How Much Reserve Fuel: Quantifying the Maximal Energy Cost of System Disturbances
Ram Padmanabhan, Craig Bakker, Siddharth Abhijit Dinkar, Melkior Ornik
TL;DR
This work addresses quantifying the maximal extra energy a controller must expend to stabilize an LTI system in finite time when disturbances are present. It formalizes and compares nominal ($E_N^*$) and disturbed ($E_D^*$) energies, deriving an explicit upper bound on the worst-case disturbed energy using a least-squares argument and an infinite-dimensional bound, and introduces additive ($r_A$) and multiplicative ($r_M$) metrics to capture the cost of disturbances. The paper provides closed-form expressions for optimal nominal and disturbed controls, plus bounds that relate the two energies to disturbance magnitude and task hardness, and validates the approach with ADMIRE fighter jet simulations. The results offer practical fuel-budget-oriented metrics for robust finite-time stabilization tasks and highlight how task hardness and initial-condition distance influence the energy penalty from disturbances.
Abstract
Motivated by the design question of additional fuel needed to complete a task in an uncertain environment, this paper introduces metrics to quantify the maximal additional energy used by a control system in the presence of bounded disturbances when compared to a nominal, disturbance-free system. In particular, we consider the task of finite-time stabilization for a linear time-invariant system. We first derive the nominal energy required to achieve this task in a disturbance-free system, and then the worst-case energy over all feasible disturbances. The latter leads to an optimal control problem with a least-squares solution, and then an infinite-dimensional optimization problem where we derive an upper bound on the solution. The comparison of these energies is accomplished using additive and multiplicative metrics, and we derive analytical bounds on these metrics. Simulation examples on an ADMIRE fighter jet model demonstrate the practicability of these metrics, and their variation with the task hardness, a combination of the distance of the initial condition from the origin and the task completion time.