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Energy-Aware Performance Evaluation of Nonlinear Mechatronic Systems Under Matched-Tracking Conditions
Bhanuka Dayawansa
TL;DR
Trajectory metrics do not account for actuator effort, particularly in nonlinear systems where stiffness and dissipation depend on state. The paper introduces matched-performance energy-aware evaluation combining tracking error with cumulative actuator energy and finds that nonlinear systems require more energy despite comparable tracking accuracy.
Problem
Trajectory-based metrics describe tracking deviation but do not account for actuator effort, allowing systems with similar tracking responses to differ in energy consumption.
Method
The paper introduces a matched-performance framework combining tracking metrics with cumulative actuator energy and uses EAPI to represent both aspects.
Results
The nonlinear system requires nearly twice the actuator energy of the linear system, with relative energy increases of ∼92% to ∼107% across matched-performance pairs.
Takeaways & Limitations
Including actuator energy reveals performance differences that trajectory-based metrics alone may not capture under closely matched tracking conditions.
Takeaways & Limitations
RMSE remains limited because it depends only on trajectory deviation and excludes actuator input, velocity, and energy-related quantities.
Abstract
from arXiv · showhide
Trajectory-tracking metrics such as root-mean-square error (RMSE), overshoot, and settling time are widely used to evaluate control performance in mechatronic systems. However, these measures describe output tracking alone and do not account for the actuator effort required to produce the observed motion. This limitation becomes more pronounced in nonlinear systems, where stiffness and dissipation depend on the system state. This paper examines how these effects influence actuator energy under similar tracking conditions. An energy-aware evaluation framework is introduced that combines tracking error with cumulative actuator energy and enables comparison between systems with approximately matched performance. A simple index (EAPI) is used to capture both aspects in a single measure. Simulation results for linear and nonlinear systems under proportional-derivative control show that comparable tracking accuracy can correspond to significantly different actuator energy. The nonlinear system consistently requires more energy across matched operating points, reflecting the influence of nonlinear stiffness and friction. These results suggest that trajectory-based metrics alone may not fully capture differences in system effort, and that including energy provides a more informative basis for performance evaluation.
I. INTRODUCTION
Trajectory metrics describe how closely mechatronic outputs follow a reference but omit actuator effort, a limitation that is especially relevant for nonlinear systems. The paper introduces matched-performance, energy-aware evaluation to expose energetic differences hidden by similar tracking responses.
- Conventional metrics such as rise time, settling time, overshoot, and RMSE quantify trajectory-tracking behavior.
- Similar tracking responses can require significantly different actuator energy because nonlinear systems couple energy storage, dissipation, state, and input.
- The evaluation framework combines tracking metrics with cumulative actuator energy under matched, identical tracking conditions.
- The matched-error comparison framework exposes energy differences hidden by trajectory-only metrics, while EAPI serves as a compact reporting index.
- The nonlinear model retains amplitude-dependent stiffness and nonlinear dissipation, which influence energy storage and loss during system evolution.
B. Energy Balance
The energy-balance formulation identifies actuator input as a power contribution and damping mechanisms as energy losses. This provides the basis for relating system dynamics to energetic behavior.
- Differentiating the system energy and substituting the dynamics yields an energy-balance expression.
- Actuator input contributes power u ˙x, while viscous damping and Coulomb friction dissipate energy.
- The balance distinguishes energy supplied by the actuator from dissipation during system evolution.
C. Actuator Energy
Actuator energy is evaluated from absolute actuator power integrated over time, complementing trajectory-error measures. The resulting framework jointly evaluates tracking accuracy and energetic cost.
- Cumulative actuator energy integrates absolute actuator power over the interval [0, T].
- Absolute power measures total actuator effort independent of direction and is suited to limited energy recovery, actuator loading, and thermal effects.
- RMSE depends only on trajectory deviation and excludes actuator input, velocity, and energy-related quantities.
- EAPI jointly evaluates normalized tracking error and actuator energy using a weighting factor λ.
- The combined index captures energetic differences even when trajectory-based metrics appear similar.
III. ENERGETIC DIFFERENCES UNDER NONLINEAR DYNAMICS
The analysis compares linear and nonlinear systems whose tracking performance is similar while examining differences in stored mechanical energy. Positive nonlinear stiffness adds an amplitude-dependent energy component relative to the linear case.
- The analysis examines how nonlinear dynamics influence actuator energy requirements relative to linear systems under similar tracking performance.
- The nonlinear and corresponding linear systems are compared using their associated stored mechanical energies.
- For α > 0, nonlinear stiffness adds an amplitude-dependent energy component that increases with displacement magnitude.
- The additional nonlinear energy component contributes to higher stored energy levels than in the linear case.
B. Dissipative Effects
The nonlinear system dissipates additional energy through friction and contributes extra actuator work through nonlinear stiffness and friction, even under comparable motion.
- Dissipative Effects: Energy balances for the nonlinear and linear systems are compared over [0, T] to isolate differences in stored and dissipated energy.The comparison is formed by integrating both systems’ energy expressions and subtracting them.
- Dissipative Effects: The nonlinear system’s dissipated energy includes cẋ^2 and Fc|ẋ|, whereas the linear system includes only cẋ^2.These terms are integrated over the interval [0, T].
- Dissipative Effects: For Fc > 0, Coulomb friction adds non-negative dissipation that increases with motion duration and velocity magnitude.The actuator must compensate for this additional dissipated energy.
- Dissipative Effects: Nonlinear stiffness and friction introduce additional contributions to actuator work absent from the linear system.Both right-hand-side terms are non-negative for α > 0 and Fc > 0.
D. Interpretation Under Comparable Tracking
Meaningful energetic comparison requires approximately matched tracking performance, because different accuracy levels can themselves change actuator effort. Under similar tracking accuracy, nonlinear storage and dissipation can still produce substantially different energy use.
- Interpretation Under Comparable Tracking: Nonlinear stiffness adds amplitude-dependent energy storage αx4, while Coulomb friction adds dissipation Fc|ẋ|.These internal mechanisms can change actuator energy even when tracking accuracy is similar.
- Interpretation Under Comparable Tracking: More accurate tracking may naturally require greater control effort, making unmatched energy comparisons difficult to attribute to system dynamics.The framework therefore evaluates energy under approximately equivalent tracking conditions.
- Interpretation Under Comparable Tracking: For a system Σi, tracking and energetic measures are defined together to support performance-constrained comparison.The measures combine tracking behavior with actuator energy.
- Interpretation Under Comparable Tracking: Actuator energy is compared only when tracking performance is sufficiently close within a prescribed tolerance.This condition reduces the role of tracking-quality differences in explaining energy differences.
B. Parameterized Controller Exploration
The controller exploration generates achievable tracking-energy operating points for linear and nonlinear systems, then pairs points with similar tracking error for comparison.
- Parameterized Controller Exploration: A PD controller maps controller parameters to tracking-error and energy values.This creates operating points for exploring the tracking-energy trade-off.
- Parameterized Controller Exploration: Sweeping controller gains separately for each system yields sets of achievable operating points.The sets show how tracking performance and actuator energy vary across controller configurations.
- Parameterized Controller Exploration: Each linear operating point θL is paired with the nonlinear point θN that minimizes tracking-error difference subject to the matching condition.The procedure compares points with similar tracking accuracy.
D. Energetic Comparison
The energetic comparison evaluates matched operating points and shows that the nonlinear system accumulates substantially more actuator energy than the linear system despite similar output trajectories.
- Energetic Comparison: The percentage increase ΓE quantifies additional actuator energy required by the nonlinear system under comparable tracking conditions.It is computed from the nonlinear and linear energies of a matched pair.
- Energetic Comparison: The simulation compares linear and nonlinear systems under similar tracking conditions using proportional-derivative control.MATLAB ode45 is used over T = 8 s with the stated system parameters and gain ranges.
- Energetic Comparison: Cumulative actuator energy is computed from the proposed actuator-energy formulation during system execution.Fig. 1 presents the cumulative energy evolution for both systems.
- Energetic Comparison: Approximately 350 energy units accumulate for the nonlinear system versus about 200 units for the linear system by simulation end.The nonlinear system accumulates energy more rapidly over time.
- Energetic Comparison: Similar output trajectories still require more control effort in the nonlinear system because its additional stiffness and friction increase stored energy and dissipation.Trajectory similarity alone therefore does not characterize nonlinear energetic behavior.
B. Phase-Space Energetic Distortion
Phase portraits reveal amplitude-dependent energetic distortion in the nonlinear system, while matched-performance evaluation separates actuator-energy differences from tracking accuracy. The strongest matched pair shows substantially higher nonlinear energy despite nearly identical RMSE.
- B. Phase-Space Energetic Distortion: Phase-space trajectories are approximately elliptical for the linear system but asymmetric and amplitude-dependent for the nonlinear system.The nonlinear deformation is consistent with a modified energy-function geometry.
- B. Phase-Space Energetic Distortion: Matched-performance pairing approximately equalized RMSE so energetic differences primarily reflected intrinsic nonlinear dynamics rather than tracking quality.Operating points were paired within a prescribed RMSE matching tolerance across controller-gain sweeps.
- B. Phase-Space Energetic Distortion: 106.8% higher actuator energy occurred for the nonlinear system at nearly identical tracking error.The matched pair had nonlinear RMSE = 0.14089 and Energy = 402.690.
D. Matched-Performance Comparison
Under matched tracking conditions, nonlinear systems consistently require more actuator energy and produce higher EAPI values than linear systems. EAPI exposes energetic differences that trajectory-based metrics alone may miss.
- Energy comparison: ∼92% to ∼107% higher energy across matched-performance pairs means the nonlinear system requires nearly twice the actuator energy.The trend holds across multiple matched pairs rather than a single controller choice.
- EAPI comparison: EAPI combines tracking error with actuator energy to compare systems under matched operating conditions.The framework uses matched performance to separate energy differences from differences in tracking accuracy.
- EAPI comparison: The nonlinear system produced larger EAPI values than the linear system across all matched operating conditions.For the strongest matched-performance case, EAPI was 0.60905 for the linear system and 0.85627 for the nonlinear system, a 40.59% increase.
- Interpretation: Because matched systems retain nearly identical RMSE values, trajectory-only evaluation can miss the nonlinear system’s higher energetic cost.The proposed energy-aware metric provides greater discriminatory capability than classical trajectory-only evaluation in this comparison.
- Interpretation: Across all tested pairs, energy and EAPI increased for the nonlinear system despite similar tracking performance.These results support including energy alongside trajectory metrics for a more complete performance comparison.