Approximate Energetic Resilience of Nonlinear Systems under Partial Loss of Control Authority

TL;DR

This work develops an energetic resilience framework to quantify the maximal additional control energy required when a nonlinear system loses authority over part of its actuators while achieving fixed-time reachability. It introduces an energy-based resilience metric $r_A(t_f,R)$ and derives tractable bounds by first analyzing linear driftless systems, then extending to general nonlinear dynamics via a mean-control approximation and bounding techniques. For linear driftless cases, the authors obtain closed-form expressions and tight bounds, including a precise result for single-actuator loss. In nonlinear settings, approximate energies and worst-case bounds are proposed and validated through simulations on driftless, linear, and wind-affected nonlinear models, showing the bounds are informative with modest conservatism. The framework enables resource planning and resilience assessment under faults or adversarial inputs, and points to future work on multi-actuator losses and resilient controller design.

Abstract

In this paper, we quantify the resilience of nonlinear dynamical systems by studying the increased energy used by all inputs of a system that suffers a partial loss of control authority, either through actuator malfunctions or through adversarial attacks. To quantify the maximal increase in energy, we introduce the notion of an energetic resilience metric. Prior work in this particular setting does not consider general nonlinear dynamical systems. In developing this framework, we first consider the special case of linear driftless systems and recall the energies in the control signal in the nominal and malfunctioning systems. Using these energies, we derive a bound on the energetic resilience metric. For general nonlinear systems, we first obtain a condition on the mean value of the control signal in both the nominal and malfunctioning systems, which allows us to approximate the energy in the control. We then obtain a worst-case approximation of this energy for the malfunctioning system, over all malfunctioning inputs. Assuming this approximation is exact, we derive bounds on the energetic resilience metric when control authority is lost over one actuator. A set of simulation examples demonstrate that the metric is useful in quantifying the resilience of the system without significant conservatism, despite the approximations used in obtaining control energies for nonlinear systems.

Figures (4)

• Figure 1: Illustration of energetic resilience metric as a function of distance of the initial condition from the origin for the linear driftless model of an underwater robot. The actual difference in energies $E_T-E_N$ does not vary significantly with different choices of uncontrolled inputs $u_{uc}$, and is hence not shown here.
• Figure 2: Difference between nominal and malfunctioning energies, compared to the upper bound on the metric $r_A(t_f, R)$, for a linear model of a fighter jet. The thinner dashed lines plot $E_T-E_N$ for a variety of uncontrolled inputs $u_{uc}$.
• Figure 3: Difference between nominal and malfunctioning energies, compared to the upper bound on the metric $r_A(t_f, R)$, for a nonlinear wind-affected model. The thinner dashed lines plot $E_T-E_N$ for a variety of uncontrolled inputs $u_{uc}$.
• Figure 4: Sensitivity of metric to different amplitudes of wind disturbance.

Theorems & Definitions (23)

• Definition 1
• Definition 2: Nominal Energy
• Definition 3: Malfunctioning Energy
• Definition 4: Total Energy
• Definition 5: Worst-case Total Energy
• Definition 6: Energetic Resilience
• Proposition 1
• proof
• Proposition 2
• proof