KPCA for Thrust Vectoring Systems Exhibiting Singular Points

Abstract

This paper considers a class of thrust vectoring systems, which are nonlinear, overactuated, and time-invariant. We assume that the system is composed of two subsystems and there exist singular points around which the linearized system is uncontrollable. Furthermore, we assume that the system is stabilizable through a two-level control allocation. In this particular setting, we cannot do much with the linearized system, and a direct nonlinear control approach must be used to analyze the system stability. Under adequate assumptions and a suitable nonlinear continuous control-allocation law, we can prove uniform asymptotic convergence of the points of equilibrium using Lyapunov input-to-state stability and the small gain theorem. This control allocation, however, requires the design of an allocated mapping and introduces two exogenous inputs. In particular, the closed-loop system is cascaded, and the output of one subsystem is the disturbance of the other, and vice versa. In general, it is difficult to find a closed-form solution for the allocated mapping; it needs to satisfy restrictive conditions, among which Lipschitz continuity to ensure that the disturbances eventually vanish. Additionally, this mapping is in general nontrivial and non-unique. In this paper, we propose a new kernel-based predictive control allocation to substitute the need for designing an analytic mapping, and assess if it can produce a meaningful mapping ``on-the-fly" by solving online an optimization problem. The simulations include three examples, which are the manipulation of an object through an unmanned aerial vehicle in two and three dimensions, and the control of a surface vessel actuated by two azimuthal thrusters.

Figures (8)

• Figure 1: Cascaded closed-loop system \ref{['eq:S1_controlled']}, \ref{['eq:S2_controlled']}.
• Figure 2: UAV manipulating an object in two dimensions.
• Figure 3: Comparison between the optimal mapping \ref{['eq:optimal_mapping']} and the Lipschitz continuous mapping \ref{['eq:control_law4']}. We see that the optimal but discontinuous mapping cannot stabilize the system.
• Figure 4: Example 1. One-step command following using $\alpha_\mathrm{d} = \pi/2,$$\kappa_\mathrm{w}=1$ without the kernel term, with the kernel but without the bell curve weight, and with the kernel using the bell curve. It is shown that the penalization of the attitude deviation from the kernel space is crucial to stabilize the system. Furthermore, note that the inclusion of the bell curve yields a faster and more efficient response, as it establishes an effective region where the kernel deviation penalty is activated.
• Figure 5: UAV manipulating an object in three dimensions.

Theorems & Definitions (4)

• Remark 2
• Remark 5
• Remark 8
• Theorem 9