Engine Sensitivity in Semi-Automatic Calibration

• Engine Sensitivity is defined as the local responsiveness of engine outputs to changes in actuator settings across the admissible domain.
• It plays a central role in calibrating engines by directing adaptive measurement refinement and balancing fuel efficiency with emission constraints.
• Leveraging finite differences and interpolation error, sensitivity estimates enable targeted calibration strategies that reduce bench time and improve drivability.

Searching arXiv for the specified paper and closely related calibration work to ground the article. arxiv_search.^^^^^^^^2query2^^^^^^^) OR title:\2^^^^^^^^"Semi-automatically optimized calibration of internal combustion engines\"","max_results":5,"sort_by":"relevance"}) arxiv_search.search({"^^^^^^^^2query2 Engine sensitivity, in the context of semi-automatic calibration of internal combustion engines, is the local or regional responsiveness of measured engine outputs to changes in actuator settings over the admissible domain. In the formulation of "Semi-automatically optimized calibration of internal combustion engines" (&&&^^^^2query2^^^^&&&), the engine is represented by a measurement map PRESERVED_PLACEHOLDER_^^^^2query2^^^^ from actuator settings to measured outputs, and sensitivity of output PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ to actuator $j$ is the local slope $\partial y_i/\partial u_j$. Because the method operates on a discretized domain with quasi-stationary measurements, sensitivity is not estimated through explicit Jacobian construction, but is inferred from local interpolation error and from directional finite differences obtained by “cross-measurements.” This notion is central to adaptive measurement refinement, emission-constrained calibration, and the construction of a drivable, fuel-optimal actuator map (&&&^^^^2query2^^^^&&&).

^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^. Definition and calibration context

In this framework, outputs include torque, fuel consumption, fuel mass flow, specific fuel consumption/BSFC, limits of measurand ranges such as temperatures and pressures, and emissions including CO, HC, NOx, PM, and PN. The actuator set comprises injected fuel quantity (IF), revolution frequency (RF, controlled), rail pressure (RP), air filling (AF), turbine geometry (TG), main timing (MT), pilot injection (PI), and pilot timing (PT). Sensitivity therefore refers to the dependence structure of these outputs on this actuator vector over the admissible actuator box $U \subset \mathbb{R}^m$ (&&&^^^^2query2^^^^&&&).

The calibration objective is to minimize fuel consumption subject to drivability and emission constraints over driving cycles. Sensitivity matters because high-sensitivity regions require denser measurement to capture the behavior of $F$ accurately, and because emission-constrained calibration benefits from identifying actuator directions that most strongly reduce a constrained pollutant. The method also imposes a discrete drivability constraint limiting actuator variations between neighboring operating points; this acts as a Lipschitz-like bound and restricts how aggressively local sensitivity can be exploited (&&&^^^^2query2^^^^&&&).

The operation field $\mathrm{OP}$ is the set of admissible frequency–torque pairs generated by feasible actuation, and the manifold map $M:\mathrm{OP}\to\mathbb{R}^m$ returns actuator settings for a given operating point. In this setting, engine sensitivity is not a standalone diagnostic quantity; it is a decision signal for where to measure, how to refine the discretization, and how to trade fuel economy against emissions and drivability.

2. Role in the semi-automatic calibration workflow

The semi-automatic workflow comprises adaptive measurement refinement, data cleaning, discretized search/optimization, and incorporation of actuator–behavior dependence and emission bounds (&&&^^^^2query2^^^^&&&). Sensitivity enters first in measurement planning. An actuator-space grid is constructed, its dual graph is built, and measurement ramps are routed stochastically toward low-density regions. Refinement is then triggered either symmetrically, when local fits are poor, or asymmetrically, when a particular actuator direction most effectively improves emissions.

The measurement plan generation uses a graph whose nodes are grid boxes and whose edges connect adjacent boxes. Nodes are weighted by reciprocal data density and edges by combined density. A target box is chosen randomly with probability proportional to reciprocal density, and a shortest path is computed with Dijkstra. In each box on that path, a random point is chosen and successive points are connected by measurement ramps, with ramp lengths determined by maximal actuator variation speeds and a fixed measurement frequency of $1$ Hz (&&&^^^^2query2^^^^&&&).

During execution, quasi-stationary sweeps are performed. Measurements are stored only when sufficiently different from prior values, while a minimal storage frequency is enforced. Ramps are aborted or rerouted near admissible-range boundaries, and hysteresis is treated by dynamic correction heuristics. After acquisition, data cleaning constructs a threshold graph on actuator points and iteratively removes high-degree nodes until only isolated points remain. The reduced set $F_{\mathrm{red}}$ is then used for fitting and optimization, while the full set remains available for density weighting (&&&^^^^2query2^^^^&&&).

Sensitivity also governs gap closing. Once a preliminary ILP solution identifies empty stacks, local models based on Newton interpolation or secants are used to predict useful points, and randomized measurements are seeded around those predictions. This suggests that sensitivity is not confined to local slope estimation; it also shapes how the measurement campaign allocates bench time to poorly resolved or constraint-active regions.

3. Mathematical formulation

The engine behavior is modeled as

PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2query2^^^^

The operation field is discretized into PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2id:(Burggraf et al., 2018) OR title:\2^^^^ rectangles PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\22^^^^, and a stack PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^3 collects all measured points falling into rectangle PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^4 (&&&^^^^2query2^^^^&&&).

Emission constraints are defined on a driving cycle PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^5 for each pollutant PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^6 with limit PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^7: PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^8 Drivability is imposed through actuator-wise bounds PRESERVED_PLACEHOLDER_^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^9: $j$^^^^2query2^^^^

After discretization, resistance-time weights $j$^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ are derived from the driving cycle and fuel-optimization weights $j$2 from the optimization curve. The discrete emission constraints become

$j$3

For neighboring rectangles, drivability is written as

$j$4

The optimization is formulated as an ILP with binary decision variables $j$5 and prey values $j$6: $j$7 subject to one representative per stack or a placeholder if none is feasible, along with drivability and emission constraints. The stack constraint is

$j$8

the drivability incompatibility constraint is

$j$9

and the emission constraints are

$\partial y_i/\partial u_j$^^^^2query2^^^^

where placeholders $\partial y_i/\partial u_j$^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ incur penalty values $\partial y_i/\partial u_j$2 (&&&^^^^2query2^^^^&&&).

4. Quantification of sensitivity

The paper quantifies sensitivity through two principal mechanisms. The first is directional finite differences. In asymmetric refinement, a new measurement that attains the best emission value in its stack triggers “cross-measurements,” in which each actuator is varied individually to identify the direction with the largest effect on pollutant $\partial y_i/\partial u_j$3. The local sensitivity is estimated by

$\partial y_i/\partial u_j$4

and the direction with largest magnitude is selected for targeted refinement (&&&^^^^2query2^^^^&&&).

The second mechanism is interpolation error. Symmetric refinement uses second-order Newton interpolation. If $\partial y_i/\partial u_j$5 and $\partial y_i/\partial u_j$6 is the local interpolant built from $\partial y_i/\partial u_j$7 neighbors, refinement is triggered when

$\partial y_i/\partial u_j$8

This threshold exceedance is interpreted as evidence of strong local curvature or nonsmoothness, and the corresponding box is split along all active actuator dimensions (&&&^^^^2query2^^^^&&&).

The method therefore uses local interpolation error as a proxy for high gradients or nonsmoothness and uses localized slope probing for targeted emission improvement. It explicitly does not use variance-based global methods such as Sobol indices. A plausible implication is that the method is designed for operational calibration rather than for full uncertainty attribution: sensitivity is valued primarily for its utility in adaptive measurement and constrained optimization, not for exhaustive decomposition of variance.

5. Robustness, data quality, and constraint handling

Data handling is integral to how sensitivity estimates remain usable in practice. Threshold-graph pruning removes redundant measurements and yields a pseudo-random cloud aligned with refined grid geometry. Measurement storage rules avoid oversampling nearly identical points, while extrapolation of latent measurands is used to preempt admissible-range violations. If a measurement violates admissible ranges $\partial y_i/\partial u_j$9, the ramp is interrupted and subsequent ramps may be rerouted from the last valid point (&&&^^^^2query2^^^^&&&).

Hysteresis and latency are treated by dynamic corrections based on the assumption that local clusters within boxes are smooth and possess identifiable time constants. The paper assumes continuity of $U \subset \mathbb{R}^m$^^^^2query2^^^^ but allows nondifferentiability. Robustness is then obtained by relying on measured values rather than surrogate-only predictions, refining wherever interpolation error is large, imposing drivability constraints that prevent overreaction to noisy sensitivity estimates, and using placeholders with conservative penalties so that gap-closing can only improve emission sums (&&&^^^^2query2^^^^&&&).

This architecture constrains the practical use of sensitivity. Large local slopes do not automatically justify large actuator changes, because drivability bounds restrict neighboring map discontinuities. Sensitivity is therefore operationalized within a bounded feasible set rather than treated as an unconstrained optimization direction.

6. Practical behavior on the diesel-engine example

The demonstrated system is an AVL-based diesel engine model with compression ignition, turbocharger, pilot injection, and variable turbine geometry, using $U \subset \mathbb{R}^m$^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ actuators and $U \subset \mathbb{R}^m$2 measurands. The operation field is discretized with $U \subset \mathbb{R}^m$3 in each axis, producing $U \subset \mathbb{R}^m$4 stacks. In the full calibration, all actuators except PI are dynamic across their full ranges, measurements are acquired once per second, and maximal actuator variation speeds constrain ramp lengths (&&&^^^^2query2^^^^&&&).

Several actuator sensitivities are highlighted in the results. Injection quantity (IF) strongly governs torque in the compression ignition model and heavily influences BSFC surfaces. Pilot injection and pilot timing (PI/PT) “drastically reduces” NOx by elevating chamber temperature prior to main injection, so the sensitivity of NOx to PT is leveraged particularly at low-speed and low-torque points. Air filling (AF) and turbine geometry (TG) influence boost and manifold pressure and thereby combustion completeness and emissions. Main timing (MT), described as analogous to SI spark timing, shifts combustion phasing and affects cylinder pressure development and NOx (&&&^^^^2query2^^^^&&&).

The observed trade-offs are explicit. Calibrations for RANDOM versus NEDC show lower NOx over large parts of the operation field for RANDOM, but the RANDOM solution has $U \subset \mathbb{R}^m$5–$U \subset \mathbb{R}^m$6 higher specific fuel consumption on most points compared to NEDC. Emissions outside the cycle-covered region are $U \subset \mathbb{R}^m$7–$U \subset \mathbb{R}^m$8 higher if not explicitly optimized there, which indicates that sensitivity-driven targeting is concentrated where the weights $U \subset \mathbb{R}^m$9 are nonzero. The AF actuator map with drivability constraints differs from the one without drivability constraints, directly illustrating how local sensitivity exploitation is bounded by $F$^^^^2query2^^^^ (&&&^^^^2query2^^^^&&&).

Quantitatively, a basic calibration produced a preliminary NEDC solution with average fuel consumption of $F$^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ L/^^^^2id:(Burggraf et al., 2018) OR title:\2query2query2^^^^ km, though with high CO, HC, and NOx integrals because emissions were not constrained. Closing gaps by interpolation-guided measurements required $F$2 hours. In the full calibration, NEDC EURO 4 was reached in $F$3 h, NEDC EURO 5 in $F$4 h, and dedicated RANDOM EURO 5 in $F$5 h. Compared with uniform grids, the adaptive method required $F$6–$F$7 fewer measurements to reach comparable quality (&&&^^^^2query2^^^^&&&).

7. Computational characteristics, limitations, and scope

The search strategy combines density-weighted Dijkstra routing, deterministic refinement triggers, repeated ILP feasibility checks, and placeholder-supported gap closing. For interpolation, nearest-neighbor selection is performed by brute force at complexity $F$8. With at most $F$9k points and $\mathrm{OP}$^^^^2query2^^^^, this corresponds to about $\mathrm{OP}$^^^^2id:(Burggraf et al., 2018) OR title:\2^^^^ billion elementary operations, which the paper characterizes as manageable off-line relative to engine bench time. For $\mathrm{OP}$2 and $\mathrm{OP}$3, the ILP contains $\mathrm{OP}$4 equalities, about $\mathrm{OP}$5 inequalities, $\mathrm{OP}$6 stack variables, and $\mathrm{OP}$7k–$\mathrm{OP}$8k measurement decision variables (&&&^^^^2query2^^^^&&&).

The method has clear limitations. The map $\mathrm{OP}$9 is assumed continuous but may be nondifferentiable, so interpolation quality can degrade near sharp transitions. Quasi-stationary ramps introduce hysteresis and latency that dynamic correction can mitigate but not eliminate. Discretized emission weights capture resistance time while ignoring traversal order, so fine temporal dynamics are abstracted. Sensitivity quantification remains local and directional, and thresholds for storage and refinement are heuristic rather than numerically specified (&&&^^^^2query2^^^^&&&).

The paper states that the pipeline generalizes to other engines and actuator sets; adding EGR or spark timing would fit naturally as additional actuators and measurands. It also notes that sensitivity analysis could be enhanced by finer grids, more sophisticated local models such as Gaussian process surrogates, gradient-based estimation where differentiability permits, and global screening methods such as the Morris method. This suggests that, within the paper’s formulation, engine sensitivity is best understood as an adaptive, measurement-driven construct that links local engine physics, bench-time allocation, and constrained discrete optimization into a single calibration methodology (&&&^^^^2query2^^^^&&&).