Tangent Hyperplanes VII: Residual-Aware Control

Tangent Hyperplanes VII: Residual-Aware Control

tangent-space

Using residuals as control and validation signals instead of ignoring them.

Residuals as Signals

Part 4 treated residuals as the finite-scale gap between nonlinear evolution and tangent-space prediction. This final compact part connects that gap to decisions about model validity and task error.

This page distills Residual-Aware Control into the canonical reading path while preserving the original critique: residual analysis is useful only when state estimation, model scope, and validation are honest.

Quantitative Bound

For a smooth system, the second-order Taylor remainder gives a local bound of the form

\[ \|r(t)\| \leq C\left(\|\delta x(t_0)\|^2 + \|\delta u\|_\infty^2\right) \]

over a fixed finite horizon, provided the perturbation family stays in a common smooth domain. State and input norms need declared scales. The constant $C$ depends on regional Hessian bounds, the actual deviation enclosure, and propagation through the variational dynamics. A sampled Hessian is an estimate, not a certified bound over a neighborhood.

More explicitly, define actual deviation $e$, first-order comparison $\eta$, and remainder $r=e-\eta$ using the same input experiment. Then

\[ \dot r=Ar+N,\qquad r(t)=\int_{t_0}^t\Phi(t,s)N(s)\,ds. \]

The Taylor forcing $N$ depends on the actual joined state/input deviation. Dropping its propagation or replacing that deviation with a linear prediction requires additional justification. The advanced article gives an exact scalar counterexample where the unpropagated expression fails.

A residual-aware controller does not merely know that this bound exists. It uses an estimate of the residual to decide when to:

accept a local linear rollout;
shrink the step size;
re-linearize;
switch from LQR-style feedback to MPC or another robust method;
flag the model as outside its stated region.

Residual-Triggered Control Logic

A conservative decision rule can be written as

\[ \rho(t) = \frac{\|r(t)\|}{\epsilon_r}, \]

where $\epsilon_r$ is the residual tolerance for the current task. Then:

if $\rho(t) \ll 1$, the specified remainder is small relative to that budget;
if $\rho(t) \approx 1$, inspect the comparison and consider refinement, a smaller proposed perturbation, or relinearization;
if $\rho(t) > 1$, that error budget has been exceeded.

The numerical threshold must be tied to a task metric and uncertainty model. A small remainder alone does not establish feasibility, stability, or an entire trust region. Measurement innovations additionally contain model and observation errors; they do not uniquely identify Taylor remainder.

Residuals Are Not Always Bad

Residuals can expose a local approximation’s limitations. They are not universal measurements of intrinsic curvature: a coordinate change can introduce a nonlinear field on a flat state line. The control question is how a defined discrepancy propagates to the task and whether a feasible response is available.

Adaptive DDP Pattern

A proposed residual-aware optimization loop must keep rollout, derivatives, cost quadrature, constraints, and the time mesh consistent:

Linearize and solve the local quadratic problem.
Roll out the nonlinear dynamics.
Measure the difference between predicted and realized state updates.
Compare the residual with a Hessian-derived or empirical tolerance.
Accept only after evaluating nonlinear cost and feasibility; otherwise shrink the update or revise the approximation.

If refining the mesh, preserve terminal time, transfer controls consistently, reintegrate the nominal trajectory, and recompute derivatives and quadrature. An inner fixed-mesh solve does not prove convergence of arbitrary mesh adaptation. Integration steps, optimizer nodes, and physical controller updates are different quantities. Switching controllers additionally requires a feasible, timely action and an appropriate stability argument; hysteresis alone is insufficient.

For local forcing bound $q$ and propagator bound $e^{a(t-s)}$, the zero-initial-error comparison is $R(h)=q(e^{ah}-1)/a$, with limit $qh$ at $a=0$. Its interval rule uses a logarithm, not a dimensionally invalid square root of time. Accumulated error from preceding intervals must also be propagated.

Tube MPC Interpretation

In tube MPC, feedback must keep possible errors inside a stated tube around the nominal trajectory. A bound on nonlinear remainder can contribute to its disturbance set, together with model and estimation uncertainty:

\[ \mathcal{T}(t) = \{x : \|x - x_{\text{nom}}(t)\| \leq \eta(t)\}, \]

where $\eta(t)$ must enclose reachable errors. A Taylor-validity radius alone does not establish containment. For a discrete local error model, a sufficient fixed-set condition is $A_KE\oplus W\subseteq E$, together with admissible initial error, feedback inputs, and tightened nominal constraints. A nonlinear or hybrid application needs the corresponding reachable/invariant-set and feasibility checks.

Golf Swing Optimization

For golf-swing optimization, residual-aware language is attractive because the motion couples earlier state preparation to a brief impact event:

residuals can indicate where a local tangent model is losing predictive authority;
residual thresholds can guide when to re-linearize a simulated swing;
differences between simulated and measured motion should not be attributed to “failed residual management” without accounting for soft tissue, muscle activation, measurement noise, and model error.

For a smooth terminal output with derivative $C_T$, the leading contribution of a remainder generated at time $s$ is weighted by $C_T\Phi(T,s)$. This connects proximal motion, actuation, compliance, and the remaining correction time to clubface, velocity, and strike-location errors. Event-based outputs also require contact-time and reset sensitivity. Ball launch predictions need a collision model; fewer computation nodes alone do not establish a better swing.

Critique to Preserve

Human movement residuals are not automatically geometric curvature. They may come from unmodeled compliance, neural delays, marker noise, inverse-dynamics ambiguity, or bad parameter identification. Residual-aware control is a useful engineering pattern, not a diagnosis of biological strategy unless the measurement pipeline supports that claim.

Relationship to the Advanced Sources

The longer article contains the complete propagated-bound derivation, independently checked scalar examples, and explicit application protocols. Its lay explanation uses the same comparison and evidence boundaries. No quadrotor speedup, ATRIAS fall reduction, or human neural strategy is established by those examples.

Series Summary

The seven-part compact path now reads as:

geometry: tangent spaces and exact infinitesimal structure;
dynamics: variational equations along a reference trajectory;
control: LQR, DDP, and iLQR on the moving tangent system;
residuals: finite-scale approximation, propagation, and conditional quadratic error;
contraction: differential stability certificates;
hybrid systems: saltation maps and mode-change scope;
residual-aware control: runtime trust-region and validation signals.

Together, these pages provide the single rendered reading path. The older advanced articles are retained as reference sources for derivations, examples, and critique notes.

Citation

BibTeX citation: Download .bib

@online{untitled,
  author = {},
  title = {Tangent {Hyperplanes} {VII:} {Residual-Aware} {Control}},
  url = {https://affinedrift.com/articles/tangent-hyperplanes-series/part-7-residual-aware.html},
  langid = {en}
}

For attribution, please cite this work as:

“Tangent Hyperplanes VII: Residual-Aware Control.” n.d. https://affinedrift.com/articles/tangent-hyperplanes-series/part-7-residual-aware.html.

--- css: /css/nonlinearity.css title: "Tangent Hyperplanes VII: Residual-Aware Control" description: "Using residuals as control and validation signals instead of ignoring them." aliases: - ../tangent-hyperplane-contraction/chapters/07-high-dimensional-applications.html - ../tangent-hyperplane-contraction/chapters/08-implementation-and-validation.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/Landing_Integral_Superposition.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/01_Integral_Superposition_I_Fundamentals.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/02_Integral_Superposition_II_Variational_Dynamics.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/03_Integral_Superposition_III_Physical_Interpretations.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/04_Integral_Superposition_IV_Residuals_and_Curvature.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Integral_Superposition_Series_Package/A_Appendix_Integral_Exactness.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Advanced_Integral_Expansions_Package/03_Control_Synthesis_as_Integral_Superposition.html categories: - tangent-space --- ## Residuals as Signals [Part 4](./part-4-residuals-curvature.html) treated residuals as the finite-scale gap between nonlinear evolution and tangent-space prediction. This final compact part connects that gap to decisions about model validity and task error. This page distills [Residual-Aware Control](../tangent-hyperplane-articles/Advanced/Residual-Aware_Control.html) into the canonical reading path while preserving the original critique: residual analysis is useful only when state estimation, model scope, and validation are honest. ## Quantitative Bound For a smooth system, the second-order Taylor remainder gives a local bound of the form $$ \|r(t)\| \leq C\left(\|\delta x(t_0)\|^2 + \|\delta u\|_\infty^2\right) $$ over a fixed finite horizon, provided the perturbation family stays in a common smooth domain. State and input norms need declared scales. The constant $C$ depends on regional Hessian bounds, the actual deviation enclosure, and propagation through the variational dynamics. A sampled Hessian is an estimate, not a certified bound over a neighborhood. More explicitly, define actual deviation $e$, first-order comparison $\eta$, and remainder $r=e-\eta$ using the same input experiment. Then $$ \dot r=Ar+N,\qquad r(t)=\int_{t_0}^t\Phi(t,s)N(s)\,ds. $$ The Taylor forcing $N$ depends on the actual joined state/input deviation. Dropping its propagation or replacing that deviation with a linear prediction requires additional justification. The advanced article gives an exact scalar counterexample where the unpropagated expression fails. A residual-aware controller does not merely know that this bound exists. It uses an estimate of the residual to decide when to: - accept a local linear rollout; - shrink the step size; - re-linearize; - switch from LQR-style feedback to MPC or another robust method; - flag the model as outside its stated region. ## Residual-Triggered Control Logic A conservative decision rule can be written as $$ \rho(t) = \frac{\|r(t)\|}{\epsilon_r}, $$ where $\epsilon_r$ is the residual tolerance for the current task. Then: - if $\rho(t) \ll 1$, the specified remainder is small relative to that budget; - if $\rho(t) \approx 1$, inspect the comparison and consider refinement, a smaller proposed perturbation, or relinearization; - if $\rho(t) > 1$, that error budget has been exceeded. The numerical threshold must be tied to a task metric and uncertainty model. A small remainder alone does not establish feasibility, stability, or an entire trust region. Measurement innovations additionally contain model and observation errors; they do not uniquely identify Taylor remainder. :::{.callout-note} ## Residuals Are Not Always Bad Residuals can expose a local approximation's limitations. They are not universal measurements of intrinsic curvature: a coordinate change can introduce a nonlinear field on a flat state line. The control question is how a defined discrepancy propagates to the task and whether a feasible response is available. ::: ## Adaptive DDP Pattern A proposed residual-aware optimization loop must keep rollout, derivatives, cost quadrature, constraints, and the time mesh consistent: 1. Linearize and solve the local quadratic problem. 2. Roll out the nonlinear dynamics. 3. Measure the difference between predicted and realized state updates. 4. Compare the residual with a Hessian-derived or empirical tolerance. 5. Accept only after evaluating nonlinear cost and feasibility; otherwise shrink the update or revise the approximation. If refining the mesh, preserve terminal time, transfer controls consistently, reintegrate the nominal trajectory, and recompute derivatives and quadrature. An inner fixed-mesh solve does not prove convergence of arbitrary mesh adaptation. Integration steps, optimizer nodes, and physical controller updates are different quantities. Switching controllers additionally requires a feasible, timely action and an appropriate stability argument; hysteresis alone is insufficient. For local forcing bound $q$ and propagator bound $e^{a(t-s)}$, the zero-initial-error comparison is $R(h)=q(e^{ah}-1)/a$, with limit $qh$ at $a=0$. Its interval rule uses a logarithm, not a dimensionally invalid square root of time. Accumulated error from preceding intervals must also be propagated. ## Tube MPC Interpretation In tube MPC, feedback must keep possible errors inside a stated tube around the nominal trajectory. A bound on nonlinear remainder can contribute to its disturbance set, together with model and estimation uncertainty: $$ \mathcal{T}(t) = \{x : \|x - x_{\text{nom}}(t)\| \leq \eta(t)\}, $$ where $\eta(t)$ must enclose reachable errors. A Taylor-validity radius alone does not establish containment. For a discrete local error model, a sufficient fixed-set condition is $A_KE\oplus W\subseteq E$, together with admissible initial error, feedback inputs, and tightened nominal constraints. A nonlinear or hybrid application needs the corresponding reachable/invariant-set and feasibility checks. ## Golf Swing Optimization For golf-swing optimization, residual-aware language is attractive because the motion couples earlier state preparation to a brief impact event: - residuals can indicate where a local tangent model is losing predictive authority; - residual thresholds can guide when to re-linearize a simulated swing; - differences between simulated and measured motion should not be attributed to "failed residual management" without accounting for soft tissue, muscle activation, measurement noise, and model error. For a smooth terminal output with derivative $C_T$, the leading contribution of a remainder generated at time $s$ is weighted by $C_T\Phi(T,s)$. This connects proximal motion, actuation, compliance, and the remaining correction time to clubface, velocity, and strike-location errors. Event-based outputs also require contact-time and reset sensitivity. Ball launch predictions need a collision model; fewer computation nodes alone do not establish a better swing. :::{.callout-warning} ## Critique to Preserve Human movement residuals are not automatically geometric curvature. They may come from unmodeled compliance, neural delays, marker noise, inverse-dynamics ambiguity, or bad parameter identification. Residual-aware control is a useful engineering pattern, not a diagnosis of biological strategy unless the measurement pipeline supports that claim. ::: ## Relationship to the Advanced Sources The longer article contains the complete propagated-bound derivation, independently checked scalar examples, and explicit application protocols. Its [lay explanation](../tangent-hyperplane-articles/Advanced/Residual-Aware_Control_LAYMAN.html) uses the same comparison and evidence boundaries. No quadrotor speedup, ATRIAS fall reduction, or human neural strategy is established by those examples. ## Series Summary The seven-part compact path now reads as: 1. geometry: tangent spaces and exact infinitesimal structure; 2. dynamics: variational equations along a reference trajectory; 3. control: LQR, DDP, and iLQR on the moving tangent system; 4. residuals: finite-scale approximation, propagation, and conditional quadratic error; 5. contraction: differential stability certificates; 6. hybrid systems: saltation maps and mode-change scope; 7. residual-aware control: runtime trust-region and validation signals. Together, these pages provide the single rendered reading path. The older advanced articles are retained as reference sources for derivations, examples, and critique notes. ## Related Articles ::: {.callout-note title="See Also"} - **[Tangent-Space Series](../../pages/tangent-hyperplanes.html)** — The series hub: full reading order, provenance, and companion material - **[Superposition in Affine Control Systems](../superposition.html)** — The dropped higher-order terms that residual-aware control treats as trust-region signals - **[Tangent Hyperplanes VI: Hybrid Systems](part-6-hybrid.html)** — The previous part of the series :::