Dynamic input allocation in presence of periodic exogenous signals: spline-based and sensitivity functions approaches

Modern engineering systems are increasingly characterized by overactuation, where multiple actuators influence the same output variable. This redundancy offers opportunities for performance enhancement, robustness, and fault tolerance, yet it also poses the fundamental challenge of allocating actuator effort in a manner that improves efficiency without degrading closed-loop behavior. This thesis develops and analyzes novel strategies for Dynamic Control Allocation (DCA) in systems with input redundancy, with particular emphasis on nonlinear dynamics and periodic operating conditions. The overarching objective is to design allocation schemes that operate within the nullspace of the input-output map of the plant, thereby preserving the plant output while reshaping actuator commands to optimize steady-state performance. This property of output invisibility allows allocation to be introduced as an auxiliary mechanism on top of existing feedback controllers, enriching system behavior without altering regulation or tracking performance. Building upon this principle, the thesis advances DCA along three interconnected directions: the development of spline-based internal models for allocation, the application of DCA to complex overactuated platforms, and the extension of allocation to nonlinear systems subject to periodic exogenous signals. The first contribution lies in the introduction of polynomial and spline-type internal models for allocation signals. By moving beyond constant or purely sinusoidal structures, the proposed spline-based allocator offers enhanced expressiveness and flexibility, enabling more effective redistribution of actuator effort in steady state. Theoretical guarantees ensure stability, invisibility, and optimality, while numerical studies confirm superior performance compared with earlier approaches, particularly in the presence of time-varying references. The second contribution is an application-oriented study of allocation strategies on a highly nonlinear aerial vehicle equipped with tilting–twisting propellers. Two distinct DCA methods an instantaneous allocator and a periodic spline-based allocator are compared under both idealized and realistic actuator dynamics. The results highlight a fundamental trade-off: while the instantaneous allocator is simple and optimal in the absence of dynamics, the spline-based method achieves smoother and more robust performance when actuator dynamics are non-negligible. This investigation demonstrates the feasibility of deploying DCA in demanding aerospace scenarios and clarifies the relative advantages of different allocation paradigms. The third contribution extends DCA to nonlinear systems subject to periodic exogenous signals. The proposed framework combines gradient-based optimization with parameter-dependent sensitivity dynamics, thereby extending control allocation beyond conventional point-wise (instantaneous) schemes. This formulation systematically incorporates periodicity in both the reference trajectories and the cost functionals, allowing the allocator to exploit the temporal structure of the problem rather than optimizing solely at each instant. A key feature of the proposed method is the preservation of the output invisibility property, ensuring that the introduction of allocation dynamics does not interfere with the nominal closed-loop system output guaranteed by the baseline controller. The approach is implemented and validated on a nonlinear two-mass spring–damper system, which serves as an academic benchmark demonstrating the effectiveness, stability, and practical relevance of the proposed sensitivity-based dynamic allocation framework. Taken together, these developments establish a coherent and versatile framework for dynamic control allocation in both linear and nonlinear systems. By combining rigorous analysis with illustrative applications, the thesis demonstrates how allocation can be leveraged to reduce actuator usage, improve efficiency, and enhance robustness, all while ensuring that closed-loop output remains unaffected. The contributions pave the way toward experimental validation on robotic and aerial platforms, integration with high-level motion planning, and extensions to distributed and fault-tolerant control architectures. In this way, the work advances dynamic control allocation from a primarily theoretical construct to a practical methodology for managing redundancy in complex modern systems.