used for control design purposes. Unpredictable wind gusts resulting in unpredictable aerodynamic effects complicate this robustness issue even more. In the next section, more details clarifying these control difficulties will be provided, along with associated solutions proposed in the literature.
1.5 Survey on the control of RWVTOL vehicles
RWVTOL vehicles have long attracted the automatic community due to a number of challenges associated with these systems. Control difficulties are directly related to the fact that these vehicles are underactuated and strongly nonlinear, that their envelop of flight can be very large, and that they are usually subjected to uncertainties and unmodeled dynamics, etc.. Control design methods for these vehicles have been investigated in both contexts of linear and nonlinear control systems. In what follows a panorama of these methods is sketched.
1.5.1 Linear control using linear approximations
In the context of linear control theory, the control design for helicopters and other RWV-TOL vehicles is based on linear approximations along trim trajectories9. Commonly, the
9. Trajectories for whichv˙= 0,ω˙ = 0.
Survey on the control of RWVTOL vehicles 27 direct application of linear approximations leads to the use of Euler angles {φ, θ, ψ} for attitude parametrization, allowing for the decoupling of the linearized system into four Single-Input-Single-Output (SISO) systems associated with the regulation of a single vari-able (longitudinal position or velocity, lateral position or velocity, yaw, and altitude) (see e.g. (Prouty, 2002), (Pflimlin et al., 2007a), (Peddle et al., 2009), etc.). To be more pre-cise, let us recall an example case reported in (Pflimlin et al., 2007a) for the hovering control mode of a symmetric ducted fan tailsitter in the absence of wind gusts. The sys-tem under consideration takes the form of Syssys-tem (1.2) withFB and ΓB given by (1.11) (see Section 1.4.2). By assuming that the vehicle’s translational and rotational velocities are relatively small (i.e. near hover flight), all aerodynamic drag and lift forces, which are proportional to the square of the vehicle’s velocities, can be neglected. In turn, the momentum drag (a particularity of ducted fan tailsitters) can be approximated by−Qv, with Qk5e√
Denoting vIx˙ and using Euler angles for attitude parametrization, in first-order ap-proximations the system is equivalent to the four following SISO linear systems:
• The longitudinal channel
From here, all kinds of linear control design techniques can be applied to each channel.
Linear control methods have been applied (with more or less success) to many RWV-TOL vehicles. For instance, Proportional Integral Derivative (PID) control laws were designed for hover control and slow translational motion control for RWVTOL vehicles such as the iSTAR (Lipera et al., 2001), SLADe (Peddle et al., 2009), HoverEye (Pflim-lin et al., 2007a) ducted fan tailsitters, or a quadrotor helicopter (Castillo et al., 2005), etc.. Besides, optimal control techniques have been widely applied to these systems. Let us review some examples:
• Linear-Quadratic-Regulator (LQR) controllers (Kwakernaak and Sivan, 1972) were proposed, e.g., for the Short Range Station Keeping (SRSK) task of an assault helicopter (Rynaski, 1966), for the stabilization of the lateral position and roll angle of a quadrotor helicopter (Castillo et al., 2005), for the altitude stabilization of the Caltech ducted fan in forward flight (Teel et al., 1997), for velocity control of a tailsitter (Stone, 2004), etc..
• The Linear-Quadratic-Gaussian (LQG) control technique (Stein and Athans, 1987), which can be simply viewed as a combination of a Kalman filter with a LQR, has been investigated for helicopter control (see e.g. (Bendotti and Morris, 1995), (Mammar, 1992), (Benallegue et al., 2006), and the references therein).
• The H2 and H∞ control techniques (see e.g. (Zames, 1981), (Francis and Zames, 1984), (Doyle et al., 1989)) have also been widely used (see e.g. (Mammar and Duc, 1992), (Takahashi, 1993), (Civita et al., 2003), (Luo et al., 2003), (Prempain and Postlethwaite, 2005), and the references therein). In (Bendotti and Morris, 1995), (Mammar, 1992) the authors recorded through simulation and experimental results that H∞ controllers (defined by the authors) provided better performance and robustness than LQG controllers (also defined by the authors).
Other control design studies can also be found in the literature in the context of linear optimal control for RWVTOL vehicles. Basically, all optimal control methods first express the control problem as a mathematical optimization problem (e.g. a quadratic cost func-tion, a H2 or H∞ norm of a transfer matrix, etc.) and then find the controller to solve this. They may provide good robustness and performance. But this strongly relies on the level of exactitude of the model for control design.
The principle weakness of linear control techniques for RWVTOL vehicles lies in the fact that the control design is based on a linear approximation. Theoretically, the sta-bility of the controlled system is only guaranteed in a limited basin of attraction. A popular remedy for this issue is the gain-scheduling technique (Shamma, 1988), (Shamma and Athans, 1992). In the context of helicopter control, some gain-scheduling controllers have been proposed (see e.g. (Kadmiry and Driankov, 2004), and the references therein, etc.). Basically, this technique consists of two steps. Firstly, local linear controllers, based on linearized models about several trim configurations, are designed to cover a large oper-ating domain. Then, a global controller is obtained by interpoloper-ating,i.e. “scheduling”, the gains of the local controllers. Gain scheduling may be able to incorporate mature linear control methodologies into nonlinear control design. However, despite its wide-spread use, this technique is merely an engineering practice or a heuristic approach. For instance, the robustness, performance, and stability properties of the global gain scheduled controller are not explicitly addressed in the control design process, and therefore only extensive simulations and experiments allow to evaluate these properties.
Limitations of linear control techniques applied to strongly nonlinear systems like RWVTOL vehicles are well understood. Beside the limitation of the local stability and convergence results, other shortcomings require also close attention:
• The linearization requires a good knowledge of the equilibrium trajectory. Note that determining an equilibrium trajectory necessitates a precise model of aerodynamic effects. Since the attitude of RWVTOL vehicles can vary in large proportions as a
Survey on the control of RWVTOL vehicles 29 function of the reference trajectory, the process of identifying aerodynamics effects in a large operating domain can be both difficult and costly. Besides, unpredictable wind gusts and model uncertainties further complicate the problem of equilibrium determination.
• Due to aerodynamic perturbations, the system under consideration often operates far from the desired trajectory. In this case the linearization is no longer significant of the real dynamics of the vehicle.
• The linearization requires a minimal parametrization (e.g.Euler angles parametriza-tion) of the rotation matrix, an element of the group SO(3). This leads to sin-gularities in the representation which artificially limit the stability domain of the controllers.
• Linear controllers based on linear approximations do not make use of the symmetry properties (i.e. for a rigid body the cinematic system is invariant under the group SE(3)). Therefore, instead of simplifying the control design they often make it more complicated.
Nonlinear control design techniques allow to bypass most of these shortcomings.